Multiferroic behavior in $\mathrm{EuTi}{\mathrm{O}}_3$ films constrained by symmetry

We have elucidated the spin, lattice, charge, and orbital coupling mechanism underlying the multiferroic character in tensile-strained $\mathrm{EuTi}{\mathrm{O}}_3$ films. Symmetry determined by oxygen octahedral tilting shapes the hybridization between the Eu $4f$ and the Ti $3d$ orbitals, and this inhibits predicted Ti displacement proper ferroelectricity. Instead, phonon softening emerges at low temperatures within the pseudocube (110) plane, orthogonal to the anticipated ferroelectric polarization symmetry. Additionally, the magnetic anisotropy is determined by orbital distortion through hybridization between the Ti $3d$ and the typically isotropic $\mathrm{E}{\mathrm{u}}^{2+}4f$ states. This unique scenario demonstrates the critical role symmetry plays in the coupling of order parameters defining multiferroic behavior.

Figure 1

Reciprocal space scans (along the L direction) of half order reflections resulting from the oxygen octahedral tilt pattern. Absence of the half order (1/$2\overline{1/2}5$/2) indicates that the film has a single-type tilt domain. Cartoons illustrating (left) the ETO pseudocubic unit cell showing $G$-AFM order with all three coexisting spin-ordering interactions and (right) the biaxial strain and single oxygen-related tilt pattern imposed by the DSO (110) substrate.

Figure 2

(a) Ti $K$-edge near-edge spectra with linear polarization at 5 and 275 K. (b) Expanded plots of the spectral intensities of the $e_{\mathrm{g}}$ preedge peak contrasting the temperature dependence of polarization parallel to both in-plane [100] and out-of-plane [001] as depicted in the inset. (c) Cartoon illustration of the relative incident polarization (yellow arrows) with low-temperature phonon softening effect along [001] (cyan arrow). (d) Azimuthal-temperature dependence of $e_{\mathrm{g}}$ preedge intensity along both $e\left|\right|\left[110\right]$ and $e\left|\right|\left[\overline{1}10\right]$. (e) Cartoon displaying the relative incident geometries with the arrow indicating increased softening. (f) Temperature dependence of the $e_{\mathrm{g}}$ preedge intensity with the polarization along the [110] and $\left[\overline{1}10\right]$. (g) Temperature dependence of the in-plane film resistance. The inset shows the comparison between other strain states. (h) Cartoon showing the aggregate relative orientation of Ti phonon softening along the cube diagonal with respect to the oxygen tilting within the (110) plane.

Figure 3

(a) Temperature-dependent XRMS-scattering intensity of the ${\left(003\right)}_{\mathrm{ETO}}$ reflection at the Eu $L_{\mathrm{II}}$ edge indicative of FM spin order. The inset of the bottom left corner illustrates the horizontal-scattering experimental configuration (π-σ), and the top right shows resonant enhancement (background subtracted) above and below $T_{\mathrm{c}}$. (b) Energy scans at the ${\left(003\right)}_{\mathrm{ETO}}$ reflection in (π-π) geometry where the charge-scattering intensity can interfere with the magnetic scattering. The in- and out-of-phase interferences of magnetic and charge scatterings are dependent on the aligned spin direction. (c) Field-dependent sweeps with fixed energy in (π-π) showing clear FM hysteresis loops. Data are offset for clarity. With the applied field along the easy magnetic axis $\left[\overline{1}10\right]$, the hysteresis loop shows a minimum coercivity. Along the [010] and close to the [110], the intensity difference between positive and negative field directions decreases, and the loop broadens. The inset images illustrate, regardless of changes in the field direction by sample rotation, the spins stay within the easy axis $\left[\overline{1}10\right]$. (d) A cartoon model illustrating the relative orientation of the oxygen ${\mathrm{O}}_{\mathrm{h}}$ tilting, Ti phonon softening, and uniaxial magnetic anisotropy.