Electronic structures of CeRu$_2X_2$ ($X$ = Si, Ge) in the paramagnetic phase studied by soft X-ray ARPES and hard X-ray photoelectron spectroscopy

Soft and hard X-ray photoelectron spectroscopy (PES) has been performed for one of the heavy fermion system CeRu$_2$Si$_2$ and a $4f$-localized ferromagnet CeRu$_2$Ge$_2$ in the paramagnetic phase. The three-dimensional band structures and Fermi surface (FS) shapes of CeRu$_2$Si$_2$ have been determined by soft X-ray $h u$-dependent angle resolved photoelectron spectroscopy (ARPES). The differences in the Fermi surface topology and the non-$4f$ electronic structures between CeRu$_2$Si$_2$ and CeRu$_2$Ge$_2$ are qualitatively explained by the band-structure calculation for both $4f$ itinerant and localized models, respectively. The Ce valences in CeRu$_2X_2$ ($X$ = Si, Ge) at 20 K are quantitatively estimated by the single impurity Anderson model calculation, where the Ce 3d hard X-ray core-level PES and Ce 3d X-ray absorption spectra have shown stronger hybridization and signature for the partial $4f$ contribution to the conduction electrons in CeRu$_2$Si$_2$.