Fermi Surface Variation Near Quantum Critical Point of CeRu2(Si1-xGex )2 Studied by Resonant ARPES

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CeRu 2 Si 2 is a representative heavy-fermion (HF) compound with a paramagnetic ground state. The large specific heat coefficient γ ~ 350 mJ/mol K 2 indicates the itinerancy of the Ce 4 f electrons in the ground state, and the Kondo crossover temperature T 0 is estimated to be ~20-25 K. A magnetic ground state appears upon substitution of Ge atoms for Si atoms, i.e., CeRu 2 (Si 1- x Ge x ) 2 , corresponding to the application of a negative chemical pressure: The system is antiferromagnetic for x = 0.07-0.57 and ferromagnetic for x = 0.57-1.0 [1]. The boundary of the emergence of the antiferromagnetic ground state, namely, the critical composition x c = 0.07, at a very low temperature is considered to be the quantum critical point (QCP), where the Ce 4 f electrons may be transformed from itinerant electrons to localized electrons, possibly accompanied by variations in the shape and volume of Fermi surfaces (FSs). The de Haas van Alphen (dHvA) experiment indicated that the 4 f -itinerant FSs are observed in paramagnetic CeRu 2 Si 2 , while the FSs of ferromagnetic CeRu 2 Ge 2 are almost the same as those of LaRu 2 Ge 2 , indicating that the 4 f electrons are localized in CeRu 2 Ge 2 [2,3]. The important issue is in the point where the variation from the 4 f -itinerant FSs to the 4 f -localized FSs occurs upon the substitution of Ge atoms for Si atoms, and one of the most possible candidates is the point near QCP, i.e., around x c . Therefore, in order to verify the existence of the FS variation across x c , we have investigated the Fermi surfaces CeRu 2 Si 2 and CeRu 2 (Si 0.82 Ge 0.18 ) 2 in paramagnetic state at around T 0 by soft X-ray angle-resolved photoelectron spectroscopy (ARPES) [4]. In order to enhance the photoemission signals of the Ce 4 f electrons, we have performed an ARPES experiment in the Ce 3 d → 4 f resonance energy region. ARPES was also performed for LaRu 2 Si 2 as a reference material where the 4 f electrons do not participate in the FS formation. The experiments were performed at beamline BL23SU . Figures 1(b) and 1(c), 1(d) and 1(e), and 1(f) and 1(g) show ARPES spectra of the valence bands of LaRu 2 Si 2 , CeRu 2 Si 2 , and CeRu 2 (Si 0.82 Ge 0.18 ) 2 , respectively, along the X-Z and Γ -X directions of the Brillouin zone (BZ) illustrated in Fig. 1(a). The angle- scanning measurements were performed at photon energies h ν corresponding to the Γ -X-Z plane, i.e., h ν = 765, 860, and 855 eV ( off-resonance hereafter) for LaRu 2 Si 2 , CeRu 2 Si 2 , and CeRu 2 (Si 0.82 Ge 0.18 ) 2 , respectively. As indicated by the red broken lines, the observed dispersions of energy bands are similar between CeRu 2 Si 2 and CeRu 2 (Si 0.82 Ge 0.18 ) 2 , while the energy dispersions of LaRu 2 Si 2 are remarkably different from those of CeRu 2 Si 2 . Figures 2(a)-2(c) show off-resonance FS images of LaRu 2 Si 2 , CeRu 2 Si 2 , and CeRu 2 (Si 0.82 Ge 0.18 ) 2 , respectively, represented by the intensities of the ARPES spectra integrated near Fermi energy ( E F ) 76 (a) Γ Z Z X X X (b) (c) (d) (e) (f) (g) Intensity (arb. units) Intensity (arb. units) Intensity (arb. units) Binding Energy (eV) Binding Energy (eV) Binding Energy (eV) Binding Energy (eV) 1.2 0.8 0.4 0.0 1.2 0.8 0.4 0.0 1.2 0.8 0.4 0.0 Binding Energy (eV) 1.2 0.8 0.4 0.0 Binding Energy (eV) 1.2 0.8 0.4 0.0 1.2 0.8 0.4 0.0 Z X X Γ Γ Z X X Γ Γ Z X X Γ Γ k z k x k y CeRu 2 Si 2 CeRu 2 Si 2 LaRu 2 Si 2 LaRu 2 Si 2 CeRu 2 (Si 0.82 Ge 0.18 ) 2 CeRu 2 (Si 0.82 Ge 0.18 ) 2 Materials Science : Electronic & Magnetic Properties Fig. 1. (a) Brillouin zone for CeRu 2 (Si 1- x Ge x ) 2 . Off- resonance ARPES spectra of LaRu 2 Si 2 along (b) X-Z and (c) Γ -X directions, those of CeRu 2 Si 2 along (d) X-Z and (e) Γ -X directions, and those of CeRu 2 (Si 0.82 Ge 0.18 ) 2 along (f) X-Z and (g) Γ -X directions. Fermi Surface Variation Near Quantum Critical Point of CeRu 2 (Si 1- x Ge x ) 2 Studied by Resonant ARPES as a function of momenta ( k x , k y ). For comparison, the calculated FSs of LaRu 2 Si 2 and CeRu 2 Si 2 in the k x – k y plane [5] are shown in Figs. 2(d) and 2(e), respectively. The numbers 1-4 and 5 designate