Ab initio elasticity at finite temperature and stress in ferroelectrics

Computing the temperature and stress dependence of the full elastic constant tensor from first-principles in non-cubic materials remains a challenging problem. Here we circumvent the aforementioned challenge via the generalized quasiharmonic approximation in conjunction with the irreducible derivative approach for computing strain dependent phonons using finite difference, explicitly including dipole-quadrupole contributions. We showcase this approach in ferroelectric PbTiO$_3$ using density functional theory, computing all independent elastic constants and piezoelectric strain coefficients at finite temperature and stress. There is good agreement between the quasiharmonic approximation and the experimental lattice parameters close to 0 K. However, the quasiharmonic approximation overestimates the temperature dependence of the lattice parameters and elastic constant tensor, demonstrating that a higher level of strain dependent anharmonic vibrational theory is needed.