Behaviors Of Confined Dirac Electron States In Graphene And Silicene Under External Fields

Graphene and silicene are typical of relativistic condensed matter physics,candidates of future quantum devices for their excellent physical characteristics and ascarriers in them obey relativistic quantum mechanical equation, i.e., Dirac equation,they are platforms for quantum electrodynamics. In this thesis, to study the confinedstates of Dirac equation under electric, magnetic and mass fields, we developedsectionized series expansion method to solve Dirac equation under external fields, andinvestigate the characteristics of energy spectra of confined Dirac electron states undermagnetic and electric fields in graphene. Quantum dots and Klein tunneling effect insilicene are also discussed.First, based on sectionized series expansion method under Schr dinger equation,we are able to obtain the exact solutions of Dirac equation in various magnetic fieldswith central symmetry.We discuss the confinement of Dirac electron in inhomogeneousmagnetic fields, and taking graphene magnetic quantum dots (rings) formed byinhomogeneous magnetic fields as examples, we investigate the effect of theinhomogeneity of magnetic field om Landau levels. We show that the inhomogeneitycan lift the infinite degeneracy of Landau levels, and result in adjustable energy levelordering.When other fields such as electric field enter into Dirac eqution, Dirac equation cannot be converted into the forms like Schr dinger equation, which causes it moredifficult to solve. For these issues, firstly, we develop the sectionized series expansionmethod for decoupled Dirac equation, and obtain the positive energy spectra of Diracdonor states of graphene in magnetic fields. To acquire the positive and negative energyspectra of Coulomb impurity states in magnetic fields, we propose and implement thecoupled sectionized series expansion method to Dirac equation, and can obtain the wavefunctions and complete sperctra of Dirac equation under electric, magnetic and massfields. In the derivation of the asymptotic solution to Dirac equation, we obtain thecriterion of confined-deconfined Dirac electron states under external fields. Byemploying this method, we investigate the spectrum structure of Coulomb impuritystates under magnetic field in graphene, and its relations with electron mass, magneticfield, Coulomb field and angular momentums. By comparing the energy spectra of the valleys K and K ’, we show that the valley degeneracy of Coulomb impurity statesunder magnetic field for massive Driac electron is lifted. By exact diagonalization, wediscuss the strength of valley mixing. Furthermore, in graphene, we investigate thecombined effect of uniform magnetic field and linear electric field on the confinementof Dirac electron, and concude that when the strength of electric field is smaller thanthat of magnetic field, Dirac electron is confined; otherwise, Dirac electron isdeconfined and Klein tunneling occurs.Finally, in consideration of the fact that perpendicular electric fieldthrough silicene generates mass field, we propose that inuniform perpendicularelectric field forms silicene quantum dots, where Dirac electron can be confined.By investigating the shape-size effect and magnetic field effect of this kind ofsilicene quantum dots, we demonstrate that mass field and magnetic field canmodulates confined Dirac electron states in silicene. Then, taking parabolicmass field and in-plane electric field as an example, through the evolution of thespecra, we observe the Klein tunneling effect. We find that at the transitionpoint of confinement-deconfinement of Dirac electron states, the positive energystates collapse to discrete energy levels, but the negative energy states allcollapse to zero, which is verified by analytical solution.