Multiband Superconductivity with Unexpected Deficiency of Nodal Quasiparticles in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$

Superconductivity in the heavy-fermion compound ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ is a prototypical example of Cooper pairs formed by strongly correlated electrons. For more than 30 years, it has been believed to arise from nodal $d$-wave pairing mediated by a magnetic glue. Here, we report a detailed study of the specific heat and magnetization at low temperatures for a high-quality single crystal. Unexpectedly, the specific-heat measurements exhibit exponential decay with a two-gap feature in its temperature dependence, along with a linear dependence as a function of magnetic field and the absence of oscillations in the field angle, reminiscent of multiband full-gap superconductivity. In addition, we find anomalous behavior at high fields, attributed to a strong Pauli paramagnetic effect. A low quasiparticle density of states at low energies with a multiband Fermi-surface topology would open a new door into electron pairing in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$.

Figure 1

(a) Electronic specific heat of an $S$-type ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ single crystal divided by temperature, $C_{\mathrm{e}}/T$, as a function of $T$ measured in $H\parallel [100]$. The dashed line is a fit to the low-$T$ part of the $C_{\mathrm{e}}(T)$ data at 0 T using the BCS formula $C_{\mathrm{e}}(T)=A\exp (-{\mathrm{\Delta }}_0/T)+{\gamma }_{0}T$. (b) Temperature variation of $\delta C_{\mathrm{sc}}(T,H)/{\gamma }^{*}T+1$ at $H=0$ and a best fit to the two-gap BCS model (solid line). Here, $\delta C_{\mathrm{sc}}(T,H)=C_{\mathrm{e}}(T,H)-C_{\mathrm{e}}(T,3\mathrm{T})$ and ${\gamma }^{*}=0.84\mathrm{J}/(\mathrm{mol}{\mathrm{K}}^2)$ are introduced to satisfy entropy balance in the BCS framework. The gradual increase of the normal-state $C_{\mathrm{e}}/T$ upon cooling is included in $\delta C_{\mathrm{sc}}(T,H)$. The contribution of each gap to the total specific heat is also shown. (c) Field-temperature phase diagram for $H\parallel [100]$ determined by the specific-heat (circles) and magnetization (triangles) measurements. A contour plot of $\delta C_{\mathrm{sc}}(T,H)/T$ in the superconducting state is shown using the data from (a). Anomalous $\delta C_{\mathrm{sc}}>0$ behavior that can be ascribed to the Pauli paramagnetic effect is clearly seen in the high-$H$ and low-$T$ region below 0.1 K.

Figure 2

The ratio $C_{\mathrm{e}}/T$ and its slope $k_H=d(C_{\mathrm{e}}/T)/d({\mu }_{0}H)$ as a function of magnetic field applied parallel to the [100] and [001] directions at 0.06 (circles), 0.1 (squares), and 0.2 K (triangles). No hysteresis is found with increasing and decreasing fields. Inset shows $C_{\mathrm{e}}/T(\varphi )$ at 0.1 K in 0.7 T.

Figure 3

(a) Band structure of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ calculated using the $\mathrm{LDA}+U$ method. (b) The calculated Fermi surfaces colored by the magnitude of the Fermi velocity $v_{\mathrm{F}}$. (c) The total density of states and the partial density of states for the three bands. The Fermi level corresponds to $E=0$. Here “hole1” and “hole2” are a small hole ring and a connected hole sheet located around the $Z$ point, respectively, and “ele” is a tubular electron sheet located around the $X$ point.

Figure 4

(a) Magnetization $M$ of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ in $H\parallel [100]$ at 70 mK. The solid line represents the equilibrium magnetization $M_{\mathrm{eq}}$ obtained by averaging over increasing and decreasing field sweeps. (b) Graphs of $M_{\mathrm{eq}}-{\chi }_{\mathrm{n}}H$ at 0.07, 0.12, 0.2, 0.3, 0.4, and 0.5 K, where ${\chi }_{\mathrm{n}}H$ is the paramagnetic contribution. The inset shows temperature dependence of the Maki parameter ${\kappa }_2$ evaluated from the specific-heat (circles) and the magnetization (triangles) using the relations $d(M_{\mathrm{eq}}-{\chi }_{\mathrm{n}}H)/dH{|}_{H_{\mathrm{c}2}}=1/4\pi \beta (2{\kappa }_2^2-1)$ and $(\mathrm{\Delta }C/T){|}_{T_{\mathrm{c}}}={(d{H}_{\mathrm{c}2}/dT)}^2/4\pi \beta (2{\kappa }_2^2-1)$. Here, $\mathrm{\Delta }C$ is the jump in the specific heat at $T_{\mathrm{c}}(H)$ from Fig. 1. The value $\beta =1.16$ assumes a triangular vortex lattice. The slope of ${dH}_{\mathrm{c}2}/dT$ is estimated using the data in Fig. 1. (c) Magnetization measured during a zero-field cooling process, $M_{\mathrm{ZFC}}$, at 1, 1.5, 1.8, 3, and 10 T after subtracting the value $M_0$ of $M_{\mathrm{ZFC}}$ at 0.7 K.