Complex polarization ordering in ${\text{PbTiO}}_3$ nanowires: A first-principles computational study

Based on parameter-free density-functional-theory calculations, we demonstrate the possibility of nonrectilinear curling vortex electric dipole configurations in ${\text{PbTiO}}_3$ nanowires. We predict that the critical size for the genesis of the vortex polarization instability (with an axial toroidal moment) is $16\text{Å}$. We also report previously unknown phase transitions between the nonrectilinear vortex and conventional rectilinear axial polarization configurations mediated by strain and surface terminations. The ability to switch reversibly between the vortex (clockwise/counterclockwise) and axial (positive/negative) polarization states may open up transformative technological possibilities.

Figure 1

(Color online) (a) Schematic view of nonrectilinear vortex polarization configuration in a $48\text{Å}\times 48\text{Å}\times 48\text{Å}$ ${\text{PbZr}}_{0.4}{\text{Ti}}_{0.6}{\text{O}}_3$ nanodot as predicted by first-principles-based effective Hamiltonian techniques (based on Ref. 10). (b) DFT-predicted closure domain structure in ultrathin epitaxial ${\text{SrRuO}}_3/{\text{BaTiO}}_3/{\text{SrRuO}}_3$ ferroelectric capacitor (as reported in Ref. 14). (c) Influence of the axial strain and surface terminations on the favored nonrectilinear vortex and rectilinear axial polarization configurations in ${\text{PbTiO}}_3$ nanowires, based on the findings of the present work.

Figure 2

(Color online) Schematic illustration showing construction of an infinitely long $2\times 2$ nanowire with its axis along $z$ direction. Either a Pb-centered (top) or a Ti-centered (bottom) perovskite unit cell can be used to construct the nanowire, which results in ${\text{TiO}}_2$- or PbO-terminated lateral sidewalls, respectively. The nanowire with either termination will display alternating layers of PbO and ${\text{TiO}}_2$ planes along the axial directions.

Figure 3

(Color online) Unit-cell-decomposed dipole moments (panel a) and in-plane displacements of individual atoms with respect to a paraelectric reference state in PbO (panels b and c) and ${\text{TiO}}_2$ (panels d–f) transverse planes for a $4\times 4$ ${\text{TiO}}_2$-terminated nanowire at zero axial strain. Panels g–l, same as panels a–f, but for a $4\times 4$ PbO-terminated nanowire at 3.7% axial compressive strain. Dotted lines represent the boundary of each individual unit cell in the wire cross section.

Figure 4

(Color online) (a) Toroidal moment [gray (red) triangles, using the left axis] and axial polarization [black (blue) solid squares, using the right axis] as a function of axial strain for the $4\times 4$ ${\text{TiO}}_2$-terminated nanowire. Black (blue) hollow squares represent values of axial polarization calculated using actual Born effective charges (obtained through density functional perturbation theory) instead of the bulk Born effective charges. In its relaxed state (zero-axial strain) the nanowire bears a significant amount of toroidal moment $\left(0.79\times {10}^{-12}\text{C}/\text{cm}\right)$ with negligible axial polarization. Application of axial tensile strain of about 3% leads to a switching from the vortex to the axial-polarized state (shown in inset). (b) Same as (a), but for the $4\times 4$ PbO-terminated nanowire. The relaxed state of the nanowire is axially polarized with zero toroidal moment, however a compressive axial strain of about 3.5% leads to a phase transition from the axial to a vortex polarized state. The phase transition in both nanowires occur abruptly at a $c$ value of $3.87\text{Å}$, as indicated in a and b.

Figure 5

(Color online) Plot of coefficients ${\left|C_i\right|}^2$ against phonon mode frequencies, obtained by zone center phonon mode analysis of the reference paraelectric states, for the relaxed $4\times 4$ ${\text{TiO}}_2$-terminated nanowire (left) and the $4\times 4$ PbO-terminated nanowire at 3.7% axial compressive strain (right). Imaginary and real frequencies are plotted in gray (red) and black (blue) on negative and positive $x$ axis, respectively.