Atomic and electronic structure of domains walls in a polar metal

Polar metals counterintuitively bring two well-known phenomena into coexistence, namely, bulk polar displacements, and an electronic Fermi surface giving rise to metallic conduction. However, little is known about the polar domains or domain walls in such materials. Using atomic resolution electron microscopy imaging combined with first principles density functional theory, we show that uncharged head-to-tail walls, and “charged” head-to-head and tail-to-tail walls can exist in the bulk of such crystals of polar metals ${\mathrm{Ca}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$, where both structural changes at the wall as well as electrostatic considerations define the wall nature. Significant built-in potentials of 30–170 meV are predicted at such walls.

Figure 1

(a) [110] view of ${\mathrm{Ca}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ with a magnified perovskite bilayer showing the Ca displacements (black arrows) which are used to define polar displacement vector ${\mathbf{v}}_{\mathrm{P}}$ and angle ${\theta }_P$. The Ca, Ru, and O atoms are represented by blue, gray, and red circles, respectively. ADF-STEM images along the [110] zone of (b) H-H, (c) T-T, and (d) H-T ${90}^{\circ }$ domain walls with orientation of ${\mathbf{v}}_{\mathrm{P}}$ indicated by the yellow arrows and the overlaid red and blue features. The yellow dashed line in (d) is a visual guide to the H-T ${90}^{\circ }$ domain wall. Zoomed in regions of the wall show below panels (b)–(d) depict the characteristic polar displacements of the Ca atoms across the domain walls. (e)–(g) Maps of ${\theta }_P$ calculated from the ADF-STEM images of H-H, T-T, and H-T ${90}^{\circ }$ domain walls with DFT derived values overlaid in the black boxes. Scale bars are 2 nm each.

Figure 2

(a) DFT calculated supercells of (a) head-to-tail and (b) head-to-head (H-H) and tail-to-tail (T-T) ${90}^{\circ }$ domain walls in ${\mathrm{Ca}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$, with the magnified area in (a) showing the [110] Ru-O-Ru bond angle. The two perovskite bilayers of the supercell structure are labeled $L1$ and $L2$. (c) and (d) Line profiles of the [110], $\left[1\overline{1}0\right]$, and [001] Ru-O-Ru bond angles across H-T, H-H, and T-T domain walls corresponding to the structures in (a) and (b), respectively. Note the strong difference in the [110] Ru-O-Ru bond angles in $L1$ and $L2$ for the H-H and T-T ${90}^{\circ }$ domain walls.

Figure 3

(a) ADF-STEM image of an intersection of CaO 180° and ${90}^{\circ }$ domain walls forming a $T$ junction, which transforms an H-T ${90}^{\circ }$ wall into a T-T ${90}^{\circ }$ wall. The direction of in-plane polarization is indicated by the yellow arrows and overlaid color (blue left, red right). (b) Maps of ${\theta }_P$ showing the characteristic oscillatory pattern for the H-T ${90}^{\circ }$ wall, confirming its presence at the $T$ junction. Inset, schematic of the domain structure at the $T$ junction. (c) ADF-STEM imaging of a second $T$ junction at the intersection of an H-T and H-H ${90}^{\circ }$ domain wall and (d) the corresponding ${\theta }_P$ maps of the intersection. Scale bars are 2 nm.

Figure 4

ADF-STEM image along ${90}^{\circ }$ of a domain wall $T$ junction in ${\mathrm{Ca}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$ with an illustration of the observed domain configuration (below). (b) High resolution ADF-STEM image taken from the area left (L) of the ${90}^{\circ }$ domain wall in (a) with the blue and red overlay indicating the polarization orientation. Note, the $n\ne 2$ defect layer is used as reference guide. (c) Magnified images of the CaO 180° domain wall taken from the yellow bow in (b) showing a reversal of the Ca polar chevron pattern within the perovskite layer. (d) ADF-STEM images taken from the area to the right of the ${90}^{\circ }$ domain wall. (e) Zoomed in image taken from the yellow box in (d) showing that the polar displacements are no longer in-plane confirming the presence of a continuous ${90}^{\circ }$ domain wall across the $T$ junction.

Figure 5

(a) Projected DOS for the H-T and H-H (T-T) ${90}^{\circ }$ domain walls. (b) Number of states at the Fermi level, $g\left(E_F\right)$, for the H-T (square) and H-H (T-T) (circle) ${90}^{\circ }$ domain walls. (c) Calculated macroscopic average electrostatic potential with the approximate screening length λ indicated. The inset illustrates the impact of geometric and electrostatic effects on the total DOS in (a) for the different domain walls.

Figure 6

(a) Change in the number of states at $E_F,g\left(E_F\right)$, and change in the Fermi level as a function of the ${\mathrm{RuO}}_6$ rotation $\left({X_2}^{+}\right)$ and tilt $\left({X_3}^{-}\right)$ modes amplitude in bulk ${\mathrm{Ca}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$. (b) Total DOS for several amplitudes of the ${X_2}^{+}$ and ${X_3}^{-}$ modes spanning the ground states crystal structure $\left({X_2}^{+}+{X_3}^{-}=100\%\right)$ to no rotation and tilt $\left({X_2}^{+}+{X_3}^{-}=0\%\right)$.

Figure 7

Calculated local (blue) and macroscopic (red) average potential for the H-H and T-T ${90}^{\circ }$ domain walls with introduction of vacuum in the calculation.