Charged domain walls in improper ferroelectric hexagonal manganites and gallates

Ferroelectric domain walls are attracting broad attention as atomic-scale switches, diodes, and mobile wires for next-generation nanoelectronics. Charged domain walls in improper ferroelectrics are particularly interesting as they offer multifunctional properties and an inherent stability not found in proper ferroelectrics. Here we study the energetics and structure of charged walls in improper ferroelectric ${\mathrm{YMnO}}_3,{\mathrm{InMnO}}_3$, and ${\mathrm{YGaO}}_3$ by first-principles calculations and phenomenological modeling. Positively and negatively charged walls are asymmetric in terms of local structure and width. The wall width scales with the amplitude of the primary structural order parameter and the coupling strength to the polarization, reflecting that polarization is not the driving force for domain formation. We introduce general rules for how to engineer $n$- and $p$-type domain wall conductivity based on the domain size, polarization, and electronic band gap. This opens the possibility of fine tuning the local transport properties and designing $p\text{−}n$-junctions for domain wall-based nanocircuitry.

Figure 1

Energetics of charged domain walls. (a) Crystal structure of hexagonal $R{\mathrm{MnO}}_3$ in the $P{6}_{3}cm$ space group, illustrating the $R$ cation corrugation and $\text{Mn}{\mathrm{O}}_5$ tilt. Changes in $\mathrm{\Phi }$ with position of the trimerization center $R1$ and direction of cation are illustrated for ${\alpha }^{+}$ (top) and ${\gamma }^{-}$ (bottom) in (b). Red and blue arrows show the tilt direction of the trigonal bipyramids above and below the $R$ cation layer, respectively. (c) Sketch of topologically protected domain vortex surrounded by the six domains with respect to $R$ cation corrugation and $\mathrm{\Phi }$, and (d) mapped energy landscape with respect to $(Q,\mathrm{\Phi })$, indicating the energy path for $\mathrm{\Delta }\mathrm{\Phi }={60}^{\circ }$ or $\mathrm{\Delta }\mathrm{\Phi }={180}^{\circ }$ DWs. (e) Calculated DW formation energies for $1\times 1\times 6$ supercells of ${\mathrm{YMnO}}_3$ with $\mathrm{\Delta }\mathrm{\Phi }={180}^{\circ }\left({\alpha }^{-}|{\alpha }^{+}\right)$, and $\mathrm{\Delta }\mathrm{\Phi }={60}^{\circ }\left({\alpha }^{+}|{\beta }^{-}\right)$, $\left({\alpha }^{-}|{\gamma }^{+}\right)$, and $\left({\beta }^{-}|{\gamma }^{+}\right)$ configurations. Inset figures illustrate the resulting local structure at the head-to-head (top) and tail-to-tail (bottom) DW.

Figure 2

Crystal structure evolution. Mode amplitudes across ferroelectric DWs in ${\mathrm{YMnO}}_3,{\mathrm{YGaO}}_3$ and ${\mathrm{InMnO}}_3$, (a) trimerization amplitude $Q$, (b) trimerization phase $\mathrm{\Phi }$, and (c) polarization $P$ with (solid lines) and without (dashed lines) dielectric screening of the depolarizing field.

Figure 3

Local crystal structure across charged DWs. Layer-resolved (a) phase $\mathrm{\Phi }$ with fitted profile from Eq. (7), (b) amplitude ${\alpha }_A,R$ cation displacements, (c) $\mathrm{\Delta }{z}_{R1}$, (d) $\mathrm{\Delta }{z}_{R2}$, and (e) polarization $P$ across ${\mathrm{YMnO}}_3,{\mathrm{InMnO}}_3$, and ${\mathrm{YGaO}}_{3}1\times 1\times 6$ supercells with $\mathrm{\Delta }\mathrm{\Phi }={60}^{\circ }\left({\alpha }^{-}|{\beta }^{+}\right)$ configuration. Dash-dotted horizontal lines show bulk $\mathrm{\Phi },{\alpha }_A,\mathrm{\Delta }{z}_{Ri}$, and $P$ values from DFT relaxed 30-atom unit cells. Dashed vertical lines illustrate the position of the tail-to-tail (left) and head-to-head (right) DWs. Yellow shading represents $P_{\downarrow }$ domain $\left({\alpha }^{-}\right)$, pink shading $P_{\uparrow }$ domain $\left({\beta }^{+}\right)$.

Figure 4

Electrostatic potential. (a) Macroscopically averaged electrostatic potential across $1\times 1\times 6\mathrm{\Delta }\mathrm{\Phi }={60}^{\circ }\left({\alpha }^{-}|{\beta }^{+}\right)$ supercells of ${\mathrm{YMnO}}_3$ (black), ${\mathrm{InMnO}}_3$ (red), and ${\mathrm{YGaO}}_3$ (blue). (b) Electrostatic potential for increasing DW distance in $\mathrm{\Delta }\mathrm{\Phi }={60}^{\circ }\left({\alpha }^{-}|{\beta }^{+}\right){\mathrm{YMnO}}_3$. The dashed vertical lines in (b) represent the position of the tail-to-tail (left) and head-to-head (right) walls for the different supercell sizes.

Figure 5

Electrostatic breakdown. (a) Polarization and (b) charge compensation at the walls as a function of domain width for ${\mathrm{YMnO}}_3,{\mathrm{InMnO}}_3$, and ${\mathrm{YGaO}}_3$. We note that the kink in ${\mathrm{YGaO}}_3$ is due to the saturation of the polar mode due to small domain size versus the increase due to screening. DFT calculated band gaps were used in the calculations.

Figure 6

Local electronic structure. Calculated atomically resolved local electronic density of states for $1\times 1\times 6\mathrm{\Delta }\mathrm{\Phi }={60}^{\circ }\left({\alpha }^{-}|{\beta }^{+}\right)$ supercells of (a) ${\mathrm{YMnO}}_3$, (b) ${\mathrm{InMnO}}_3$, and (c) ${\mathrm{YGaO}}_3$, illustrating the changes in the Fermi level position across the supercells. The top row gives the local DOSes at the head-to-head walls, the second row the local DOSes at the center of the ${\beta }^{+}$ domains, and the bottom row the local DOSes at the tail-to-tail walls.