Terahertz conductivity of heavy-fermion systems from time-resolved spectroscopy

The Drude model describes the free-electron conduction in simple metals, governed by the freedom that the mobile electrons have within the material. In strongly correlated systems, however, a significant deviation of the optical conductivity from the simple metallic Drude behavior is observed. Here, we investigate the optical conductivity of the heavy-fermion system $\mathrm{Ce}{\mathrm{Cu}}_{6-x}{\mathrm{Au}}_x$, using time-resolved, phase-sensitive terahertz spectroscopy. The terahertz electric field creates two types of excitations in heavy-fermion materials: First, the intraband excitations that leave the heavy quasiparticles intact. Second, the resonant interband transitions between the heavy and light parts of the hybridized conduction band that break the Kondo singlet. We find that the Kondo-singlet-breaking interband transitions do not create a Drude peak, while the Kondo-retaining intraband excitations yield the expected Drude response. This makes it possible to separate these two fundamentally different correlated contributions to the optical conductivity.

Figure 1

(a) The reflected THz signal from the quantum-critical compound, $\mathrm{Ce}{\mathrm{Cu}}_{5.9}{\mathrm{Au}}_{0.1}$, at a sample temperature of 20 K. (b) A schematic of the hybridized band structure showing the intraband excitations and interband transitions as a result of terahertz excitation. Note that intraband excitations are possible due to the presence of impurity or Umklapp scattering, like in simple metals, even though the momentum transfer by the THz radiation is negligible. Wave vectors ${\mathrm{k}}_{\mathrm{F},0}$ and ${\mathrm{k}}_{\mathrm{F},\mathrm{large}}$ correspond to small and large Fermi surface, respectively [29]. (c) The instantaneous part [blue-shaded region in (a)] resulting from the intraband excitations. (d) The delayed correlated part [red-shaded region in (a)] of the signal featuring interband transitions. The delay time (${\tau }_{\mathrm{K}}^{*}=6$ ps) corresponds to a Kondo lattice temperature of $T_{\mathrm{K}}^{*}=8$ K.

Figure 2

(a),(b) The spectra obtained by Fourier transforming the instantaneous and delayed THz responses of the heavy-fermion compound ($\mathrm{Ce}{\mathrm{Cu}}_6$) at sample temperatures of 81 K and 2 K. The reference spectrum is the signal from the Pt mirror. (c),(d) The reflectivity curves at 81 K and 2 K, corresponding to the instantaneous and delayed THz responses. (e) The real part of the conductivity obtained from the instantaneous response shows a heavy-Fermi-liquid behavior (green dashed curve) in the low-frequency region at 2 K and a peak corresponding to the CEF resonance at higher frequencies. (f) The real part of the conductivity obtained from the correlated delayed response shows the low-frequency Kondo resonance and the high-frequency CEF resonance. A smooth transfer of the correlated spectral weight from the CEF resonance at high temperatures to the Kondo resonance at low temperatures is highlighted by the arrows in the yellow-shaded regions.

Figure 3

(a) The real part of the optical conductivity of $\mathrm{Ce}{\mathrm{Cu}}_6$ obtained from the instantaneous response as a function of sample temperature. The green-shaded region indicates the low-frequency region where the Drude-like free-electron response at high temperatures deviates to a heavy-Fermi-liquid behavior at low temperatures. The gray-shaded region indicates the region where the first CEF resonance starts appearing as the temperature is reduced below 10 K. The green-dashed lines are the Drude fitting curves. (b) The low-frequency region in (a) for a few selected temperatures. (c),(d) The change in DC resistivity and scattering rate, obtained from low-frequency Drude fit, normalized to the value at 145 K. (e) The real part of the optical conductivity of $\mathrm{Ce}{\mathrm{Cu}}_6$ obtained from the delayed correlated response as a function of sample temperature. Being a pure correlated response, a single-peak Lorentz function (blue-dashed curves) with a fixed linewidth of 0.12 THz is used to fit the low-frequency region (green-shaded region). The CEF resonance (in gray-shaded region) shows the characteristic temperature dependence where the spectral weight from the CEF resonance is transferred to the Kondo resonance on reducing the temperature. (f) The temperature-dependent Lorentz functions used to reproduce the low-frequency conductivity response [all curves are colored as in (e)]. (g) The temperature-dependent amplitude of the singl peak Lorentz functions. Note that these reproduce the heavy-fermion Kondo-spectral-weight behavior of Ref. [28].

