The Monte Carlo Study Of The Anderson Model In The Disordered System

In the paper that Absence of Diffusion in certain Random Lattices, P. W. Anderson proved that if there is random ups and downs in the potential field, the electron would not travel in the whole solid but in the local area when the degree of disorder is large enough. Or in other words, the extended states will turn into the localized states. The localization caused by disorder is called"The Anderson transition".The Anderson model is a simple tight-binding model. Disorder refers to the case that the energy at every point is random. Hamiltonian could be expressed as ijiIn the above formula,ε_i stands for the energy of electron on the grid , which meets certain probability distribution (for example,ε_i equally distributed between W/2 and W/2 ). V_ij refer to the energy required for a electron to jump from j to i . In this paper we set V_ij to be for the jump between nearest neighbors, and to be zero for other cases. In addition to the above diagonal disorder, we can assume thatε_i is a constant and V_ij is in disorder (not diagonal disorder) or both are in disorder.Considering the complex nature of disordered systems, one can establish some simple models to reflect the physical essence. In the single electron approximate, one-dimensional disorder Anderson model have been widely applied. From the one-dimensional model there have come out many meaningful conclusions. But after all, one-dimensional model is a rather simplified model, which has obvious limitations when studying the actual system in disorder. Two-dimensional disordered systems included basic principle of the disordered system, which will be well applied in solving practical problems. However, it is rather complex and difficult to deal with the two-dimensional disordered systems in mathematics. Theoretical results have been controversial.In this paper, based on the work of predecessors, we study of the factors that influence the localization length. Starting from the Anderson model of one-dimensional disordered systems, we study the law that governs the change of the localization length of two- and three-dimensional disordered systems, using transfer matrix method and Monte Carlo simulation method. Some significant conclusions have been made as the following: (1) Localization phenomenon of the electron states will appears with the increasing of the degree of disorder; (2) For the two-dimensional disordered system, there exist quasi-extended states in the energy band centre, and quasi-metal behavior appears inside the system; (3) For the three-dimensional disordered system, with the increasing of the degree of disorder, the width of the energy band will stretch to some extent. (4) There exist both of the localized states and the extended states in three-dimensional disordered system.All of the above works have offered more information to the disordered system localization theory.