Impurity scattering rate and coherence factor in vortex core of sign-reversing $s$-wave superconductors

We investigate the impurity scattering rates for quasiparticles in vortex cores of sign-reversing $s$-wave superconductors as a probe to detect the internal phase difference of the order parameters among different Fermi surfaces. The impurity scattering rates and coherence factors are related to quasiparticle interference effect by the scanning tunneling microscopy and spectroscopy technique. With use of the Born and Kramer-Pesch approximations for the Andreev bound states, we show that the sign-reversed forward scatterings are dominant in vortex cores. Owing to the coherence factor in vortex cores of $\pm s$-wave superconductors, the impurity scattering rate of the Andreev bound states has a characteristic distribution on the Fermi surfaces. For comparison, the impurity scattering rates in vortex cores of $s$-wave and $d$-wave superconductors are also discussed.

Figure 1

Schematic figure of the effective Fermi surface.

Figure 2

Schematic figures of (a) the forward and (b) the backward scatterings around a single vortex core in the low energy.

Figure 3

(Color online) Two hole Fermi surfaces ${\alpha }_1,{\alpha }_2$ and two electron Fermi surfaces ${\beta }_1,{\beta }_2$.

Figure 4

(Color online) $\mathit{q}$ dependence of the normalized impurity scattering rate ${\tilde{\Gamma }}_{\mathit{k},\mathit{k}+\mathit{q}}$ in the isotropic $s$-wave superconductors. The energy is $ϵ=0.3{\Delta }_{\infty }$.

Figure 5

(Color online) $\mathit{q}$ dependence of the normalized impurity scattering rate ${\tilde{\Gamma }}_{\mathit{k},\mathit{k}+\mathit{q}}$ in the $d$-wave superconductors $d\left(\mathit{k}\right)=-\left(k_x^2-k_y^2\right)/\left(k_x^2+k_y^2\right)$. The energy is $ϵ=0.3{\Delta }_{\infty }$.

Figure 6

(Color online) Schematic figures of the sign of the pair functions for (a):$d$-wave and (b):$\pm s$-wave symmetries.

Figure 7

(Color online) The arrow denotes the sign-reversed forward scattering in $d$-wave case.

Figure 8

(Color online) $\mathit{q}$-dependence of the normalized impurity scattering rate ${\tilde{\Gamma }}_{\mathit{k},\mathit{k}+\mathit{q}}$ in the isotropic $\pm s$-wave superconductors. The energy is $ϵ=0.3{\Delta }_{\infty }$.

Figure 9

(Color online) Energy dependence of the normalized energy width $\tilde{\gamma }\left(ϵ,n\right)$ with $n=4$ in the isotropic $\pm s$-wave and $s$-wave superconductors.

Figure 10

(Color online) Fermi velocity dependence of the normalized energy width $\tilde{\gamma }\left(ϵ,n\right)$ with $n=4$ in the isotropic $\pm s$-wave and $s$-wave superconductors. The energy is $ϵ=0.1\Delta$.

Figure 11

(Color online) $\mathit{q}$-dependence of the normalized impurity scattering rate ${\tilde{\Gamma }}_{\mathit{k},\mathit{k}+\mathit{q}}$ in the isotropic $\pm s$-wave superconductors. The energy is $ϵ=0.3{\Delta }_{\infty }$. The amplitude of the Fermi velocity $v_{F,n}$ on $\alpha$ Fermi surface $\left(n=1,2\right)$ is $v_{F,n}=0.8v_{F,m}$ $\left(m=3,4\right)$.