Confined Dirac Electron Structures Of Monolayer And Bilayer Graphene In External Fields

Recently, Dirac electron materials become a hot topic in condensed matter physics. In this thesis, we study confined Dirac electronic structures and their modulations by external fields in monolayer and bilayer graphene. The relativistic behaviors of confined Dirac electron are revealed and compared with the traditional Schr? dinger electron. We developed and improved the sectionized series expansion method(SSE) of our group to provide an effective method to solve eigenvalues of coupled differential equations in complex external fields.First, we study effects of hydrogenic impurity on monolayer graphene magnetic quantum dots. The impurity breaks the symmetry of spectra in quantum dots. The positive and negative energy states are attracted and repelled by the impurity, respectively. The zero-energy levels become nondegenerate and their corresponding states split into hole-like states. Varying magnetic field and dot size, we discussed the competitions between impurity and magnetic dot. The effects of impurity can also be confirmed by the corresponding electron probability densities and binding energies.Second, we address atomic collapse and its magnetic modulation in gapped monolayer graphene. Introducing a finite core radius, we can obtain the complete supercritical Coulomb impurity spectra and accurate wave functions. The behaviors of atomic collapse in graphene are compared with those in the superheavy atom. When a magnetic field is applied, atomic collapse can be effectively restrained, and the magnetic modulations are remarkably different in valley K and K'. The Landau levels also exhibit different changes in supercritical and subcritical impurity potentials. By studying mass and core-size effects, we found that gapped graphene is an ideal platform to observe atomic collapse and its magnetic modulation.Third, we study the magnetic quantum dot and ring in bilayer graphene. The four-dimensional SSE method is firstly used to solve the Landau levels in bilayer graphene, which can confirm that the method is precise. Then we obtained accurate spectra of quantum dot and ring formed by perpendicular inhomogeneous magnetic fields. The Landau levels(except zero-energy levels) become nondegenerate and split into discrete angular momentum states. We discussed the magnetic and size effects, combining with the electron probability densities. These results are different from those of traditional quantum dots and rings.Finally, we address the quantum capacitance of a bilayer graphene device in the presence of Rashba spin-orbit interaction(SOI) by applying external magnetic fields and interlayer biases. Quantum capacitance reflects the mixing of the spin-up and spin-down states of Landau levels, and can be effectively modulated by the gate and interlayer bias. We discussed the magnetic oscillation of quantum capacitance in detail. Rashba SOI is subject to form beating patterns and interlayer bias can notably suppress beating patterns. We also explored the thermal effects on quantum capacitance under different temperature, which is useful for the application of device. These results are agree with the corresponding experiments.