Crystal fields occur due to a potential difference between chemically different atomic species. In Van-der-Waals heterostructures such fields are naturally present perpendicular to the planes. It has been realized recently that twisted graphene multilayers provide powerful playgrounds to engineer electronic properties by the number of layers, the twist angle, applied electric biases, electronic interactions and elastic relaxations, but crystal fields have not received the attention they deserve.Here we show that the bandstructure of large-angle twisted double bilayer graphene is strongly modified by crystal fields. In particular, we experimentally demonstrate that twisted double bilayer graphene, encapsulated between hBN layers, exhibits an intrinsic bandgap. By the application of an external field, the gaps in the individual bilayers can be closed, allowing to determine the crystal fields. We find that crystal fields point from the outer to the inner layers with strengths in the bottom/top bilayer E b = 0.13 V/nm ≈ −Et = 0.12 V/nm. We show both by means of first principles calculations and low energy models that crystal fields open a band gap in the groundstate. Our results put forward a physical scenario in which a crystal field effect in carbon substantially impacts the low energy properties of twisted double bilayer graphene, suggesting that such contributions must be taken into account in other regimes to faithfully predict the electronic properties of twisted graphene multilayers. arXiv:1910.10524v2 [cond-mat.mes-hall] 4 Nov 2019 a c FIG. 1. a) Two twisted, AB-stacked bilayer graphene (BG) sheets. The electrostatic potential of the outer layers is different from the potential in the inner layers. This leads to crystal fields Et = −E b pointing in opposite direction in the top and bottom BG. In the experiment, the two bilayer systems are encapsulated in hBN which reduces the strength of the crystal fields compared to vacuum. b) The TDBG band structure consists of Brillouin-zones of the top and bottom BG rotated with respect to each other. For large twist angles θ, the bands of the top and bottom layer intersect at energies large compared to the Fermi energies of the individual layers. Therefore, at typical Fermi energies, the individual BG band structures remain intact.The crystal fields open a single-particle gap in both layers. c) In such a structure we observe a gap at zero density and zero external field in a resistance versus density trace R(ntot). We show traces for two devices, device 1 is further discussed in the main text and further measurements of device 2 and device 3 are shown in the Supplemental Material.

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Fermi nesting Correlated states have been shown to emerge in bilayer and trilayer graphene with the two-dimensional layers at just the right angle with respect to each other. Key to the enhanced importance of interactions are the so-called moiré electronic bands that form in such systems. Rickhaus et al . explored a related system of two graphene bilayers twisted with respect to each other. The twist angle was set so that the layer coupling was strong enough to form moiré bands but weak enough for the carrier concentration in the top and bottom bilayer to be controlled separately. Doping the top bilayer with electrons and the bottom bilayer with holes, the researchers created a correlated state with nested Fermi surfaces. —JS

Control over minivalley polarization and interlayer coupling is demonstrated in double bilayer graphene twisted with an angle of 2.37°. This intermediate angle is small enough for the minibands to form and large enough such that the charge carrier gases in the layers can be tuned independently. Using a dual-gated geometry we identify and control all possible combinations of minivalley polarization via the population of the two bilayers. An applied displacement field opens a band gap in either of the two bilayers, allowing us to even obtain full minivalley polarization. In addition, the carriers, formerly separated by their minivalley character, are mixed by tuning through a Lifshitz transition, where the Fermi surface topology changes. The high degree of control over the minivalley character of the bulk charge transport in twisted double bilayer graphene offers new opportunities for realizing valleytronics devices such as valley valves, filters, and logic gates.

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