Divergence of the Magnetic Grüneisen Ratio at the Field-Induced Quantum Critical Point in ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$

The heavy-fermion metal ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ is studied by low-temperature magnetization $M(T)$ and specific-heat $C(T)$ measurements at magnetic fields close to the quantum critical point ($H_c=0.06\mathrm{T}$, $H\perp c$). Upon approaching the instability, $dM/dT$ is more singular than $C(T)$, leading to a divergence of the magnetic Grüneisen ratio ${\Gamma }_{\mathrm{mag}}=-(dM/dT)/C$. Within the Fermi-liquid regime, ${\Gamma }_{\mathrm{mag}}=-G_r(H-H_c^{\mathrm{fit}})$ with $G_r=-0.30\pm 0.01$ and $H_c^{\mathrm{fit}}=(0.065\pm 0.005)\mathrm{T}$ which is consistent with scaling behavior of the specific-heat coefficient in ${\mathrm{YbRh}}_2({\mathrm{Si}}_{0.95}{\mathrm{Ge}}_{0.05}{)}_2$. The field dependence of $dM/dT$ indicates an inflection point of the entropy as a function of magnetic field upon passing the line $T^{\star }(H)$ previously observed in Hall and thermodynamic measurements.

Figure 1

Magnetization divided by field $M/H$ of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ as a function of temperature. Inset: Inverse of $M/H$ vs temperature. The dashed line indicates linear fit for low-temperature region.

Figure 2

(a) Temperature derivative of the magnetization as $-(dM/dT)/T$ (left axis), (b) electronic specific heat as $C(T)/T$, and (c) magnetic Grüneisen ratio ${\Gamma }_{\mathrm{mag}}$ vs temperature (on a logarithmic scale) for ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ in magnetic fields applied perpendicular to $c$ axis. Volume thermal expansion of ${\mathrm{YbRh}}_2({\mathrm{Si}}_{0.95}{\mathrm{Ge}}_{0.05}{)}_2$ as $-\beta /T$ at zero field is shown for comparison in (a) (right axis) [12]. Inset in (a): Expanded plot for $-(dM/dT)/T$ vs $T$ for $H=0.3\mathrm{T}$. The arrow indicates the position of the maximum. Inset in (b): ${\Gamma }_{\mathrm{mag}}$ vs temperature for ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ on a log-log plot. The solid lines represent ${\Gamma }_{\mathrm{mag}}(T)\propto T^{-ϵ}$ with $ϵ=0.7$ and 2.0 for low- and high-temperature regions, respectively. Inset in (c): Field dependence of saturated ${\Gamma }_{\mathrm{mag}}$, as $T\to 0$. The solid line indicates $-G_r(H-H_c^{\mathrm{fit}}{)}^{-1}$ with $G_r=-0.3$ and $H_c^{\mathrm{fit}}=0.065\mathrm{T}$.

Figure 3

Magnetization difference divided by temperature increment $-\Delta M/\Delta T$ (see text) vs magnetic field for ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ at $T=0.08\mathrm{K}$ (black circles), 0.33 K (red open squares), 0.75 K (blue open circles) and 1.5 K (green squares). Inset: $H\mathrm{\text{−}}T$ phase diagram of ${\mathrm{YbRh}}_2{\mathrm{Si}}_2$ with the peak positions in $-\Delta M/\Delta T$ (black solid circles) and $T^{\star }(H)$ (gray solid line) as determined from transport and thermodynamic properties (red or gray symbols) [22, 23]. The black and blue or dark gray lines represent the phase boundary of the antiferromagnetic (AFM) ground state and crossover to the Fermi-liquid (FL) regime, respectively [22].