Magnetoelastic coupling in $R\mathrm{Ti}{\mathrm{O}}_3$ $(R=\mathrm{La},\mathrm{Nd},\mathrm{Sm},\mathrm{Gd},\mathrm{Y})$ investigated with diffraction techniques and thermal expansion measurements

A detailed analysis of the crystal structure in $R\mathrm{Ti}{\mathrm{O}}_3$ with $R=\mathrm{La}$, Nd, Sm, Gd, and Y reveals an intrinsic coupling between orbital degrees of freedom and the lattice which cannot be fully attributed to the structural deformation arising from bond-length mismatch. The $\mathrm{Ti}{\mathrm{O}}_6$ octahedra in this series are all irregular with the shape of the distortion depending on the rare-earth ionic radius. These octahedron distortions vary more strongly with temperature than the tilt and rotation angles. Around the Ti magnetic ordering all compounds exhibit strong anomalies in the thermal-expansion coefficients, these anomalies exhibiting opposite signs for antiferromagnetic and ferromagnetic compounds. Furthermore, the strongest effects are observed in materials close to the magnetic crossover from antiferromagnetic to ferromagnetic order.

Figure 1

(Color online) Crystal structure of $R\mathrm{Ti}{\mathrm{O}}_3$ with corner-sharing $\mathrm{Ti}{\mathrm{O}}_6$ octahedra in $Pbnm$ notation. Dashed lines show the cubic unit cell of the undistorted structure, and solid lines show the orthorhombic unit cell. Green: Ti ions. Blue: O ions. Red: $R$ ions.

Figure 2

(a) The octahedral tilt $\Theta$ and the rotation $\Phi$ versus the tolerance factor $t$ for various $d^0$ systems with perovskite structure [space-group number 62, $Pbnm$ (Ref. 41)]. (b) Difference of both basal $M\text{−}\mathrm{O}\left(2\right)$ bond lengths and difference of both O(2)-O(2) bond lengths versus $t$. (Open symbols denote the values for $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$ and $\mathrm{Y}\mathrm{Ti}{\mathrm{O}}_3$; lines are linear fits to the data.)

Figure 3

(Color online) (a)–(c) Orthorhombic lattice parameters of $R\mathrm{Ti}{\mathrm{O}}_3$ ($R=\mathrm{La}$, Nd, Sm, Gd, and Y) as a function of temperature. (d) $c$ parameter scaled to the room-temperature value. (e) Orthorhombic splitting $\epsilon =(a-b)∕(a+b)$. The arrows indicate the magnetic transition temperatures $T_{N1}∕T_C$, and the stars mark the values for $\epsilon$ measured by powder neutron diffraction. Open circles denote the values for the second ${\mathrm{Nd}}_{1-x}\mathrm{Ti}{\mathrm{O}}_{3+\delta }$ sample with a different stoichiometry and $T_N=81\mathrm{K}$.

Figure 4

Results of our powder neutron diffraction measurements on $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$ at different temperatures. The first three points were taken from Ref. 5. The others were measured on the same sample. (a) $\mathrm{Ti}{\mathrm{O}}_6$ octahedral tilt $\Theta$ and rotation $\Phi$. (b) Differences of the O(2)-O(2) bond lengths. (c) Differences of the Ti-O bond lengths. (d) Differences of the O(2)-O(2) bond lengths versus $\epsilon$. The approximately linear dependence of $\Delta$ (O(2)-O(2) bond length) and $\epsilon$ underlines that $\epsilon$ is a good indicator for the octahedral distortion $\Delta$ (O(2)-O(2) bond length). (Lines are a guide to the eyes.)

Figure 5

Thermal expansion $\alpha =1∕l\partial l∕\partial T$ for (a) $\mathrm{Nd}\mathrm{Ti}{\mathrm{O}}_3$ parallel to $b$ and $c$, (b) $\mathrm{Sm}\mathrm{Ti}{\mathrm{O}}_3$ parallel to $a$, and (c) $\mathrm{Y}\mathrm{Ti}{\mathrm{O}}_3$ parallel to $a$ and $c$.

Figure 6

(Color online) (a) Thermal expansion $\alpha =1∕l\partial l∕\partial T$ of $\mathrm{Y}\mathrm{Ti}{\mathrm{O}}_3$ for various magnetic fields as a function of temperature $(H\Vert a)$. (b) Magnetostriction $\Delta L\left(H\right)∕L$ of $\mathrm{Y}\mathrm{Ti}{\mathrm{O}}_3$ as a function of applied magnetic field $H\Vert a$.

Figure 7

(Color online) Magnetic phase diagram for $R\mathrm{Ti}{\mathrm{O}}_3$ and ${\mathrm{La}}_{1-x}{\mathrm{Y}}_x\mathrm{Ti}{\mathrm{O}}_3$. $T_C$ and $T_N$ are plotted as a function of the rare-earth ionic radius. Squares: $T_C$, $T_N$ for $R\mathrm{Ti}{\mathrm{O}}_3$ from this work. Triangles: values for $R\mathrm{Ti}{\mathrm{O}}_3$ from Greedan (Ref. 33). Crosses: values for $R\mathrm{Ti}{\mathrm{O}}_3$ from Katsufuji et al. (Ref. 55). Circles: values for ${\mathrm{La}}_{1-x}{\mathrm{Y}}_x\mathrm{Ti}{\mathrm{O}}_3$ from Okimoto et al. (Ref. 3). Stars: values for ${\mathrm{La}}_{1-x}{\mathrm{Y}}_x\mathrm{Ti}{\mathrm{O}}_3$ from Goral et al. (Ref. 56). The insets show the principal octahedral deformations of the end members.

Figure 8

(Color online) Results from single-crystal x-ray diffraction measurements at room temperature (open circles). (a) Apical tilt $\Theta$ and basal rotation $\Phi$ of the $\mathrm{Ti}{\mathrm{O}}_6$ octahedra of $R\mathrm{Ti}{\mathrm{O}}_3$. (b) Ti-O-Ti bond angles. (c) $R\text{−}\mathrm{O}$ bond lengths [see plot (g) for notation]. (d) O(2)-O(2) bond lengths. (e) Ti-O bond lengths. (f) Ratio of O(2)-O(2) bond lengths and ratio of Ti-O(2) bond lengths [see also plot (h)]. (g) Displacement of the $R$ ion from the symmetry position ($0,0,1∕4$). (h) $R\text{−}\mathrm{Ti}$ bond lengths. (i) Oxygen coordination of the $R^{3+}$ ion with the numbering for plot $\left(c\right)$. (j) $\mathrm{Ti}{\mathrm{O}}_6$—black arrows indicate both O(2)-O(2) bonds which are about parallel to $a$ or $b$, and white arrows indicate the two Ti-O(2) bonds which are rotated by $\Phi$ towards $a$ or $-b$.

Figure 9

(a) Orthorhombic splitting $\epsilon$ of ${\mathrm{La}}_{1-x}{\mathrm{Y}}_x\mathrm{Ti}{\mathrm{O}}_3$ $(x=0.50)$ compared with the other $R\mathrm{Ti}{\mathrm{O}}_3$ with similar rare-earth ionic radii. (b) Inverse magnetic susceptibility $1∕\chi$ of ${\mathrm{La}}_{0.5}{\mathrm{Y}}_{0.5}\mathrm{Ti}{\mathrm{O}}_3$ compared with those in $\mathrm{La}\mathrm{Ti}{\mathrm{O}}_3$ and in $\mathrm{Y}\mathrm{Ti}{\mathrm{O}}_3$.