High-resolution photoemission study on low-$T_K$ Ce systems: Kondo resonance, crystal field structures, and their temperature dependence

We present a high-resolution photoemission study on the strongly correlated Ce compounds $\mathrm{Ce}{\mathrm{Cu}}_6$, $\mathrm{Ce}{\mathrm{Cu}}_2{\mathrm{Si}}_2$, $\mathrm{Ce}{\mathrm{Ru}}_2{\mathrm{Si}}_2$, $\mathrm{Ce}{\mathrm{Ni}}_2{\mathrm{Ge}}_2$, and $\mathrm{Ce}{\mathrm{Si}}_2$. Using a normalization procedure based on a division by the Fermi-Dirac distribution we get access to the spectral density of states up to an energy of $5k_{B}T$ above the Fermi energy $E_F$. Thus we can resolve the Kondo resonance and the crystal field (CF) fine structure for different temperatures above and around the Kondo temperature $T_K$. The CF peaks are identified with multiple Kondo resonances within the multiorbital Anderson impurity model. Our theoretical $4f$ spectra, calculated from an extended noncrossing approximation (NCA), describe consistently the observed photoemission features and their temperature dependence. By fitting the NCA spectra to the experimental data and extrapolating to low temperatures, $T_K$ can be extracted quantitatively. The resulting values for $T_K$ are in excellent agreement with the results from bulk sensitive measurements, e.g., inelastic neutron scattering.

Figure 1

Left: Theoretical $4f$ spectrum from NCA calculations based on the SIAM for $T=11\mathrm{K}$ and the model parameters given in the caption of Fig. 5. The hatched region indicates the “photoemission” region below $E_F$. The inset shows the near-$E_F$ region with Kondo resonance $\left(\mathsf{D}\right)$ and crystal field features $(\mathsf{C},\mathsf{E})$. $\mathsf{B}$ and $\mathsf{F}$ are the spin-orbit satellites $(J=5∕2)$, $\mathsf{A}$ is the ionization peak. For finite $U$ the two-electron state $f^2$ would appear far right of the displayed energy range at $\approx U-\mid {ϵ}_f\mid$. Right: Sketch of the energy level scheme of the SIAM for the conduction-band states ($c$ states) with bandwidth $D$ and for the impurity ($f$ states) with crystal field and spin-orbit splittings.

Figure 2

Theoretical $4f$ spectrum from NCA calculations based on SIAM for different temperatures. The displayed energy range covers the Kondo resonance close to zero and the crystal field structures at approximately $-10\mathrm{meV}$ and $20\mathrm{meV}$. The inset shows the Kondo resonance at low temperatures, i.e., at $T=0.1T_K$ (unitarity limit).

Figure 3

Photoemission spectrum of $\mathrm{Ce}{\mathrm{Cu}}_6$ (left panel: He ${\mathrm{II}}_{\alpha }$ with $h\nu =40.8\mathrm{eV}$; right panel: He ${\mathrm{I}}_{\alpha }$ with $h\nu =21.22\mathrm{eV}$). Closed circles represent the raw data ($\Delta E=5.4\mathrm{meV}$, $T=16\mathrm{K}$), mainly consisting of the tail of the KR. The SO partner $(J=7∕2)$ appears at a higher binding energy of $\approx 250\mathrm{meV}$ given in the inset $(\Delta E\approx 15\mathrm{meV})$. Open circles represent the normalized data using the experimentally broadened FDD, the shaded area marks the unreliable spectral range above $5k_{B}T$. The solid line represents the fitted NCA spectral function with ${ϵ}_f=-1.05\mathrm{eV}$, $D=2.8\mathrm{eV}$, CF splittings of the $J=5∕2$ sextet ${\Delta }_{\mathit{CF}}=7.2∕13.9\mathrm{meV}$, SO splitting ${\Delta }_{\mathit{SO}}=250\mathrm{meV}$, and hybridization $V=116\mathrm{meV}$.

Figure 4

Temperature dependence of the experimental (PES) and calculated $4f$ spectra of $\mathrm{Ce}{\mathrm{Cu}}_6$ after the application of the normalization procedure. The PES data are taken with He ${\mathrm{II}}_{\alpha }$ (left panel) and He ${\mathrm{I}}_{\alpha }$ radiation (right panel). The used NCA model parameters are given in the caption of Fig. 3. The insets show the raw spectra, normalized to the same intensity at a binding energy of $\approx 100\mathrm{meV}$.

Figure 5

Photoemission spectra of $\mathrm{Ce}{\mathrm{Cu}}_2{\mathrm{Si}}_2$ (He ${\mathrm{II}}_{\alpha }$, $h\nu =40.8\mathrm{eV}$). Left panel: normalization at $T=11\mathrm{K}$ (cf. Fig. 3); the inset shows an extended energy range $(\Delta E\approx 15\mathrm{meV})$ including the SO $(J=7∕2)$ excitation at $\approx 270\mathrm{meV}$ below $E_F$ taken with He ${\mathrm{II}}_{\alpha }$ radiation (open circles) and with He ${\mathrm{I}}_{\alpha }$ (closed circles). Right panel: temperature dependence of the experimental (He ${\mathrm{II}}_{\alpha }$) and the calculated $4f$ spectral function of $\mathrm{Ce}{\mathrm{Cu}}_2{\mathrm{Si}}_2$ after the normalization. Model parameters: ${ϵ}_f=-1.57\mathrm{eV}$, $D=4.3\mathrm{eV}$, ${\Delta }_{\mathit{CF}}=32∕37\mathrm{meV}$, ${\Delta }_{\mathit{SO}}=270\mathrm{meV}$, and $V=200\mathrm{meV}$. The inset shows the raw data normalized to the same intensity at a binding energy of $\approx 100\mathrm{meV}$.

