Coupled structural distortions, domains, and control of phase competition in polar ${\mathrm{SmBaMn}}_2{\mathrm{O}}_6$

Materials with coupled or competing order parameters display highly tunable ground states, where subtle perturbations reveal distinct electronic and magnetic phases. These states generally are underpinned by complex crystal structures, but the role of structural complexity in these phases often is unclear. We use group-theoretic methods and first-principles calculations to analyze a set of coupled structural distortions that underlie the polar charge and orbitally ordered antiferromagnetic ground state of $A$-site ordered ${\mathrm{SmBaMn}}_2{\mathrm{O}}_6$. We show that these distortions play a key role in establishing the ground state and stabilizing a network of domain wall vortices. Furthermore, we show that the crystal structure provides a knob to control competing electronic and magnetic phases at structural domain walls and in epitaxially strained thin films. These results provide new understanding of the complex physics realized across multiple length scales in ${\mathrm{SmBaMn}}_2{\mathrm{O}}_6$ and demonstrate a framework for systematic exploration of correlated and structurally complex materials.

Figure 1

Experimental phase diagram of the $A$-site ordered $R{\mathrm{BaMn}}_2{\mathrm{O}}_6$ family ($R=\mathrm{rare}\mathrm{earth}$), adapted from Refs. [15, 16]. Here CO = charge order, OO = orbital order, PM = paramagnetic, FM = ferromagnetic, AFM = antiferromagnetic.

Figure 2

Coupled ground-state charge, orbital, and spin order of ${\mathrm{SmBaMn}}_2{\mathrm{O}}_6$. The ${\mathrm{Mn}}^{3+}$/${\mathrm{Mn}}^{4+}$ CO forms a checkerboard in the $ab$ plane. The ${\mathrm{Mn}}^{3+}$ sites host a $d_{x^2-r^2}/d_{y^2-r^2}$ OO, which form stripes parallel to the $b$ axis. The spins order in a CE-type AFM pattern, where they form FM-coupled zigzags in the $ab$ plane (indicated by light blue and red). These zigzags are AFM coupled to each other within the $ab$ plane and also along $c$. The crystallographic unit cell is shown by the dashed green box, and the CE-AFM magnetic unit cell is shown by the black box. The green arrows indicate the setting of the orthorhombic axes $a$ and $b$ relative to the tetragonal axes.

Figure 3

Structural distortions that contribute to the ground-state crystal structure. (a) High-symmetry reference structure $P4/mmm$. The ground-state $P{2}_{1}am$ structure decomposes into four structural distortions that transform like irreps of $P4/mmm$: (b) an out-of-phase octahedral tilt that transforms like $M_5^{-}$, (c) a breathing distortion coupled to the ${\mathrm{Mn}}^{3+}$/${\mathrm{Mn}}^{4+}$ charge order that transforms like $M_4^{+}$, (d) a polar mode that transforms like ${\mathrm{\Gamma }}_5^{-}$, and (e) a set of displacements that form in stripes along $b$ and transform like ${\mathrm{\Sigma }}_2$. In (c) and (d), the distortion amplitudes are artificially increased for better visualization. For clarity, in (e) only the apical oxygen displacements are highlighted with arrows, although all atoms displace from their high-symmetry positions. The green arrows indicate the setting of the orthorhombic axes $a$ and $b$ relative to the tetragonal axes.

Figure 4

Understanding how structural distortion couplings establish the ground state. The energy surface in the space defined by the $M_5^{-}$, $M_4^{+}$, and ${\mathrm{\Sigma }}_2$ distortions is shown in (a). The triplet ($Q_T/{\tilde{Q}}_T$, $Q_b/{\tilde{Q}}_b$, $Q_s/{\tilde{Q}}_s$) indicates the combination of $M_5^{-}$, $M_4^{+}$, and ${\mathrm{\Sigma }}_2$ distortions frozen into $P4/mmm$ along a given branch. The color and symbol for each curve in each panel indicate the distortion amplitude that is changing along the curve, so only one distortion is changing along any curve in any panel. In the center panel, the curves discussed in the main text are shown with solid lines, and are labeled by the coupling terms that are active. Here, $\tilde{Q}$ are the distortion amplitudes obtained from the $\mathrm{DFT}+U$-relaxed $P{2}_{1}am$ structure with A-AFM. The stripe distortion order parameter $(s_1,s_2)$ is fixed along the (0.88, 0.47) direction, which is the direction obtained in the $\mathrm{DFT}+U$-relaxed $P{2}_{1}am$ A-AFM state. (b) Shows energy surfaces obtained by freezing in the ${\mathrm{\Gamma }}_5^{-}$ distortion alone (green circles), with fixed $Q_s={\tilde{Q}}_s$ (cyan triangles), and with fixed $Q_T={\tilde{Q}}_T$ and $Q_b={\tilde{Q}}_b$ (magenta squares). For all calculations, the lattice parameters are fixed to those of $P4/mmm$ with A-AFM order, and A-AFM order is imposed (see Appendix pp5 for details).

