Figure 3.

(a) Polarization as a function of atomic shifts from the centrosymmetric configuration (λ = 0) to the LDA-optimized configuration (λ = 1), in BaTiO₃ with vacancy. Square symbols are the raw data directly obtained from the Berry phase calculations. Empty circles are the result after shifting the raw data by an integer number of P_quan, showing a continuous change of P with respect to λ. (b) Polarization as a function of atomic shifts from the centrosymmetric configuration (λ = 0) to the LDA-optimized configuration (λ = 1), in BaTiO₃ with , , or vacancy. The polarizations in (b) are obtained by shifting the raw data by an integer number of P_quan. For perfect bulk BaTiO₃ without vacancy, the first-principles calculated value of polarization is −0.21 C·m⁻² (i.e., the star symbol at λ = 1).