Interband superconductivity: Contrasts between Bardeen-Cooper-Schrieffer and Eliashberg theories

The recently discovered iron pnictide superconductors apparently present an unusual case of interband-channel pairing superconductivity. Here we show that in the limit where the pairing occurs within the interband channel, several surprising effects occur quite naturally and generally: different density of states on the two bands leads to several unusual properties, including a gap ratio which behaves inversely to the ratio of density of states; the weak-coupling limits of the Eliashberg and the BCS theories, commonly taken as equivalent, in fact predict qualitatively different dependence of the ${\Delta }_1/{\Delta }_2$ and $\Delta /T_c$ ratios on coupling constants. We show analytically that these effects follow directly from the interband character of superconductivity. Our results show that in the interband-only pairing model the maximal gap ratio is $\sqrt{N_2/N_1}$ as strong-coupling effects act only to reduce this ratio. Our results show that pnictide BCS calculations must use renormalized coupling constants to get accurate results. Our results also suggest that if the large experimentally reported gap ratios (up to a factor 2) are correct, the pairing mechanism must include more intraband interaction than what is usually assumed.

Figure 1

(Color online) The ratio of the gap functions in an interband-pairing case, as a function of ${\lambda }_{\text{eff}}$, for the BCS (dashed line) and Eliashberg-Einstein (line) spectrum and spin-fluctuation (triangle) spectrum cases. The dotted line represents numerical Eliashberg-Einstein spectrum results in which the mass-renormalization parameter has been artificially taken as 1, showing that the difference between BCS and Eliashberg is mainly a mass-renormalization effect. The dashed-dotted line (lower part, main panel, and upper inset) is the renormalized BCS (RBCS) (Ref. 8) approximation to the Eliashberg numerical results. Inset: analytic approximations to numerical results; diamonds are BCS, Eq. (5), and circles are Eliashberg, Eq. (15).

Figure 2

(Color online) $\Delta \left(T=0\right)/T_c$ ratios are shown as a function of the overall coupling constant.

Figure 3

(Color online). (Left) The behavior of the Eliashberg $\Delta \left(0\right)/T_c$ ratios as a function of the ratio of coupling constants. (Right) The behavior of $T_c$ in this case. For both cases ${\lambda }_{\text{eff}}$ is fixed at 1.