Figure 4

(a) The real part of the optical conductivity of $\mathrm{Ce}{\mathrm{Cu}}_{5.9}{\mathrm{Au}}_{0.1}$ obtained from the instantaneous response as a function of sample temperature. The green-shaded region indicates the low-frequency region where the Drude-like free-electron response at high temperatures deviates to a non-Fermi-liquid behavior at low temperatures. The gray-shaded region indicates the region where the first CEF resonance starts appearing as the temperature is reduced below 10 K. The green-dashed lines are the Drude fitting curves. (b) The low-frequency region in (a) for a few selected low temperatures, where the yellow-shaded regions mark the deviations from Drude-like free-electron to non-Fermi-liquid behavior. (c),(d) The change in DC resistivity and scattering rate, obtained from low-frequency Drude fit, normalized to the value at 287 K. (e) The real part of the optical conductivity of $\mathrm{Ce}{\mathrm{Cu}}_{5.9}{\mathrm{Au}}_{0.1}$ obtained from the delayed correlated response as a function of sample temperature. Being a pure correlated response, a single-peak Lorentz function (blue-dashed curves) with a fixed linewidth of 0.3 THz is used to fit the low-frequency region (green-shaded region). (f) The temperature-dependent Lorentz functions used to reproduce the low-frequency conductivity response [all curves are colored as in (e)]. (g) The temperature-dependent amplitude of the single-peak Lorentz functions. Note that these reproduce the quantum-critical Kondo-spectral-weight behavior of Ref. [28].

Figure 5

The low-temperature normalized DC resistivity for (a) heavy-fermion $\mathrm{Ce}{\mathrm{Cu}}_6$ and (b) quantum-critical $\mathrm{Ce}{\mathrm{Cu}}_{5.9}{\mathrm{Au}}_{0.1}$ compounds. The red-dashed curve represents the corresponding low-temperature $T^2$ and $T$ fitting of the resistivity data, in accordance with Fermi-liquid and non-Fermi-liquid behavior, respectively. (c) $\omega /T$ scaling plot of the optical conductivity (real part) for $\mathrm{Ce}{\mathrm{Cu}}_{5.9}{\mathrm{Au}}_{0.1}$ taken from the instantaneous response, showing approximate scaling collapse in the range $1\lesssim \hslash \omega /k_{\mathrm{B}}T\lesssim 9$. The exponent of the $\hslash \omega /k_{\mathrm{B}}T$ dependence is 1.3. The inset shows the standard deviation of the scaling fit in dependence on the exponent $\alpha$.

Figure 6

(a) The real part of the optical conductivity of $\mathrm{Ce}{\mathrm{Cu}}_5{\mathrm{Au}}_1$ obtained from the instantaneous response as a function of sample temperature. (b) The low-frequency conductivity response for a few selected temperatures [all curves are colored as in (a)]. (c) The normalized frequency-averaged resistivity obtained from (a) as a function of temperature, normalized with respect to the value at 180 K.

Figure 7

The real part of THz optical conductivity of Pt film (used as reference in our measurements) for a set of temperatures. The curves are offset for clarity.

Figure 8

The imaginary part of the THz optical conductivity of (a) $\mathrm{Ce}{\mathrm{Cu}}_6$ and (b) $\mathrm{Ce}{\mathrm{Cu}}_5{\mathrm{Au}}_1$ samples obtained from the instantaneous response as a function of the sample temperature.

Figure 9

The imaginary part of THz optical conductivity of $\mathrm{Ce}{\mathrm{Cu}}_{5.9}{\mathrm{Au}}_{0.1}$ obtained from (a) the instantaneous response and (b) the delayed response as a function of sample temperature.

Figure 10

The evolution of normalized lattice Kondo temperature, $T_{\mathrm{K}}^{*}$ as a function of sample temperature. $T_{\mathrm{K}}^{*}$ is obtained from the position of the envelope of the delayed pulse as sketched in the inset. Throughout the temperature range, the Kondo temperature remains constant.