Field-Angle-Dependent Specific Heat Measurements and Gap Determination of a Heavy Fermion Superconductor ${\mathrm{URu}}_2{\mathrm{Si}}_2$

To identify the superconducting gap structure in ${\mathrm{URu}}_2{\mathrm{Si}}_2$, we perform field-angle-dependent specific heat measurements for the two principal orientations in addition to field rotations, and a theoretical analysis based on microscopic calculations. The Sommerfeld coefficient $\gamma (H)$’s in the mixed state exhibit a distinctly different field dependence. This comes from point nodes and the substantial Pauli paramagnetic effect of ${\mathrm{URu}}_2{\mathrm{Si}}_2$. These two features combined give rise to a consistent picture of superconducting properties, including a possible first order transition of $H_{c2}$ at low temperatures.

Figure 1

$H$-dependence of low-$T$ $\gamma (H)$ for $H\parallel c$ (●) and $H\parallel a$ (○), calculated by quasiclassical theory [8]. We plot $\gamma (H)/{\gamma }_0$ vs $H/H_{c2}$ for the line node on the equator (a) and point nodes on the poles (b), as schematically shown there. ${\gamma }_0$ denotes the Sommerfeld coefficient in the normal state.

Figure 2

$C(H,\theta )$ of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ at $T=0.34\mathrm{K}$. The field-rotational plane is perpendicular to [010] (a) and to [001] (b) directions. Inset shows $C/T$ vs $T$ plot at $H=0$.

Figure 3

(a) Field variation of $C(H)$ of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ at $T=0.34\mathrm{K}$ for $H\parallel a$ (open circles) and $H\parallel c$ (solid circles). Solid lines are guides to the eyes. (b) $H$-dependent part of $C$ as a function of $H/H_{c2}$. The dotted line is a linear extrapolation of the low-field part for $H\parallel c$. (c) $H$-variation of $C$ in the normal state at 1.52 K for $H\parallel a$ and $H\parallel c$. (d) $T$-variation of $M/H$ in the normal state below 25 K for $H\parallel a$ and $H\parallel c$.

Figure 4

$H$-dependence of $\gamma (H)$ for $H\parallel a$ in the linear point node case $\varphi (\mathbf{k})=\sin \theta$ (a) and the quadratic point node case $\varphi (\mathbf{k})={sin}^2\theta$ (b). Solid circles are the experimental data in Fig. 3b. $\mu =0$ (A), 0.85 (B), 1.7 (C) in (a). $\mu =0$ (A), 0.6 (B), 1.0 (C) in (b).

Figure 5

The same as in Fig. 4, but for $H\parallel c$. $\mu =0$ (A), 1.6 (B), 2.5 (C) in (a). $\mu =0$ (A), 1.3 (B) in (b).