The Study Of The Surface State Of The Low Dimensional System

Study on surface states(edge states) is an old but interesting field. Because of the existence of the surface, translational invariant symmetry is destroyed, there are unique physical and chemical properties on the surface. There are a lot of applications on this field, for example, the epitaxial on crystals, electronic devices, optoelectronic de-vices and so on. Emerged in recent years the quantum Hall effect, quantum spin Hall effect, topological insulators, optical waveguide, etc. all related to the study on the surface. In this essay, we give an in-depth discussion on1) the non-existence of the edge states in semi-infinite armchair edged graphene,2) the inhomogeneous hopping interaction on the zigzag graphene nanoribbon, and the effect of inhomogeneous inter-action, compared with that in the bulk, on the edge states of the semi-infinite zigzag edged graphene sheet,3) the edge states in the similar model with1-D strong-weak coupling binary material.In1st Chapter, we outline the research on the progress of surface states. Begin with the emergence of hall effect, hall spin effect, topological insulators, at first we give a brief review of them in the chronological list, and then give an introduction on each of them respectively. As we know, graphene has a unique energy spectra, and new physical phenomena provided by it play an important role in the above fields, besides, it is one of our researches, so we would like to show the energy spectra of it as the second part; finally, we give an introduction on the surface electronic states, including the surface electric states and the their detections, and the Tamm state&Shockley state in details, and at last the surface plasmon, which is one of our topic in the near future.In2nd Chapter, we study the homogeneous graphene system. We employ the transfer matrix method in homogenous semi-infinite armchair edged graphene sheet, and give an analytical proof of the non-existence of the edge states in such a system rigorously, and besides, we give a study review on this topic by others, who employ other ways.In3rd Chapter, we study the inhomogeneous semi-infinite zigzag edged graphene model. At first, we show a tight-binding solution of the energy spectra in the4chains zigzag graphene nanoribbon, with the homogeneous next-nearest-neighbor interaction and inhomogeneous nearest-neighbor interaction; then we study a model, similar to those by Guinea, Pereira, and we just consider about a different hopping interaction at the edge in our model, and try to find out the condition for the existence of the edge states, including both the zero-energy and non-zero-energy edge states.In4th Chapter, we study the edge states in a semi-infinite1D photonic model, similar to a binary metal-dielectric coupling system. We consider a defect at the edge, and try to find out the relation between the parameters and the edge states in this system.