| In the past decades, the First-Principle Methods based on density functional theory (DFT), plane wave basic sets and pseudopotential have been very successful in the study of many-body problems. We use local density approximation (LDA) in the exchange-correlation potential. The bulk modulus ofâ…¡-â…¢2-â…¥4 semiconductor have larger values. The value of bulk modulus is larger, the compressibilities is lower. For the bulk modulus ofâ…¡-â…¢2-â…¥4 semiconductor,we have not enough data, so we need computing the bulk modulus using the first-principle methods. In the thesis, we perform the first-principles investigation for a family of a compoundsâ…¡-â…¢2-â…¥4. The properties of two possible structures, defect chalcopyrite(DC) and defect famatinite(DF) are both calculated. We extend Cohen's empirical formula for zinc-blende compounds to this family of compounds. The relation between the energy and volume is fitted to the integrated form of third-order Birch-Murnaghan's equation of state. The bulk modulus(B), pressure derivative(B') and equilibrium volume(Vo) can be extracted from the equation. The two different structures DC and DF have similar bulk moduli. For example, the bulk modulus is found to increase as Te is substituted with Se and S, respectively. This trend is due to the change of bond length and bond energy. There are little experimental results available for the bulk moduli. D. Errandonea et al. recently had high-pressure x-ray diffraction study on the structure and phase transitions of ZnGa2Se4 up to 23GPa. They found that it exhibits a defect-stannite(DF) structure up to 15.5 GPa and in the range from 15.5 GPa to 18.5 GPa the low-pressure phase coexists with a high-pressure phase, which remains stable up to 23GPa. The high pressure phase is assigned to a defect NaCl-type structure, they obtained Bo=50(2) GPa and B0'=4. The bulk modulus of a binary zinc-blende compound can be estimated by Cohen's empirical formula. The original empirical formula B is appropriate for zinc-blende solids. Alloys and more complex structures with tetrahedral bonding may also be estimated in an approximate way by taking average nearest-neighbor distances. We extend the formula toâ…¡-â…¢2-â…¥4 compounds:for bulk modulus B in GPa and the nearest-neighbor distance di in A. we only include three bonds because one tetrahedrally bonded atom is replaced by a vacancy. We sum them up with a factor 1/4 for each tetrahedral bond. The coefficient is chosen analogous to II-VI semiconductors. The trend is understandable if we consider the covalency increases going to the upper rows of the Periodic Table. Because of the larger contribution of covalency to B, the loss of covalent bond charge can reduce B. The different structures DF and DC have almost the same bulk modulus from the extended Cohen's empirical formula, mainly because they have similar electronic structures and bond-lengths. For DF structure, our ab initio results agree reasonably well with the empirical model. One might apply the extended empirical formula to estimate the bulk moduli ofâ…¡-â…¢2-â…¥4 semiconductors. For DC structure, we must use the empirical formula carefully. The experimental results are needed to verify all the theoretical conclusions.
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