Ideal barriers to polarization reversal and domain-wall motion in strained ferroelectric thin films

The ideal intrinsic barriers to domain switching in $c$-phase ${\text{PbTiO}}_3$ (PTO), ${\text{PbZrO}}_3$ (PZO), and ${\text{PbZr}}_{1-x}{\text{Ti}}_x{\text{O}}_3$ (PZT) are investigated via first-principles computational methods. The effects of epitaxial strain on the atomic structure, ferroelectric response, barrier to coherent domain reversal, domain-wall energy, and barrier to domain-wall translation are studied. It is found that PTO has a larger polarization, but smaller energy barrier to domain reversal, than PZO. Consequentially the idealized coercive field is over two times smaller in PTO than PZO. The Ti-O bond length is more sensitive to strain than the other bonds in the crystals. This results in the polarization and domain-wall energy in PTO having greater sensitivity to strain than in PZO. Two ordered phases of PZT are considered, the rocksalt structure and a (100) PTO/PZO superlattice. In these simple structures we find that the ferroelectric properties do not obey Vergard’s law, but instead can be approximated as an average over individual five-atom unit cells.

Figure 1

(Color online) Double-well potential of PTO at zero in-plane strain. The ferroelectric state (minimum sketched at right) has point-group symmetry $C_{4v}$; the paraelectric state (saddle point sketched at center) has $D_{4h}$.

Figure 2

(Color online) Lattice-constant ratio $c/a$ versus applied epitaxial strain, obtained by relaxing in the ferroelectric (a) or paraelectric (b) state.

Figure 3

(Color online) Polarization (a) and energy-barrier height (b) versus applied epitaxial strain in tetragonal PTO-PZO systems.

Figure 4

(Color online) Ideal coercive field versus epitaxial strain as determined from Eq. (2) using the data from Fig. 3.

Figure 5

(Color online) Hysteresis curves calculated from Eq. (1) at zero epitaxial strain.

Figure 6

(Color online) Domain-wall energy and barrier to domain-wall motion in strained (a) PTO and (b) PZO films. The energy barrier is scaled $2\times$.

Figure 7

(Color online) (a) Domain-wall energy and (b) barrier to domain-wall motion in a strained (100) PZT superlattice. (c) Sketch of energy landscape emerging from the calculations.

Figure 8

(Color online) Partial PDFs for Ti-O (left) and Zr-O (right) distances. Top, middle, and lower panels show applied epitaxial strains of $ϵ=-1.2\%$ (compressive), 0, and $+1.2\%$ (tensile), respectively.

Figure 9

(Color online) Partial PDF of Pb-O distances. Top, middle, and lower panels show applied epitaxial strains of $ϵ=-1.2\%$, 0, and $+1.2\%$, respectively.

Figure 10

(Color online) Partial PDF of Pb-Ti and Pb-Zr distances. Top, middle, and lower panels show applied epitaxial strains of $ϵ=-1.2\%$, 0, and $+1.2\%$, respectively.

Figure 11

(Color online) Decomposition of polarization on a plane-by-plane basis across domain wall. Left, PTO; right, PZO. Top, Pb-centered walls; bottom, Ti- or Zr-centered walls. Symbols denote PbO (circles), ${\text{TiO}}_2$ (squares), and ${\text{ZrO}}_2$ (triangles) planes.

Figure 12

(Color online) Atomic displacements across domain walls. Left, PTO; right, PZO. Top, Pb-centered walls; bottom, Ti- or Zr-centered walls. Symbols denote Pb (circles), Ti (squares), Zr (triangles), O in the $xz$ or $yz$ plane (pluses), and O in the $xy$ plane (crosses).

Figure 13

(Color online) Dependence of domain-wall energies on in-plane lattice constant. Solid symbols: PTO and PZO. Open symbols: PZT, plotted versus average lattice constant (connected by dashed lines) or versus local lattice constant (connected by solid lines).