Relationship between crystal structure and multiferroic orders in orthorhombic perovskite manganites

We use resonant and nonresonant x-ray diffraction measurements in combination with first-principles electronic structure calculations and Monte Carlo simulations to study the relationship between crystal structure and multiferroic orders in the orthorhombic perovskite manganites, $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ ($R$ is a rare-earth cation or Y). In particular, we focus on how the internal lattice parameters (Mn-O bond lengths and Mn-O-Mn bond angles) evolve under chemical pressure and epitaxial strain, and the effect of these structural variations on the microscopic exchange interactions and long-range magnetic order. We show that chemical pressure and epitaxial strain are accommodated differently by the crystal lattice of $\text{o-}R{\mathrm{MnO}}_3$, which is key for understanding the difference in magnetic properties between bulk samples and strained films. Finally, we discuss the effects of these differences in the magnetism on the electric polarization in $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$.

Figure 1

Crystal structure of $\text{o-}R{\mathrm{MnO}}_3$. Green spheres indicate $R^{3+}$ cations; purple, ${\mathrm{Mn}}^{3+}$ cations; and red, ${\mathrm{O}}^{2-}$ anions. Panel (a) shows the view in the $bc$ plane, and panel (b) shows it in the $ab$ plane (only Mn and O ions are shown).

Figure 2

$d\text{−}p\text{−}d$ superexchange paths within the $ab$ planes in $\text{o-}R{\mathrm{MnO}}_3$. (a) $e_g\text{−}{p}_{\sigma }\text{−}{e}_g$ superexchange paths. The cooperative JT distortion of ${\mathrm{MnO}}_6$ octahedra favors the ordering of the $e_g$ orbitals such that an occupied $e_g$ orbital (colored) on one Mn site overlaps with an empty $e_g$ orbital (white) on the neighboring Mn site via the $p_{\sigma }$ state of oxygen. (b) $t_{2g}\text{−}{p}_{\pi }\text{−}{t}_{2g}$ superexchange paths.

Figure 3

Phase diagram of bulk $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$, following Ref. [34]. Borders are drawn based on magnetic susceptibility and electric polarization measurements conducted on powders. The area on the left of the dashed line in the E-AFM phase represents $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ for which conflicting data on measurements of the magnetism and ferroelectricity was reported. Labels at the top of the image indicate the $R$ ion associated with the radii indicated on the lower horizontal axis. Increased orange shading indicates higher predicted electric polarization.

Figure 4

Magnetic modulation vector $q_b$ as function of radius of the $R$ ion. Literature data for bulk samples (single crystals and powders) [17, 36, 37, 39, 40, 41, 42, 43] of $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ are presented as empty circles. In Refs. [17, 36, 37, 39, 40, 42, 43], $q_b$ values were determined based on neutron diffraction measurements. Reference [41] is a study of the mixed crystal systems ${\mathrm{Eu}}_{1-x}{\mathrm{Y}}_x{\mathrm{MnO}}_3$ and their $q_b$ values were extracted from XRD measurements of lattice modulation at low temperatures. Films (literature data from Refs. [21, 44, 45] and our new measurements) are indicated by filled circles. The dashed line indicates the trend for bulk materials, and the green solid line is that for relaxed films. The blue line shows the discrepancy between bulk samples and strained films with the same $R$ ion. $q_b$ is in reciprocal lattice units.

Figure 5

Magnetic intensity from reciprocal space scans along the [010] direction, taken using resonant x-ray diffraction at the Mn $L_3$ edge. Data are from [010]-oriented films of orthorhombic ${\mathrm{TbMnO}}_3$, ${\mathrm{HoMnO}}_3$, and ${\mathrm{LuMnO}}_3$. The widths of the peaks include contributions from the limited thickness, the limited probe depth at resonance, and the correlation length of the AFM order. $q_b$ is in reciprocal lattice units.

Figure 6

Heisenberg ($J_c$, $J_{ab}$, $J_a$, $J_b$, $J_{\mathrm{diag}}$, $J_{3nn}$), biquadratic ($B_c$ and $B_{ab}$), and four-spin ring ($K_c$ and $K_{ab}$) exchange interactions considered in the model Hamiltonian of Eq. (3). The 40-atom $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ supercell ($1\times 2\times 1$ of the 20-atom unit cell) containing eight Mn ions (purple spheres, the lighter spheres indicate Mn ions in neighboring cells) is shown ($R$ and O ions are not shown).