the hole FSs and the electron FS, respectively. Figures 2(f) and 2(g) show the calculated three-dimensional (3D) images of the large hole FS of band 4 of CeRu 2 Si 2 and LaRu 2 Si 2 , respectively, indicating a significant volume variation due to the participation of the Ce 4 f electrons in the FS formation [5]. A large FS surrounding the Z point was experimentally detected in every compound. While the experimental large FS of LaRu 2 Si 2 has a square shape and agrees well with the calculated hole FS of band 4, the experimental large FS of CeRu 2 Si 2 has a circular shape and agrees well with the calculated electron FS of band 5. The large FS of CeRu 2 (Si 0.82 Ge 0.18 ) 2 is almost identical to that of CeRu 2 Si 2 . A residual drawback in this assignment is that the hole FS of band 4 in Fig. 2(e), which is considered to be the heaviest FS branch including the largest contribution of the Ce 4 f electrons [5], is not observed in both CeRu 2 Si 2 and CeRu 2 (Si 0.82 Ge 0.18 ) 2 . Therefore, angle-scanning measurements were performed at the Ce 3 d → 4 f resonant energy of h ν = 881 eV ( on-resonance ) for CeRu 2 Si 2 and CeRu 2 (Si 0.82 Ge 0.18 ) 2 to enhance the signals of the 4 f electrons. The resonance energy corresponds to k z = 0.29 π / c and 0.37 π / c for CeRu 2 Si 2 and CeRu 2 (Si 0.82 Ge 0.18 ) 2 , respectively. Figures 3(a) and 3(b) show on-resonance FS images of CeRu 2 Si 2 and CeRu 2 (Si 0.82 Ge 0.18 ) 2 , respectively. Surprisingly, the large circular FS surrounding the Z point in the off-resonance images becomes invisible in the on-resonance images. On the other hand, another FS appears only inside the square BZ boundary surrounding the Z point in both CeRu 2 Si 2 and CeRu 2 (Si 0.82 Ge 0.18 ) 2 . The size and shape of this FS are comparable to those of the calculated hole FS of band 4 at k z = 0.3 π / c away from the Γ -Z-X plane, shown by red lines in Figs. 3(a) and 3(b), and thus the observed FS is assigned to be the FS of band 4, which includes the largest contribution of the Ce 4 f electrons. Thus, the assignment of the FSs in Figs. 2 and 3 is totally consistent. The observation of the FS including a strong 4 f contribution in both CeRu 2 Si 2 and CeRu 2 (Si 0.82 Ge 0.18 ) 2 indicates that the 4 f -delocalized regime is extended beyond the critical composition x c . The absence of the FS change near x c is strongly implied in the paramagnetic state below T 0 , and it can be considered that the spin-density-wave model is suitable for the quantum criticality of CeRu 2 (Si 1- x Ge x ) 2 . 77 5 4 1-3 4 1-3 5 k y (Å -1 ) 1.5 High Low 1.5 2.0 2.5 3.0 k y (Å -1 ) k y (Å -1 ) 1.0 1.0 -1.0 -1.5 0.5 0.5 -0.5 0.0 1.5 1.5 2.0 2.5 3.0 1.0 1.0 -1.0 -1.5 0.5 0.5 -0.5 0.0 0.0 0.0 1.5 1.5 2.0 2.5 3.0 1.0 1.0 -1.0 -1.5 0.5 0.5 -0.5 0.0 0.0 Γ X X Γ Z (a) (d) (b) (e) (c) (f) (g) Z CeRu 2 Si 2 LaRu 2 Si 2 LaRu 2 Si 2 LaRu 2 Si 2 CeRu 2 Si 2 CeRu 2 Si 2 k x (Å -1 ) CeRu CeRu 2 (Si (Si 0.82 0.82 Ge Ge 0.18 0.18 ) 2 (b) (a) k y (Å -1 ) k x (Å -1 ) 1.5 1.5 2.0 2.5 3.0 1.0 1.0 –1.0 –1.5 0.5 0.5 –0.5 0.0 0.0 k x (Å -1 ) 1.5 1.5 2.0 2.5 3.0 High Low 1.0 1.0 –1.0 –1.5 0.5 0.5 –0.5 0.0 0.0 CeRu CeRu 2 Si Si 2 CeRu CeRu 2 (Ru (Ru 0.82 0.82 Si Si 0.18 0.18 ) 2 Tetsuo Okane SPring-8 / JAEA E-mail: okanet@spring8.or.jp References [1] M. Sugi et al .: Phys. Rev. Lett. 101 (2008) 056401. [2] H. Aoki et al .: Phys. Rev. Lett. 71 (1993) 2110. [3] C.A. King et al .: Physica B. 171 (1991) 161. [4] T. Okane, T. Ohkochi, Y. Takeda, S.-i. Fujimori, A. Yasui, Y. Saitoh, H. Yamagami, Y. Matsumoto, M. Sugi, N. Kimura, T. Komatsubara and H. Aoki: Phys. Rev. Lett. 102 (2009) 216401 . [5] H. Yamagami et al .: J. Phys. Soc. Jpn. 61 (1992) 2388; J. Phys. Soc. Jpn. 62 (1993) 592. Fig. 2. Off-resonance FS images of LaRu 2 Si 2 (a) , CeRu 2 Si 2 (b) , and CeRu 2 (Si 0.82 Ge 0.18 ) 2 (c) in the k x - k y plane, compared with calculated FS images of LaRu 2 Si 2 (d) and CeRu 2 Si 2 (e) [5]. BZ is illustrated by broken lines. Calculated 3D images of hole FS of band 4 of CeRu 2 Si 2 (f) and LaRu 2 Si 2 (g) around Z point (body center) [5]. Fig. 3. On-resonance FS images in k x - k y planes of (a) CeRu 2 Si 2 and (b) CeRu 2 (Si0.82 Ge 0.18 ) 2 , compared with calculated hole FS of band 4 at k z = 0.3 π / c away from Γ -Z-X plane, represented by red lines [5].