Theory of fluctuations in a two-band superconductor: ${\mathrm{MgB}}_2$

A theory of fluctuations in two-band superconductor ${\mathrm{MgB}}_2$ is developed. Since the standard Ginzburg-Landau (GL) approach fails in description of its properties, we generalize it based on the microscopic theory of a two-band superconductor. Calculating the microscopic fluctuation propagator, we build up the nonlocal two-band GL functional and the corresponding time-dependent GL equations. This allows us to calculate the main fluctuation observables such as fluctuation specific heat and conductivity.

Figure 1

Upper part: Graphic representation of Dyson equation (4) for the fluctuation propagator $L_{\alpha \beta }$ (wavy line). The Greek letters indicate the band indices. The black dot represents the coupling-constants matrix $-g_{\alpha \beta }$, the loop represents the polarization operator ${\Pi }_{\gamma \delta }$, and the shaded triangle represents the scattering vertex $C_{\gamma \gamma }$. Without interband scattering the polarization operator is diagonal ${\Pi }_{\gamma \delta }={\Pi }_{\gamma \gamma }{\delta }_{\gamma \delta }$ Lower part: Equation for the scattering vertex $C_{\alpha \alpha }$. The dashed line is the impurity scattering line $1∕2\pi {\nu }_{\alpha }{\tau }_{\alpha }$. We omit the band index, because without interband scattering this equation is diagonal with respect to the band indices.

Figure 2

(Color online) Temperature dependencies of the paraconductivity components (solid lines). Left panel shows the in-plane component in units of $e^2∕\left(32\hslash {\xi }_{z,1}\right)$ [function $\kappa \left(ϵ\right)∕\sqrt{ϵ}$, see Eqs. (40, 43)]. This curve also gives the fluctuation specific heat in units $1∕\left(8\pi {\xi }_{1,x}^2{\xi }_{1,z}\right)$ [see Eq. (39)]. Right panel shows the $z$-axis component in units of $e^2{\xi }_{z,1}∕\left(32\hslash {\xi }_{x,1}^2\right)$ calculated from Eq. (44). Both components were calculated using parameters $S_{12}=0.035$, $\tilde{\theta }=0.377$, and $r=300$. For comparison we also show the Ginzburg-Landau asymptotics (dotted lines) and $\sigma$-band contribution to paraconductivity (dashed lines). One can see that the in-plane component diverges more slowly and the $z$-axis component diverges more quickly than the paraconductivity in a single-band anisotropic superconductor. In the ${\sigma }_z$ plot we indicate three characteristic regions with different functional dependencies of fluctuation parameters.