Evidence for electron localisation in a moir\'e-of-moir\'e superlattice
The localisation of electrons in a lattice potential is an quantum-mechanical phenomenon and is often associated with remarkable physical properties of solids involving electron spins, electric polarisations and topological effects. In particular, even a small amount of distortion of the lattice potential can localise otherwise-delocalised quantum states in low-dimensional electron systems, dramatically influencing their thermodynamic properties and charge-transport behaviour. Study of such electron localisation induced by an aperiodic lattice potential remains exceptionally challenging in solid-state systems, since extrinsic disorders can trivially trap electrons in potential minima near disorders, obscuring the underlying quantum-mechanical origin of localisation phenomena. Van der Waals heterostructures can provide an alternative route for explorations of the phenomena via the emergence of superlattice potentials generated by rotating and stacking individual layers. Here, we report strong signatures of electron localisation in helical trilayer graphene, where the interplay of two moir\'e patterns gives rise to a moir\'e-of-moir\'e superlattice with distinct regions of moir\'e-periodic and moir\'e-aperiodic potentials. Remarkably, our measurements reveal the presence of double moir\'e-induced bands and high-order Brown-Zak oscillations, which are direct reflections of the periodic region with two constituent moir\'e patterns, and a superimposed anomalous hysteretic signal attributable to the aperiodic region. The data strongly suggest that electron wave functions are partially localised driven by the loss of a periodic lattice potential. Our work provides insight into the effects of spatially inhomogeneous lattice potentials on the low-dimensional electronic states and introduces a promising approach to control electron localisation for practical applications in solid-state devices.