The discovery of superconductivity and correlated insulating states in twisted bilayer graphene (tBG) has spurred significant research. Understanding the underlying mechanisms requires investigating the normal-metal state transport, especially near the magic angle. Fifteen years ago, it was predicted that twisting two graphene sheets would create a moiré superlattice potential. Reducing the twist angle increases the Bloch period, bridging the length scales between materials and cold atom systems. This moiré potential modifies the electronic structure, reducing Fermi velocity and bandwidth, thereby enhancing electron-electron interactions. The moiré band closest to charge neutrality remains Dirac-like but with reduced Fermi velocity. The reduced bandwidth is quantified by the separation of van Hove singularities (VHS). Numerical studies confirmed the long-wavelength continuum picture, although experimental confirmation was initially debated. The continuum model predicts a vanishing Fermi velocity at magic angles, with recent work suggesting only the first magic angle is stable. This work introduces a new critical angle (θcr ~1.15°) where the Fermi velocity equals the phonon velocity, leading to a vanishing phonon contribution to resistivity and a significant resistivity increase for small angular deviations. The focus of this work is on carrier transport theory in the metallic regime, not the superconducting or correlated insulator states. While a phonon mechanism for superconductivity and possibilities of Mott insulator or Wigner crystal for the correlated insulator are considered, this paper centers on the metallic regime.

Previous research anticipated the impact of a slight relative rotation of two graphene sheets, creating a moiré superlattice potential that significantly affects electronic structure. This was explored through theoretical predictions and confirmed via numerical ab initio studies. Experimental validation, however, initially remained controversial. The Bistritzer and MacDonald model predicted a vanishing Fermi velocity at magic angles, assuming rigid lattices, later refined by models incorporating lattice relaxation. Several experimental studies focusing on local spectroscopy explored the moiré-induced electronic structure. The vanishing bandwidth, rather than the Fermi velocity, appears more relevant to the observed strongly correlated physics. Prior work also predicted that charged impurities would dominate resistivity at low carrier densities and temperatures, with gauge phonons dominating in most experimental regimes. This crossover was already known in monolayer graphene but occurs at much higher temperatures. Two experimental transport studies focusing on the metallic regime produced different conclusions regarding the dominant scattering mechanisms, one suggesting a Planckian mechanism and another favoring phonon scattering.

This study employs a Boltzmann transport theory, extending previous work on monolayer graphene. Several key extensions are implemented for tBG: (1) inclusion of both intraband and interband scattering processes; (2) consideration of finite-temperature dynamical screening of electron-phonon matrix elements; (3) going beyond the linear Dirac dispersion to account for nonlinear lattice effects near the VHS. The methodology includes both intraband and interband electron-phonon scattering processes. The finite-temperature dynamical screening of the electron-phonon matrix element is crucial, especially for low twist angles where only antisymmetric gauge phonon modes survive. Beyond the linear Dirac dispersion, the van Hove singularity is included to model resistivity saturation with temperature. The theory explicitly addresses the geometric enhancement of the gauge phonon mode. The dynamical polarizability at finite temperature is calculated semi-analytically (a novel aspect of this study). The static screening approximation is compared with the fully dynamically screened result, finding it closer to the correct result than the unscreened approximation. The role of the Fermi velocity (vF) relative to the phonon velocity (cph) defines two regimes. The intraband scattering regime (vF > cph) uses established theory, showing Bloch-Grüneisen behavior. The interband scattering regime (vF < cph), dominant near the magic angle, has a different behavior, including a linear-in-T resistivity even below the Bloch-Grüneisen temperature. An effective two-band Hamiltonian is used to capture the VHS effects. The Planckian model, as a competing mechanism, is also implemented using a phenomenological model with resistivity proportional to kBT, analyzed within both Dirac and two-band models. Lattice relaxation effects are incorporated using two models: a continuum elasticity theory and a molecular dynamics approach. These models provide relaxed atomic positions which are used to calculate the moiré coupling parameters via Fourier transformation. These parameters are then input into the continuum Hamiltonian for band structure calculations, providing a renormalized Fermi velocity. Fitting procedures are detailed, using various data sets with different twist angles, temperatures, and carrier densities for both electron-phonon and Planckian models.

The study finds that phonon scattering, not a Planckian mechanism, is the dominant scattering mechanism in the metallic regime of tBG. The phonon-mediated theory accurately predicts the weak density dependence observed experimentally, unlike the Planckian theory which shows much stronger density dependence. Both theories predict linear-in-T resistivity at low temperatures and saturation at high temperatures, but the origins of saturation differ: in the phonon model, it is set by the electronic bandwidth; in the Planckian model, it is intrinsic and independent of band structure. The linear-in-T resistivity at low temperature is explained by interband scattering, which persists to lower temperatures than expected in typical Fermi liquids. The crossover between vF < cph and vF > cph, occurring at a critical angle θc, leads to sharp dips in resistivity that span several orders of magnitude. These dips are attributed to the drastic change in the scattering phase space when the Fermi velocity crosses the phonon velocity. Lattice relaxation effects are analyzed using two different models and found to influence the quantitative details but not the overall qualitative agreement between the theory and the experimental results. Fitting procedures validate the phonon theory over the Planckian model. The fitted Fermi velocity and coupling parameters are consistent with theoretical expectations for the phonon model, while the Planckian model produces inconsistencies including violations of the Planckian bound (C ≤ 1). The study shows good quantitative agreement between the electron-phonon scattering theory and the experimental data across a range of parameters, including temperature, carrier density, and twist angle. This encompasses the low-temperature non-monotonic behavior and the linear-in-T behavior at intermediate temperatures.

The findings strongly support the phonon scattering mechanism as the primary driver of transport in the metallic regime of tBG. The detailed theory successfully accounts for several previously unexplained experimental observations, including the linear-in-T resistivity at low temperatures, the saturation of resistivity at high temperatures, and the weak density dependence at intermediate temperatures. The identification of the critical angle θc and the significant resistivity variations near it offer a new understanding of the transport properties of tBG. This emphasizes the importance of careful experimental distinctions between phonon-related effects and other possible phenomena such as Mott insulation or superconductivity. The model provides a complete theoretical framework for tBG transport in the metallic regime, integrating both impurity and phonon scattering mechanisms.

This work provides a comprehensive theory of carrier transport in twisted bilayer graphene's metallic regime, emphasizing phonon scattering as the dominant mechanism. The theory explains previously puzzling experimental observations and makes predictions testable in future experiments. Future research could explore the implications of the critical angle and the interplay of phonon and impurity scattering in different regimes of tBG.

While this study presents a comprehensive theoretical framework, potential limitations include the approximations made in the models, such as the effective two-band Hamiltonian and the relaxation-time approximation within the Boltzmann transport formalism. The accuracy of the models is also contingent on the precision of the input parameters, including phonon velocities and coupling constants, which may have some uncertainties. The focus is on the metallic regime, excluding a comprehensive analysis of the correlated insulating and superconducting phases.