Kondo scaling of $4f$-electron states and the Kondo singlet breakdown in heavy fermions

Abstract

The low-energy spin- and charge-sensitive thermodynamic properties of a broad range of strongly correlated 4f-electron systems follow Kondo scaling, with a characteristic Kondo temperature, $T_K$. While the theory is known for thermodynamic properties and high-energy spectroscopies of Kondo materials, the surface sensitivity of electron spectroscopy limits the extent to which Kondo scaling can be quantitatively verified. In this study, bulk-sensitive photon-in photon-out temperature-dependent resonant inelastic X-ray scattering (RIXS), in combination with single-impurity Anderson model (SIAM) calculations, is used to provide quantitative evidence of low- and high-energy Kondo scaling in CeSi$_2$. RIXS Ce M$_5$-edge spectra show a clear decrease in the occupancy of the $f^0$ state as temperature increases accompanied by an increase of the spectral weight of the $f^1\underline L^1$ state, in good agreement with the SIAM calculations. The results demonstrate the breakdown of the Kondo singlet state, coupled with thermal occupation of the low-lying first-excited magnetic states. The RIXS data reveal a temperature evolution of the $f^n$ spectral weights, which is in stark contrast to that extracted from photoemission and inverse photoemission spectroscopies. This study provides an accurate spectroscopic method to determine the Kondo energy $k_B$$T_K$ that is consistent with thermodynamic measurements, and highlights soft X-ray RIXS as a quantitative bulk probe of low- and high-energy-scale hybridization effects in strongly correlated materials.

Figures (5)

• Figure 1: (a) Ce $M_{4,5}$ edge XAS spectra of CeSi$_2$ measured at T = 15 K. (b) RIXS spectra of CeSi$_2$ taken with the incident photon energy tuned to $h \nu = 882.5 eV$ (label A in the XAS spectrum) and $h\nu = 887.5 eV$ (label B in the XAS spectrum).
• Figure 2: Schematic energy level diagram of a cerium Kondo system showing initial, intermediate, and final states of RIXS. The blue and red arrows represent transitions for $T = 0$ and $T\gg T_K$, respectively. The blue and red schematic spectra represent spectra for $T = 0$ and $T\gg T_K$, respectively. For $h\nu =A$, at $T = 0$, the initial state is dominated by the ground state while for $T\gg T_K$ it is dominated by the first excited state. The arrows indicate transitions which explain the temperature dependence of $f^1$ and $f^2$ final states via both $d^{9}f^{2}$ and $d^{9}f^{3}$ in the intermediate state. Similarly, for $h\nu =B$ the arrows indicate transitions which explain the temperature dependence of $f^1$ and $f^0$ final states via the $d^{9}f^{1}$ in the intermediate state. Spin-orbit states are omitted for clarity.
• Figure 3: Scattering angle dependence of the CeSi$_2$ RIXS intensity at h$\nu$ = B with $\pi$ polarization. The inset shows the evolution of the integrated $f^0$ intensity.
• Figure 4: Temperature dependent RIXS spectra of CeSi$_2$ with the incident photon energy h$\nu$ = A. The $f^2$ final state is temperature independent.
• Figure 5: (a) Temperature ($T$) dependent RIXS spectra of CeSi$_2$ with the incident photon energy h$\nu$ = B. A significant $T$-dependence of the $f^0$ final state is observed. (b) Calculated $T$-dependent RIXS spectra with $\pi$ polarization using the SIAM. The insets show an enlarged view of the $f^1$ peaks and the evolution of the integrated $f^0$ intensity with temperature and its fitting using the Boltzmann weight of the local moment $\beta(T)$ (see SM).