Electronic and magnetotransport properties of twisted bilayer graphene in the presence of external electric and magnetic fields

We investigate extensively the electronic and transport properties of twisted bilayer graphene when subjected to both an external perpendicular electric field and a magnetic field. Using a basic tight-binding model, we show the flat electronic band properties as well as the density of states (DOS), both without and with the applied electric field. In the presence of an electric field, the degeneracy at the Dirac points is lifted and a nonmonotonic behavior is found in the energy shift between the Dirac points of each layer, especially for twist angles below ${3}^{\ensuremath{\circ}}$. We also study the behavior of the Landau levels (LL) spectra for different twist angles within a very low energy range. These LL spectra get modified under the influence of the external electric field. Moreover, we calculate the dc Hall conductivity (${\ensuremath{\sigma}}_{xy}$) for a very large system using the kernel polynomial method. Interestingly, as we tune the twist angle from a higher to a lower value, ${\ensuremath{\sigma}}_{xy}$ undergoes a transition from a half-integer to an integer quantum Hall effect, i.e., the value of ${\ensuremath{\sigma}}_{xy}$ shifts from $\ifmmode\pm\else\textpm\fi{}4(n+1/2)\phantom{\rule{0.28em}{0ex}}(2{e}^{2}/h)$ ($n$ is an integer) to $\ifmmode\pm\else\textpm\fi{}2n\phantom{\rule{0.28em}{0ex}}(2{e}^{2}/h)$ around a small twist angle of $\ensuremath{\theta}=2.{005}^{\ensuremath{\circ}}$. At this angle, ${\ensuremath{\sigma}}_{xy}$ acquires a Hall plateau at zero Fermi energy. However, the behavior of ${\ensuremath{\sigma}}_{xy}$ remains unaltered when the system is exposed to the electric field, particularly at the magic angle where the bands in both layers can hybridize, and strong interlayer coupling plays a crucial role.