Continuity of Ce $4f$ electronic structure across the quantum critical point: A resonant photoemission study on $\mathrm{Ce}{\mathrm{Ni}}_{1-x}{\mathrm{Co}}_x{\mathrm{Ge}}_2$

Ce $3d\text{−}4f$ and $4d\text{−}4f$ resonant photoemission spectroscopies have been performed on the heavy-fermion compound $\mathrm{Ce}{\mathrm{Ni}}_{1-x}{\mathrm{Co}}_x{\mathrm{Ge}}_2$, where the ground-state properties systematically change from the magnetic $(0⩽x<0.3)$ to nonmagnetic $(0.3<x⩽1.0)$ regime via the quantum critical point (QCP, $x=0.3$). Co-substitution dependence of the bulk Ce $4f$ electronic structure shows gradual evolution of Kondo resonance at the Fermi level together with the reduction of the Ce $4f^0$ final state in agreement with the single impurity Anderson model (SIAM). The SIAM analysis shows that the Kondo temperature and specific-heat coefficient change continuously from the weakly hybridized $\mathrm{Ce}\mathrm{Ni}{\mathrm{Ge}}_2$ to strongly hybridized $\mathrm{Ce}\mathrm{Co}{\mathrm{Ge}}_2$. These indicate that the Ce $4f$ electronic structure of $\mathrm{Ce}{\mathrm{Ni}}_{1-x}{\mathrm{Co}}_x{\mathrm{Ge}}_2$ changes continuously through the QCP.

Figure 1

(Color online) Ce $4f$ spectra of the experiments (circles) and its fitting results of the NCA calculation (Ref. 23) (lines) in Ce $3d\text{−}4f$ (left) and Ce $4d\text{−}4f$ (right) RPE for $\mathrm{Ce}{\mathrm{Ni}}_{1-x}{\mathrm{Co}}_x{\mathrm{Ge}}_2$. The inset shows the same ones in Ce $3d\text{−}4f$ RPE with enlarged scale.

Figure 2

(Color online) (a) The off-RPE spectra and the $3d$ PDOS (Co and Ni) for $x=0$, 1, obtained from Ce $3d\text{−}4f$ RPE and the corresponding LDA band calculation, (Ref. 25), respectively. (b) Hybridization matrix elements of bulk contribution used for the NCA calculation. (Ref. 23). Ce $4f$ spectra of $\mathrm{Ce}\mathrm{Co}{\mathrm{Ge}}_2$ and the NCA fitting curves, Gaussian convoluted with the instrumental energy resolution, for (c) Ce $3d\text{−}4f$ RPE and (d) Ce $4d\text{−}4f$ RPE.

Figure 3

(Color online) (a) Hybridization matrix intensity at $E_F$ and Ce $4f$ electron number evaluated from NCA calculation (Ref. 23) for the bulk contribution. (b) The Kondo temperature, $T_K$, and (c) the specific heat coefficient, $\gamma$, obtained from PE and the SH measurement. (Ref. 20).