Kondo-hole substitution in heavy fermions: Dynamics and transport

Kondo-hole substitution is a unique probe for exploring the interplay of interactions, $f$-electron dilution, and disorder in heavy-fermion materials. Within the diluted periodic Anderson model, we investigate the changes in single-particle dynamics as well as response functions, as a function of Kondo-hole concentration ($x$) and temperature. We show that the spectral weight transfers due to Kondo-hole substitution have characteristics that are different from those induced by temperature; the dc resistivity crosses over from a highly nonmonotonic form with a coherence peak in the $x\to 0$ limit to a monotonic single-impurity-like form that saturates at low temperature. The thermopower exhibits a characteristic maximum as a function of temperature, the value of which changes sign with increasing $x$, and its location is shown to correspond to a low energy scale of the system. The Hall coefficient also changes sign with increasing $x$ at zero temperature and is highly temperature dependent for all $x$. As $x$ is increased beyond a certain $x_c$, the Drude peak and the mid-infrared peak in the optical conductivity vanish almost completely; a peak in the optical scattering rate melts and disappears eventually. We discuss the above-mentioned changes in the properties in terms of a crossover from coherent, Kondo lattice behavior to single-impurity-like, incoherent behavior with increasing $x$. A comparison of theory with experiments carried out for the dc resistivity and the thermopower of ${\mathrm{Ce}}_{1-x}{\mathrm{La}}_x{\mathrm{B}}_6$ yields good agreement.

Figure 1

Low energy scale ${\omega }_L\left(x\right)$ varying with Coulomb interaction $U$ for different substitution values of $x$. The hybridization is chosen to be $V^2=0.4$, and the conduction band center is at ${\epsilon }_c=0.5$.

Figure 2

Local $f$ spectral function varying with scaled frequency $\omega /{\omega }_L\left(x\right)$ for substitution values $x=0.3$, $x=0.5$, $x=0.9$, and $x=0.95$. The model parameters are the same as in Fig. 1.

Figure 3

Hybridization as a function of absolute frequency $\omega /t_{*}$ for various substitution ($x$) values. The parameters are $U=5.23,V^2=0.4,n_f=0.98,n_c=0.53$.

Figure 4

Expanded view of the low-frequency region of the local $f$ DOS. Inset: Variation of low energy scale with Kondo-hole concentration.

Figure 5

Main panel: Resistivity per $f$ site as a function of scaled temperature, $T/{\omega }_L$ (model parameters are $U\simeq 5.11,V^2=0.6,n_f\simeq 0.98,n_c\simeq 0.59$). Inset: The strong-coupling single-impurity Anderson model (SIAM) resistivity compared with concentration value $x=0.95$. Model parameters for SIAM are $U=5.11,V^2=0.2,{\epsilon }_c=0.5$.

Figure 6

Resistivity as a function of temperature for three conduction electron occupancies and various concentration values. The coherence peak is seen to disappear beyond $x\gtrsim 1-n_c$, which is roughly $0.7,0.4$, and $0.2$ for $n_c\sim 0.30$ (top panel), $n_c\sim 0.60$ (middle panel), and $n_c\sim 0.83$ (bottom panel), respectively. The model parameters are $V^2=0.6$, $U\sim 5.20$.

Figure 7

Upper panel: Thermopower vs $T/{\omega }_L$ (model parameters are $U\sim 5.23,V^2=0.4,n_f\sim 0.98,n_c\sim 0.53$). Lower panel: Thermopower vs $T/t^{*}$ in dilute limit, i.e., $x=0.99$ with $U\sim 7.17,V^2=0.4,n_f\sim 0.97,n_c\sim 0.52$. Inset: Thermopower of SIAM for $U\sim 5.80,V^2=0.4$.

Figure 8

Hall coefficient $R_H\left(T\right)$ vs temperature $T$ scaled by low energy scale ${\omega }_L(x=0)$. The model parameters are $U\sim 5.35,V^2=0.6$, and ${\epsilon }_c=0.5$ for which the occupancies are $n_f=1.0$ and $n_c=0.55$.

Figure 9

Hall angle vs temperature $T$ scaled by low energy scales ${\omega }_L$ at $x=0$ (model parameters are same as for Fig. 8).

Figure 10

Top panel: Zero-temperature optical conductivity as a function of $\omega /t*$ for various Kondo-hole concentrations. Bottom panel: Band dispersion for various $x$ values. The model parameters are $U=5.11,V^2=0.6,n_f\simeq 0.98$, and $n_c\simeq 0.59$.

Figure 11

False-color contour plot of the single-particle dispersion, $D^{\mathit{\text{CPA}}}({\epsilon }_{\mathbf{k}},\omega )=-\mathrm{Im}{G}_c^{\mathit{\text{CPA}}}({\epsilon }_{\mathbf{k}},\omega )/\pi$ for concentration values $x=0.1$, $x=0.34$, $x=0.5$, and $x=0.7$ (from top to bottom). The model parameters are same as Fig. 10.

Figure 12

Zero-temperature optical scattering rate as a function of $\frac{\omega }{{\omega }_L}$ (${\omega }_L$ is low energy scale at $x=0$) for various disorder strengths. The model parameters are $U=5.32,V^2=0.6,n_f\simeq 0.97$, and $n_c\simeq 0.43$.

Figure 13

Main panel: Optical scattering rate for $x=0.45$ and various temperatures, shown as fractions of the low energy scale. In the inset, the resistivity vs scaled temperature is shown, also for $x=0.45$. The other model parameters are $V^2=0.6,{\epsilon }_c=0.7,\eta \simeq 0$, $U\simeq 5.32$, $n_f=0.97$, and $n_c=0.43$.

Figure 14

Comparison of theory with experiment for ${\mathrm{Ce}}_x{\mathrm{La}}_{1-x}$. Left panel: Theoretically computed resistivity vs $T$ for various $x$. In right panel, experimental data for ${\mathrm{Ce}}_x{\mathrm{La}}_{1-x}{\mathrm{B}}_6$ by Sato et al. [10].

Figure 15

Comparison of experiment with theory. Top panel: Experimental data for ${\mathrm{Ce}}_x{\mathrm{La}}_{1-x}{\mathrm{B}}_6$ by Kim et al. [49]. Lower panel: Theoretically computed thermopower for various $x$.