Structural and energetic properties of domains in PbTiO${}_3$/SrTiO${}_3$ superlattices from first principles

We report first-principles calculations, within the density functional theory, on the structural and energetic properties of 180${}^{\circ }$ stripe domains in (PbTiO${}_3$)${}_n$/(SrTiO${}_3$)${}_n$ superlattices. For the explored periodicities ($n=3$ and 6), we find that the polydomain structures compete in energy with the monodomain phases. Our results suggest the progressive transition, as a function of $n$, from a strong to a weak electrostatic coupling regime between the SrTiO${}_3$ and PbTiO${}_3$ layers. Structurally, they display continuous rotation of polarization connecting 180${}^{\circ }$ domains. A large offset between [100] atomic rows across the domain wall and huge strain gradients are observed. The domain wall energy is very isotropic, depending very weakly on the stripe orientation.

Figure 1

Differences in energies between polydomain, monodomain, and nonpolar configurations in (3$\mid$3) PbTiO${}_3$/SrTiO${}_3$ superlattices, as a function of the domain period $N_x$. Total energies of supercells are given per five-atom perovskite unit cell. Circles represent the configurations where the AFD modes are not allowed ($N_y=1$), while squares represent configuration with condensed AFD modes ($N_y=2$). Diamond indicates a configuration where the DW lies along the $\langle 110\rangle$ direction, also allowing for the condensation of AFD modes. The monodomain phases have been labeled as in Ref. 8, where a full analysis of these configurations is provided. In the nonpolar configuration, the AFD distortions have been considered. All energies are given with respect to the most stable monodomain configuration.

Figure 2

(a) Schematic representation of the center of a domain in a (3$\mid$3) PbTiO${}_3$/SrTiO${}_3$ superlattice (see region embodied by a bracket in Fig. 4). Atoms are represented by balls: Sr in yellow, Ti in green, O in blue, and Pb in grey. In panels (b)–(d) we represent the amplitude of the rotations (squares) and tiltings (diamonds) of each TiO${}_6$ octahedra: (b) at the center of a domain in the polydomain configuration with $N_x=12$, (c) in the ground state monodomain phase (with polarization in the PbTiO${}_3$ layer pointing close to the perovskite unit cell diagonal, see Ref. 8), and (d) in a monodomain phase with polarization lying along [001].

Figure 3

Local polarization profile of polydomain structures in (PbTiO${}_3$)${}_n$/(SrTiO${}_3$)${}_n$ superlattices with (a) $n=3$ and (b) $n=6$. The PbTiO${}_3$ and SrTiO${}_3$ are depicted as gray and white regions respectively. Red dashed squares in the SrTiO${}_3$ layers mark the position where antivortices are formed.

Figure 4

Schematic view of the $(3\mid 3)$ superlattice with $N_x=12$ and $N_y=1$ indicating how local values of the magnitudes plotted in Figs. 5 and 7 are defined. Red (blue) lines represent local values of in-plane, $a$, (out-of-plane, $c$) lattice constants, measured from the in-plane (out-of-plane) distance between equivalent cations of the same chemical species in consecutive unit cells along the $x$ ($z$) direction. Magnitudes with subscript 1 (2) indicate unit cells centered on a $\left[001\right]$ AO (TiO${}_2$) atomic plane. Local polarization is marked with arrows. Black dotted lines indicate the offset between [100] atomic rows to the left and right of the domain walls, defined as the relative vertical shift of A-cations in a given atomic plane. Bracket at the bottom of the up domain indicates the position of its center, where the values plotted as empty symbols in Figs. 5 and 7 are obtained. Finally, domain walls are represented by dashed lines.

Figure 5

Left panels: layer-by-layer out-of-plane polarization, $P_z$, inferred from the Born effective charges and the atomic displacements for (a) a $(3\mid 3)$ and (c) a $(6\mid 6)$ superlattice. Right panels: layer-by-layer tetragonality for (b) a $(3\mid 3)$ and (d) a $(6\mid 6)$ superlattice. Empty symbols represent values at the center of an up domain, while filled symbols correspond to averaged values (root mean square in the case of polarization) along the [100] direction.

Figure 6

Schematic representation of the distortion induced by the domain structure in (a) bulk PbTiO${}_3$, (b) PbTiO${}_3$ thin films and (c) PbTiO${}_3$/SrTiO${}_3$ superlattices. (a) In bulk, displacements of Pb cations cause an offset between [100] atomic rows across the DW. (b) In thin films, in addition to the offset between domains, rotation of the polarization near the interface is responsible of a nonvanishing strain gradient $\frac{\partial {\varepsilon }_{11}}{\partial z}$. (c) In the case of the PbTiO${}_3$/SrTiO${}_3$ superlattices, the offset and modulation of the strain field in the PbTiO${}_3$ layer (in grey) propagates into the SrTiO${}_3$ (in white).

Figure 7

Left panels: local layer-by-layer offset between $\left[100\right]$ atomic rows to the left and right of the DW for (a) a $(3\mid 3)$ and (c) a $(6\mid 6)$ superlattice. Right panels: local in-plane strain across the center of an up domain in (b) a $(3\mid 3)$ and (d) a $(6\mid 6)$ superlattice. A large nondiagonal component of the strain gradiend, $\frac{\partial {\varepsilon }_{11}}{\partial z}$ can be observed close to the interfaces.