$4f$-Derived Fermi Surfaces of ${\mathrm{CeRu}}_2({\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x{)}_2$ near the Quantum Critical Point: Resonant Soft-X-Ray ARPES Study

Angle-resolved photoelectron spectroscopy in the Ce $3d\to 4f$ excitation region was measured for the paramagnetic state of ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$, ${\mathrm{CeRu}}_2({\mathrm{Si}}_{0.82}{\mathrm{Ge}}_{0.18}{)}_2$, and ${\mathrm{LaRu}}_2{\mathrm{Si}}_2$ to investigate the changes of the $4f$ electron Fermi surfaces around the quantum critical point. While the difference of the Fermi surfaces between ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ and ${\mathrm{LaRu}}_2{\mathrm{Si}}_2$ was experimentally confirmed, a strong $4f$-electron character was observed in the band structures and the Fermi surfaces of ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ and ${\mathrm{CeRu}}_2({\mathrm{Si}}_{0.82}{\mathrm{Ge}}_{0.18}{)}_2$, consequently indicating a delocalized nature of the $4f$ electrons in both compounds. The absence of Fermi surface reconstruction across the critical composition suggests that SDW quantum criticality is more appropriate than local quantum criticality in ${\mathrm{CeRu}}_2({\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x{)}_2$.

Figure 1

Schematic phase diagram for QCP of HF metal [1]. It is illustrated against temperature versus control parameter $\delta$ which represents the strength ratio of Kondo effect to RKKY interaction. $T_N$, $T_{\mathrm{FL}}$, and $T_0$ designate the Néel temperature, the onset temperature of Fermi liquid behavior, and the crossover temperature where Kondo screening sets in, respectively. Blue and red broken lines represent a crossover from the $4f$-delocalized FSs to the $4f$-localized FSs in the SDW QCP model and the LQC model, respectively.

Figure 2

(a) BZ for ${\mathrm{CeRu}}_2({\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x{)}_2$. Red and blue planes signify the measured plane by the angle-scanning and the $h\nu$-scanning measurements, respectively. Off-resonance ARPES spectra of ${\mathrm{LaRu}}_2{\mathrm{Si}}_2$ along the $X\mathrm{\text{−}}Z$ (b) and $\Gamma \mathrm{\text{−}}X$ (c) directions, those of ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ along the $X\mathrm{\text{−}}Z$ (d) and $\Gamma \mathrm{\text{−}}X$ (e) directions, and those of ${\mathrm{CeRu}}_2({\mathrm{Si}}_{0.82}{\mathrm{Ge}}_{0.18}{)}_2$ along the $X\mathrm{\text{−}}Z$ (f) and $\Gamma \mathrm{\text{−}}X$ (g) directions. Red broken lines indicate dispersive energy bands.

Figure 3

Off-resonance FS images of ${\mathrm{LaRu}}_2{\mathrm{Si}}_2$ (a), ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ (b), and ${\mathrm{CeRu}}_2({\mathrm{Si}}_{0.82}{\mathrm{Ge}}_{0.18}{)}_2$ (c) in the $k_x\mathrm{\text{−}}{k}_y$ plane, compared with calculated FSs of ${\mathrm{LaRu}}_2{\mathrm{Si}}_2$ (d) and ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ (e) [7]. In (d) and (e), numbers designate the label of each FS, the same as used in Refs. 21, 22, and red and blue lines represent the hole FS of band 4 and the electron FS of band 5, respectively. The BZ is illustrated by broken lines. Calculated 3D images of the hole FS of band 4 of ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ (f) and ${\mathrm{LaRu}}_2{\mathrm{Si}}_2$ (g) around the $Z$ point (body center) [7].

Figure 4

FS images of ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ (a), ${\mathrm{CeRu}}_2({\mathrm{Si}}_{0.82}{\mathrm{Ge}}_{0.18}{)}_2$ (b), and ${\mathrm{LaRu}}_2{\mathrm{Si}}_2$ (c) in the $k_{xy}\mathrm{\text{−}}{k}_z$ plane. The BZ is indicated by blue broken lines. Calculated 3D images of the electron FS of band 5 of ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ (d) and ${\mathrm{LaRu}}_2{\mathrm{Si}}_2$ (e) around the $\Gamma$ point (body center) [7].

Figure 5

On-resonance ARPES spectra of ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ (a) and ${\mathrm{CeRu}}_2({\mathrm{Si}}_{0.82}{\mathrm{Ge}}_{0.18}{)}_2$ (b) along the line corresponding to the $X\mathrm{\text{−}}Z$ direction. Images of the band dispersion near $E_F$ obtained from the on-resonance ARPES spectra of ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ (c) and ${\mathrm{CeRu}}_2({\mathrm{Si}}_{0.82}{\mathrm{Ge}}_{0.18}{)}_2$ (d). Red squares in (a)–(d) signify the $E_F$-crossing point of the strongly $c\mathrm{\text{−}}f$ hybridized bands. On-resonance FS images in the $k_x\mathrm{\text{−}}{k}_y$ plane of ${\mathrm{CeRu}}_2{\mathrm{Si}}_2$ (e) and ${\mathrm{CeRu}}_2({\mathrm{Si}}_{0.82}{\mathrm{Ge}}_{0.18}{)}_2$ (f), compared with the calculated hole FS of band 4 at $k_z=0.3\pi /c$ away from the $\Gamma \mathrm{\text{−}}Z\mathrm{\text{−}}X$ plane, represented by red lines [7].