Mechanism of covalency-induced electric polarization within the framework of maximally localized Wannier orbitals

It has been well established that covalency significantly enhances the electric polarization produced by the ionic displacement for ferroelectric perovskite transition metal oxides (TMO). Furthermore, recent experimental and theoretical works on the organic ferroelectrics TTF-CA $(\text{tetrathiafulvalene}-p-\text{chloranil})$ have revealed that the covalency-induced polarization is one to two orders of magnitude larger than that of the ionic polarization and that the two contributions are in the opposite direction. Here we propose a formulation to analyze the detailed mechanism of the covalency-induced polarization within the framework of maximally localized Wannier orbitals and apply it to an organic exotic ferroelectrics TTF-CA and typical ferroelectric perovskite TMOs, ${\mathrm{BaTiO}}_3$, and ${\mathrm{PbTiO}}_3$. This formulation discriminates three components in the electronic contribution to the polarization. The first one corresponds to the point charge model, the second to the intra-atomic or molecular polarization, and the third comes from the electron transfer between unit cells. The framework of the present formulation is the same as the one proposed by Bhattacharjee and Waghmare [Phys. Chem. Chem. Phys. 12, 1564 (2010)], but we give a more explicit expression of each component and discuss fundamental aspects of the formulation.

Figure 1

Mechanism for the electronic polarization induced by the atomic displacement. The upper half is for the paraelectric phase and shows the schematic density of states of the AB compound in which the $a$ state of atom A has a lower energy than the $b$ state of atom B. The red broken arrows show the $a-b$ hybridization. The lower half shows the case of the ferroelectric phase. The pair with a shortened (elongated) atomic distance has stronger (weaker) hybridization. The change in the hybridization induces the electron transfer indicated by the horizontal red arrows. The leftward (rightward) electron transfer produces positive (negative) polarization.

Figure 2

The left two are the $3z^2-r^2$ type MLWO of the Ti-$3d$ conduction band and the right two are the $p_z$ type MLWO of the O-$2p$ valence band for ${\mathrm{PbTiO}}_3$. The upper (lower) panel is for the paraelectric (ferroelectric) phase. Oxygen atoms displace leftward in the ferroelectric phase. In the background lattice, large black spheres are Pb and small red spheres are O. Ti is hidden by O or wave function. The yellow (blue) part is the positive (negative) part of the MLWO.

Figure 3

The left two are the MLWOs of the LUMO band and the right two are those of the HOMO band for TTF-CA. The upper (lower) panel is for the paraelectric (ferroelectric) phase. In the ferroelectric phase, CA displaces leftward as indicated by green arrows. The yellow (blue) part is the positive (negative) part of the MLWO.

Figure 4

(a) The electron transfer shown in the lower panel of Fig. 1. Here and also in the following, the transferred charge ($-e\times$ amount of electron transfer) is shown. (b) The electron transfer shown in (a) can be converted to a scheme having a combination of the electron flow and the electron transfer within a unit cell. (c) Another possible scheme equivalent to scheme (b). (d) A more appropriate picture for the electron transfer in bulk systems.

Figure 5

(a) The original expansion coefficients for CA and TTF for $\lambda =1$ versus the lattice points. (b) Those defined by Eqs. (38) and (39).

Figure 6

Electronic processes associated with the molecular displacement shown by green arrows. The red (blue) arrows show the major (minor) processes of electron flow. The amount of charge transfer is shown. Note that anion (CA) moves leftward producing the positive polarization, while electrons move rightward producing the negative polarization.

Figure 7

The red (blue) arrows show the major (minor) electron-flow processes in ${\mathrm{BaTiO}}_3$. The upper left for O-$2p$ band, the lower left for Ba-$5p$ band, and the right for O-$2s$ band. Amount of transferred charge is shown. The rightward (leftward) arrows produce the negative (positive) polarization.

Figure 8

Schematic explanation on the mechanism of the polarization in the $\mathrm{Pb}-5d$ and $\mathrm{O}-2s$ band complex (upper panel) and in the $\mathrm{O}-2p$ and $\mathrm{Pb}-6s$ band complex (lower panel). In both band complexes, the direction of the electron transfer is rightward in the Pb bands, producing the negative polarization while, that in the O bands is leftward producing the positive polarization. The light blue and dark blue curvy arrows indicate the electron flow processes.

Figure 9

Examples of the MLWOs with their center at O1 of ${\mathrm{PbTiO}}_3$. The left side shows those of the O-$2s$ bands and the middle and right sides are for those of the O-$2p$ bands. The upper panel (lower) is for the paraelectric (ferroelecteric) phase. The changes in MLWOs associated with the ferroelectric transition produce the electron flow contributions to the polarization via Pb orbitals. At the same time, the appearance of the net intra-atomic polarization at the Pb sites can also be seen. In the background lattice, large black spheres are Pb and small red spheres are O. Ti is hidden by O or wave function. The yellow (blue) part is the positive (negative) part of the MLWO.

Figure 10

Examples of the MLWOs with their center at Pb of ${\mathrm{PbTiO}}_3$. The left and middle sides show those of the Pd-$5d$ bands and the right side is for those of the Pb-$6s$ bands. The upper panel (lower) is for the paraelectric (ferroelecteric) phase. The changes in MLWOs associated with the ferroelectric transition produce the electron flow contributions to the polarization via O orbitals. In the background lattice, large black spheres are Pb and small red spheres are O. Ti is hidden by O or wave function. The yellow (blue) part is the positive (negative) part of the MLWO.