Anisotropic $c\ensuremath{-}f$ Hybridization in the Ferromagnetic Quantum Critical Metal ${\mathrm{CeRh}}_{6}{\mathrm{Ge}}_{4}$

Anisotropic $c - f$ Hybridization in the Ferromagnetic Quantum Critical Metal ${CeRh}_{6} {Ge}_{4}$

Yi Wu, Yongjun Zhang, Feng Du, Bin Shen, Hao Zheng, Yuan Fang, Michael Smidman, Chao Cao, Frank Steglich, Huiqiu Yuan, Jonathan D. Denlinger, and Yang Liu

Phys. Rev. Lett. 126, 216406 – Published 28 May 2021

Abstract

Heavy fermion compounds exhibiting a ferromagnetic quantum critical point have attracted considerable interest. Common to two known cases, i.e., ${CeRh}_{6} {Ge}_{4}$ and ${YbNi}_{4} P_{2}$ , is that the $4 f$ moments reside along chains with a large interchain distance, exhibiting strong magnetic anisotropy that was proposed to be vital for the ferromagnetic quantum criticality. Here, we report an angle-resolved photoemission study on ${CeRh}_{6} {Ge}_{4}$ in which we observe sharp momentum-dependent $4 f$ bands and clear bending of the conduction bands near the Fermi level, indicating considerable hybridization between conduction and $4 f$ electrons. The extracted hybridization strength is anisotropic in momentum space and is obviously stronger along the Ce chain direction.The hybridized $4 f$ bands persist up to high temperatures, and the evolution of their intensity shows clear band dependence. Our results provide spectroscopic evidence for anisotropic hybridization between conduction and $4 f$ electrons in ${CeRh}_{6} {Ge}_{4}$ , which could be important for understanding the electronic origin of the ferromagnetic quantum criticality.

Received 19 December 2020
Revised 5 May 2021
Accepted 6 May 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.216406

Physics Subject Headings (PhySH)

Electronic structure Kondo effect Quantum phase transitions

Ferromagnets Heavy-fermion systems

Angle-resolved photoemission spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yi Wu¹, Yongjun Zhang^1,2, Feng Du¹, Bin Shen¹, Hao Zheng¹, Yuan Fang¹, Michael Smidman¹, Chao Cao³, Frank Steglich ^1,4, Huiqiu Yuan^1,5,6,*, Jonathan D. Denlinger⁷, and Yang Liu ^1,5,†

¹Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou 310058, China
²Institute for Advanced Materials, Hubei Normal University, Huangshi 435002, China
³Department of Physics, Hangzhou Normal University, Hangzhou 311121, China
⁴Max Planck Institute for Chemical Physics of Solids, Dresden 01187, Germany
⁵Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310058, China
⁶State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310058, China
⁷Advanced Light Source, E.O. Lawrence Berkeley National Lab, Berkeley, California 94720, USA

^*Corresponding author. hqyuan@zju.edu.cn
^†Corresponding author. yangliuphys@zju.edu.cn

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Vol. 126, Iss. 21 — 28 May 2021

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Figure 1
(a) Crystal structure of ${CeRh}_{6} {Ge}_{4}$ . The light red plane indicates the possible cleavage surface. (b) Bulk Brillouin zone (BZ) and the momentum axes defined in this paper. $k_{z}$ is defined along the bulk [010] direction. (c) Large-range scan showing core levels (inset is a full-scale view). (d) Photon-energy dependent scan (30–150 eV) along the in-plane $k_{x}$ direction at $E_{F}$ . An inner potential of $\sim 12 eV$ is used for the conversion to $k_{z}$ . The data is plotted using a color code legend shown at the bottom. (e) $k_{x} - k_{y}$ FS map at 90 eV ( $k_{z} \sim 0$ ). The black hexagons in (d) and squares in (e) indicate the bulk BZ boundaries, and the black dots in (e) indicate high symmetry momenta points. (f) Energy-momentum dispersions at representative $k_{z}$ cuts, together with the localized $4 f$ calculation shown in the first BZ (dashed blue curves). High symmetry momenta points ( $Γ$ , K, M) and bands crossing $E_{F}$ ( $α$ , $β$ , $γ$ , $η$ ) are labeled.
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Figure 2
(a) Resonant ARPES spectrum (top) and the integrated energy distribution curves (EDCs) in comparison with off-resonant data (bottom). (b) Constant energy maps at $E_{F}$ (top panels) and $- 0.5 eV$ (bottom panels) at 90 eV (left, $k_{z} \sim 0$ ) and 122 eV (right, $k_{z} \sim 0.25 b^{*}$ ). The calculations are shown in the bottom right BZ (blue dots). The black squares with dots indicate the bulk BZ boundaries [similar to Fig. 1]. (c) Comparison of the 80 and 122 eV spectra along $k_{x}$ at the same $k_{z}$ and $k_{y}$ ( $k_{z} \sim 0.25 b^{*}$ , $k_{y} = 0$ ). The rightmost panel shows the conduction bands from the localized $4 f$ calculation (red curves) and the expected $4 f^{1}$ peaks (diffuse blue lines): their hybridization leads to the dispersive $4 f$ bands observed experimentally.
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Figure 3
(a) Resonant ARPES spectra at 7 and 120 K ( $k_{y} = 0$ ). (b),(c) Temperature evolution of EDCs corresponding to green [F2 peak, (b)] and blue [F1 peak, (c)] dashed lines in (a). (d),(e) Background-subtracted peak height $Δ H$ [defined in (b)] as a function of temperature for F2 (d) and F1 (e), normalized to the values at 120 K.
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Figure 4
(a),(b) Resonant ARPES spectra across the F2 peak along directions perpendicular ( $k_{x}$ , a) and parallel ( $k_{y}$ , b) to the Ce chains. (c),(d) ARPES spectra in (a),(b) divided by RC-FDD. Purple diamonds (blue triangles) with error bars indicate extracted peak positions from EDC (momentum-distribution curve) analysis. White dotted lines are model fittings. The arrows in (b),(d) highlight the hybridization gap ( $= 2 V$ ) along $k_{y}$ .
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Physical Review Letters

Anisotropic c−f Hybridization in the Ferromagnetic Quantum Critical Metal CeRh6Ge4

Yi Wu, Yongjun Zhang, Feng Du, Bin Shen, Hao Zheng, Yuan Fang, Michael Smidman, Chao Cao, Frank Steglich, Huiqiu Yuan, Jonathan D. Denlinger, and Yang Liu

Phys. Rev. Lett. 126, 216406 – Published 28 May 2021

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Anisotropic $c - f$ Hybridization in the Ferromagnetic Quantum Critical Metal ${CeRh}_{6} {Ge}_{4}$