Impact of the ground-state $4f$ symmetry for anisotropic $cf$ hybridization in the heavy-fermion superconductor ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$

We report the ground-state symmetry of Ce $4f$ states in the heavy-fermion superconductor ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$, yielding anisotropic $cf$ hybridization between Ce $4f$ states and conducting electrons. By analyzing linear dichroism in soft x-ray absorption and core-level hard x-ray photoemission spectra, the $4f$ symmetry is determined as $\mathrm{\Sigma }$-type ${\mathrm{\Gamma }}_7$, promoting predominant hybridization with the conducting electrons originating from the Ge site. The band structures probed by soft x-ray angle-resolved photoemission indicate that Ge $4p$ components contribute to band renormalization through anisotropic hybridization effects, suggesting that control of the electronic structures of a Ge orbital helps to achieve the exotic phenomena in ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$.

Figure 1

Linearly polarized XAS spectra of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ at Ce $M_{4,5}$ edges for a photon incidence angle $\theta$ = ${70}^{\circ }$ with $s$- and $p$-polarization, together with the spectral simulations assuming the ground state in the ${\mathrm{\Gamma }}_7$ (${\alpha }^2$ = 0.4 and 0.81) symmetry and that in the ${\mathrm{\Gamma }}_6$ symmetry.

Figure 2

(a) XAS and XMCD spectra at Ni $L_{2,3}$ and Ce $M_{4,5}$ edges of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$. (b) Magnetic susceptibility of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$. Dotted data indicate the experimental results [43] with the simulation (solid lines) using the parameters in Table 1 (see the Appendix).

Figure 3

(a) Ce $3d$ core-level spectrum together with Ni $2p$ states. (b) Ce $3d_{5/2}$ state (solid lines) with the subtracted background (dashed lines).

Figure 4

(a),(b) Ce $3d_{5/2}$ core-level photoemission spectra with $s$- and $p$-polarization, together with the spectral simulations and multiplet calculations assuming the ground-state symmetry of $\mathrm{\Sigma }$-type ${\mathrm{\Gamma }}_7$. The photon incidence and photoelectron emitting angles are ${15}^{\circ }$ and ${45}^{\circ }$, respectively. The azimuthal angle ($\varphi$) is set to be $0^{\circ }$ (a) and ${45}^{\circ }$ (b) for the crystal $c\text{-axis}$. The experimental geometries are illustrated in the insets. Note that the spectra obtained in the $0^{\circ }$ configuration ($\mathrm{\Delta }E\sim 400$ meV) in (a) were recorded with slightly better energy resolution than that for the ${45}^{\circ }$ configuration ($\mathrm{\Delta }E\sim 550$ meV) in (b). The dashed lines in (a) and (b) are guides to the eye to stress the multiplet structures. (c) Azimuthal ($\varphi$) dependence of the LD spectra obtained from the experiment and the spectral simulations with $\mathrm{\Sigma }$- and $\mathrm{\Pi }$-types.

Figure 5

The schematic view of charge distribution for $\mathrm{\Sigma }$- and $\mathrm{\Pi }$-type ${\mathrm{\Gamma }}_7$ orbital of the Ce $4f$ states on ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ in the $ab$-plane of the tetragonal lattice in (a) and (b), respectively, together with the crystal structure of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ in (c).

Figure 6

(a) Ce $3d\text{−}4f$ resonance photoemission spectra of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$. The inset shows the x-ray absorption spectrum of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ with the marker indicating the Ce $3d\text{−}4f$ resonance condition. The $4f$ PDOS convoluted by the Fermi-Dirac function, and the energy resolution is plotted in the bottom. (b),(c) ARPES intensity plots of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ obtained by the photon energies of 744 and 881.1 eV, respectively.

Figure 7

FS slices in the Z-$\mathrm{\Gamma }$-X planes of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ and ${\mathrm{LaNi}}_2{\mathrm{Ge}}_2$ obtained by the $k_x-k_y$ mapping at $h\nu =651$ (714) eV and 705 eV, respectively. (a)–(c) Momentum distributions of ARPES intensity in the Z-centered (a) and $\mathrm{\Gamma }$-centered (b),(c) plane of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ (a),(b) and ${\mathrm{LaNi}}_2{\mathrm{Ge}}_2$ (c), which are obtained by integrating the photoelectron intensity from the Fermi level and $-0.2$ eV in the unoccupied side. The dashed lines represent the BZ boundaries. (d),(e) $k_{\mathrm{F}}$ plots for ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ (d) and ${\mathrm{LaNi}}_2{\mathrm{Ge}}_2$ (e). The dots with error bars represent the $k_{\mathrm{F}}$'s estimated from each specific angle slice obtained by the 651 and 714 eV data together with the high statistics measurements along the high-symmetry lines. The solid lines are guides to the eye of Fermi surfaces following the experimentally evaluated $k_{\mathrm{F}}$'s. The dashed lines represent the BZ boundaries.

Figure 8

(a),(b) ARPES intensity plot of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ along the Z-X direction recorded with $h\nu$ = 651 (a) and 714 (b) eV. (c),(d) Same as (a),(b) of ${\mathrm{LaNi}}_2{\mathrm{Ge}}_2$ recorded with $h\nu =643$ eV (c) and 705 eV (d). Note that the ARPES data obtained at the different photon energies are measured at different polar angles to measure the same symmetry lines in the Brillouin zone of the body-centered-tetragonal lattice. (e),(f) MDCs of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$ (solid lines) and ${\mathrm{LaNi}}_2{\mathrm{Ge}}_2$ (dashed lines) with $h\nu \sim 650$ eV (e) and 710 eV (f), which is extracted from (a)–(d).

Figure 9

Calculated band structures of ${\mathrm{LaNi}}_2{\mathrm{Ge}}_2$. The size of the dots indicates the degree of the orbital components of La $5d$, Ni $3d$, and Ge $4p$ states.

Figure 10

Circular polarized ARPES spectra of ${\mathrm{LaNi}}_2{\mathrm{Ge}}_2$ recorded with $h\nu$ = 643 and 705 eV in (a)–(d) and (e)–(h), respectively. ARPES intensity plots are obtained with right (${\mu }_{-}$) and left (${\mu }_{+}$) circular polarized photons along the Z-X direction in (a),(e) and (b),(f). (c),(g) Intensity difference (${\mu }_{-}-{\mu }_{+}$) of circularly polarized ARPES spectra. (d),(h) MDCs with right and left circular polarized photons.

Figure 11

Specific heat of ${\mathrm{CeNi}}_2{\mathrm{Ge}}_2$. Solid line indicates the simulation based on the CEF model. Dots denote the experimental results [25].