Crossband versus intraband pairing in superconductors: Signatures and consequences of the interplay

We analyze the paradigmatic competition between intraband and crossband Cooper-pair formation in two-band superconductors, neglected in most works to date. We derive the phase-sensitive gap equations and describe the crossover between the intraband-dominated and the crossband-dominated regimes, delimited by a “gapless” state. Experimental signatures of crosspairing comprise notable gap splitting in the excitation spectrum, non-BCS behavior of gaps versus temperature, as well as changes in the pairing symmetry as a function of temperature. The consequences of these findings are illustrated on the examples of ${\mathrm{MgB}}_2$ and ${\mathrm{Ba}}_{0.6}{\mathrm{K}}_{0.4}{\mathrm{Fe}}_2{\mathrm{As}}_2$.

Figure 1

The relevance of crosspairing is illustrated based on the band structure of (a) bulk ${\mathrm{MgB}}_2$ [37] and (b) 6-monolayer ${\mathrm{MgB}}_2$ [38]. Only $\sigma$ bands close to the $\mathrm{\Gamma }$ point of the Brillouin zone are shown, with chemical potential $\mu =500$ meV and energy scale of the pairing $\mathrm{\Omega }=75$ meV. In (b), each interior monolayer contributes a pair of holelike bands ${\sigma }_1$ and ${\sigma }_2$, and the surface band is denoted by $S^{\prime }$. The (purple) overlapping shadows project the momentum states where crossband pairing between opposite momenta states among the $\sigma$ bands is feasible.

Figure 2

Effect of crosspairing for the same phase. (a) Excitation gaps with their (b) corresponding pair amplitudes as a function of ${\lambda }_{33}$ at $T=4.2$ K. The three miniplots above (a) show the density of states for ${\lambda }_{33}=0.1$, 0.15, and 0.2, illustrating the behavior in the intraband-dominated regime, gapless state, and the crosspairing dominated regime, respectively. (c) Mean-field critical temperature vs ${\lambda }_{33}$.

Figure 3

Energy dispersions corresponding to Fig. 2, for ${\lambda }_{33}$ increased towards the gapless state.

Figure 4

Superconducting gaps of bulk ${\mathrm{MgB}}_2$ as a function of temperature, for intraband pairing only (solid lines), and in the presence of weak crosspairing (dashed lines).

Figure 5

(a) Excitation gaps ${\mathrm{\Delta }}_{\pm }$, and (b) real part of the pair amplitudes ${\mathrm{\Delta }}_i^R$ as a function of temperature, for parameters of $(\alpha ,\beta )$ bands in $\left(\mathrm{BaK}\right){\mathrm{Fe}}_2{\mathrm{As}}_2$, with nominal $s_{\pm }$ antiphase and in the presence of crosspairing. Data points with error bars are the experimentally measured gaps of ${\mathrm{Ba}}_{0.6}{\mathrm{K}}_{0.4}{\mathrm{Fe}}_2{\mathrm{As}}_2$ [16]. For microscopic parameters reproducing the experimentally measured gaps at zero temperature (see text), an $s_{\pm }\to s_{++}$ transition is found at 36 K, corresponding to the critical temperature of the measured gaps at the $\mathrm{\Gamma }$ point of ${\mathrm{Ba}}_{0.6}{\mathrm{K}}_{0.4}{\mathrm{Fe}}_2{\mathrm{As}}_2$ [16].