Beurling densities of regular spectra of self-similar spectral measure with consecutive digit sets

Beurling density plays a key role in the study of frame-spectrality of normalized Lebesgue measure restricted to a set. Accordingly, in this paper, the authors study the $s$-Beurling densities of regular maximal orthogonal sets of a class of self-similar spectral measures, where $s$ is the Hausdorff dimension of its support and obtain their exact upper bound of the densities

Gate-Tunable Tunneling Resistance in Graphene/Topological Insulator Vertical Junctions

Graphene-based vertical heterostructures, particularly stacks incorporated with other layered materials, are promising for nanoelectronics. The stacking of two model Dirac materials, graphene and topological insulator, can considerably enlarge the family of van der Waals heterostructures. Despite well understanding of the two individual materials, the electron transport properties of a combined vertical heterojunction are still unknown. Here we show the experimental realization of a vertical heterojunction between Bi2Se3 nanoplate and monolayer graphene. At low temperatures, the electron transport through the vertical heterojunction is dominated by the tunneling process, which can be effectively tuned by gate voltage to alter the density of states near the Fermi surface. In the presence of a magnetic field, quantum oscillations are observed due to the quantized Landau levels in both graphene and the two-dimensional surface states of Bi2Se3. Furthermore, we observe an exotic gate-tunable tunneling resistance under high magnetic field, which displays resistance maxima when the underlying graphene becomes a quantum Hall insulator