The Transport Properties Of The Two-dimensional Billiards

Nanostructure is an artificial micro-structure, which scales in the nanometer range. Recent years, many people have observed a series of interesting physical phenomena in this nanostructure. With lithography and crystal growth technology maturing, arbitrary production of small-scale shaped nanostructures has become a reality in recent years. In the last decade of the twentieth century, semiconductor materials and semiconductor manufacturing process technology have developed significantly, so the semiconductor nanostructures become the ideal model of electronic conduction in the micro-cavity structure. The theoretical model of which we call two-dimensional electron gas system is also called quantum billiard system.Quantum billiards in comparison with the thin metal films have the advantage of lower electron density at low temperatures, but also by changing the field strength to control electron density. Lower electron density will have greater long-Fermi (which can achieve 40nm) and large electron mean free path (which can achieve 10μm), it provides space for the movement of electrons and the interaction between the electrons can be ignored, and the effect of impurity particle and phonon scattering is little. The electrons which move freely are the classical particle, so the conductance of transport properties will be very convenient to study in quantum billiards.Generally, the boundaries of two-dimensional quantum billiard system determine the nature of the shape, as long as change the geometry of billiards, we can better control the movement of particles in the system. When the shape of billiards is regular, for example, circular billiards, square billiards and ellipse billiard systems, these systems are capable of undergoing integration, in these billiards the movement of particles is regular. When we study the system, we can not find the analytical solution, particles moving in these billiards are irregular and show a chaotic characteristic, these systems are not integral system, such as stadium billiards, Sinai billiards. Generally, the analysis results can not be obtained for the quantum chaos systems, one can only obtain the numerical results by the numerical method to solve the Schr?dinger equation, because the numerical method requires large amounts of datum and parameters, and the algorithm is extremely complicated, we only get the approximate solution.In the last century, with the study of various methods of quantum billiards produced, Landauer conjecture the conductance was obtained by calculating the transfer matrix, and the sum of all transmission coefficients. Then Du Mengli, Delos and other researchers in the study of Rydberg atoms optical absorption spectra in strong electromagnetic fields, they took out the closed orbit theory based on the Gutzwiller trace formula. This theory promoted the development of quantum billiards, and it provided the principle to study the correspondence between classical physics and quantum physics in quantum billiard system, the closed orbit theory is called the only bridge between the classical world and quantum world. The classical physics and quantum mechanics correspondence has gone through a long history. When the quantum mechanics was born, Plank and Einstein was puzzled by the disagreement between black-body radiation and classical physics. Then Hersenberg established a quantum method to understand classical mechanics, and the quantum system was finished when the Schr?dinger equation was provided by Schr?dinger. Later Gutzwiller developed the semi-classical method and the closed orbit theory was established by Du and J.B. Delos. When the quantum phase-space theory was put forward, the classical-quantum correspondence was developed quickly.In this issue, we start from the study of quantum and classical correspondence using the semi-classical orbit theory, and studied the transport properties of the two-dimensional circular quantum billiard in weak magnetic fields. We obtain the transmission coefficients of conductivity by the interaction of eigenfunctions and semi-classical approximation Green's function. At the same time, we obtained quantum spectra though Fourier transform to the transfer matrix element. Then we calculate classical trajectories of the electron, and compare the quantum spectrum peak positions of transmission matrix element with the length of electronic classical orbit. The results show that the peak positions coincide well with the track length,and prove that the semi-classical theory is the bridge connecting classical and quantum physics. We not only discuss the model of lead levels but also studied the case of vertical wires, and observed that they have similar properties.In recent years, some of the difficulties are encountered in classical mechanics, and solved when we use the fractal theory. Self-similarity is an important property of fractal theory, which means that the local shape curve is similar to the overall shape, although magnify many times, the shape of the curve remain the same. The important is that fractal theory is a bridge connecting portion and the whole. It made people realize that the relationship between the whole and the parts not linear but nonlinear.We studied the circular and annular billiards using the fractal and phase space methods, found that the electron motion in circular billiards is regular movement, the chaos does not exist. However, annular systems are irregular billiards, the electron motion is chaotic, so we get some properties of chaotic.