Grüneisen Ratio Divergence at the Quantum Critical Point in ${\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{u}}_{6-x}{\mathrm{A}\mathrm{g}}_x$

The heavy-fermion system ${\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{u}}_{6-x}{\mathrm{A}\mathrm{g}}_x$ is studied at its antiferromagnetic quantum critical point, $x_c=0.2$, by low-temperature ($T\ge 50\mathrm{m}\mathrm{K}$) specific heat, $C(T)$, and volume thermal expansion, $\beta (T)$, measurements. Whereas $C/T\propto \log (T_0/T)$ would be compatible with the predictions of the itinerant spin-density-wave (SDW) theory for two-dimensional critical spin fluctuations, $\beta (T)/T$ and the Grüneisen ratio, $\Gamma (T)\propto \beta /C$, diverge much weaker than expected, in strong contrast to this model. Both $C$ and $\beta$, plotted as a function of the reduced temperature $t=T/T_0$ with $T_0=4.6\mathrm{K}$, are similar to what was observed for ${\mathrm{Y}\mathrm{b}\mathrm{R}\mathrm{h}}_2({\mathrm{S}\mathrm{i}}_{0.95}{\mathrm{G}\mathrm{e}}_{0.05}{)}_2$ ($T_0=23.3\mathrm{K}$), indicating a striking discrepancy to the SDW prediction in both systems.

Figure 1

Specific heat as $C/T$ vs $T$ (on a logarithmic scale) for different ${\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{u}}_{6-x}{\mathrm{A}\mathrm{g}}_x$ polycrystals. The inset shows the evolution of the antiferromagnetic phase transition temperature $T_N$ vs $x$ as derived from specific heat (squares: this study; circles [11]) and electrical resistivity (triangles: [21]) results.

Figure 2

(a) Electronic specific heat of ${\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{u}}_{5.8}{\mathrm{A}\mathrm{g}}_{0.2}$ as $C_{el}/T$ vs $T$ (on a logarithmic scale) for $B=0$ and differing magnetic fields. At $B>0$, $C_{el}$ is obtained after subtraction of the Cu nuclear specific heat contribution $C_n\propto B^2/T^2$ [21]. (b) Volume thermal expansion coefficient $\beta$ of the same sample studied in (a) as $\beta /T$ vs $\log T$. $\beta ={\alpha }_1+{\alpha }_2+{\alpha }_3$ with ${\alpha }_i$ being the linear thermal expansion coefficients along the three perpendicular directions of the sample. The solid lines indicate logarithmic temperature dependences in $C_{el}(T)/T$ and $\beta (T)/T$.

Figure 3

Temperature dependence of the Grüneisen ratio $\Gamma =(V_{\mathrm{m}\mathrm{o}\mathrm{l}}/{\kappa }_T)(\beta /C)$ with molar volume $V_{\mathrm{m}\mathrm{o}\mathrm{l}}=6.37\times {10}^{-5}{\mathrm{m}}^3{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ and isothermal compressibility ${\kappa }_T=1\times {10}^{-11}{\mathrm{P}\mathrm{a}}^{-1}$ [24]. The inset shows the same data on a logarithmic temperature scale. The solid line represents $-\log (T)$ dependence.

Figure 4

Scaling of the low-$T$ specific heat of ${\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{u}}_{5.8}{\mathrm{A}\mathrm{g}}_{0.2}$ and ${\mathrm{Y}\mathrm{b}\mathrm{R}\mathrm{h}}_2({\mathrm{S}\mathrm{i}}_{0.95}{\mathrm{G}\mathrm{e}}_{0.05}{)}_2$ [7, 20] as $(C/T-E)T_0$ vs $\log (T/T_0)$ according to Sereni et al. [23]. For ${\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{u}}_{5.8}{\mathrm{A}\mathrm{g}}_{0.2}$, $T_0=4.6\mathrm{K}$ and $E=0.105\mathrm{J}/{\mathrm{K}}^2\mathrm{m}\mathrm{o}\mathrm{l}$; for ${\mathrm{Y}\mathrm{b}\mathrm{R}\mathrm{h}}_2({\mathrm{S}\mathrm{i}}_{0.95}{\mathrm{G}\mathrm{e}}_{0.05}{)}_2$, $T_0=23.3\mathrm{K}$ and $E=0.066\mathrm{J}/{\mathrm{K}}^2\mathrm{m}\mathrm{o}\mathrm{l}$. The solid line represents scaling function $7.2\log (T_0/T)$ observed in several other HF systems [23].

Figure 5

Volume thermal expansion as $\beta /T$ vs normalized temperature $T/T_0$ (on a logarithmic scale) for ${\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{u}}_{5.8}{\mathrm{A}\mathrm{g}}_{0.2}$ ($T_0=4.6\mathrm{K}$, right axis) and ${\mathrm{Y}\mathrm{b}\mathrm{R}\mathrm{h}}_2({\mathrm{S}\mathrm{i}}_{0.95}{\mathrm{G}\mathrm{e}}_{0.05}{)}_2$ [20] ($T_0=23.3\mathrm{K}$, left axis). The arallel solid lines indicate the logarithmic temperature dependence of $\beta (T)/T$.