Ab initio approach to photostriction in classical ferroelectric materials

The change of shape under illumination by visible light, called photostriction, is investigated in the classical ferroelectrics barium titanate and lead titanate. By means of the $\mathrm{\Delta }\mathrm{SCF}$ method, the use of first-principle calculations confirms that the converse piezoelectric effect is the main driving force of the photostriction of the polar axis in those materials. As a result, when compared to barium titanate and bismuth ferrite, lead titanate is a better photostrictive material in the direction of the polar axis, due to its larger longitudinal piezoelectric constant. On the other hand, in directions transverse to the polar axis, photoinduced electronic pressure can also become a sizable contribution that can either compete or cooperate with the piezoelectric effect, depending on the transitions involved. A simple Landau model is further developed and shows reasonable qualitative agreement with results from $\mathrm{\Delta }\mathrm{SCF}$ calculations, which is promising for a fast screening of materials with high photostrictive effects.

Figure 1

(a) Regular Kohn-Sham calculation to find the ground state electronic density: From an input potential $V_{\mathrm{in}}$, the now known Kohn-Sham Hamiltonian can be diagonalized. Its lowest eigenstates (“orbitals”) are then filled with electrons, and a new potential is calculated based on the latter electronic distribution. The output potential is then plugged in as input potential until convergence to a reasonable accuracy is reached. (b) $\mathrm{\Delta }\mathrm{SCF}$ calculation: at filling, not all low energy lying eigenstates are filled, but some electrons are imposed to occupy excited orbitals. An output potential is then calculated based on this electronic distribution, and the density is self-consistently solved under the constraint of certain occupation numbers.

Figure 2

Band structure of ${\mathrm{PbTiO}}_3$ in the $P4mm$ ferroelectric (a) and $Pm\overline{3}m$ paraelectric (b) phases and ${\mathrm{BaTiO}}_3$ in the paraelectric $Pm\overline{3}m$ (c) and ferroelectric $R3m$ (d) states. Definition of $k$ points used can be found in Ref. [39]. Arrows represent the different “transitions” considered here.

Figure 3

Evolution of the $a$ (left panel) and $c$ (right panel) lattice constants in ferroelectric lead titanate for $Z\to Z$ (black squares), $Z\to \mathrm{\Gamma }$ (blue circles), $X\to Z$ (green triangles), and $X\to \mathrm{\Gamma }$ (red diamonds).

Figure 4

10% isosurface of the amplitude of the monoelectronic wave functions for states in the valence band located at $X$ and $Z$ in the Brillouin zone, and in the conduction band at $Z$ and $\mathrm{\Gamma }$, in tetragonal lead titanate.

Figure 5

(left panel) The pseudocubic lattice constant $a$ of rhombohedral barium titanate decreases with increasing concentration of photoexcited electrons $n_e$, for both transitions. (right panel) Concurrently, the pseudocubic angle $\alpha$ increases.

Figure 6

Change of lattice constant of cubic ${\mathrm{BaTiO}}_3$ (red) and cubic ${\mathrm{PbTiO}}_3$ (blue) for different transitions.

Figure 7

Isosurface of the wave-function amplitude of the valence band Kohn-Sham orbital at $X$ and $\mathrm{\Gamma }$ for cubic ${\mathrm{BaTiO}}_3$ (left) and ${\mathrm{PbTiO}}_3$ (right).

Figure 8

Change of lattice constant $\delta a$ and $\delta c$ with increasing number of photoexcited electrons $n_e$ in tetragonal lead titanate calculated from $\mathrm{\Delta }\mathrm{SCF}$ calculations, for LDA and PBESol functional.

Figure 9

$\mathrm{\Delta }\mathrm{SCF}$ calculations for two different $k$ meshes, and for a $2\times 2\times 2$ supercell (SC) calculated using a $9\times 9\times 9$ $k$ mesh, performed in ${\mathrm{PbTiO}}_3$ in its $P4mm$ phase.

Figure 10

Change of polarization with the increasing number of photoexcited electrons (left panel) along the tetragonal axis in $P4mm$ lead titanate and (right panel) along the pseudocubic $\left[001\right]$ direction of $R3m$ ${\mathrm{BaTiO}}_3$.

Figure 11

Change of pseudocubic lattice constant in rhombohedral ferroelectric ${\mathrm{BaTiO}}_3$, as calculated in $\mathrm{\Delta }\mathrm{SCF}$ (filled symbols), and as estimated from the converse piezoelectric effect (open symbols).

Figure 12

Relative change of (left panel) basal lattice constant and (right panel) tetragonal lattice constant with increasing concentration of electrons in the conduction band for a $Z\to Z$ transition in lead titanate, for (filled black squares) $\mathrm{\Delta }\mathrm{SCF}$ calculations, (empty black diamonds) estimates from the change of polarization and the converse piezoelectric effect from Eq. (2), (blue empty circles) $\mathrm{\Delta }\mathrm{SCF}$ calculations with frozen atomic positions, (red dashed line) deformation potential in the Landau model, (green dashed dotted line) piezoelectric effect induced by the photoinduced electric field in the Landau model, (yellow line).

Figure 13

Relative change of (left panel) basal lattice constant and (right panel) tetragonal lattice constant with increasing concentration of electrons in the conduction band for a $X\to Z$ transition in ${\mathrm{PbTiO}}_3$, for (filled black squares) $\mathrm{\Delta }\mathrm{SCF}$ calculations, (empty black diamonds) estimates from the change of polarization and the converse piezoelectric effect from Eq. (2), (blue empty circles) $\mathrm{\Delta }\mathrm{SCF}$ calculations with frozen atomic positions, (red dashed line) deformation potential in the Landau model, (green dashed dotted line) piezoelectric effect induced by the photoinduced electric field in the Landau model, (yellow line).

Figure 14

(left panel) Change of pseudocubic lattice constant and (right panel) change of pseudocubic angle for different concentrations of photoexcited carriers in $R3c$ ${\mathrm{BiFeO}}_3$ (BFO), $P4mm$ ${\mathrm{PbTiO}}_3$ (PTO), and $R3m$ ${\mathrm{BaTiO}}_3$ (BTO). For PTO, only the change of lattice constant along the tetragonal axis $\left[001\right]$ is plotted.