Figure 7

Theoretically optimized structural parameters of strained films and bulk $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ vs the radius of the $R$ cation: Panels (a) and (b) give the Mn-O-Mn bond angles within the $ab$ planes (IP angle) and along the $c$ direction (OP angle), respectively; panels (c)–(e) show short ($s$), medium ($m$), and long ($l$) Mn-O bond lengths of the ${\mathrm{MnO}}_6$ octahedra, respectively; and panel (f) shows the lengths of the O(1)-O(2) bridges [see Fig. 1] within the $ab$ planes. Bulk samples are shown by empty circles and strained films by filled circles. For $\mathrm{o}\text{−}{\mathrm{LuMnO}}_3$, the triangles denote calculations for hypothetical films which are compressively strained in the $ac$ plane by the same amount but in the opposite direction as the experimentally measured tensile strained films. $\mathrm{o}\text{−}{\mathrm{LuMnO}}_3$ 26-nm film and the corresponding inverse case are highlighted in gray; 104-nm film and the inverse case are shown in black. Compressive strain within the $ac$ planes of the $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ films is shown by the violet color; tensile strain is indicated by the blue color. The dashed lines connecting the data points for bulk $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ are guides to the eye.

Figure 8

Orbital mixing angles $\theta$ in strained films and bulk $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ vs the radius of the $R$ cation $r_R$. Bulk samples are shown by empty circles, and experimentally measured strained films by filled circles. For $\mathrm{o}\text{−}{\mathrm{LuMnO}}_3$, the triangles denote calculations for hypothetical films which are compressively strained in the $ac$ plane by the same amount but in the opposite direction as the experimentally measured tensile strained films. The 26-nm $\mathrm{o}\text{−}{\mathrm{LuMnO}}_3$ film and the corresponding inverse case are highlighted in gray; the 104-nm film and the inverse case are in black. Compressive strain within the $ac$ planes of the $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ films is shown by the violet color; tensile strain is indicated by the blue color. The dashed lines connecting the data points for bulk $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ are guides to the eye.

Figure 9

Calculated exchange coupling constants vs the radius of the $R$ cation: Panels (a)–(d) show the Heisenberg exchanges $J_c$, $J_{ab}$, $J_b$, and $J_{3nn}$, respectively; panel (e) shows the biquadratic in-plane exchanges $B_{ab}$, and panel (f) shows the four-spin ring couplings $K_c$. Bulk samples are indicated by the empty circles, and experimentally measured strained films are shown by the filled circles. For $\mathrm{o}\text{−}{\mathrm{LuMnO}}_3$, the triangles denote the hypothetical films which are compressively strained in the $ac$ plane by the same amount but opposite direction as experimentally measured tensile strained films. The $\mathrm{o}\text{−}{\mathrm{LuMnO}}_3$ 26-nm film and the corresponding inverse case are highlighted in gray, and the 104- and inv104-nm cases are in black. Compressive strain within the $ac$ planes of the $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ films is shown by the violet color, and tensile strain is given by the blue color. The dashed line connecting the data points for the bulk $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$ is used to guide the eye.

Figure 10

Experimentally determined and calculated modulation vectors of the ground-state magnetic phases in bulk and strained $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$. Panel (a) shows $q_b$ for bulk $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$, and panel (b) shows that for the films of $\mathrm{o}\text{−}R{\mathrm{MnO}}_3$. Black circles indicate experimentally determined (exp.) $q_b$, and purple circles denote calculated $q_b$ (MC). The gray circle in panel (b) indicates the experimentally measured $q_b$ in the 26-nm film of ${\mathrm{LuMnO}}_3$, and the green circle shows the calculated $q_b$ for this film; the $q_b$ values for the 104-nm ${\mathrm{LuMnO}}_3$ film are shown with the usual black (measured) or purple (calculated with MC) circles (note that the purple circle at $q_b=0.5$ is obscured by the green circle). $q_b$ is in reciprocal lattice units.

Figure 11

Theoretically identified lowest energy magnetic phases in 26- and 104-nm films of $\mathrm{o}\text{−}{\mathrm{LuMnO}}_3$. Panel (a) shows the magnetic unit cell for H-AFM order (crystallographic $\mathrm{o}\text{−}{\mathrm{LuMnO}}_3$ unit cell is doubled along the $b$ axis); panel (b) shows the magnetic unit cell for I-AFM order (crystallographic unit cell is doubled along $b$ and $c$ directions). Mn ions are highlighted in purple; red arrows indicate Mn spins. Lu and O ions are not shown.