$k$ Dependence of the Crystal-Field Splittings of $4f$ States in Rare-Earth Systems

The occupation, energy separation, and order of the crystal-field-split $4f$ states are crucial for the understanding of the magnetic properties of rare-earth systems. We provide the experimental evidence that crystal-field-split $4f$ states exhibit energy dispersion in momentum space leading to variations of energy spacings between them and even of their energy sequence across the Brillouin zone. These observations were made by performing angle-resolved photoemission experiments on ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ and properly simulated within a simple model based on results obtained by inelastic neutron scattering experiments and band structure calculations. Our findings should be generally applicable to rare-earth systems and have considerable impact on the understanding of magnetism and related phenomena.

Figure 1

ARPES spectra taken from the Si-terminated surface of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ along the $\overline{\Gamma }\mathrm{\text{−}}\overline{X}$ direction of the surface BZ using 45 eV photons and linear vertical (a), circular left (b), circular right (c), and linear horizontal (d) polarization.

Figure 2

(a) Valence bands deriving from subsurface Rh atoms calculated for a Si-terminated slab of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ along the $\overline{\Gamma }\mathrm{\text{−}}\overline{X}$ direction of the surface BZ. Yb $4fs$ were treated as quasicore states. The dot size scales with the Rh $4d$ contributions. (b) Dispersion of the crystal-field-split Yb $4f$ states simulated within the hybridization model.

Figure 3

Experimental (upper panels) and simulated (lower panels) ARPES spectra from the Si-terminated surface of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ taken along the $\overline{\Gamma }\mathrm{\text{−}}\overline{X}$ direction (a) and along parallel directions shifted from the $\overline{\Gamma }$ point by about 0.03 (b) and 0.06 (c) of the $[\overline{\Gamma }\mathrm{\text{−}}\overline{X}]$ distance.