Numerical Instability Of The Transfer Matrix Method And Transport Properties Of Graphene

Transfer matrix method has wide applications in physics. The method can be used in the studies of Ising model, quantum spin systems, electrical transport, the band calculations of periodic systems, periodic and aperiodic chains, electromagnetic wave propagation, optical propagation in multilayer system, elastic wave propagations and so on. Transfer matrix method also has wide applications in other fields, such as en-gineering and economics. However, the method has the shortcoming of its own, i.e., the numerical instability. This is a problem that the data memory of computer will overflow when the number of transfer steps is beyond a critical value. In this thesis, we present a new improved transfer matrix method algorithm, which can overcome the numerical instability effectively. It will be shown that the new improved method has the advantage that the computing time is the zeroth order of the system length that we deal with. The thesis also gives the comparison between the new improved method and other numerical instability avoiding method, such as the scattering matrix method, the computing time of which is the first order of the system length.Graphene is a single layer of graphite. Owing to its strict 2D material properties, peculiar relativistic electron behaviors and great potential for applications, graphene has stirred great interest and thorough studies since its realization in 2004. In this thesis, we study the effect of a single vacancy to the transport properties of metallic armchair graphene nanoribbon. It is found that the different position of the vacancy can induce very different transport behavior response. Further more, we will explain such response behavior from the point view of local density of states. Then we discuss the transport properties of graphene nanoconstriction, and study the role of localized edge state at the zigzag boundary to the whole structure. At last, we study the the transport properties of T-type graphene structure systematically, both armchair edged and zigzag edged.The thesis will be organized as follows:Chapter One introduces the necessary formulae and basic knowledge that we use in the thesis, such as the Landauer formula, tight-binding model and finite difference method. And then the basic properties of graphene is introduced.Chapter Two will explain what is transfer matrix, and what's the origin of numer- ical instability? And then put forward the new improved transfer matrix method. As an example of application, we use the new improved method to study the spin polarization distribution of a bar system with Rashba spin-orbital interaction. With the comparison of new improved method and scattering matrix method, we show that the computing time of new improve method is the zeroth order of system length, whereas the scattering matrix method is the first order.In Chapter Three, we first study the effect of a single vacancy to the transport properties of metallic armchair graphene nanoribbon, and make the investigation from the point view of local density of states distributions. Then we discuss the transport properties of graphene nanoconstriction to study how the localized edge state at the zigzag boundary affect the transport properties of the whole structure.In Chapter Four, we study the the transport properties of T-type graphene structure systematically, both armchair edged and zigzag edged. It is shown that for the armchair edged one, there is a perfect linear response between the shift of conductance gap and the applied gate voltage, which reveals a good tunability of the conductance gap by the gate voltage.At last, Chapter Five is the appendix. The main formulae of scattering matrix are given. These formulae will be used in chapter two.