The Fractal Study Of Transport Properties In Mesoscopic Billiard

Mesoscopic billiard system is a kind of billiard system that is between macro and microsystem, which is the ideal model for studying the mesoscopic device. The boundary ofmesoscopic billiard system determines its mechanical properties in a large degree, so it has aspecial advantage in researching classical chaos, semiclassical and quantum mechanics.Usually the mesoscopic billiard displays the following properties: periodic or chaos of thesystem; the chaos is the soft chaos or hard chaos. In the second chapter we use the phasespace to qualitatively analyze the periodic and chaos of the system. If there are periodic areasand chaotic areas in phase space, the corresponding chaos is the soft chaos. If there arechaotic areas in phase space only, the corresponding chaos is the hard chaos. The system westudy is generally the soft chaos.Fractal theory is an important branch theory of nonlinear science, the main study of thenonlinear system is about the self similarity phenomenon. In the research, fractal is oftenlinked together with Chaos. In some forms, fractal can be regarded as a means to research thechaos, to help people recognize the whole from the part of chaos. Some chaos is a kind ofstrange attractor; this strange attractor has an irregular, aperiodic, perplexing, self-similarstructure characteristic, which is fractal. When we study the actual billiard ball device,electron can not be confined to a specific closed region, and the system must be an open one,so we need to study the transport problem of the particles in the micro cavity. There will be afar-reaching significance to study the transport properties of the micro cavity structuresimulated by billiards. So, in this paper, we select an open billiard, which is the theory modelof the mesoscopic devices, to research its transport properties.With the development of lithography and the growth of crystal in recent twenty years, it ispossible to produce all kinds scale and random shape billiard system. Compared with the thinmetal membrane, the electron density of this theoretical model is relatively low, under lowtemperature, this kind of electron has big Feimi wavelength (up to 40nm), and big electronmean free path (up to 10um), meanwhile the interactions between electrons can be ignored. Inaddition, the interactions between electron and impure particles are very small, so this kind ofelectron can be regarded as classical particles. The similarity of quantum transport characteristics in all kinds of scale is induced by fractal structure in the underlying classicalmixed phase space. So, according to the classic theory, it is possible to study the fractal oftransport properties in the mesoscopic billiard by numerical calculation. In this paper, weselect an open Sinai billiard system as the theoretical model, and study the fractal of transportproperties by numerical statistics for the first time. In addition, we use box counting methodto calculate the fractal dimension, which is discussed in detail to prove fractal properties.This paper has four chapters. The first chapter is the summary, we mainly introduce somebasic concepts, such as fractal, mesoscopic, fractal dimension, and that the significance of theselected topic is mentioned. In the second chapter, we introduce the dynamics properties of aseries of quantum billiards using the Poincare section in the phase space, for the sake offinding the corresponding relationship between quantum and classical mechanics. In the thirdchapter, we have studied the quarter Sinai billiards using quantum and classical methods,analyzed the fractal of transport properties with numerical calculation, and discussed theeffect of the gate voltage for the system through the fractal dimension. In addition, we haveselected an open Sinai billiard as a model to study the fractal properties, and analyze thechaos of the system and the effect of the openings for the transport with the fractal dimension.The fourth chapter is about the summaries and the prospects.