Thermodynamic study of gap structure and pair-breaking effect by magnetic field in the heavy-fermion superconductor ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$

This paper presents the results of specific-heat and magnetization measurements, in particular their field-orientation dependence, on the first discovered heavy-fermion superconductor ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ ($T_{\mathrm{c}}\sim 0.6$ K). We discuss the superconducting gap structure and the origin of the anomalous pair-breaking phenomena, leading, e.g., to the suppression of the upper critical field $H_{\mathrm{c}2}$, found in the high-field region. The data show that the anomalous pair breaking becomes prominent below about 0.15 K in any field direction, but occurs closer to $H_{\mathrm{c}2}$ for $H\parallel c$. The presence of this anomaly is confirmed by the fact that the specific-heat and magnetization data satisfy standard thermodynamic relations. Concerning the gap structure, field-angle dependencies of the low-temperature specific heat within the $ab$ and $ac$ planes do not show any evidence for gap nodes. From microscopic calculations in the framework of a two-band full-gap model, the power-law-like temperature dependencies of $C$ and $1/T_1$, reminiscent of nodal superconductivity, have been reproduced reasonably. These facts further support multiband full-gap superconductivity in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$.

Figure 1

Magnetic field dependence of the equilibrium magnetization $M_{\mathrm{eq}}$ at various temperatures in (a) $H\parallel a$ and (b) $H\parallel c$. Insets show raw magnetization curves measured at 0.07 and 0.2 K.

Figure 2

Field dependence of the magnetization at several temperatures, where the paramagnetic contribution is subtracted, for (a) $H\parallel a$ and (b) $H\parallel c$. (c) and (d) Differential-susceptibility data obtained from (a) and (b), respectively. In all the figures, the magnetic field is normalized by $H_{\mathrm{c}2}$ at 0.07 K.

Figure 3

Temperature dependence of $M/H$ in the zero-field-cooling process for (a) $H\parallel a$ and (b) $H\parallel c$.

Figure 4

Temperature dependence of $C_{\mathrm{e}}/T$ at various fields in (a) $H\parallel a$ and (b) $H\parallel c$. Here, the nuclear specific-heat contribution is subtracted. Inset in (a) is a semi-log plot of the zero-field $C_{\mathrm{e}}\left(T\right)$ data. The dashed line is a fit by using the BCS function in the low-temperature regime.

Figure 5

Field dependence of $C_{\mathrm{e}}/T$ at 60 mK for $H\parallel a$ (open) and $H\parallel c$ (closed). Lower inset shows an enlarged plot in the low-field regime. Upper inset shows the slope of $C_{\mathrm{e}}\left(H\right)/T$ ($\delta \gamma /\delta H$; dashed lines) and the coefficient of the $T^2$ term in the $M(T,H)$ data (${\beta }_2$; circles).

Figure 6

Field angle $\varphi$ dependence of $C_{\mathrm{e}}/T$ measured under a rotating magnetic field within the $ab$ plane at (a) 0.1 and (b) 0.06 K. Here, $\varphi$ is an azimuthal angle between the magnetic field and the $a$ axis. Dashed lines are fits to the data using $C_{\mathrm{e}}\left(\varphi \right)=C_0+C_H[1-A_{4}cos\left(4\varphi \right)]$, where $C_0$ and $C_H$ are the zero-field and field-dependent parts of $C_{\mathrm{e}}$. (c) Field-angle-resolved $C_{\mathrm{e}}/T$ at 0.06 K in a magnetic field rotated within the $ac$ plane ($\varphi ={90}^{\circ }$) as a function of the field angle $\theta$ measured relative to the $c$ axis, as illustrated in the center-bottom inset. For clarity, the data measured in the interval $-{30}^{\circ }\le \theta \le {100}^{\circ }$ (solid symbols) are plotted repeatedly (open symbols; $\theta$ is converted to $-\theta$ and ${180}^{\circ }-\theta$).

Figure 7

Temperature dependencies of (a) the thermodynamic critical field $H_{\mathrm{c}}$, (b) the upper critical field $H_{\mathrm{c}2}$, (c) the slope of $H_{\mathrm{c}2}\left(T\right)$, and (d) the Maki parameter ${\kappa }_2$, determined from specific-heat and the magnetization measurements.

Figure 8

Two-band full-gap analyses of (a) $C\left(T\right)$ and (b) $1/T_1\left(T\right)$ by using model A. Analyses of (c) $C\left(T\right)$ and (d) $1/T_1\left(T\right)$ by using models B and C. The calculated results are represented by lines. Solid circles in (c) are experimental data of $C_{\mathrm{e}}/\gamma T$. Open circles in (a) represent modified specific-heat data, $[C_{\mathrm{e}}(T,0\mathrm{T})-C_{\mathrm{e}}(T,3\mathrm{T})]/{\gamma }^{*}T+1$, which was introduced in Ref. [17]. Circles in (b) and (d) are experimental data of $1/T_1$ taken from the previous report [12] on polycrystalline ${\mathrm{CeCu}}_{2.05}{\mathrm{Si}}_2$. Solid and dashed lines in (b) represent the calculated results with $\eta /k_{\mathrm{B}}{T}_{\mathrm{c}}=0.01$ and 0.1, respectively. Solid and dashed lines in (d) are the calculated results by models B and C with $\eta /k_{\mathrm{B}}{T}_{\mathrm{c}}=0.1$, respectively. Models B and C provide the same results of $C\left(T\right)$. A dot-dashed line in (d) is the calculated result by model C with temperature-dependent smearing factor, $\eta \left(T\right)/k_{\mathrm{B}}{T}_{\mathrm{c}}=0.1+0.5{(T/T_{\mathrm{c}})}^2$.