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Three-dimensional (3D) simulations of ferroelectric hysteresis and butterfly loops have been carried out based on solving the time dependent Ginsburg-Landau equations by using a finite volume method. The influence of externally mechanical loadings with a tensile strain and a compressive strain on the hysteresis and butterfly loops has been studied numerically. The 3D ferroelectric domain formation and its evolution have also been presented in the paper. The simulation results successfully reveal the macroscopically nonlinear response to the applied stresses and electric field. 77.80.Dj, 78.20.Bh Ferroelectricity is one of the fastest growing fields during the past several decades. Interest in this field is attributable to the increasing numbers of practical applications in micro-electromechanical systems, microwave devices, and memory PACS:

This investigation aims at an Nb-doped -TiAl intermetallic compound system in which part of Ti or Al atoms are substituted by Nb atoms. The structural parameters, the energy band structures, the electronic densities of states and the elastic constants of Nb-doped -TiAl intermetallic compound are calculated and studied by using the first-principles method based on the density functional theory and other physical theory. The first-principle calculations presented here are based on electronic density-functional theory framework. The ultrasoft pseudopotentials and a plane-wave basis set with a cut-off energy of 320 eV are used. The generalized gradient approximation refined by Perdew and Zunger is employed for determining the exchange-correlation energy. Brillouin zone is set to be within 333 k point mesh generated by the Monkhorst-Pack scheme. The self-consistent convergence of total energy is at 1.010-6 eV/atom. In view of geometry optimization, it is shown that doping with Nb can change the structural symmetry of the -TiAl intermetallic compound. The calculated formation energies indicate that the formation energy of the system in which Ti atom is replaced by Nb atom is smaller than that of Al atom replaced by Nb atom. Accordingly, they tend to substitute Ti atom when Nb atoms are introduced into the -TiAl system. The calculated band structures of Nb-doped -TiAl system show that they all have metallic conductivities, which implies that the brittleness of -TiAl intermetallic compound could be tailored by Nb-doping. The partial densities of states of the Nb-doped and pure -TiAl systems indicate that the intensity of covalent bond between Ti atom and Nb atom is weaker than covalent bond between Ti atom and Al atom while the Ti atoms are replaced by Nb atoms in the -TiAl system. What is more, the density of states near Fermi energy increases after Al atoms has been replaced by Nb atoms in the -TiAl system. This is an important factor for improving the ductility of -TiAl intermetallic compound. The calculated elastic constants, bulk modulus and shear modulus of Nb-doped -TiAl systems indicate that the ductility and the fracture strength of Nb-doped -TiAl system are both better than those of pure -TiAl system, especially in the system where part of Al atoms are replaced by Nb atoms. The plastic deformation capacity of Nb-doped -TiAl system is thus improved comparatively.

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