Trilayer graphene with a twisted middle layer has recently emerged as a new platform exhibiting correlated phases and superconductivity near its magic angle. A detailed characterization of its electronic structure in the parameter space of twist angl
e $theta$, interlayer potential difference $Delta$, and top-bottom layer stacking $tau$ reveals that flat bands with large Coulomb energy vs bandwidth $U/W > 1$ are expected within a range of $pm 0.2^{circ}$ near $theta simeq1.5^{circ}$ and $theta simeq1.2^{circ}$ for $tau_{rm AA}$ top-bottom layer stacking, between a wider $1^{circ} sim 1.7^{circ}$ range for $tau_{rm AB}$ stacking, whose bands often have finite valley Chern numbers thanks to the opening of primary and secondary band gaps in the presence of a finite $Delta$, and below $theta lesssim 0.6^{circ}$ for all $tau$ considered. The largest $U/W$ ratios are expected at the magic angle $sim 1.5^{circ}$ when $|Delta| sim 0$~meV for AA, and slightly below near $sim 1.4^{circ}$ for finite $|Delta| sim 25$~meV for AB stackings, and near $theta sim 0.4^{circ}$ for both stackings. When ${tau}$ is the saddle point stacking vector between AB and BA we observe pronounced anisotropic local density of states (LDOS) strip patterns with broken triangular rotational symmetry. We present optical conductivity calculations that reflect the changes in the electronic structure introduced by the stacking and gate tunable system parameters.
Rhombohedral $N = 3$ trilayer graphene on hexagonal boron nitride (TLG/BN) hosts gate-tunable, valley-contrasting, nearly flat topological bands that can trigger spontaneous quantum Hall phases under appropriate conditions of the valley and spin pola
rization. Recent experiments have shown signatures of C = 2 valley Chern bands at 1/4 hole filling, in contrast to the predicted value of C = 3. We discuss the low-energy model for rhombohedral N-layer graphene (N = 1, 2, 3) aligned with hexagonal boron nitride (hBN) subject to off-diagonal moire vector potential terms that can alter the valley Chern numbers. Our analysis suggests that topological phase transitions of the flat bands can be triggered by pseudomagnetic vector field potentials associated to moire strain patterns, and that a nematic order with broken rotational symmetry can lead to valley Chern numbers that are in agreement with recent Hall conductivity observations.
We investigate the band structure of twisted monolayer-bilayer graphene (tMBG), or twisted graphene on bilayer graphene (tGBG), as a function of twist angles and perpendicular electric fields in search of optimum conditions for achieving isolated nea
rly flat bands. Narrow bandwidths comparable or smaller than the effective Coulomb energies satisfying $U_{textrm{eff}} /W gtrsim 1$ are expected for twist angles in the range of $0.3^{circ} sim 1.5^{circ}$, more specifically in islands around $theta sim 0.5^{circ}, , 0.85^{circ}, ,1.3^{circ}$ for appropriate perpendicular electric field magnitudes and directions. The valley Chern numbers of the electron-hole asymmetric bands depend intrinsically on the details of the hopping terms in the bilayer graphene, and extrinsically on factors like electric fields or average staggered potentials in the graphene layer aligned with the contacting hexagonal boron nitride substrate. This tunability of the band isolation, bandwidth, and valley Chern numbers makes of tMBG a more versatile system than twisted bilayer graphene for finding nearly flat bands prone to strong correlations.
We predict that antiferromagnetic bilayers formed from van der Waals (vdW) materials, like bilayer CrI$_3$, have a strong magnetoelectric response that can be detected by measuring the gate voltage dependence of Faraday or Kerr rotation signals, tota
l magnetization, or anomalous Hall conductivity. Strong effects are possible in single-gate geometries, and in dual-gate geometries that allow internal electric fields and total carrier densities to be varied independently. We comment on the reliability of density-functional-theory estimates of interlayer magnetic interactions in van der Waals bilayers, and on the sensitivity of magnetic interactions to pressure that alters the spatial separation between layers.
Flatbands with extremely narrow bandwidths on the order of a few mili-electron volts can appear in twisted multilayer graphene systems for appropriate system parameters. Here we investigate the electronic structure of a twisted bi-bilayer graphene, o
r twisted double bilayer graphene, to find the parameter space where isolated flatbands can emerge as a function of twist angle, vertical pressure, and interlayer potential differences. We find that in twisted bi-bilayer graphene the bandwidth is generally flatter than in twisted bilayer graphene by roughly up to a factor of two in the same parameter space of twist angle $theta$ and interlayer coupling $omega$, making it in principle simpler to tailor narrow bandwidth flatbands. Application of vertical pressure can enhance the first magic angle in minimal models at $theta sim 1.05^{circ}$ to larger values of up to $theta sim 1.5^{circ}$ when $ P sim 2.5$~GPa, where $theta propto omega/ upsilon_{F}$. Narrow bandwidths are expected in bi-bilayers for a continuous range of small twist angles, i.e. without magic angles, when intrinsic bilayer gaps open by electric fields, or due to remote hopping terms. We find that moderate vertical electric fields can contribute in lifting the degeneracy of the low energy flatbands by enhancing the primary gap near the Dirac point and the secondary gap with the higher energy bands. Distinct valley Chern bands are expected near $0^{circ}$ or $180^{circ}$ alignments.
We investigate the bandwidth compression due to out of plane pressure of the moire flatbands near charge neutrality in twisted bilayer graphene for a continuous range of small rotation angles of up to $sim2.5^{circ}$. The flatband bandwidth minima an
gles are found to grow linearly with interlayer coupling {omega} and decrease with Fermi velocity. Application of moderate pressure values of up to 2.5 GPa achievable through a hydraulic press should allow accessing a flatband for angles as large as $sim 1.5$^{circ}$ instead of $sim 1 circ$ at zero pressure. This reduction of the moire pattern length for larger twist angle implies an increase of the effective Coulomb interaction scale per moire cell by about 50% and enhance roughly by a factor of $sim 2$ the elastic energy that resists the commensuration strains due to the moire pattern. Our results suggest that application of pressure on twisted bilayer graphene nanodevices through a hydraulic press will notably facilitate the device preparation efforts required for exploring the ordered phases near magic angle flatbands.
We investigate the electronic structure of the flat bands induced by moire superlattices and electric fields in nearly aligned ABC trilayer graphene-boron nitride interfaces where Coulomb effects can lead to correlated gapped phases. Our calculations
indicate that valley-spin resolved isolated superlattice flat bands that carry a finite Chern number $C = 3$ proportional to layer number can appear near charge neutrality for appropriate perpendicular electric fields and twist angles. When the degeneracy of the bands is lifted by Coulomb interactions these topological bands can lead to anomalous quantum Hall phases that embody orbital and spin magnetism. Narrow bandwidths of $sim10$ meV achievable for a continuous range of twist angles $theta lesssim 0.6^{circ}$ with moderate interlayer potential differences of $sim$50 meV make the TLG/BN systems a promising platform for the study of electric-field tunable Coulomb interaction driven spontaneous Hall phases.
