Linear Versus Nonlinear Electro-Optic Effects in Materials

Two schemes are proposed to compute the nonlinear electro-optic (EO) tensor for the first time. In the first scheme, we compute the linear EO tensor of the structure under a finite electric field, while we compute the refractive index of the structure under a finite electric field in the second scheme. Such schemes are applied to $\mathrm{Pb}(\mathrm{Zr},\mathrm{Ti}){\mathrm{O}}_3$ and ${\mathrm{BaTiO}}_3$ ferroelectric oxides. It is found to reproduce a recently observed feature, namely, why $\mathrm{Pb}({\mathrm{Zr}}_{0.52}{\mathrm{Ti}}_{0.48}){\mathrm{O}}_3$ adopts a mostly linear EO response while ${\mathrm{BaTiO}}_3$ exhibits a strongly nonlinear conversion between electric and optical properties. Furthermore, the atomistic insight provided by the proposed ab initio scheme reveals the origin of such qualitatively different responses, in terms of the field-induced behavior of the frequencies of some phonon modes and of some force constants.

Figure 1

Clamped EO coefficients as a function of electric field applied along the [111] direction in (a) PZT and (b) BTO, respectively.

Figure 2

Mode decomposition of the clamped EO coefficient ${\mathcal{R}}_{33}^{\eta }$ in (a) PZT and (b) BTO. Phonon frequency for selected modes at the $\mathrm{\Gamma }$ point of the first Brillouin zone with the insets corresponding to the eigenvector of $A_1^{(3)}$ and $A_1^{(2)}$ modes at zero field in (c) PZT and (d) BTO, respectively. The inverse of the square of the phonon frequency, ${\omega }^{-2}$, as a function of electric field in (e) PZT and (f) BTO. The inset of panel (f) zooms in the data for electric field between $3.08\times {10}^8\mathrm{V}/\mathrm{m}$ and $3.16\times {10}^8\mathrm{V}/\mathrm{m}$.

Figure 3

Force constants at the $\mathrm{\Gamma }$ point of the first Brillouin zone as a function of electric field for (a) Ti/Zr-Ti/Zr, (c) O-O and (e) Ti/Zr-O bonds in PZT; and (b) Ti-Ti, (d) O-O, and (f) Ti-O bonds in BTO.