Disorder-induced transition between $s_{\pm }$ and $s_{++}$ states in two-band superconductors

We have reexamined the problem of disorder in two-band superconductors, and shown, within the framework of the $T$-matrix approximation, that the suppression of $T_c$ can be described by a single parameter depending on the intraband and interband impurity scattering rates. $T_c$ is shown to be more robust against nonmagnetic impurities than would be predicted in the trivial extension of Abrikosov-Gor'kov theory. We find a disorder-induced transition from the $s_{\pm }$ state to a gapless and then to a fully gapped $s_{++}$ state, controlled by a single parameter—the sign of the average coupling constant $\langle \lambda \rangle$. We argue that this transition has strong implications for experiments.

Figure 1

Critical temperature for various $\tilde{\sigma }$ and $\eta$ as a function of the effective interband scattering rate ${\tilde{\Gamma }}_{ab}$ for the same parameters. Note that curves for different sets of $\tilde{\sigma }$ and $\eta$ overlap and fall onto one of the three universal curves depending on the $\langle \lambda \rangle$. $N_b/N_a=2$, coupling constants for illustrative purpose are chosen for $\langle \lambda \rangle >0$ as $({\lambda }_{aa},{\lambda }_{ab},{\lambda }_{ba},{\lambda }_{bb})=(3,-0.2,-0.1,0.5)$, for $\langle \lambda \rangle =0$ as $(2,-2,-1,1)$, and for $\langle \lambda \rangle <0$ as $(1,-2,-1,1)$.

Figure 2

Critical temperature $T_c$ and ${\tilde{\Delta }}_{\alpha n}$ (in units of $T_{c0}$) for first, second, and third Matsubara frequencies $n$ at $T=0.04T_{c0}$, $\tilde{\sigma }=0.5$, $\eta =1$, and $N_b/N_a=2$ and $\langle \lambda \rangle >0$. The coupling constants are chosen as in Fig. 1.

Figure 3

(a) Density of states $N_{\text{tot}}\left(\omega \right)/N$ vs ${\Gamma }_a/T_{c0}$ and $\omega /T_{c0}$ for $\langle \lambda \rangle >0$ and impurity parameters $\tilde{\sigma }=0.5$, $\eta =1$, $N_b/N_a=2$, $N=N_a+N_b$. (b) Total superfluid density $1/{\left({\omega }_p{\lambda }_L\right)}^2$ vs ${\Gamma }_a/T_{c0}$ and $T/T_{c0}$, where ${\omega }_p$ is the total plasma frequency and ${\lambda }_L$ is the London penetration depth.