Interplay between magnetic order at Mn and Tm sites alongside the structural distortion in multiferroic films of $o\text{−}\mathrm{TmMn}{\mathrm{O}}_3$

We employ resonant soft x-ray diffraction to individually study the magnetic ordering of the Mn and the Tm sublattices in single-crystalline films of orthorhombic $\left(o\text{−}\right)\mathrm{TmMn}{\mathrm{O}}_3$. The same magnetic ordering wave vector of $\left(0q0\right)$ with $q\approx 0.46$ is found for both ionic species, suggesting that the familiar antiferromagnetic order of the Mn ions induces a magnetic order on the Tm unpaired $4f$ electrons. Indeed, intensity variations of magnetic reflections with temperature corroborate this scenario. Calculated magnetic fields at the Tm sites are used as a model magnetic structure for the Tm, which correctly predicts intensity variations at the Tm resonance upon azimuthal rotation of the sample. The model allows ruling out a $bc$-cycloid modulation of the Mn ions as the cause for the incommensurate ordering, as found in $\mathrm{TbMn}{\mathrm{O}}_3$. The structural distortion, which occurs in the ferroelectric phase below $T_C$, was followed through nonresonant diffraction of structural reflections forbidden by the high-temperature crystal symmetry. The $\left(0q0\right)$ magnetic reflection appears at the Mn resonance well above $T_C$, indicating that this reflection is sensitive also to the intermediate sinusoidal magnetic phase. The model presented suggests that the Tm $4f$ electrons are polarized well above the ferroelectric transition and are possibly not affected by the transition at $T_C$. The successful description of the induced order observed at the Tm resonance is a promising example for future element-selective studies in which “spectator” ions may allow access to previously unobtainable information about other constituent ions.

Figure 1

Experimental geometry of the RSXD experiment on a film sample. The scattering plane contains the momentum transfer $\mathit{Q}$ and the directions of the incoming and outgoing beam, $\vec{k}$ and ${\vec{k}}^{\prime }$, respectively. $\theta$ is the Bragg angle and ${\chi }_0$ is the angle between $\mathit{Q}$ and the film surface normal $\widehat{n}$. $\mathrm{\Psi }$ is an angle of sample rotation around the $\mathit{Q}$ direction. The bottom arrow (in green) indicates the $a$-axis direction within the film plane when $\mathrm{\Psi }=0$.

Figure 2

Temperature dependence of the $\left(0q0\right)$ reflection taken at the $\mathrm{Mn}{L}_3$ edge (643 eV) with incoming $\pi$-polarized light. Sample: 400-nm-thick [010]-oriented film. Inset: intensity of the $\left(0q0\right)$ reflection as a function of incoming photon energy, measured with $\pi$- and with $\sigma$-polarized incoming light, at 10 K. The sample was oriented here with the $c$ axis perpendicular to the scattering plane, suppressing the signal produced with incoming $\sigma$ light, as expected (see main text).

Figure 3

(a) Temperature dependence of integrated intensities. Blue and black circles represent the $\left(0q0\right)$ reflection measured with RSXD at the $\mathrm{Mn}{L}_3$ and $\mathrm{Tm}{M}_5$ edges, respectively. These reflections were measured in the same experiment. The solid line is a fit of the Tm RSXD dataset to an induced order moment [Eq. (2), see main text] using the Mn RSXD dataset to account for the temperature variation in the magnetic field. The red triangles are the (0 3 3) reflection measured with XRD. (b) Temperature dependence of the modulation parameter $q$, from the Mn $L_3$ and $\mathrm{Tm}{M}_5$ edges (measured in the same experiment).

Figure 4

Variation of integrated intensity of the $\left(0q0\right)$ reflection upon azimuthal rotation, measured at the $\mathrm{Mn}{L}_3$ edge at 16 K, using $\sigma$ and $\pi$ linearly polarized light. Solid lines are calculations of expected intensity from Eqs. (1) and (A1).

Figure 5

Depiction of the commensurate canted E-type model, projected onto the $bc$ plane. E1 and E2 are the two equivalent E-type domains (labels as in Ref. [7]). Mn ions (large spheres) are shown with empty and filled (red) arrows, representing the collinear and $c$-axis canted Mn moments. The canting angle $\varphi$ between them is indicated. The Tm ions (small spheres) are presented in bright (green) and dark (purple) shades indicating the local field magnitudes $m_1$ and $m_2$ at the Tm sites (see Table 1). Dashed boxes indicate the chemical unit cell.

Figure 6

Reciprocal space scan along the $\left(0k0\right)$ direction, taken at the $\mathrm{Tm}{M}_5$ edge using incoming $\sigma$- and $\pi$-polarized light at 16 K.

Figure 7

Energy dependences of the $\left(0q0\right)$ and $\left(010\right)$ reflections measured at 10 K, using linearly polarized incoming light, of either $\pi$ (black symbols) or $\sigma$ (red symbols) polarizations.

Figure 8

Temperature dependence of the $\left(0q0\right)$ reflection measured at the $\mathrm{Tm}{M}_5$ absorption edge. Inset: a reciprocal space map of the $\left(0q1\right)$ reflection at 10 K, measured on a [110]-oriented film using $\pi$-polarized light.

Figure 9

Variation in RSXD intensity from the $\left(0q0\right)$ and $\left(0q1\right)$ reflections upon azimuthal rotation, measured at 10 K at the Tm $M_5$ edge (∼1460 eV) with incoming $\sigma$- or $\pi$-polarized linear light. Data are normalized by the sum of intensities from $\sigma$ and $\pi$ light (see figure legend). Triangles and circles correspond to data from [110]-oriented or [010]-oriented samples, respectively. Solid lines are calculations based on Eqs. ((1), (A2), and (A3). Dashed lines are the equivalent calculation of an induced Tm order from a Mn $bc$ cycloid.

Figure 10

Temperature dependence of the structural (0 3 3) reflection, forbidden by the high-temperature space group.