Graphene superlattices in van der Waals heterostructures

The technological development of graphene has generated new high-quality systems offer access to the earlier inaccessible extremes of quantum physics. When graphene is placed on an atomically flat substrate with hexagonal lattice with a close lattice constant, such as boron nitride (hBN), and their crystalline axes are aligned, a long-wavelength perfectly periodic moir\'{e} pattern forms for electrons in graphene. Various regimes of possible moir\'{e} minibands at zero magnetic field [Phys. Rev. B 87, 245408 (2013); Phys. Rev. B~88, 205418; Phys. Rev. B 88, 155415 (2013); New J. Phys, 15, 123009 (2013)] and strong magnetic field [Nature 497, 594 (2013), Nature Physics 10, 525 (2014); Phys Rev B 89, 075401 (2014)] will be discussed. Experimentally available magnetic fields are enough to provide flux $\varphi $ through the moir\'{e} superlattice cell comparable to the magnetic flux quantum $\varphi_{0}$ and reach the regime of fractal Hofstadter spectra. As a result, a single device can offer a multiplicity of two-dimensional electron systems, realised at rational flux values $\varphi = \varphi_{0}$, $\varphi_{0}$/2, 2$\varphi_{0}$/3, etc., each with its own intricate topological properties, including quantum Hall effect physics related to the effective Landau levels emerging from these magnetic minibands at the nearby range of magnetic fields.