Mixed State of a Dirty Two-Band Superconductor: Application to ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$

We investigate the vortex state in a two-band superconductor with strong intraband and weak interband electronic scattering rates. Coupled Usadel equations are solved numerically, and the distributions of the pair potentials and local densities of states are calculated for two bands at different values of magnetic fields. The existence of two distinct length scales corresponding to different bands is demonstrated. The results provide qualitative interpretation of recent scanning tunneling microscopy experiments on vortex structure imaging in ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$.

Figure 1

Local density of state for $\pi$ band at different points of vortex lattice unit cell at $h=0.1$ for ${\mathcal{D}}_1={\mathcal{D}}_2=1$. Inset illustrates the circular cell approximation and shows points at which the spectra are presented.

Figure 2

Spatial dependencies of pair potentials [(a) and (b)] and partial DOS at $E=0$ [(c) and (d)] for isolated vortex for two ratios $D_1/D_2$: $D_1=0.2D_2$ and $D_1=D_2$.

Figure 3

Field dependencies of maximum pair potentials [(a) and (b)] and averaged DOS at $ϵ=0$ [(c) and (d)] for two ratios $D_1/D_2$: $D_1=0.2D_2$ and $D_1=D_2$.

Figure 4

Ratio of the apparent vortex size ${\xi }_v$ as defined in Fig. 2c to the coherence length ${\xi }_{c2}$ extracted from the upper critical field, ${\xi }_{c2}\equiv \sqrt{{\Phi }_0/(2\pi H_{c2})}$ plotted vs ratio of the length scales in two bands, ${\xi }_2/{\xi }_1\equiv \sqrt{{\mathcal{D}}_2/{\mathcal{D}}_1}$. According to Ref. 24 for ${\mathrm{M}\mathrm{g}\mathrm{B}}_2$ single crystal the ratio ${\xi }_v/{\xi }_{c2}$ is around 3.