Figure 5

Visualizing coupled structural domains and revealing competing phases. (a) Structural domains within one orthorhombic twin of ${\mathrm{SmBaMn}}_2{\mathrm{O}}_6$. The domains (red dots) are points within the two-dimensional space defined by the ${\mathrm{\Sigma }}_2$ stripe order-parameter directions ($s_1,s_2$) for that twin. The domains in the top right and lower left quadrants have octahedral tilt $Q_T$, while the domains in the other two quadrants have tilt $-Q_T$. The pink (blue) regions denote regions with CO/breathing distortion $+Q_b$ ($-Q_b$), and the hatched (solid) regions indicate regions with polarization $Q_P$ ($-Q_P$). (b) Paths between domains, indicated by gray lines, with the barrier structures labeled by colored circles: $Pbam$ (green), $Pmam$ (white), and $Pbmm$ (purple). Extending to the four-dimensional order-parameter space defined by the full ${\mathrm{\Sigma }}_2$ order parameter ($s_1,s_2,s_3,s_4$), we can construct closed paths that represent (c) threefold and (d) fourfold domain wall vortices. Here, the thick orange lines indicate paths between domains in opposite orthorhombic twins (twin walls), the orange circle is the $C2mm$ barrier for this path.

Figure 6

Phase control with epitaxial strain. Total energy versus strain for several structural and magnetic phases of ${\mathrm{SmBaMn}}_2{\mathrm{O}}_6$. The $0\%$ strain is defined relative to the CE-AFM lattice constants. The dashed line indicates the strain value at which the energy of the CE-AFM $P{2}_{1}am$ and FM $Cmmm$ phases are energetically degenerate. Negative and positive strain values correspond to compressive and tensile strains, respectively.

Figure 7

Energy barrier between competing phases. Total energy versus constrained spin angle for bulk and compressively strained ${\mathrm{SmBaMn}}_2{\mathrm{O}}_6$. The angle that the left reference spin (in the red box) makes with the horizontal axis defines the spin angle $\theta$, so $\theta =0^{\circ }$ and ${90}^{\circ }$ correspond to CE-AFM and FM, respectively. The dark gray arrow indicates the direction of the hypothetical magnetic field.

Figure 8

Total energy of ${\mathrm{SmBaMn}}_2{\mathrm{O}}_6$ as a function of (a) $U$ (with fixed $J=1.2\mathrm{eV}$), and (b) $J$ (with fixed $U=4\mathrm{eV}$). Total energy as a function of $U$ for (c) ${\mathrm{NdBaMn}}_2{\mathrm{O}}_6$ and (d) ${\mathrm{LaBaMn}}_2{\mathrm{O}}_6$. For (c) and (d), $J=1.2\mathrm{eV}$. All energies are referenced to the $Cmmm$ A-AFM energy.

Figure 9

Relationship between sublattices, the stripe distortion, and orthorhombic twins. Here, red and blue represent sublattices A and B, respectively. The red (blue) arrows represent the apical oxygen displacements on sublattice A (B), all other displacements that contribute to the stripe distortion are suppressed for clarity. With the setting of the orthorhombic axes $a$ and $b$ relative to the tetragonal axes shown, the two orthorhombic twins can be distinguished via different settings of the space group (a) $P{2}_{1}am$ and (b) $Pb{2}_{1}m$. The distortion forms stripes of upward and downward displacements perpendicular to the long axis, so in twin $P{2}_{1}am$ the stripes lie along $b$ and in twin $Pb{2}_{1}m$ they lie along $a$. The black dashed line indicates the crystallographic unit cell.

Figure 10

Atomic displacements that contribute to the stripe distortion. The left column shows distortions present on sublattice A only: (a) $a^{-}{a}^{-}{c}^0$ octahedral tiltlike distortion and cation displacements, (b) $a^0{a}^0{c}^{+}$ octahedral rotationlike distortion, and (c) Jahn-Teller distortion. These distortions transform like the $(a,0,0,0)$ direction of ${\mathrm{\Sigma }}_2$ and establish a structure with symmetry $Pmam$. The middle column shows distortions on sublattice B only: (d) $a^{-}{a}^{-}{c}^0$ octahedral tiltlike distortion and cation displacements, (e) $a^0{a}^0{c}^{+}$ octahedral rotationlike distortion, and (f) Jahn-Teller distortion, which transform like the $(0,a,0,0)$ direction of ${\mathrm{\Sigma }}_2$ and establish a different $Pmam$ domain. The right column (g)–(i) shows the case where equal amplitudes of these distortions are present on both sublattices, which yields structures that transform like the $(a,a,0,0)$ direction of ${\mathrm{\Sigma }}_2$ (symmetry $Pbam$). If the distortion amplitudes on the two sublattices are different [the ($a,b$,0,0) direction of ${\mathrm{\Sigma }}_2]$, then the $P{2}_{1}am$ structure is established. Displacements of the oxygen atoms are indicated by black arrows, and the Mn, Sm, and Ba displacements are shown with purple, white, and green arrows, respectively. The amplitudes of all distortions are artificially increased for clarity.

Figure 11

CO, OO, and CE-AFM order parameters. (a) The CO/OO/CE-AFM phase with ${\mathrm{Mn}}^{3+}$ and ${\mathrm{Mn}}^{4+}$ sites colored red and blue, respectively. The order parameters are shown individually for (b) OO, (c) magnetic order $L_{\mathrm{CE}}$ on the ${\mathrm{Mn}}^{3+}$ sublattice, and (d) magnetic order $X_{\mathrm{CE}}$ on the ${\mathrm{Mn}}^{4+}$ sublattice. The black dashed lines indicate the unit cell for each order parameter.

Figure 12

Evolution of the crystal structure of $R{\mathrm{BaMn}}_2{\mathrm{O}}_6$ with change to the rare-earth $R^{3+}$ ionic radius: (a) $c/a$ ratio for the $P{2}_{1}am$ CE-AFM and the $P4/mmm$ FM and A-AFM phases, and (b) the symmetry-adapted mode amplitudes for the $P{2}_{1}am$ CE-AFM phase.