Figure 1

Setup used for the XRMS experiment. The polarization of the incident light is tunable, the scattered intensity is recorded with a diode, and the electron yield current is measured on the copper sample holder. The surface normal $\vec{n}$ always lies in the scattering plane, while the perpendicular direction can be changed manually (angle $\phi$).Reuse & Permissions

Figure 2

(a) Temperature dependence of the scattering intensity for incident $\pi$-polarized radiation at the Mn L${}_2$ edge. The charge-scattering background measured above the ordering temperature has been subtracted from each measurement. Black points indicate the peak center retrieved by a Gaussian fit. (b) Relative shift of the peak-center value of ${\tau }_{\text{Mn}}$ between 40 K and its minimum for different values of the film thickness.Reuse & Permissions

Figure 3

Temperature dependence of the multiferroic order in epitaxial TbMnO${}_3$ films in complementary measurements (a) by XRMS and SQUID magnetometry (inset) and (b) by SHG. (a) The $b$ and $c$ component of the magnetic Mn order and the $\mathit{ab}$ component of the Tb order for a TbMnO${}_3$ film of 11 nm. Data were extracted from intensity measurements of $\sigma$- and $\pi$-polarized incident light at the (0 ${\tau }_{\mathrm{Mn}}$ 0) and (0 ${\tau }_{\mathrm{Tb}}$ 0) positions using Eqs. (1, 2, 3, 4). The curve in the inset shows the thermoremanent magnetization for comparison. (b) SHG intensity from the ${\chi }_{caa}$ component at 1.907 eV for a TbMnO${}_3$ film of 100 nm. The inset shows the polarization anisotropy of the SHG signal at $T=11$ K. The SHG polarization was fixed at $0^{\circ }$ or ${90}^{\circ }$ while rotating the incident-light polarization. Solid lines are fits using Eqs. (5) and (6).Reuse & Permissions

Figure 4

Sketch of the SHG geometry used for the measurements, illustrating the angle of polarization ${\phi }_{2\omega ,\omega }$ of the SHG and the fundamental light with respect to the sample axes.Reuse & Permissions

Figure 5

Spatial map of the circular XRMS dichroism measured at (0 ${\tau }_{\mathrm{Mn}}$ 0) at the Mn L edge of a TbMnO${}_3$ film of 100 nm. The sample was cooled with the x-ray beam at the “burn point” position (green ellipse approximates beam size) and measured at 11 K. The arrows indicate the electric-polarization direction associated with the respective cycloidal domain.Reuse & Permissions

Figure 6

Electric-field dependence of the SHG signal from the 100 nm TbMnO${}_3$ film at 50 K (top) and 15 K (bottom). The field was applied with noncontact electrodes along the $c$ axis. Assuming SHG contributions by the polarization, the electric field, and the crystallographic structure, a polarization hysteresis was fitted to the SHG data according to Ref. 35 (bottom, solid lines). Note that leakiness often produces pseudopolarization loops of a similar shape in pyroelectric current measurements. However, the SHG measurements are not affected by leakiness so that the polarization hysteresis in the inset is genuine.Reuse & Permissions