Evolution of Magneto-Orbital order Upon $B$-Site Electron Doping in ${\mathrm{Na}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Mn}}_7{\mathrm{O}}_{12}$ Quadruple Perovskite Manganites

We present the discovery and refinement by neutron powder diffraction of a new magnetic phase in the ${\mathrm{Na}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Mn}}_7{\mathrm{O}}_{12}$ quadruple perovskite phase diagram, which is the incommensurate analogue of the well-known pseudo-CE phase of the simple perovskite manganites. We demonstrate that incommensurate magnetic order arises in quadruple perovskites due to the exchange interactions between $A$ and $B$ sites. Furthermore, by constructing a simple mean field Heisenberg exchange model that generically describes both simple and quadruple perovskite systems, we show that this new magnetic phase unifies a picture of the interplay between charge, magnetic, and orbital ordering across a wide range of compounds.

Figure 1

(a) ${\mathrm{Pr}}_{1-x}{\mathrm{Ca}}_x{\mathrm{MnO}}_3$ phase diagram adapted from Ref. [4]. (b) ${\mathrm{Na}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Mn}}_7{\mathrm{O}}_{12}$ phase diagram based upon the present study. Both (a) and (b) have a common average $B$ site valence scale. (c) Temperature dependence of magnetic moments that compose the $\mathrm{i}p\mathrm{CE}$ structure of ${\mathrm{Na}}_{0.75}{\mathrm{Ca}}_{0.25}{\mathrm{Mn}}_7{\mathrm{O}}_{12}$. (d) Cartoons of $B$ site orbital ordering that spans average $B$-site valences from $+3.5$ to $+\mathrm{3.25.}$ Note that the purple $B\text{−}{\mathrm{Mn}}^{(4-x)+}$ orbitals indicate a likely orbital filling scheme.

Figure 2

Neutron powder diffraction data (red points) measured in bank 2 (average $2\theta =58.3°$) of the WISH diffractometer from (a) the paramagnetic, (b) CE, and c) $\mathrm{CE}+\mathrm{i}p\mathrm{CE}$ magnetic phases of ${\mathrm{Na}}_{0.75}{\mathrm{Ca}}_{0.25}{\mathrm{Mn}}_7{\mathrm{O}}_{12}$. The fitted models described in the main text are shown as solid black lines. Difference patterns ($I_{\mathrm{obs}}-I_{\text{calc}}$) are given as solid blue lines. The location of nuclear (black label) and magnetic (green label) diffraction peaks from phase I (black ticks), phase II (purple ticks), and impurity phases (orange ticks) are given at the bottom of the figure. The inset to (a) highlights the presence of two $I2/m$ nuclear phases, I and II.

Figure 3

(a) The $\mathrm{i}p\mathrm{CE}$ magnetic structure with magnetic moments artificially shown to rotate in the $ac$ plane. The origin of the incommensurate modulation has been placed at the center of the zeroth unit cell, drawn as a black solid line. For reference, the simple perovskite primitive unit cell and orbital supercell are superimposed, drawn as dotted and dashed lines, respectively. Double headed arrows denote $b$-axis stacking of moments that are antialigned within the same unit cell. Single headed arrows denote ferromagnetic stacking along $b$. (b) The orbital structure of the zigzag chains, with spins rotating in the true $bc$ plane. $A\text{−}{\mathrm{Mn}}^{3+}$, $B\text{−}{\mathrm{Mn}}^{4+}$, and $B\text{−}{\mathrm{Mn}}^{3+}$ moments are colored green, red, and blue, respectively. $B\text{−}{\mathrm{Mn}}^{3+}$ and $B\text{−}{\mathrm{Mn}}^{4+}$ $d_{3{\mathrm{z}}^2-{\mathrm{r}}^2}$ orbitals are shaded in blue and pink, respectively.

Figure 4

Magnetic phase diagram as determined from model Heisenberg exchange energy calculations described in the main text, and presented in detail in the Supplemental Material [19]. The vertical dashed line demarks the value of $J^{AB}/\tilde{J}$ determined from the experimental values of $k_x$ and $k_z$ found in all measured $\mathrm{i}p\mathrm{CE}$ phases.