High-order fractal states in graphene superlattices.

Graphene superlattices were shown to exhibit high-temperature quantum oscillations due to periodic emergence of delocalized Bloch states in high magnetic fields such that unit fractions of the flux quantum pierce a superlattice unit cell. Under these conditions, semiclassical electron trajectories become straight again, similar to the case of zero magnetic field. Here, we report magnetotransport measurements that reveal second-, third-, and fourth-order magnetic Bloch states at high electron densities and temperatures above 100 K. The recurrence of these states creates a fractal pattern intimately related to the origin of Hofstadter butterflies. The hierarchy of the fractal states is determined by the width of magnetic minibands, in qualitative agreement with our band-structure calculations.

Electron-Hole Tunneling Revealed by Quantum Oscillations in the Nodal-Line Semimetal HfSiS.

We report a study of quantum oscillations in the high-field magnetoresistance of the nodal-line semimetal HfSiS. In the presence of a magnetic field up to 31 T parallel to the c axis, we observe quantum oscillations originating both from orbits of individual electron and hole pockets, and from magnetic breakdown between these pockets. In particular, we reveal a breakdown orbit enclosing one electron and one hole pocket in the form of a "figure of eight," which is a manifestation of Klein tunneling in momentum space, although in a regime of partial transmission due to the finite separation between the pockets. The observed very strong dependence of the oscillation amplitude on the field angle and the cyclotron masses of the orbits are in agreement with the theoretical predictions for this novel tunneling phenomenon.

Tuning the valley and chiral quantum state of Dirac electrons in van der Waals heterostructures.

Chirality is a fundamental property of electrons with the relativistic spectrum found in graphene and topological insulators. It plays a crucial role in relativistic phenomena, such as Klein tunneling, but it is difficult to visualize directly. Here, we report the direct observation and manipulation of chirality and pseudospin polarization in the tunneling of electrons between two almost perfectly aligned graphene crystals. We use a strong in-plane magnetic field as a tool to resolve the contributions of the chiral electronic states that have a phase difference between the two components of their vector wave function. Our experiments not only shed light on chirality, but also demonstrate a technique for preparing graphene's Dirac electrons in a particular quantum chiral state in a selected valley.