Electronic and magnetic properties of two dimensional crystals

In the last few years, two dimensional crystals have become available for experimental studies. Good examples of such systems are monolayers and bilayers of graphene and monolayers of transition metal dichalcogenides such as MoS₂ and WSe₂. The availability of two dimensional crystals has encouraged physicists to study the electronic and magnetic properties of such systems. This thesis adds to the theoretical knowledge about electronic and magnetic properties of two dimensional crystals with the focus on graphene and MoS₂. As a general theme in this thesis, we calculate how in general these systems interact with electric and magnetic fields and what their response is to such stimuli. In particular, we have studied the response of monolayer graphene to an in-plane electric field. We have also looked at spin-orbit coupling effects that arise from applying perpendicular or in-plane external electric fields, especially their consequences for transport properties of bilayer graphene. We investigated the electronic properties of charge carriers confined in a mesoscopic ring structure using a gate voltage in bilayer graphene. We also showed how spin-orbit coupling can affect the electrical properties of such rings. We found how spin-orbit coupling can affect the transport properties in bilayer graphene. We also investigated the RKKY or indirect exchange coupling between magnetic moments in monolayer MoS₂ through calculating wave vector dependent spin susceptibility. We examined the electronic properties of electrons and holes confined electrostatically into a bilayer graphene ring. We presented an analytical solution for finding energy levels in the ring. We showed that the magnetic field dependence of the lowest energy level with fixed angular momentum in bilayer graphene rings, in contrast to usual semiconductor quantum rings, is not parabolic but displays an asymmetric “Mexican hat“. We found that introducing spin-orbit coupling in the ring can flatten this Mexican hat. We studied the effect of an orbital Rashba type effect, induced by an in-plane electric field in monolayer graphene. Using perturbation theory, we showed that this term can affect the energy levels in a crossed electric and magnetic field such that the electron and hole levels repel each other. We calculated the AC transport of monolayer graphene in the linear-response regime and showed that taking the orbital Rashba term into account casts doubt on the universality of the minimum conductivity of monolayer graphene. We studied the effect of spin-orbit coupling in transport properties of bilayer graphene systems by calculating tunnelling through npn and np junctions. We showed that at sufficiently large spin-orbit strength, normal transmission through a barrier which is forbidden in bilayer graphene becomes finite. We predict that in a weak Rashba spin-orbit regime, outgoing electrons show signals which are spin polarized. We also showed that considering spin-orbit coupling only in the barrier of an npn junction can invert the spin of the incoming electrons. Finally, we obtained analytical expressions for the wave vector-dependent static spin susceptibility of monolayer transition metal dichalcogenides, considering both the electron-doped and hole-doped cases. These results are then applied to the calculations of physical observables of monolayer MoS₂. We claculated that the hole-mediated RKKY exchange interaction for in-plane impurity-spin components decays with a different power law from what is expected for a two-dimensional Fermi liquid. In contrast, we calculated that the out-of-plane spin response shows the conventional long-range behaviour.