Momentum dependence of $4f$ hybridization in heavy-fermion compounds: Angle-resolved photoemission study of ${\mathrm{YbIr}}_2{\mathrm{Si}}_2$ and ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$

Heavy-fermion compounds ${\mathrm{YbIr}}_2{\mathrm{Si}}_2$ and ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ were studied by means of angle-resolved photoemission (PE). The splittings and dispersions of the $4f^{13}$ bulk and surface PE signals in the region of the expected energy crossings of the $4f$ final states and the valence bands in the Brillouin zone are observed. The experimental results are explained in terms of a simplified periodic Anderson model by a momentum dependence of the electron hopping matrix element between the $4f$ and valence-band states.

Figure 1

(Color online) Crystal structure of $\mathrm{Yb}{T}_2{\mathrm{Si}}_2$. Orthogonal unit vectors ${\mathbf{n}}_x$, ${\mathbf{n}}_y$, and ${\mathbf{n}}_z$ are set as described in the text.

Figure 2

Angle-integrated PE spectra of mainly Si terminated ${\mathrm{YbIr}}_2{\mathrm{Si}}_2$ taken at $25\mathrm{eV}$ (hatched) and $110\mathrm{eV}$ (dots) photon energy (bottom). Subspectra reflect results of a least-squares fit using atomic $\mathrm{Yb}4f^{12}$ and $4f^{13}$ final-state multiplets (Ref. 43). Corresponding data of mainly Si (middle) and Yb (top) terminated ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$.

Figure 3

(Color online) Angle-resolved PE spectra of ${\mathrm{YbIr}}_2{\mathrm{Si}}_2$ taken along the $\overline{\Gamma }\text{−}\overline{M}$ direction (bottom) and the two parallel to $\overline{\Gamma }$-$\overline{M}$ directions shifted (as shown in Fig. 9) by $\mathrm{\Delta }{\mathbf{k}}_b=0.19{\mathrm{\AA }}^{-1}$ (middle) and $\mathrm{\Delta }{\mathbf{k}}_a=0.50{\mathrm{\AA }}^{-1}$ (top) relative to $\overline{\Gamma }$-$\overline{M}$, respectively. $\Theta$ is polar detection angle. The spectra are normalized each to its maximum intensity.

Figure 4

(Color online) Angle-resolved PE spectra of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ taken along the $\overline{\Gamma }$-$\overline{M}$ direction. The spectra are normalized each to its integral intensity. Note, that the surface-to-bulk ratio of the $4f$ signal is large indicating surface termination by an almost closed Yb layer (Ref. 44).

Figure 5

(Color online) Angle-resolved PE spectra of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ taken along the $\overline{\Gamma }$-$\overline{M}$ direction at $h\nu =110\mathrm{eV}$. The spectra are normalized each to its integral intensity.

Figure 6

(Color online) Band structure calculated along the $\Gamma$-$M$ direction of the BZ for the nine-atomic layer ${\mathrm{YbIr}}_2{\mathrm{Si}}_2$ slab (left-hand side) and for the slab, where Yb atoms are substituted by Ba in the surface layer and by La in the bulk (right-hand side). Experimental data measured at $h\nu =25\mathrm{eV}$ are shown in a color-scale representation. Lighter colors correspond to higher PE intensities.

Figure 7

Angle-resolved PE spectra of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ taken along the $\overline{\Gamma }$-$\overline{M}$ direction at different photon energies. To emphasize band dispersions the spectra were normalized to equal intensities of all measured PE features. Lighter colors correspond to higher PE signals.

Figure 8

Angular dependence of the $4f_{7∕2}$ PE intensities in $g$ (solid line) and $d$ (dashed line) channels from different $\mu$ states for the experimental geometry used in the present work. Radial matrix elements $D_l$ were set to 1 for the respective channel and to 0 for the other channel.

Figure 9

(Color online) Results of the model PAM calculations performed for different slices of a paraboloidlike valence band. Calculated intensities show the $f$ character of the hybrid bands describing qualitatively the scenarios observed in Fig. 3.

Figure 10

(Color online) Angle-resolved $4f$ emission of ${\mathrm{YbIr}}_2{\mathrm{Si}}_2$ simulated by means of the simplified PAM: contribution from the bulk (top, multiplied by factor 5), the surface layer (middle) and the total intensity distribution in the PE spectra (bottom).