Effect of the Vortices on the Nuclear Spin Relaxation Rate in the Unconventional Pairing States of the Organic Superconductor $(\mathrm{TMTSF}{)}_2{\mathrm{PF}}_6$

This Letter theoretically discusses quasiparticle states and nuclear spin relaxation rates $T_1^{-1}$ in the quasi-one-dimensional superconductor $(\mathrm{TMTSF}{)}_2{\mathrm{PF}}_6$ under a magnetic field applied parallel to the conduction chains. We study the effects of Josephson-type vortices on $T_1^{-1}$ by solving the Bogoliubov–de Gennes equation for $p$-, $d$- or $f$-wave pairing interactions. In the presence of line nodes in pairing functions, $T_1^{-1}$ is proportional to $T$ in sufficiently low temperatures because quasiparticles induced by vortices at the Fermi energy relax spins. We also try to identify the pairing symmetry of $(\mathrm{TMTSF}{)}_2{\mathrm{PF}}_6$.

Figure 1

(a) Fermi surface in the Q1D tight-binding model [$E(\mathbf{k})=0$] in the $k_a\mathrm{\text{−}}{k}_b$ plane. $k_c$ dependence of the Fermi surface is negligible in this scale of the figure. (b) The gap function ${\Delta }_0\psi (k_a)$ along the Fermi surface for $p$-, $d$-, and $f$-wave pairing symmetries at $T=0$ and $H=0$.

Figure 2

The LDOS $N(E,\mathbf{r})$ at the site nearest to the vortex core is presented at $T=0$ for the $p$ wave in (a), the $d$ wave in (b), and the $f$ wave in (c). The LDOS in the vortex state is plotted with solid lines. The results in the uniform state at zero field and those in the normal state are shown with the dashed and dotted lines, respectively.

Figure 3

Spatial profiles of the pair amplitudes and supercurrents in a unit cell are shown for the $d$-wave symmetry at $T=0$. In (a), the amplitudes of the pair potentials ($|{\Delta }_i|$) are presented. In (b), the amplitudes of the $b$ component of the screening current [$J_b(\mathbf{r})$] are presented.

Figure 4

The nuclear spin relaxation rate $T_1^{-1}$ is shown as a function of temperature for (a) $p$-wave, (b) $d$-wave, and (c) $f$-wave symmetries. Solid lines denote $T_1^{-1}$ at the lattice site nearest to the vortex core. The results for the uniform state at $H=0$ and those in the normal state are plotted with dashed and dotted lines, respectively. Insets in (b) and (c) show the double logarithmic chart of the results.