Emergent Superstructural Dynamic Order due to Competing Antiferroelectric and Antiferrodistortive Instabilities in Bulk ${\mathrm{EuTiO}}_3$

Microscopic structural instabilities of ${\mathrm{EuTiO}}_3$ single crystals were investigated by synchrotron x-ray diffraction. Antiferrodistortive (AFD) oxygen octahedron rotational order was observed alongside Ti derived antiferroelectric distortions. The competition between the two instabilities is reconciled through a cooperatively modulated structure allowing both to coexist. The combination of electric and magnetic fields increases the population of the modulated AFD order, illustrating how the origin of the large magnetoelectric coupling derives from the dynamic equilibrium between AFD and polar instabilities.

Figure 1

(a) Reciprocal space map in $H$ and $K$ of the antiferrodistortive octahedral order reflection at ($3/2$ $1/2$ $5/2$) presenting the modulation satellites and a cartoon depiction illustrating the oxygen octahedral rotations. (b) $L$ scan through the (4 4 2) Bragg reflection and an atomic model of the titanium displacement at 5.2 K. Broad intensity is observed at (4 4 $2\pm 1/2$) corresponding to the local AFE order. (c) An atomic model of the modulated AFD order and the corresponding AFE order. The arrows show the possible local electric polarization order but the actual titanium shift direction is not determined from this measurement. (d) $H$ and $K$ scans around ($3/2$ $3/2$ $1/2$). In both scans, the m-AFD reflections are allowed due to $H$, $K\ne L$ (e) $L$ scan around ($3/2$ $3/2$ $1/2$). The m-AFD reflection is forbidden as $H=K$ at 2 K.

Figure 2

(a) The $L$ scans around ($7/2$ $5/2$ $1/2$) right after cooling and 17 hours later. The intensity of the modulated AFD order increases slowly over a long time. (b) The temperature dependence of the AFD intensity at ($3/2$ $1/2$ $5/2$) and the resistance of the sample. The response of resistance variation by temperature changes around 160 K where the AFD order occurs. (c) Temperature dependence of the $K$ scans across ($3/2$ $1/2$ $5/2$) in color scale. (d) The temperature dependence of integrated intensities of normal and modulated AFD order reflections. The normal AFD order disappears around 160 K while the modulated AFD order persists up to 285 K.

Figure 3

(a) Energy scans of the magnetic reflection ($3/2$ $3/2$ $1/2$) in $\sigma -\pi$ geometry and fluorescence. The resonance enhancement is shown at the Eu $L_{II}$ absorption edge. (b) $H$ scans around magnetic reflection ($3/2$ $3/2$ $1/2$) and charge Bragg peak (2 2 0). The width of the magnetic peak is comparable to the width of the charge peak. (c) $H$ scans around ($3/2$ $3/2$ $1/2$) reflection in $\sigma -\sigma$ and $\sigma -\pi$ polarization geometries. Only magnetic intensity from $G$-type magnetic order is shown in $\sigma -\pi$ geometry while the AFD order reflections are seen in $\sigma -\sigma$ geometries. (d) Temperature dependence of the magnetic intensity and the critical exponent fitting curve. Inset shows the SQUID measurement of magnetization versus temperature curve with 100 Oe.

Figure 4

(a) The magnetic field effects on the modulated AFD order at ($7/2$ $5/2$ q) reflection and AFM order at ($3/2$ $3/2$ $1/2$) magnetic reflection. The red circles represent the intensity of modulated AFD order without $E$-field application. The blue squares are measurements with $E$-field, $0.67\times {10}^5\mathrm{V}/\mathrm{cm}$. A 10% intensity change is shown around the saturation field ($\sim 0.7\mathrm{T}$) with $E$-field application. The green triangles are intensities of the $G$-AFM order, which decreases as the magnitude of magnetic field increases and disappears with magnetic moment saturation. The paired arrows show the response of the magnetic moment orientation by external magnetic field. (b) The configuration of the electric and magnetic field directions according to the crystallographic geometry.