Effects of ballistic and diffusive motion of quasiparticles on spectral properties around a vortex in a two-band superconductor

Motivated by the recent results on impurity effects in ${\mathrm{MgB}}_2$, we present a theoretical model for a two-band superconductor in which the character of quasiparticle motion is ballistic in one band and diffusive in the other. We apply our model to calculate the electronic structure in the vicinity of an isolated vortex. We assume that superconductivity in the diffusive $\left(\pi \right)$ band is induced by that in the clean $\left(\sigma \right)$ band, as suggested by experimental evidence for ${\mathrm{MgB}}_2$. We focus our attention to the spatial variation of the order parameter, the current density, and the vortex core spectrum in the two bands. Our results indicate that the coupling to the $\pi$ band can lead to the appearance of additional bound states near the gap edge in the $\sigma$ band that are absent in the single-band case.

Figure 1

(Color online) Magnitude of the order parameter, $\mid {\Delta }_{\sigma ,\pi }\left(x\right)\mid$, in the $\sigma$ and $\pi$ bands, as a function of coordinate $x$ on a path through the vortex center, at $T=0.1T_c$: (a) for different ratios ${\xi }_{\pi }∕{\xi }_{\sigma }$ and fixed strength of the induced $\pi$-band gap, parametrized by the mixing ratio (see text), $\rho =0.3$; (b) for different $\rho$ and fixed ${\xi }_{\pi }∕{\xi }_{\sigma }=5$. Weak effective repulsion due to the Coulomb interaction is assumed in the subdominant ${\lambda }^{\left(1\right)}$ pairing channel, parametrized by $\Lambda =-0.1$ (see text).

Figure 2

(Color online) (a) Local density of states (LDOS) for the $\pi$ band, as a function of energy $ϵ$ for different distances $r$ from the vortex center. The spectra are shifted in a vertical direction for convenience. (b) LDOS at the chemical potential, $ϵ=0$, as a function of $r$ for different $\rho$ and fixed $T=0.5T_c$, ${\xi }_{\pi }∕{\xi }_{\sigma }=3$.

Figure 3

(Color online) (a) LDOS for the $\sigma$ band as a function of energy $ϵ$ for different distances $r$ from the vortex center. (b) The development of extra bound states near the gap edge in the $\sigma$ band for different parameter combinations. The bound states develop for sufficiently large mixing ratio $\rho$ and coherence length ratio ${\xi }_{\pi }∕{\xi }_{\sigma }$. (c) Angle-resolved LDOS for quasiparticles with momentum along the $x$ axis at positions along the $y$ axis in the $\sigma$ band, showing the bound states as a function of $y$. The dispersion of the bound states as a function of $y$ is also shown in (d) for $\rho =0.5$, ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, and $T=0.5T_c$. Spectra in (a)–(c) are shifted in a vertical direction for convenience.

Figure 4

(Color online) Contribution to the current density as a function of distance from the vortex center, shown separately for the $\pi$ band and the $\sigma$ band for fixed mixing ratio $\rho =0.3$ and different temperatures $T$; for (a) ${\xi }_{\pi }∕{\xi }_{\sigma }=1$, (b) ${\xi }_{\pi }∕{\xi }_{\sigma }=3$, and (c) ${\xi }_{\pi }∕{\xi }_{\sigma }=5$. In (d) the $\sigma$-band contribution to the total current density is shown as a function of radial coordinate, for $T=0.1T_c$, $\rho =0.3$, and for different values of ${\xi }_{\pi }∕{\xi }_{\sigma }$.