Superconducting gap function in the organic superconductor (TMTSF)${}_2$ClO${}_4$ with anion ordering; First-principles calculations and quasiclassical analysis for angle-resolved heat capacity

We calculate angle-dependent heat capacity in a low magnetic field range on the basis of the Kramer-Pesch approximation together with an electronic structure obtained by first-principles calculations to determine a superconducting gap function of (TMTSF)${}_2$ClO${}_4$ through its comparisons with experiments. The present comparative studies reveal that a nodal $d$-wave gap function consistently explains the experimental results for (TMTSF)${}_2$ClO${}_4$. It is especially emphasized that the observed unusual axis asymmetry of the angle dependence eliminates the possibility of $s$-wave and nodeless $d$-wave functions. It is also found that the directional ordering of ClO${}_4$ anions does not have any significant effects on the Fermi surface structure contrary to the previous modelings since the two Fermi surfaces obtained by the band calculations almost cross within the present full accuracy in first-principles calculations.

Figure 1

Crystal structure for (TMTSF)${}_2$ClO${}_4$.

Figure 2

Band structure for (TMTSF)${}_2$ClO${}_4$.

Figure 3

Fermi surfaces for (TMTSF)${}_2$ClO${}_4$. Inset: Closeup of the Fermi-surface crossing, sliced at $k_z=0$. The mesh describes $k$ points actually used in the Fermi-surface calculation.

Figure 4

Angular dependence of the heat capacity rotating magnetic fields on the $a$-$b^{\prime }$ plane in the case of the $s$-wave gap function.

Figure 5

Gap functions on the Fermi surfaces in the case of the nodeless $d$-wave gap function.

Figure 6

Angular dependence of the heat capacity rotating magnetic fields on the $a$-$b^{\prime }$ plane in the case of the nodeless $d$-wave gap function.

Figure 7

Schematic figure about the nodal line in the case of the nodal $d$-wave gap function I.

Figure 8

Angular dependence of the heat capacity rotating magnetic fields on the $a$-$b^{\prime }$ plane in the case of the nodal $d$-wave gap functions I, II, and III.

Figure 9

Angular dependence of the heat capacity rotating magnetic fields on the $a$-$b^{\prime }$ plane in the case of the nodal $d$-wave gap function II.

Figure 10

Tight-binding fitted band structure (solid curves). The squares denote the first-principle band structure shown in Fig. 2.

Figure 11

Angular dependence of the partial heat capacity rotating magnetic fields on the $a$-$b^{\prime }$ plane in the case of the nodal $d$-wave gap function I at $T=0.05\Delta$. $N_i$ is the partial heat capacity on the $i$th band.