Interplay of epitaxial strain and rotations in PbTiO${}_3/$PbZrO${}_3$ superlattices from first principles

We present first-principles calculations of the structural phase behavior of the [1:1] ${\mathrm{PbTiO}}_3/{\mathrm{PbZrO}}_3$ superlattice and the ${\mathrm{PbTiO}}_3$ and ${\mathrm{PbZrO}}_3$ parent compounds as a function of in-plane epitaxial strain. A symmetry analysis is used to identify the phases and clarify how they arise from an interplay between different structural distortions, including out-of-plane and in-plane polar modes, rotation of oxygen octahedra around out-of-plane or in-plane axes, and an antipolar mode. Symmetry-allowed intermode couplings are identified and used to elucidate the nature of the observed phase transitions. For the minimum-period [1:1] ${\mathrm{PbTiO}}_3/{\mathrm{PbZrO}}_3$ superlattice, we identify a sequence of three transitions that occur as the in-plane lattice constant is increased. All four of the phases involve substantial oxygen octahedral rotations, and an antipolar distortion is important in the high-tensile-strain phase. Inclusion of these distortions is found to be crucial for an accurate determination of the phase boundaries.

Figure 1

Sketch of distortions of type (a) ${\mathrm{P}}_{001}$, (b) ${\mathrm{P}}_{110}$, (c) ${\mathrm{R}}_{001}$, (d) ${\mathrm{R}}_{110}$, and (e) ${\mathrm{A}}_{\overline{1}10}$ in the PTO/PZO superlattice.

Figure 2

Calculated properties of PTO as a function of epitaxially constrained in-plane lattice constant. (a) Minimum-energy structure: octahedral rotations (black) of type ${\mathrm{R}}_{110}$ ($*$); polarizations (red) of type ${\mathrm{P}}_{001}$ ($\circ$) and ${\mathrm{P}}_{110}$ ($\times$). (b) Total energy (black squares) and $c/a$ ratio (red triangles).

Figure 3

Calculated properties of PZO as a function of epitaxially constrained in-plane lattice constant. (a) Minimum-energy structure: octahedral rotations (black) of type ${\mathrm{R}}_{110}$ ($*$) and ${\mathrm{R}}_{001}$ ($+$); polarizations (red) of type ${\mathrm{P}}_{001}$ ($\circ$) and ${\mathrm{P}}_{110}$ ($\times$). (b) Total energy (black squares) and $c/a$ ratio (red triangles).

Figure 4

Calculated properties of PTO/PZO superlattice as a function of epitaxially constrained in-plane lattice constant. (a) Minimum-energy structure: octahedral rotations (black) of type ${\mathrm{R}}_{110}$ ($*$) and ${\mathrm{R}}_{001}$ ($+$); polarizations (red) of type ${\mathrm{P}}_{001}$ ($\circ$) and ${\mathrm{P}}_{110}$ ($\times$); antipolar distortions of type ${\mathrm{A}}_{\overline{1}10}$ (filled red circles). (b) Total energy (black squares) and $c/a$ ratio (red triangles). Heavy and light vertical dotted lines indicate phase boundaries determined from our work and from Ref. 7, respectively.