Recent experiments indicate that AA-stacked bilayer graphenes (BLG) could exist. Since the energy bands of the AA-stacked BLG are different from both the monolayer and AB-stacked bilayer graphenes, different integer quantum Hall effect in the AA-stacked graphene is expected. We have therefore calculated the quantized Hall conductivity σxy and also longitudinal conductivity σxx of the AA-stacked BLG within the linear response Kubo formalism. Interestingly, we find that the AAstacked BLG could exhibit both conventional insulating behavior (theν = 0 plateau) and chirality for |μ| < t, whereν is the filling factor (ν = σxyh/e 2 ),μ is the chemical potential, and t is the interlayer hopping energy, in striking contrast to the monlayer graphene (MLG) and AB-stacked BLG. We also find that for |μ| =where n1 = 1, 2, 3, · · ·, n2 = 2, 3, 4, · · · and n2 > n1, the Hall conductivity is quantized as σxy = ± 4e 2 h n, n = 0, 1, 2, · · ·, if |μ| < t and σxy = ± 4e 2 h n, n = 1, 2, 3, · · ·, if |μ| > t.theν = ±4(n1 + n2)n plateaus are absent, where n = 1, 2, 3, · · ·, in comparison with the ABstacked BLG within the two-band approximation. We show that in the low-disorder and highmagnetic-field regime, σxx → 0 as long as the Fermi level is not close to a Dirac point, where Γ denotes the Landau level broadening induced by disorder. Furthermore, when σxy is plotted as a function ofμ, aν = 0 plateau appears acrossμ = 0 and it would disappear if the magnetic field B = πt 2 /N ehυ 2 F , N = 1, 2, 3, · · ·. Finally, the disappearance of the zero-Hall conductivity plateau is always accompanied by the occurence of a 8e 2 /h-step atμ = t.

We study the transport properties of a graphene ferromagnet-insulator-superconductor (FIS) junction within the Blonder-Tinkham-Klapwijk formalism by solving spin-polarized Dirac-Bogoliubovde-Gennes equation. In particular, we calculate the spin-polarization of tunneling current at the I-S interface and invesigate how the exchange splitting of the Dirac fermion bands influences the characteristic conductance oscillation of the graphene junctions. We find that the retro and specular Andreev reflections in the graphene FIS junction are drastically modified in the presence of exchange interaction and that the spin-polarization (PT ) of tunneling current can be tuned from the positive to negative value by bias voltage (V ). In the thin-barrier limit, the conductance G of a graphene FIS junction oscillates as a function of barrier strength χ. Both the amplitude and phase of the conductance oscillation varies with the exchange energy Eex. For Eex < EF (Fermi energy), the amplitude of oscillation decreases with Eex. For E c ex > Eex > EF , the amplitude of oscillation increases with Eex, where E c ex = 2EF + U0 (U0 is the applied electrostatic potential on the superconducting segment of the junction). For Eex > E c ex , the amplitude of oscillation decreases with Eex again. Interestingly, a universal phase difference of π/2 in χ exists between the G − χ curves for Eex > EF and Eex < EF . Finally, we find that the transitions between retro and specular Andreev reflections occur at eV = |EF − Eex| and eV = Eex + EF , and hence the singular behavior of the conductance near these bias voltages results from the difference in transport properties between specular and retro Andreev reflections.

Josephson effect ever stimulated a lot of activity in quantum Hall bilayer exciton condensates since the observation of Josephson-like tunneling and recently even receives great attention in optically excited exciton-polariton condensates. The intense interest is originated from that Josephson effect can be regarded as one of the most striking manifestations of exciton or exciton-polariton condensates. Following the proposal by Park and Das Sarma [Phys. Rev. B 74, 035338 ( 2006)], we study the excitonic Josephson junction with relative phases induced by interlayer tunneling currents. Our results indicate the novel coupling of edge tunneling currents [Phys. Rev. Lett. 109, 156802 (2012)] can be explained by tunneling-current induced Josephson effect and is a very robust evidence for exciton condensation. Also, we furthermore suggest to detect Josephson current by measuring induced magnetic field of a ring-shape excitonic Josephson junction.