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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, or 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.
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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 angles 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 construct a phenomenological scattering theory for the triangular network of valley Hall states that arises in twisted bilayer graphene under interlayer bias. Crucially, our network model includes scattering between different valley Hall states within the same valley and spin. We show that in the absence of forward scattering, symmetries reduce the network model to a single parameter that interpolates between a nested Fermi surface and flatbands, which can be understood in terms of one-dimensional chiral zigzag modes and closed triangular orbits, respectively. We demonstrate how unitarity and symmetry constrain the couplings between zigzag modes, which has important implications on the nature of interference oscillations observed in experiments.
When twisted to angles near 1{deg}, graphene multilayers provide a new window on electron correlation physics by hosting gate-tuneable strongly-correlated states, including insulators, superconductors, and unusual magnets. Here we report the discovery of a new member of the family, density-wave states, in double bilayer graphene twisted to 2.37{deg}. At this angle the moire states retain much of their isolated bilayer character, allowing their bilayer projections to be separately controlled by gates. We use this property to generate an energetic overlap between narrow isolated electron and hole bands with good nesting properties. Our measurements reveal the formation of ordered states with reconstructed Fermi surfaces, consistent with density-wave states, for equal electron and hole densities. These states can be tuned without introducing chemical dopants, thus opening the door to a new class of fundamental studies of density-waves and their interplay with superconductivity and other types of order, a central issue in quantum matter physics.
Topological insulators realized in materials with strong spin-orbit interactions challenged the long-held view that electronic materials are classified as either conductors or insulators. The emergence of controlled, two-dimensional moire patterns has opened new vistas in the topological materials landscape. Here we report on evidence, obtained by combining thermodynamic measurements, local and non-local transport measurements, and theoretical calculations, that robust topologically non-trivial, valley Chern insulators occur at charge neutrality in twisted double-bilayer graphene (TDBG). These time reversal-conserving valley Chern insulators are enabled by valley-number conservation, a symmetry that emerges from the moire pattern. The thermodynamic gap extracted from chemical potential measurements proves that TDBG is a bulk insulator under transverse electric field, while transport measurements confirm the existence of conducting edge states. A Landauer-Buttiker analysis of measurements on multi-terminal samples allows us to quantitatively assess edge state scattering and demonstrate that it does not destroy the edge states, leaving the bulk-boundary correspondence largely intact.
We have studied the dielectric screening of electric field which is induced by a gate voltage in twisted double bilayer graphene by using a sample with a mismatch angle of about 5 degrees. In low temperature magnetotransport measurements, quantum oscillations of magnetoresistance originating from two bands with different carrier density were observed. The behavior of the carrier densities with respect to the total carrier density were distinct from that of the AB-stacked tetralayer graphene. The carrier density ratio was theoretically analyzed in terms of the model that the induced charge decays exponentially with distance with a screening length {lambda}. The estimated {lambda} was slightly larger than that of AB-stacked graphene, which would possibly reflect the difference in the inter-plane distribution of probability of the wave function.
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