Systematics of x-ray resonant scattering amplitudes in $R{\mathrm{Ni}}_2{\mathrm{Ge}}_2$ ($R=\mathrm{Gd}$,Tb,Dy,Ho,Er,Tm): The origin of the branching ratio at the $L$ edges of the heavy rare earths

An investigation of the x-ray resonant magnetic scattering (XRMS) across the series of heavy rare-earth elements, coupled with first-principles calculations, has revealed that spin-orbit coupling in the $5d$ band plays a critical role in the systematics of the XRMS and x-ray magnetic circular dichroism branching ratio at the $L$ edges of magnetic rare-earth compounds.

Figure 1

Integrated intensities of $\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill L\pm \tau \hfill \end{array}\right)$ reflections measured at the Gd $L_3$ edge for (a) $\left[{\mathrm{Gd}}_{1∕4}{\mathrm{Tb}}_{1∕4}{\mathrm{Dy}}_{1∕4}{\mathrm{Ho}}_{1∕4}\right]{\mathrm{Ni}}_2{\mathrm{Ge}}_2$ and (b) $\left[{\mathrm{Gd}}_{1∕3}{\mathrm{Er}}_{1∕3}{\mathrm{Tm}}_{1∕3}\right]{\mathrm{Ni}}_2{\mathrm{Ge}}_2$. The solid lines represent the fits to the data with $I=A{(-z_1\cos \theta +z_3\sin \theta )}^2∕\sin 2\theta$, where $A$ is a scaling factor, and $z_1$ and $z_3$ are the components of the magnetic moment unit vector along the $\mathbf{a}$ and $\mathbf{c}$ directions, respectively.

Figure 2

XRMS energy scan through the rare-earth $L_3$ and $L_2$ edges for the $\left[{\mathrm{Gd}}_{1∕3}{\mathrm{Er}}_{1∕3}{\mathrm{Tm}}_{1∕3}\right]{\mathrm{Ni}}_2{\mathrm{Ge}}_2$ sample at the $\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 6+{\tau }_z\hfill \end{array}\right)$ magnetic reflection. The solid lines represent the fits to the data as described in the text.

Figure 3

XRMS energy scan through the rare-earth $L_3$ and $L_2$ edges for the $\left[{\mathrm{Gd}}_{1∕4}{\mathrm{Tb}}_{1∕4}{\mathrm{Dy}}_{1∕4}{\mathrm{Ho}}_{1∕4}\right]{\mathrm{Ni}}_2{\mathrm{Ge}}_2$ sample at the $\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 6+{\tau }_z\hfill \end{array}\right)$ magnetic reflection. The solid lines represent the fits to the data as described in the text.

Figure 4

The fit value of the integrated intensity of the resonant magnetic scattering, $E_0^4{(F_{11}-F_{1-1})}^2$, at the $L_2$ (open circles) and $L_3$ edges (closed circles) for both samples, plotted as a function of the square of the $4f\text{−}5d$ exchange energy (Ref. 17). The data are normalized to the integrated intensity at the Gd $L_2$ edge of the samples. The solid lines serve as a guide to the eye.

Figure 5

The XRMS branching ratio for the heavy rare-earth elements in $R{\mathrm{Ni}}_2{\mathrm{Ge}}_2$ compounds. The solid squares represent the ratio of intensities from the $L_2$ and $L_3$ edges shown in Fig. 4. The solid circles show the result of the first-principles calculations, as described in the text, ignoring SOC in the $5d$ band. The open circles show the results of the same calculations, now including the effect of spin-orbit coupling in the $5d$ band.

Figure 6

Energy dependence of the radial dipole matrix elements for spin up (upper line) and spin down (lower line) for Gd. The Fermi energy $E_F$ is located at 0.0 eV and the top of the $5d$ band is at approximately 8 eV. The matrix element is significantly larger for the antibonding states at the top of the $5d$ band.