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Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. When subject to both a magnetic field and a periodic electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recursive energy spectrum1.Known as Hofstadter's butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields1-10, and is one of the first quantum fractals discovered in physics. In the decades since its prediction, experimental attempts to study this effect have been limited by difficulties in reconciling the two length scales. Typical atomic lattices (with periodicities of less than one nanometre) require unfeasibly large magnetic fields to reach the commensurability condition, and in artificially engineered structures (with periodicities greater than about 100 nanometres) the corresponding fields are too small to overcome disorder completely11-17.Herewe demonstrate thatmoirésuperlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodicmodulationwith ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom.
(ProQuest: ... denotes formulae omitted.)
The total number of electron states per area of a completely filled Bloch band is n051/A,whereAis the area of the unit cell of the periodic potential. In a magnetic field, B, the number of states per area of each filled Landau level is given by B/w0, where w05h/e is the magnetic flux quantum (h, Planck's constant; e, magnitude of the electron charge). The quantumdescription of electrons subjected simultaneously to both a periodic electric field and a magnetic field can be simply parameterized by the dimensionless ratio w/w0, where w5BA is the magnetic flux per unit cell. The general solution, however, exhibits a rich complexity due to the incommensurate periodicities of the Bloch and Landau states18. For commensurate fields, corresponding to rational values of...