Anisotropic heavy-Fermi-liquid formation in valence-fluctuating $\alpha$-YbAlB${}_4$

$\alpha$-YbAlB${}_4$ is the locally isostructural polymorph of $\beta$-YbAlB${}_4$, the first example of a Yb-based heavy fermion superconductor that exhibits pronounced non-Fermi-liquid behavior above $T_c$. Interestingly, both $\alpha$-YbAlB${}_4$ and $\beta$-YbAlB${}_4$ have strongly intermediate valence. Our single-crystal study of the specific heat, magnetization, and resistivity has confirmed the Fermi-liquid ground state of $\alpha$-YbAlB${}_4$ in contrast to the quantum criticality observed in $\beta$-YbAlB${}_4$. Both systems exhibit Kondo lattice behavior with the characteristic temperature scale $T^{*}\sim 8$ K in addition to a valence-fluctuation scale $\sim 200$ K. Below $T^{*}$, $\alpha$-YbAlB${}_4$ forms a heavy-Fermi-liquid state with an electronic specific heat coefficient $\gamma \sim 130$ mJ/mol K${}^2$ and a large Wilson ratio greater than 7, which indicates ferromagnetic correlation between Yb moments. A large anisotropy in the resistivity, which is one of the largest in heavy fermions, suggests that the hybridization between 4$f$ and conduction electrons is much stronger in the $ab$ plane than along the $c$ axis, indicating this is an excellent system to study for revealing the anisotropic hybridization effects. The strongly anisotropic hybridization as well as the large Wilson ratio is the key to understanding the unusual Kondo lattice behavior and heavy-fermion formation in mixed-valence compounds.

Figure 1

(a) Magnetic part ($f$-electron contribution) of the specific heat $C_m$ plotted as $C_m/T$ versus $T$ for both $\beta$- and $\alpha$-YbAlB${}_4$ under zero field. For the $\beta$ phase $C_m/T$ shows a $\ln T$ dependence for 0.2 K $<T<$ 20 K.[9] The dash-dotted line is the fit of the results to $C_m/T=S_0/T_0\ln (T_0/T)$ (see the text). Also shown are $C/T$ of $\alpha$-LuAlB${}_4$ and $C_m/T$ of the intermediate-valence cubic system YbAl${}_3.$[18] (b) Magnetic part of the entropy $S_m$, which was obtained by integrating $C_m/T$. In $\alpha$-YbAlB${}_4$, a constant value of 127 mJ/K${}^2$mol is assumed below 0.4 K. Above 10 K, $C_m/T$ in $\alpha$-YbAlB${}_4$ merges with the ln$T$ behavior of $\beta$-YbAlB${}_4$. Thus the data for $\beta$-YbAlB${}_4$shifted to take the same value as that for $\alpha$-YbAlB${}_4$ at 20 K. The dashed line is obtained from the $\ln T$ fitting used in (a). The inset shows the low $T$ part. (c) Temperature dependence of the dc susceptibility $\chi =M/B$ measured in a field along the $ab$ plane and $c$ axis for both $\beta$- and $\alpha$-YbAlB${}_4$. Also shown is $\chi$ of the intermediate-valence cubic system YbAl${}_3$.[18]

Figure 2

Temperature dependence of the inverse susceptibility ${\chi }^{-1}=B/M$ under the field along the $ab$ plane and $c$ axis. Solid and dashed lines are Curie-Weiss fits above 150 K for $\alpha$- (open circles) and $\beta$-YbAlB${}_4$ (open squares), respectively. The inset shows the low-temperature part of ${\chi }^{-1}$ under the field along the $c$ axis.

Figure 3

Temperature dependence of the in-plane and $c$-axis resistivity ${\rho }_{ab}$ and ${\rho }_c$ of $\alpha$-YbAlB${}_4$ and ${\rho }_{ab}$ of $\beta$-YbAlB${}_4$. The magnetic part of the resistivity ${\rho }_m$ is obtained by subtracting the nonmagnetic contribution estimated by ${\rho }_{ab}$ of $\alpha$- and $\beta$-LuAlB${}_4$ (solid and dash-dotted lines, respectively) or ${\rho }_c$ of $\alpha$-LuAlB${}_4$ (dashed line). The inset shows the low $T$ part of ${\rho }_{ab}$ and ${\rho }_c$.

Figure 4

(a) Temperature dependence of the ratios ${\rho }_{ab}$/${\rho }_c$ and $\Delta {\rho }_{ab}$/$\Delta {\rho }_c$. Here $\Delta \rho$ is defined by $\Delta \rho \equiv \rho -{\rho }_0$ (see the text). (b) Temperature derivative of the resistivity $d\rho /dT$.

Figure 5

(a) Temperature dependence of $\Delta \rho \equiv \rho -{\rho }_0$. The inset shows $\Delta \rho$ versus $T^2$. Note that $\Delta {\rho }_c$ for $\alpha$-YbAlB${}_4$ is multiplied by a factor of 13 for clarity. The arrow indicates $T_F$ = 240 mK estimated for ${\rho }_c$ of $\alpha$-YbAlB${}_4$by using the resistivity exponent $\alpha$ (see the text). (b) Resistivity exponent $\alpha$ defined by $\Delta \rho =A{T}^{\alpha }$ (see the text).

Figure 6

The $T^2$ coefficient $A$ of the resistivity versus the $T$-linear coefficient of the specific heat $\gamma$ for both $\alpha$- and $\beta$-YbAlB${}_4$ as well as for other Ce- and Yb-based heavy fermions.[40, 41, 42] The open circles denote Kondo lattice systems or the systems with crystal field ground-state degeneracy $N=2$. The closed circles denote mixed-valence systems, Ce systems with $N=6$, and Yb systems with $N=8$. The solid line indicates the original Kadowaki-Woods ratio $A/{\gamma }^2=1.0$ $\times {10}^{-5}$ $\mu \Omega$ cm (K mol/mJ)${}^2$. The dashed line corresponds to $A/{\gamma }^2=4.0\times {10}^{-7}$ $\mu \Omega$ cm (K mol/mJ)${}^2$, which is the typical value in transition metals.[43] The data of $\beta$-YbAlB${}_4$ at $B=2$ T $\parallel c$ (Refs.7 and9) is also shown.