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supercell simulation and the statistical supercell method - a powerful means to calculate the electronic structure of imperfect crystals Abstract Imperfect crystals contain defects, dopants or disorder. Defects exist natively in the process of the crystal growth and produce a great effect on the crystal properties. Dopants are artificially put into a crystal to widen its use fields. They can be considered as a special kind of defects. Disorder appears in mixed crystals, which constitute a large number of high-technology materials. Owing to the increasingly extensive use of imperfect crystals, the investigations on them become a very important issue in the condensed matter. However, theoretical researches on them are confronted with gigantic difficulty because of the loss of periodic boundary condition. We have to use differently approximate methods to calculate different objects of study. For imperfect crystals with point defects, the cluster method and the supercell simulation are two methods in common use. Both methods have their strong and weak points. The cluster method can provide relative positions of defect levels, but it is unable to determine their concrete positions with respect to the top of valence band or to the bottom of conduction band. The supercell simulation approach can establish the defect level positions. But, in order to reduce the affection of periodic boundary condition, the size of supercell must be taken as large as possible. At the same time, the computational load increases dramatically. We pose two following rules for the best results of calculations in this thesis. (1) Due to the existence of defects, the chemically same atoms in the supercell may have differently effective charges. They should be classified again in the light of their symmetry and their distances from the defect center. More useful information can be obtained if so. (2) The effective charge of an atom far from the defect center should be almost the same as the same kind atom in the host. Only in this way can the effect of the periodic boundary condition on the defect center guarantee to be omitted and the calculating results be trustworthy. If it is not so, we have to enlarge the size of the supercell in use. For mixed crystal systems, on the other hand, the rigid-band approximation (RBA), the virtual crystal approximation (VCA) and the coherent potential approximation (CPA) have been constantly utilized in previous calculations. In these approximate methods, some average quantities are used to re-validate the periodic boundary conditions, such as the atomic number, atomic tight-binding parameters, potential parameters of atomic spheres, and even density of states. Usually, they are suitable to mixed crystals with metallic bond but not to those with covalent bond or ionic bond. In this thesis, we attempt a new method, the statistical supercell method, to calculate the electronic structure of a mixed crystal. This new method divides the calculations into two steps, first to calculate the electronic structure of micro-crystals appearing in a mixed crystal system and then to include the effect of disorder by means of statistical theory. This thesis is composed of five chapters. Throughout this thesis, the band structure calculations are performed by using the scalar-relativistic linear muffin-tin orbital (LMTO) method in the atomic-spheres approximation (ASA) with Hedin-Lundqvist exchange-correlation potential. LMTO-ASA method is sui