Green's-function theory of dirty two-band superconductivity

We study the effects of random nonmagnetic impurities on the superconducting transition temperature $T_c$ in a two-band superconductor, where we assume an equal-time spin-singlet $s$-wave pair potential in each conduction band and the hybridization between the two bands as well as the band asymmetry. In the clean limit, the phase of hybridization determines the stability of two states, called $s_{++}$ and $s_{+-}$. The interband impurity scatterings decrease $T_c$ of the two states exactly in the same manner when time-reversal symmetry is preserved in the Hamiltonian. We find that a superconductor with larger hybridization shows more moderate suppression of $T_c$. This effect can be explained by the presence of odd-frequency Cooper pairs, which are generated by the band hybridization in the clean limit and are broken by impurities.

Figure 1

(a) The Fermi surfaces of the two bands are illustrated on two-dimensional momentum space, where 1 and 2 denote the circular Fermi surface of the first band and that of the second band, respectively. (b) The matrix elements in Eq. (2), which connect the particle state at the first band $|1e\rangle$ with the hole state at the second band $|2h\rangle$.

Figure 2

The superconducting transition temperature in a two-band superconductor is plotted as a function of ${\xi }_0/\ell$ for $g_2/g_1=0.5$ in (a) and $g_2/g_1=0.1$ in (b). The vertical axis is normalized to the transition temperature in the clean limit $T_0$. We fix the band asymmetry at $\gamma /\left(2\pi T_0\right)=10$ and the ratio of $g_{12}/g_2=0.2$ in both (a) and (b).

Figure 3

The superconducting transition temperature in a two-band superconductor is plotted as a function of ${\xi }_0/\ell$ for several choices of $g_2/g_1$. We choose $v=T_0$ in (a) and $v=10T_0$ in (b). We fix the ratio of $g_{12}/g_1=0.02$ and $\gamma =10T_0$ in both (a) and (b).

Figure 4

The transition temperature ($T_c$) and the pair potentials (${\overline{\mathrm{\Delta }}}_1, {\overline{\mathrm{\Delta }}}_2$) for $v=\gamma =0, g_2=0.8g_1$ and $|g_{12}|=0.05g_1$. The pair potentials are calculated at $T=0.5T_c$, where ${\mathrm{\Delta }}_{1c}$ is the amplitude of ${\overline{\mathrm{\Delta }}}_1$ in the clean limit. We introduce the phases of two potentials as $g_{12}=\left|g_{12}\right|e^{i{\theta }_g}$ and that at $V_{\mathrm{imp}}{e}^{i{\theta }_{\mathrm{imp}}}$. Time-reversal symmetry is preserved at ${\theta }_g={\theta }_{\mathrm{imp}}$ in (a) and (c), whereas it is broken for ${\theta }_g\ne {\theta }_{\mathrm{imp}}$ in (b) and (d).