Magnetic order and crystalline electric field excitations of the quantum critical heavy-fermion ferromagnet $\mathrm{Ce}{\mathrm{Rh}}_6{\mathrm{Ge}}_4$

$\mathrm{Ce}{\mathrm{Rh}}_6{\mathrm{Ge}}_4$ is an unusual example of a stoichiometric heavy fermion ferromagnet, which can be cleanly tuned by hydrostatic pressure to a quantum critical point. To understand the origin of this anomalous behavior, we have characterized the magnetic ordering and crystalline electric field (CEF) scheme of this system. While magnetic Bragg peaks are not resolved in neutron powder diffraction, coherent oscillations are observed in zero-field $\mu \mathrm{SR}$ below $T_{\mathrm{C}}$, which are consistent with in-plane ferromagnetic ordering consisting of reduced Ce moments. From analyzing the magnetic susceptibility and inelastic neutron scattering, we propose a CEF-level scheme which accounts for the easy-plane magnetocrystalline anisotropy, where the low lying first excited CEF exhibits significantly stronger hybridization than the ground state. These results suggest that the orbital anisotropy of the ground state and low-lying excited state doublets are important for realizing anisotropic electronic coupling between the $f$ and conduction electrons, which gives rise to the highly anisotropic hybridization observed in photoemission experiments.

Figure 1

(a) Neutron powder diffraction pattern of $\mathrm{Ce}{\mathrm{Rh}}_6{\mathrm{Ge}}_4$ at 4 K, for one pair of detector banks on the WISH diffractometer. The results from a Rietveld refinement are also displayed. (b) Difference between the patterns taken at 0.27 K and 4 K for three pairs of banks. The scattering angles for each of the detector banks are displayed in the panels. Note that the strong features around $d=1.8$ and 2.1 Å correspond to the (200) and (111) reflections of the Cu sample holder. (c) Zero-field $\mu \mathrm{SR}$ spectra of $\mathrm{Ce}{\mathrm{Rh}}_6{\mathrm{Ge}}_4$ at two temperatures below $T_{\mathrm{C}}$, where the solid lines show the results from the fitting described in the text. (d) Temperature dependence of the internal fields (solid triangles) and Lorentzian relaxation rate $\lambda$ (open stars) obtained from fitting the $\mu \mathrm{SR}$ data.

Figure 2

(a) Temperature dependence of the inverse magnetic susceptibility of single crystal $\mathrm{Ce}{\mathrm{Rh}}_6{\mathrm{Ge}}_4$ for two field directions [17]. The solid lines show the results from fitting with a CEF model described in the text. (b) CEF-level scheme and wave functions obtained from fitting with the CEF model, where the angular distributions of the wave functions are also displayed. (c) Crystal structure of $\mathrm{Ce}{\mathrm{Rh}}_6{\mathrm{Ge}}_4$, with Ce, Rh, and Ge shown in yellow, grey, and purple, respectively. Ferromagnetic order with moments along the $a$ axis is also illustrated, as well as the proposed muon stopping sites ${\mathbf{s}}_{\mathbf{1}}$ and ${\mathbf{s}}_{\mathbf{2}}$ (orange and green spheres) and the orientation of the ground-state orbitals from the CEF model.

Figure 3

Low-angle cuts of the inelastic neutron scattering spectra of $\mathrm{Ce}{\mathrm{Rh}}_6{\mathrm{Ge}}_4$ and $\mathrm{La}{\mathrm{Rh}}_6{\mathrm{Ge}}_4$ at 7 K and 100 K for incident energies of (a) 12 meV, (b) 20 meV, (c) 38 meV, and (d) 100 meV. The integrated angular ranges are displayed in the panels.

Figure 4

Magnetic contribution to the inelastic neutron scattering intensity versus energy transfer at 7 K, for four different incident energies $E_i$. The solid line shows the calculated inelastic response for the CEF scheme in Fig. 2, where the FWHM of the quasielastic and inelastic peaks are 3.3 meV ($2T_K$), while the dashed line shows the case where the quasielastic FWHM is 3.3 meV but the inelastic peak corresponding to the first CEF excitation has a FWHM of 30 meV. Note that the slight mismatch between the data with $E_i=12$ and 20 meV is an artifact arising from slight differences in the normalization for the different incident energies.