<p>We study magneto-tunnelling between two parallel two-dimensional electron gases theoretically, where the electrons have a pseudo-spin-½ degree of freedom that is coupled to their momentum. The two-dimensional electron gases focused on in this work are single layer graphene, bilayer graphene, and single layer molybdenum disulphide. The results are derived using a linear response theory formalism in the weak tunnelling regime, and it is assumed that the electron gases are at zero temperature, with no interactions or disorder. The linear magneto-tunnelling conductance characteristics for an applied in-plane and tilted magnetic field are found to strongly depend on the pseudo-spin structure of the tunnelling matrix and the pseudo-spin's dependence on momentum. For instance, resonances in the linear magneto-tunnelling conductance are sensitive to the pseudo-spin tunnel-coupling across the barrier and how the pseudo-spin eigenstates are coupled to momentum. We discuss how measurements of the magneto-tunnelling conductance can be applied as a spectroscopic tool. We explain how to measure the pseudo-spin tunnel-coupling through least squares parameter fitting of the magneto-tunnelling conductance. We show that the parameters are interdependent, one can use the interdependency to test the consistency between theory and experiment. It is expected that measurements of pseudo-spin tunnel-coupling will be a function of the lattice structure of the double layer system, which suggests these measurements can be used as a spectroscopic tool. Additionally, we investigate in-plane electric fields in single layer graphene to see if their effects can be observed in magneto-tunnelling transport. Then, we perturbatively include the effects of electron-electron interactions in single layer graphene, and find it should dampen the linear tunnelling conductance. We investigate tunnel-coupled , parallel , single layer and bilayer graphene systems. We find that using an in-plane magnetic field, one can generate a valley polarized tunnelling current. This method is unique because it does not require manipulation of the single and bilayer graphene samples through nano-structuring, coupling to electromagnetic fields, application of mechanical strain, or the presence of defects. In particular, the valley polarization is dependent on the pseudo-spin tunnel-coupling between the single and bilayer graphene systems, and the strength of an applied in-plane magnetic field. We explicitly show through analytic derivations how an understanding of linear magneto-tunnelling transport (zero bias limit) can be used to understand non-linear magneto-tunnelling transport (finite bias).</p>
Aryl Functionalization as a Route to Band Gap Engineering in Single Layer Graphene Devices