The density of states and differential entropy per particle are analyzed for Dirac-like electrons in graphene subjected to a perpendicular magnetic field and an in-plane electric field. For comparison, the derived density of states is contrasted with the well-known case of nonrelativistic electrons in crossed magnetic and electric fields. The study considers ballistic electrons and also includes the effect of small impurity scattering. In the latter case, the limit of zero magnetic field and the so-called collapse of Landau levels in graphene are examined analytically. By comparing the results with numerical calculations on graphene ribbons, we demonstrate that the Landau state counting procedure must be modified for Dirac-like electrons, leading to a fields-dependent Landau level degeneracy factor. Additionally, it is shown that peaks in the differential entropy arise from the dispersionless surface mode localized at the zigzag edges of the ribbon.

In gapped bilayer graphene, similarly to conventional semiconductors, Coulomb impurities (such as nitrogen donors) may determine the activation energy of its conductivity and provide low-temperature hopping conductivity. However, in spite of the importance of Coulomb impurities, nothing is known about their electron binding energy Eb in the presence of gates. To close this gap, we study numerically the electron binding energy Eb of a singly charged donor in BN-enveloped bilayer graphene with the top and bottom gates at distance d and gate-tunable gap 2Δ. We show that for 10<d<200nm and 1<Δ<70meV the ratio Eb/Δ changes from 0.4 to 1.4. The ratio Eb/Δ stays so close to unity because of the dominating role of the bilayer polarization screening which reduces the Coulomb potential well depth to values ∼Δ. Still, the ratio Eb/Δ somewhat decreases with growing Δ, faster at small Δ and slower at large Δ. On the other hand, Eb/Δ weakly grows with d, again faster at small Δ and slower at large Δ. We also studied the effect of trigonal warping and found only a small reduction of Eb/Δ.

We develop a theory of magnetic breakdown (MB) near high-order saddle points in the dispersions of two-dimensional materials, where two or more semiclassical cyclotron orbits approach each other. MB occurs due to quantum tunneling between several trajectories, which leads to nontrivial scattering amplitudes and phases. We show that for any saddle point this problem can be solved by mapping it to a scattering problem in a 1D tight-binding chain. Moreover, the occurrence of magnetic breakdown on the edges of the Brillouin zone facilitates the delocalization of the bulk Landau level states and the formation of 2D orbit networks. These extended network states compose dispersive mini bands with finite energy broadening. This effect can be observed in transport experiments as a strong enhancement of the longitudinal bulk conductance in a quantum Hall bar. In addition, it may be probed in STM experiments by visualizing bulk current patterns.