Figure 6

Photoemission spectra of $\mathrm{Ce}{\mathrm{Ru}}_2{\mathrm{Si}}_2$ (He ${\mathrm{II}}_{\alpha }$, $h\nu =40.8\mathrm{eV}$). Left panel: normalization at $T=11\mathrm{K}$ (cf. Fig. 3); the inset shows an extended energy range $(\Delta E\approx 15\mathrm{meV})$ including the SO $(J=7∕2)$ excitation at $\approx 290\mathrm{meV}$ below $E_F$ taken with He ${\mathrm{II}}_{\alpha }$ radiation (open circles) and with He ${\mathrm{I}}_{\alpha }$ (closed circles). Right panel: temperature dependence of the experimental (He ${\mathrm{II}}_{\alpha }$) and the calculated $4f$ spectral function of $\mathrm{Ce}{\mathrm{Ru}}_2{\mathrm{Si}}_2$ after the normalization. Model parameters: ${ϵ}_f=-1.54\mathrm{eV}$, $D=4.2\mathrm{eV}$, ${\Delta }_{\mathit{CF}}=18∕33\mathrm{meV}$, ${\Delta }_{\mathit{SO}}=295\mathrm{meV}$, and $V=170\mathrm{meV}$. The inset shows the raw data normalized to the same intensity at a binding energy of $\approx 100\mathrm{meV}$.

Figure 7

Photoemission spectra of $\mathrm{Ce}{\mathrm{Ni}}_2{\mathrm{Ge}}_2$ (He ${\mathrm{II}}_{\alpha }$, $h\nu =40.8\mathrm{eV}$). Left panel: normalization at $T=11\mathrm{K}$; the inset shows an extended energy range $(\Delta E\approx 15\mathrm{meV})$ including the SO $(J=7∕2)$ excitation at $\approx 300\mathrm{meV}$ below $E_F$ taken with He ${\mathrm{II}}_{\alpha }$ radiation (open circles) and with He ${\mathrm{I}}_{\alpha }$ (closed circles). Right panel: temperature dependence of the experimental (He ${\mathrm{II}}_{\alpha }$) and the calculated $4f$ spectral function of $\mathrm{Ce}{\mathrm{Ni}}_2{\mathrm{Ge}}_2$ after the normalization. Model parameters: ${ϵ}_f=-1.43\mathrm{eV}$, $D=3.9\mathrm{eV}$, ${\Delta }_{\mathit{CF}}=26∕39\mathrm{meV}$, ${\Delta }_{\mathit{SO}}=275\mathrm{meV}$, and $V=209\mathrm{meV}$. The inset shows the raw data normalized to the same intensity at a binding energy of $\approx 100\mathrm{meV}$.

Figure 8

Photoemission spectra of $\mathrm{Ce}{\mathrm{Si}}_2$ (He ${\mathrm{II}}_{\alpha }$, $h\nu =40.8\mathrm{eV}$). Left panel: normalization at $T=11\mathrm{K}$; the inset shows an extended energy range $(\Delta E\approx 15\mathrm{meV})$ including the SO $(J=7∕2)$ excitation at $\approx 280\mathrm{meV}$ taken with He ${\mathrm{II}}_{\alpha }$ radiation (open circles) and with He ${\mathrm{I}}_{\alpha }$ (closed circles). Right panel: temperature dependence of the experimental (He ${\mathrm{II}}_{\alpha }$) and the calculated $4f$ spectral function of $\mathrm{Ce}{\mathrm{Si}}_2$ after the normalization. Model parameters: ${ϵ}_f=-1.35\mathrm{eV}$, $D=3.7\mathrm{eV}$, ${\Delta }_{\mathit{CF}}=25∕48\mathrm{meV}$, ${\Delta }_{\mathit{SO}}=270\mathrm{meV}$, and $V=203\mathrm{meV}$. The inset shows the raw data normalized to the same intensity at a binding energy of $\approx 100\mathrm{meV}$.

Figure 9

Left panels: NCA spectra for $\mathrm{Ce}{\mathrm{Si}}_2$ (parameters given in the captions of Fig. 8). (a) Raw and (b) convoluted with a Gaussian of $\mathrm{FWHM}=600\mathrm{meV}$. Panel (c) shows the (normalized) temperature dependence of the $f^1$ intensity as calculated from the convoluted NCA spectra in panel (b), in comparison with the results of IPES-studies (Ref. 79). Panel (d) shows the temperature dependence of the $4f$ occupation $n_f$ as resulting directly from the NCA calculations. These values are compared with the ones extracted from XAS measurements at the $\mathrm{Ce}{L}_3$ edge (Ref. 80).

Figure 10

Application of the normalization procedure of Eq. (A2) at two different temperatures. Top panel: $T=108\mathrm{K}⪢\Delta E∕k_B$ and lower panel: $9\mathrm{K}<\Delta E∕k_B$. The energy resolution was $\Delta E=5.4\mathrm{meV}$. The gray curves give the modeled photoemission spectra, including FDD and Gaussian broadening, the black solid lines represent the (broadened) intrinsic spectral functions.