Orientation of the ground-state orbital in ${\mathrm{CeCoIn}}_5$ and ${\mathrm{CeRhIn}}_5$

We present core level nonresonant inelastic x-ray scattering (NIXS) data of the heavy-fermion compounds ${\mathrm{CeCoIn}}_5$ and ${\mathrm{CeRhIn}}_5$ measured at the Ce $N_{4,5}$ edges. The higher than dipole transitions in NIXS allow determining the orientation of the ${\mathrm{\Gamma }}_7$ crystal-field ground-state orbital within the unit cell. The crystal-field parameters of the $\mathrm{Ce}M{\mathrm{In}}_5$ compounds and related substitution phase diagrams have been investigated in great detail in the past; however, whether the ground-state wave function is the ${\mathrm{\Gamma }}_7^{+}\left(x^2-y^2\right)$ or ${\mathrm{\Gamma }}_7^{-}(xy$ orientation) remained undetermined. We show that the ${\mathrm{\Gamma }}_7^{-}$ doublet with lobes along the (110) direction forms the ground state in ${\mathrm{CeCoIn}}_5$ and ${\mathrm{CeRhIn}}_5$. For ${\mathrm{CeCoIn}}_5$, however, we find also some contribution of the first excited state crystal-field state in the ground state due to the stronger hybridization of $4f$ and conduction electrons, suggesting a smaller ${\alpha }^2$ value than originally anticipated from x-ray absorption. A comparison is made to the results of existing density functional theory plus dynamical mean-field theory calculations.

Figure 1

(a) Cartoon of unit cell of $\mathrm{Ce}M{\mathrm{In}}_5$, with orbital shape of Ce in ${\mathrm{CeCoIn}}_5$ as taken from XAS for the possibility of a ${\mathrm{\Gamma }}_7^{-}$ and ${\mathrm{\Gamma }}_7^{+}$ Ce ground-state orbital. The light-blue triangle emphasizes the In2-In1-In2 triangle. (b) Weiss field hybridization function for $\mathrm{Ce}M{\mathrm{In}}_5$ from $\mathrm{DFT}+\mathrm{DMFT}$ calculations decomposed into crystal-field components of the Ce ion (red orbitals) and the out-of-plane In2 (dark-yellow dots) and in-plane In1 (yellow dots) environment, adapted from Ref. [19]. (c) Crystal-field components of the Ce ions and environment of In ions as obtained from the present NIXS experiment.

Figure 2

Nonresonant inelastic x-ray scattering (NIXS) data (dots) of CeRhIn5 (a) and CeCoIn5 (b) at the Ce $N_{4,5}$ edges for the two crystallographic directions $\vec{q}$ $\parallel$ [100] (blue) and $\vec{q}$ $\parallel$ [110] (green) at $T$ = 6 K, plus simulations (lines) for the respective ${\mathrm{\Gamma }}_7^{-}$ ground states $(xy$ orientation) (see text). The bottom panels (c) and (d) show the difference spectra $I_{\vec{q}\parallel \left[110\right]}-I_{\vec{q}\parallel \left[100\right]}$ (circles) and simulated dichroism for the respective ${\mathrm{\Gamma }}_7^{-}$ (orange lines) and ${\mathrm{\Gamma }}_7^{+}$ (gray lines) crystal-field ground states, and for a mixed ground state called scaled ${\mathrm{\Gamma }}_7^{-}$ (dark red lines) (see text).

Figure 3

Simulated difference spectra $I_{\vec{q}\parallel \left[110\right]}-I_{\vec{q}\parallel \left[100\right]}$ for ${\alpha }^2$ values of 0 and 0.5, and of 0.38 for ${\mathrm{CeRhIn}}_5$ and 0.13 for ${\mathrm{CeCoIn}}_5$.

Figure 4

Crystal-field splitting of $J$ = 5/2 multiplet of ${\mathrm{CeCoIn}}_5,{\mathrm{CeRhIn}}_5$ and for completeness of ${\mathrm{CeIrIn}}_5$ as adapted from Refs. [20, 21, 22]. For $M$ = Co and Rh the sign of the wave function is now determined which has been taken into account when drawing the $f^1$ charge densities of the respective states.