We report on ground-and excited state transport through an electrostatically defined few-hole quantum dot in bilayer graphene in both parallel and perpendicular applied magnetic fields. A remarkably clear level scheme for the two-particle spectra is found by analyzing finite bias spectroscopy data within a two-particle model including spin and valley degrees of freedom. We identify the two-hole ground-state to be a spin-triplet and valley-singlet state. This spin alignment can be seen as Hund's rule for a valley-degenerate system, which is fundamentally different to quantum dots in carbon nano tubes and GaAs-based quantum dots. The spin-singlet excited states are found to be valley-triplet states by tilting the magnetic field with respect to the sample plane. We quantify the exchange energy to be 0.35 meV and measure a valley and spin g-factor of 36 and 2, respectively.

Quantum states in graphene are 2-fold degenerate in spins, and 2-fold in valleys. Both degrees of freedom can be utilized for qubit preparations. In our bilayer graphene quantum dots, we demonstrate that the valley g-factor g v, defined analogously to the spin g-factor g s for valley splitting in a perpendicular magnetic field, is tunable by over a factor of 4 from 20 to 90, by gate voltage adjustments only. Larger g v results from larger electronic dot sizes, determined from the charging energy. On our versatile device, bipolar operation, charging our quantum dot with charge carriers of the same or the opposite polarity as the leads, can be performed. Dots of both polarities are tunable to the first charge carrier, such that the transition from an electron to a hole dot by the action of the plunger gate can be observed. Addition of gates easily extends the system to host tunable double dots.

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.

The Kondo effect is a cornerstone in the study of strongly correlated fermions. The coherent exchange coupling of conduction electrons to local magnetic moments gives rise to a Kondo cloud that screens the impurity spin. Here we report on the interplay between spin–orbit interaction and the Kondo effect, that can lead to a underscreened Kondo effects in quantum dots in bilayer graphene. More generally, we introduce a different experimental platform for studying Kondo physics. In contrast to carbon nanotubes, where nanotube chirality determines spin–orbit coupling breaking the SU(4) symmetry of the electronic states relevant for the Kondo effect, we study a planar carbon material where a small spin–orbit coupling of nominally flat graphene is enhanced by zero-point out-of-plane phonons. The resulting two-electron triplet ground state in bilayer graphene dots provides a route to exploring the Kondo effect with a small spin–orbit interaction.

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