Gap Symmetry of the Heavy Fermion Superconductor ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ at Ambient Pressure

Recent observations of two nodeless gaps in superconducting ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ have raised intensive debates on its exact gap symmetry, while a satisfactory theoretical basis is still lacking. Here we propose a phenomenological approach to calculate the superconducting gap functions, taking into consideration both the realistic Fermi surface topology and the intra- and interband quantum critical scatterings. Our calculations yield a nodeless $s^{\pm }$-wave solution in the presence of strong interband pairing interaction, in good agreement with experiments. This provides a possible basis for understanding the superconducting gap symmetry of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ at ambient pressure and indicates the potential importance of multiple Fermi surfaces and interband pairing interaction in understanding heavy fermion superconductivity.

Figure 1

(a) Fermi velocity distributions on the electron and hole Fermi surfaces of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ at ambient pressure from $\mathrm{DFT}+U$ calculations. The heavy electron Fermi surface contains mainly a corrugated-cylinder sheet near the $X$ point and a small torus. The hole Fermi surface is complex but also contains some $f$-orbital character. (b) Illustration of the intra- and interband scatterings (arrows) of the Cooper pairs. The wavy lines denote the electron pairs on the Fermi surfaces.

Figure 2

Evolution of the eigenvalues $\lambda$ of three leading solutions as functions of $r^{11}$ or $r^{12}$. Two of the solutions belong to the same representation ($A_{1g}$) of the $D_{4h}$ group. Both are mixtures of nodeless (circle) and nodal (down triangle) $s$ waves. The other solution belongs to the $B_{1g}$ representation and has a $d_{x^2-y^2}$ symmetry. The inset in (c) highlights the interplay of three solutions.

Figure 3

Illustration of typical gap structures on the Fermi surfaces for fixed $r^{11}=0.1$: (a) $d_{x^2-y^2}$ at $r^{12}=0.3$; (b) nodal $s$ at $r^{12}=0.6$; (c) nodeless $s^{\pm }$ at $r^{12}=1.0$. The color bar represents the value of the eigenvector $g(\mathbf{k})$ normalized by its maximal value. For azimuthal ($\phi$) dependence, we choose the $k_z=1.64\pi /c$ plane, while for polar ($\theta$) dependence, we choose the diagonal cut ($k_x=k_y$) for the heavy electron Fermi surface (down triangle) and the $k_x=0$ plane cut for the hole Fermi surface (circle). The torus is neglected for clarity.

Figure 4

A theoretical phase diagram for the superconductivity in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ at ambient pressure. The axes denote the relative strengths of the intra- and interband interactions. The color bar represents the ratio between the eigenvalues of the leading $A_{1g}$ and $B_{1g}$ solutions. The phase diagram is divided into three regions: $d_{x^2-y^2}$, nodal $s$, and nodeless $s^{\pm }$, illustrated with typical gap structures from Fig. 3. The solid line marks the transition between $d_{x^2-y^2}$ and nodal $s$, and the dashed line shows the crossover from nodal $s$ to nodeless $s^{\pm }$.