Electronic Thesis and Dissertation Repository


Doctor of Philosophy


Applied Mathematics


Dr. Vladimir A. Miranskyy


The focus of this thesis is the dynamical symmetry breaking in monolayer and bilayer graphene in a magnetic field and the edge excitations in these systems. Due to the spin and valley degrees of freedom, the Coulomb interaction in graphene can lead to various broken-symmetry quantum Hall phases. The transport properties of each phase are affected by the low-energy edge excitations, and understanding the edge state properties can be crucial for identifying the true nature of the ground state. We study edge states in biased bilayer graphene in a magnetic field within the four-band continuum model. The analysis is done for the semi-infinite graphene plane and for the graphene ribbon of a finite width, in the cases of zigzag and armchair edges. Exact dispersion equations for the edge states and analytic expressions for their wave functions are written in terms of the parabolic cylinder functions. The spectrum of edge states for each type of the boundary conditions is found by numerically solving the corresponding dispersion equations. The low-energy modes localized at zigzag edges are explored in detail. In the case of monolayer graphene, we study edge excitations of the nu=0 quantum Hall state within the mean-field theory with different symmetry breaking terms. The analytical expressions for the continuum (Dirac) model wave functions are obtained for the charge density wave, Kekule distortion, ferromagnetic and (canted) antiferromagnetic phases. The dispersion equations for each phase and boundary type (zigzag and armchair) are derived, numerically solved and compared to the results of the corresponding effective tight-binding model. The effect of the next-to-nearest neighbor hopping parameter on the edge state spectrum is studied and revealed to be essential. The criteria for the existence of gapless edge states are established for each phase and edge type. Finally, we study different broken-symmetry quantum Hall phases in bilayer graphene with even filling factors. The gap equation is solved in the lowest Landau level approximation using the long-range screened Coulomb potential as well as the general form of the short-range interaction terms. Phase transitions driven by changing the external electric field and tilting the magnetic field are described.