Electronic structure and spontaneous internal field around nonmagnetic impurities in spin-triplet chiral $p$-wave superconductors

The electronic structure around an impurity in spin-triplet $p$-wave superconductors is studied by the Bogoliubov–de Gennes theory on a tight-binding model, where we have chosen $\sin p_x+\mathrm{i}\sin p_y$-wave or $\sin (p_x+p_y)+i\sin (-p_x+p_y)$-wave states, which are considered to be candidates for the pairing state in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. We calculate the spontaneous current and the local density of states around the impurity and discuss the difference between the two types of pairing. We propose that it is possible to discriminate the two pairing states by studying the spatial dependence of the magnetic field around a pair of impurities.

Figure 1

(Color) Spatial structure of the pairing potential for the MN-type (left panels) and the AOKA-type (right panels) pairings. We show a region consisting of $21\times 21$ sites with the impurity located at the center. (a) Amplitude of the dominant chiral component, $\mid {\Delta }_{+}\left(\mathbf{r}\right)\mid$ or $\mid {\Delta }_{+}^{\prime }\left(\mathbf{r}\right)\mid$. (b) Amplitude of the induced opposite chiral component, $\mid {\Delta }_{-}\left(\mathbf{r}\right)\mid$ or $\mid {\Delta }_{-}^{\prime }\left(\mathbf{r}\right)\mid$. (c) Phase structure of the induced component, $\mathrm{Arg}{\Delta }_{-}\left(\mathbf{r}\right)$ or $\mathrm{Arg}{\Delta }_{-}^{\prime }\left(\mathbf{r}\right)$. The phase of the dominant chiral component is almost uniform.

Figure 2

(Color) (a) The spontaneous current $j_e\left(\mathbf{r}\right)$ and (b) the internal magnetic field distribution $H\left(\mathbf{r}\right)$ around a single impurity for the MN-type (left panels) and the AOKA-type (right panels) pairings. We show a region consisting of $11\times 11$ sites with the impurity at the center. Solid circles in (a) indicate the impurity site.

Figure 3

(Color) (a) Spectrum of the LDOS $N(E,\mathbf{r})$ at sites (15,15), (15,16), and (1,1). (b) Density plot of the LDOS $N(E\sim 0,\mathbf{r})$ at the peak energy of the spectrum near $E\sim 0$ within a unit cell of $31\times 31$ sites. The left and right panels are, respectively, for the MN-type and the AOKA-type pairings.

Figure 4

(Color) (a) The spontaneous current $j_e\left(\mathbf{r}\right)$ and (b) the internal magnetic field $H\left(\mathbf{r}\right)$ for the MN-type (left panels), and the AOKA-type (right panels) pairings when a pair of impurities is situated in the diagonal direction. We show a region of $10\times 10$ sites for (a) and a region of $20\times 20$ sites for (b) around the impurities. Solid circles in (a) indicate the impurity sites.