Thermoelectric Response Near a Quantum Critical Point of $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$ and ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$: A Comparative Study

The thermoelectric coefficients have been measured down to a very low temperature for the Yb-based heavy-fermion compounds $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$ and ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$, often considered as model systems for the local quantum criticality case. We observe a striking difference in the behavior of the Seebeck coefficient $S$ in the vicinity of their respective quantum critical point (QCP). Approaching the critical field, $S/T$ is enhanced in $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$, but drastically reduced in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. The ratio of thermopower to specific heat remains constant for $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$, but it is significantly reduced near the QCP in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. In both systems, on the other hand, the Nernst coefficient shows a diverging behavior near the QCP. The interplay between valence and magnetic quantum criticality and the additional possibility of a Lifshitz transition crossing the critical field under magnetic field are discussed as the origin of the different behaviors of these compounds.

Figure 1

(a) Temperature dependence of the Seebeck coefficient $-S(T)/T$ for two different crystals of $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$ (samples 1 and 2) under zero field and 5 T. Inset: zero-field resistivity vs $T$ plot for samples 1 and 2. (b) $-S(T)/T$ curves under several fields for sample 2. The $-S(T)/T$ curves for ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ (c) below and (d) above the critical field of 0.66 T.

Figure 2

Contour plots of the Seebeck coefficient $-S/T$ for (a) $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$ and (b) ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ in the field-temperature plane. $T_N$, $T_{\mathrm{FL}}$, and $T^{*}$ represent the Neel ordering temperature, the crossover temperature to the Fermi-liquid state, and the crossover line where the Hall coefficient shows a rapid change, respectively [18].

Figure 3

Field dependence of the Seebeck coefficient $-S/T$ at 50 mK (left axis) and the electronic specific heat coefficient ${\gamma }_0$ (right axis) [7, 8, 30] for (a) $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$ and (b) ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$. (c) The dimensionless ratio $|q|$ vs $B\mathrm{\text{−}}{B}_c$ plot for $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$ (left axis) and ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ (right axis). The dashed lines are a guide to the eye.

Figure 4

Temperature dependence of the Nernst coefficient $-\nu (T)/T$ for (a) $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$ and (b) ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ under several magnetic fields. The $-\nu (T)/T$ curves for $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$ at $B\ge 1.0\mathrm{T}$ are shifted for clarity.