Fermi Volume Evolution and Crystal-Field Excitations in Heavy-Fermion Compounds Probed by Time-Domain Terahertz Spectroscopy

We measure the quasiparticle weight in the heavy-fermion compound ${\mathrm{CeCu}}_{6-x}{\mathrm{Au}}_x$ ($x=0$, 0.1) by time-resolved terahertz spectroscopy for temperatures from 2 up to 300 K. This method distinguishes contributions from the heavy Kondo band and from the crystal-electric-field satellite bands by different terahertz response delay times. We find that the formation of heavy bands is controlled by an exponentially enhanced, high-energy Kondo scale once the crystal-electric-field states become thermally occupied. We corroborate these observations by temperature-dependent dynamical mean-field calculations for the multiorbital Anderson lattice model and discuss consequences for quantum-critical scenarios.

Figure 1

Ce $4f$ spectral density for $T\ll T_K^{*}<{\mathrm{\Delta }}_1$ (black curve) as calculated by DMFT for ${\mathrm{CeCu}}_6$. The long arrows represent the hybridization processes between Ce $4f$ electrons (red) and conduction electrons (blue dots and shaded region) generating the Kondo and CEF resonances. The energy scale around the Fermi energy ($\omega =0$) is stretched by a factor 5.

Figure 2

(a) Time traces of the terahertz electric field reflected from a ${\mathrm{CeCu}}_6$ sample at $T=50\mathrm{K}$ (red) in comparison to a Pt reference at $T=2.0\mathrm{K}$ (black) and the average of the ${\mathrm{CeCu}}_6$ time traces for temperatures between 2.0 and 5.0 K (green). (b) Evolution of the background-subtracted CEF signal ($-2.5$ to $+2.5\mathrm{ps}$) and the Kondo signal ($+3.5$ to $+8.5\mathrm{ps}$) as the temperature decreases from 290 to 2 K. The green-shaded region depicting the envelope of the Kondo signal is a solution of the nonlinear rate equation of Ref. [18] describing the relaxation of the terahertz-excited heavy-fermion system with a single local orbital. Trivial delayed reflexes originating from the terahertz generation crystal or the cryostat windows have been identified at times $t>10\mathrm{ps}$ [18].

Figure 3

(a) Temperature dependence of the momentum-integrated Ce $4f$ spectral density in ${\mathrm{CeCu}}_6$ (solid lines), as calculated by DMFT for the Anderson lattice model. The occupied spectral density at 65.0 K, $f(\omega )A_{4f}(\omega )$ (dashed line; $f(\omega )$ is the Fermi-Dirac distribution), visualizes that the spectral width, ${\mathrm{\Gamma }}_{\mathrm{CEF}}=k_B{T}_K^{(\text{high})}\approx 17\mathrm{meV}\widehat{=}200\mathrm{K}$, is not accounted for by thermal broadening at 65.0 K alone. (b) Magnitude spectrum of the CEF and Kondo responses of ${\mathrm{CeCu}}_6$. The Kondo response peaks near zero energy, and has a width of 0.56 meV ($\widehat{=}6.7\mathrm{K}$) which agrees very well with $k_B{T}_K^{*}$. The first CEF satellite is seen at 8 meV, close to the literature value [27, 28, 29]. The shaded region is beyond the spectral width of the terahertz excitation, i.e., governed by noise. The inset shows the Pt reference, equivalent to the incident terahertz spectrum $|E_{\mathrm{in}}(\omega )|$.

Figure 4

Temperature dependence of the occupied weights of the Kondo (red) and of the CEF (blue) bands as well as the sum of the two (black) for (a) ${\mathrm{CeCu}}_6$ and (b) ${\mathrm{CeCu}}_{5.9}{\mathrm{Au}}_{0.1}$. The error bars result from averaging over ten measurements for each data point.

Figure 5

DMFT band structure near ${ϵ}_F$ of a three-orbital Anderson lattice for several temperatures. The model parameter values are chosen in order to resemble ${\mathrm{CeCu}}_6$, $T_K^{*}\approx 6\mathrm{K}$ and ${\mathrm{\Delta }}_1=7\mathrm{meV}$, ${\mathrm{\Delta }}_2=13\mathrm{meV}$. The nearly vertical line represents the light conduction band. At low temperatures, $T\lesssim T_K^{*}$, the sharp, heavy Kondo and CEF satellite bands are well resolved, each one hybridizing with the conduction band. At high temperatures, $T\gtrsim {\mathrm{\Delta }}_1$, they merge to a single, heavy band of shorter lifetime, but still of significant spectral weight at ${ϵ}_F$, thus maintaining a large Fermi volume up to $T\gtrsim T_K^{(\text{high})}\gg T_K^{*}$.