Quadratic X-Ray Magneto-Optical Effect upon Reflection in a Near-Normal-Incidence Configuration at the $M$ Edges of $3d$-Transition Metals

We have observed a quadratic x-ray magneto-optical effect in near-normal-incidence reflection at the $M$ edges of iron. The effect appears as the magnetically induced rotation of $\sim 0.1°$ of the polarization plane of linearly polarized x-ray radiation upon reflection. A comparison of the measured rotation spectrum with results from x-ray magnetic linear dichroism data demonstrates that this is the first observation of the Schäfer-Hubert effect in the x-ray regime. Ab initio density-functional theory calculations reveal that hybridization effects of the $3p$ core states necessarily need to be considered when interpreting experimental data. The discovered magneto-x-ray effect holds promise for future ultrafast and element-selective studies of ferromagnetic as well as antiferromagnetic materials.

Figure 1

(a) Experimental setup for the detection of the Schäfer-Hubert effect. (b) Reflectivity in the vicinity of the iron $M$ edges at near perpendicular incidence. (c) Normalized detector signal when rotating the analyzer by an angle of $\gamma$.

Figure 2

Photon energy dependent Schäfer-Hubert rotation of the polarization plane of linearly polarized x rays in the vicinity of the Fe $M$ edges.

Figure 3

(a) Imaginary and real part of the x-ray magnetic linear dichroism measured at the Fe $M$ edges. (b) Calculated Schäfer-Hubert rotation ${\theta }_{\mathrm{SH}}$ based on (i) the XMLD transmission data of Fig. 3a (open triangles) and (ii) the XMLD asymmetry measured in reflection [gray (red) triangles]. Both data are compared to the experimentally determined ${\theta }_{\mathrm{SH}}$ values (solid circles, cf. Fig. 2).

Figure 4

(a) $j,j_z$ resolved density of states (DOS) of the $3p$ core levels of Fe. (b) Calculated Schäfer-Hubert rotation spectrum at the Fe $M$ edge, with $j,j_z$ hybridization included (solid curve) or without (dashed curve).