In this paper, various physical properties of ZnO piezoelectric crystal have been calculated by the methods of density functional theory and density functional perturbation theory. These tensors may be defined as second derivatives of an appropriately defined energy functional with respect to the atomic displacement, electric field or strain perturbations. We have calculated the electronic and mechanical properties of ZnO crystal such as band gap, elastic, compliance and piezoelectric tensors, and also bulk modulus of hexagonal phase. These calculations have been performed under boundary condition of constant electric field and zero temperature. The results show that hexagonal ZnO has a direct band gap of 0.82 eV, its elastic tensor for the frozen-ion is smaller than the relaxed-ion and the diagonal values of frozen-ion compliance tensor S are larger

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