Structural relaxation and low-energy properties of twisted bilayer graphene

Abstract

The structural and electronic properties of twisted bilayer graphene are investigated from first-principles and tight-binding approach as a function of the twist angle (ranging from the first "magic" angle θ = 1.08 • to θ = 13.17 • , with the former corresponding to the largest unit cell, comprising 11,164 carbon atoms). By properly taking into account the long-range van der Waals (vdW) interaction, we provide the patterns for the atomic displacements (with respect to the ideal twisted bilayer). The out-of-plane relaxation shows an oscillating ("buckling") behavior, very evident for the smallest angles, with the atoms around the AA stacking regions interested by the largest displacements. The out-of-plane displacements are accompanied by a significant in-plane relaxation, showing a vortexlike pattern, where the vorticity (intended as curl of the displacement field) is reverted when moving from the top to the bottom plane and vice versa. Overall, the atomic relaxation results in the shrinking of the AA stacking regions in favor of the more energetically favorable AB/BA stacking domains. The measured flat bands emerging at the first magic angle can be accurately described only if the atomic relaxations are taken into account. Quite importantly, the experimental gaps separating the flat-band manifold from the higher and lower energy bands are intimately related to out-of-plane relaxations. The stability of the relaxed bilayer at the first magic angle is estimated to be of the order of 0.5-0.9 meV per atom (or 7-10 K). Our calculations shed light on the importance of an accurate description of the vdW interaction and of the resulting atomic relaxation to envisage the electronic structure of this really peculiar kind of vdW bilayers.

Phonons in magic-angle twisted bilayer graphene

“…Moreover, as AB/BA stacked bilayer graphene is energetically more stable than the AA stacked one, carbon atoms near the AA point tend to misalign with each other, resulting in a squeeze of the AA region. In addition to the in-plane displacements, our calculations indicate that there are also out-of-plane corrugations, i.e., the periodic modulations of interlayer distance in different regions of the moiré pattern, with the distance at the AA (AB/BA) point being 3.62 Å(3.36 Å), which is consistent with previous results [93][94][95][96].…”

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

“…We study commensurate unit cells of tBLG with D 3 symmetry [28,29]. The atomic positions are relaxed using classical force fields [30][31][32][33][34]59]. For this, we use a combination of the AIREBO-Morse [60] and Kolmogorov-Crespi [61] potentials as implemented in LAMMPS [62].…”

“…The quintessential model for strongly interacting electrons is the Hubbard model, in which electrons only interact when they are on the same "site" (typically assumed to be an atom). In tBLG near the magic angle, the moiré pattern results in the emergence of eight flat bands (including a factor of 2 from spin degeneracy) near the Fermi energy which are separated from all other bands by energy gaps [26][27][28][29][30][31][32][33][34]. Starting from atomistic tight-binding approaches, Hubbard models for tBLG can be obtained by constructing Wannier functions of the flat bands [35][36][37] (note that this is not possible when a continuum model starting point is used; in that case additional bands must be included in the Wannierization procedure [38,39]).…”

An accurate tight binding model for twisted bilayer graphene describes topological flat bands without geometric relaxation

“…TBLG is a van der Waals structure that consists of two sheets of single layer graphene laid on top of each other and twisted with a relative twist angle θ. For certain "magic" values of θ, the first of which has been cited as near 1.05 • , 1 TBLG hosts correlated insulating and superconducting phases [1][2][3][4][5] and anomalous Hall effects 6,7 It is believed that these correlated phases emerge through the interplay of electronic and structural degrees of freedom [8][9][10] .…”