Divergence of the Grüneisen Ratio at Quantum Critical Points in Heavy Fermion Metals

We present low-temperature volume thermal expansion, $\beta$, and specific heat, $C$, measurements on high-quality single crystals of ${\mathrm{C}\mathrm{e}\mathrm{N}\mathrm{i}}_2{\mathrm{G}\mathrm{e}}_2$ and ${\mathrm{Y}\mathrm{b}\mathrm{R}\mathrm{h}}_2({\mathrm{S}\mathrm{i}}_{0.95}{\mathrm{G}\mathrm{e}}_{0.05}{)}_2$ which are located very near to quantum critical points. For both systems, $\beta$ shows a more singular temperature dependence than $C$, and thus the Grüneisen ratio $\Gamma \propto \beta /C$ diverges as $T\to 0$. For ${\mathrm{C}\mathrm{e}\mathrm{N}\mathrm{i}}_2{\mathrm{G}\mathrm{e}}_2$, our results are in accordance with the spin-density wave (SDW) scenario for three-dimensional critical spin fluctuations. By contrast, the observed singularity in ${\mathrm{Y}\mathrm{b}\mathrm{R}\mathrm{h}}_2({\mathrm{S}\mathrm{i}}_{0.95}{\mathrm{G}\mathrm{e}}_{0.05}{)}_2$ cannot be explained by the itinerant SDW theory but is qualitatively consistent with a locally quantum critical picture.

Figure 1

Linear thermal expansions of ${\mathrm{C}\mathrm{e}\mathrm{N}\mathrm{i}}_2{\mathrm{G}\mathrm{e}}_2$ vs temperature at $B=0$. The inset shows volume expansion as $\beta /T$ vs $T$. Solid lines are fits as specified in Table .

Figure 2

Specific heat at $B=0$ as $C/T$ vs $T$ for ${\mathrm{C}\mathrm{e}\mathrm{N}\mathrm{i}}_2{\mathrm{G}\mathrm{e}}_2$. From the raw data (dashed line at low $T$), a contribution $C_n=\alpha /T^2$ with $\alpha =102\mu \mathrm{J}\mathrm{K}/\mathrm{m}\mathrm{o}\mathrm{l}$ has been subtracted giving the low-$T$ open circles. The inset shows the $T$ dependence of the Grüneisen ratio $\Gamma =V_m/{\kappa }_T·\beta /C$, where $V_m$ and ${\kappa }_T$ are the molar volume and isothermal compressibility, respectively. Here, we use ${\kappa }_T=1.15\times {10}^{-11}{\mathrm{P}\mathrm{a}}^{-1}$ as determined from high-pressure lattice parameter measurements at room temperature [24]. The solid line is a fit as specified in Table .

Figure 3

Thermal expansion of ${\mathrm{C}\mathrm{e}\mathrm{N}\mathrm{i}}_2{\mathrm{G}\mathrm{e}}_2$ along the $a$ axis as $\alpha /T$ vs $T$ at varying magnetic fields. The inset shows the critical Grüneisen ratio ${\Gamma }^{cr}=V_m/{\kappa }_T·{\beta }^{cr}/C^{cr}$ as $\log {\Gamma }^{cr}$ vs $\log T$ (at $B=0$) with ${\beta }^{cr}=\beta (T)-bT$ and $C^{cr}=C(T)-(\gamma T+d/T^2)$ using the parameters listed in Table . The solid line represents ${\Gamma }^{cr}\propto 1/T^x$ with $x=1$.

Figure 4

Electronic specific heat as $C_{el}/T$ (left axis) and volume thermal expansion as $-\beta /T$ (right axis) vs $T$ (on a logarithmic scale) for ${\mathrm{Y}\mathrm{b}\mathrm{R}\mathrm{h}}_2({\mathrm{S}\mathrm{i}}_{0.95}{\mathrm{G}\mathrm{e}}_{0.05}{)}_2$ at $B=0$. The solid lines indicate $\log (T_0/T)$ dependences with $T_0=30\mathrm{K}$ and 13 K for $C_{el}/T$ and $-\beta /T$, respectively. The dotted line represents $-\beta /T=a_0+a_1/T$ with $a_0=3.4\times {10}^{-6}{\mathrm{K}}^{-2}$ and $a_1=1.34\times {10}^{-6}{\mathrm{K}}^{-1}$. The inset displays the log-log plot of ${\Gamma }^{cr}(T)$ with ${\Gamma }^{cr}=V_m/{\kappa }_T·{\beta }^{cr}/C^{cr}$ using ${\kappa }_T=5.3\times {10}^{-12}{\mathrm{P}\mathrm{a}}^{-1}$ [26], ${\beta }^{cr}=\beta (T)+a_{0}T$, and $C^{cr}=C_{el}(T)$. The solid and dotted lines represent ${\Gamma }^{cr}\propto 1/T^x$ with $x=0.7$ and $x=1$, respectively.