Multiple temperature scales of the periodic Anderson model: Slave boson approach

The thermodynamic and transport properties of intermetallic compounds with Ce, Eu, and Yb ions are discussed using the periodic Anderson model with an infinite correlation between $f$ electrons. At high temperatures, these systems exhibit typical features that can be understood in terms of a single-impurity Anderson or Kondo model with Kondo scale $T_K$. At low temperatures, one often finds a normal state governed by the Fermi liquid (FL) laws with characteristic energy scale $T_0$. The slave boson solution of the periodic model shows that $T_0$ and $T_K$ depend not only on the degeneracy and the splitting of the $f$ states, the number of $c$ and $f$ electrons, and their coupling but also on the shape of the conduction-electrons density of states ($c$ DOS) in the vicinity of the chemical potential $\mu$. The ratio $T_0/T_K$ depends on the details of the band structure which makes the crossover between the high- and low-temperature regimes system dependent. We show that the $c$ DOS with a sharp peak close to $\mu$ yields $T_0⪡T_K$, which explains the “slow crossover” observed in ${\text{YbAl}}_3$ or ${\text{YbMgCu}}_4$. The $c$ DOS with a minimum or a pseudogap close to $\mu$ yields $T_0⪢T_K$; this leads to an abrupt transition between the high- and low-temperature regimes, as found in ${\text{YbInCu}}_4$-like systems. In the case of ${\text{CeCu}}_2{\text{Ge}}_2$ and ${\text{CeCu}}_2{\text{Si}}_2$, where $T_0\simeq T_K$, we show that the pressure dependence of the $T^2$ coefficient of the electrical resistance, $A=\rho \left(T\right)/T^2$, and the residual resistance are driven by the change in the degeneracy of the $f$ states. The FL laws obtained for $T⪡T_0$ explain the correlation between the specific-heat coefficient $\gamma =C_V/T$ and the thermopower slope $\alpha \left(T\right)/T$ or between $\gamma$ and the resistivity coefficient $A$. The FL laws also show that the Kadowaki-Woods ratio, $R_{KW}=A/{\gamma }^2$, and the ratio $q={\text{lim}}_{\left\{T\to 0\right\}}\alpha /\gamma T$ assumes nonuniversal values due to different low-temperature degeneracies of various systems. The correlation effects can invalidate the Wiedemann-Franz law and lead to an enhancement of the thermoelectric figure of merit. They can also enhance (or reduce) the low-temperature response of the periodic Anderson model with respect to the predictions of a single-impurity model with the same high-temperature behavior as the periodic one.

Figure 1

(Color online) Schematic plot of the noninteracting $c$ DOS ${\rho }_0\left(\omega \right)$. (a) For a regular $c$ DOS and far from the electronic half-filling, we find $T_0⪡T_K$. Close to the half-filling, $T_0/T_K$ depends on the shape of the $c$ DOS around the renormalized chemical potential $\mu$: (b) ${\rho }_0\left(\omega \right)$ is nearly constant for $\omega \sim \mu$ and $T_0\sim T_K$. (c) $\mu$ is close to a minimum of ${\rho }_0\left(\omega \right)$ and $T_0⪢T_K$. Here, ${\mu }_0\approx \mu$ is the chemical potential of $n_c=n-1$ noninteracting $c$ electrons (“small Fermi surface”) and ${\mu }_L$ is the chemical potential of $n_c=n$ noninteracting $c$ electrons (“large Fermi surface”).

Figure 2

(Color online) Schematic plot of the electronic contribution to the entropy $S_{e^{-}}$ as a function of the normalized temperature $T/T_K$ for three cases: $T_0⪢T_K$ (red dash-dotted line), $T_0\approx T_K$ (blue solid line), and $T_0⪡T_K$ (green-dashed line). The dotted lines indicate the linear Fermi-liquid regime $S_{e^{-}}\left(T\right)=T/T_0$, with a rescaled slope $T_K/T_0$. For $T>T_K$, the three curves are identical, reflecting the linear contribution from the conduction band $S_{e^{-}}\left(T\right)=\text{ln}2+T/D$. The black dot refers to a standard determination of $T_K$ from the electronic entropy: $S_{e^{-}}\left(T_K\right)=x\text{ln}2$. On this schematic plot, we used $x=1$, which gives $T_K$ as defined by the slave boson $\left(r=0\right)$ solution. Experimentally, where $T_K$ is defined by the crossover, one usually takes $x=1/2$.

Figure 3

(Color online) Schematic phase diagram of the PAM. The reduced temperature $T/T_K$ is plotted versus $T_0/T_K$, which is considered as a tunable parameter that can vary with the electronic filling, magnetic field, and/or the shape of the noninteracting DOS. The crossover regime can be nonuniversal or non-Fermi liquid.

Figure 4

(Color online) Schematic phase diagram of the PAM as a function of a magnetic field $h$. The red solid line indicates $h_c\left(T\right)$ which separates the trivial $r=0$ solution from the nontrivial one $r\ne 0$.

Figure 5

(Color online) Schematic plot of the magnetization $m_z$ as a function of reduced magnetic field $h/T_K$ for $T_0⪢T_K$ (red dash-dotted line), $T_0\approx T_K$ (blue solid line), and $T_0⪡T_K$ (green-dashed line). The dotted lines indicate the initial slope in the linear-response regime, where $m_z\left(h\right)=h/T_0$. The black dot refers to the complete polarization of the local $f$ electrons, with $m_z=1/2$, which occurs at the critical field $h_c^0=T_K$.