Charge smoothening and band flattening due to Hartree corrections in twisted bilayer graphene

Abstract

Doping twisted bilayer graphene away from charge neutrality leads to an enormous buildup of charge inhomogeneities within each Moiré unit cell. Here we show, using unbiased real-space selfconsistent Hartree calculations on a relaxed lattice, that Coulomb interactions smoothen this charge imbalance by changing the occupation of earlier identified 'ring' orbitals in the AB/BA region and 'center' orbitals at the AA region. For hole doping, this implies an increase of the energy of the states at the Γ point, leading to a further flattening of the flat bands and a pinning of the Van Hove singularity at the Fermi level. The charge smoothening will affect the subtle competition between different possible correlated phases.Studies of twisted bilayer graphene have revealed a number of striking electronic properties, 1-11 and recently culminated in the discovery of a correlated insulator and superconducting phase. 12,13 Several new experiments performed since then 14-20 have revealed new correlated and topological states, and even more theoretical works have appeared 21-61 . However, there is no consensus yet on the correct low-energy effective band-structure, let alone the relevant many-body interactions.It is well known that at fillings away from charge neutrality the electronic charge piles up in the AA regions of the Moiré unit cell. [1][2][3][4]28,48,50 The Coulomb interaction will counterbalance the formation of such large charge inhomogeneities, and is expected to significantly modify the non-interacting band dispersion. Here we study the effect of the electronic interactions on AA localization via an unbiased, self-consistent Hartree calculation.Starting from a microscopic tight-binding model on a relaxed twisted bilayer at the magic angle θ = 1.08 • at fillings at and below charge neutrality, we show that the Hartree corrections indeed cause the redistribution of charges within the unit cell. The resulting charge distribution is much smoother than that found with noninteracting theories. The smoothening itself is made possible by increasing the energy of the states at the Γ point in the Brillouin zone, which causes a further substantial flattening of the (already nearly) flat bands. Our results are qualitatively similar to earlier results based on a continuum model. 48,49 Given the tight competition between different possible correlated phases, 21,24,43 the charge transfer reported here will almost certainly affect this subtle competition.In the remainder of this paper, we first introduce our model of twisted bilayer graphene and the details of our calculation. We then present and compare the charge inhomogeneities before and after the Hartree corrections. Finally, we discuss the ramifications of our result for future theoretical modelling.The model -Our starting point is a tight-binding model of twisted bilayer graphene, 28 which builds on earlier continuum models. 6 We construct twisted bilayer graphene by starting with an AB stacked bilayer and rotating the top layer around an AB site. For a commensurate twist ...

Importance of long-ranged electron-electron interactions for the magnetic phase diagram of twisted bilayer graphene

“…For the two larger twist angles, both the Hartree and tight-binding band structures exhibit Dirac cones at the K and K points. While the noninteracting tight-binding band structure does not depend on the doping level, long-ranged electron-electron interactions captured by Hartree theory give rise to a significant doping-dependent distortion of the band structure [46][47][48][49][50]. In particular, Hartree interactions result in a flattening of the doped bands.…”

“…where n j denotes the occupancy of the p z orbital on atom j and n is the average occupancy [48]. Also, W i j denotes the screened Coulomb interaction between electrons at τ i and τ j [49].…”

“…It is well known, however, that long-ranged interactions play an important role in tBLG. Using Hartree theory, several groups [46][47][48][49][50] demonstrated that long-ranged interactions result in significant changes of the electronic structure which depend sensitively on doping and twist angle. In particular, Hartree interactions result in a flattening of the doped bands (in addition to the band flattening induced by twisting) [26][27][28][29].…”

Structural relaxation and low-energy properties of twisted bilayer graphene

“…A second, unexpected, insulating phase shows up at half-filling of the FB manifold, both on the electron and on the hole side (±2 electrons with respect to charge neutrality). The correlated insulating phase is attributed to enhanced electron-electron interaction within the FBs [14,15], although some authors are highlighting the relevance of the electron-phonon interaction [16][17][18][19]. After electrostatic doping, achieved by gating the structure, unconventional superconductivity with a critical temperature ranging from 1.7 to 3 K appears in a strong pairing regime, with a phase diagram very similar to that of the underdoped cuprates, whose origin is still to be understood [17,20].…”

Time-reversal symmetry breaking versus chiral symmetry breaking in twisted bilayer graphene

“…In this Rapid Communication, we study the effects of the Coulomb interaction at the CNP of TBG near the magic angle, with the aim of discerning whether dynamical symmetry breaking takes place under different screening conditions. We adopt a tight-binding approach to make a real-space description of the system, and we resort to a Hartree-Fock approximation in order to assess the effects of the Coulomb interaction [36,38,[52][53][54][55]. This approach allows us to treat the on-site Hubbard and long-range Coulomb interaction on the same footing.…”