Electronic specific heat of two-band layered superconductors: Analysis within the generalized two-band $\alpha$ model

The generalized two-band α model of superconductivity is consistently developed by taking into account the interband electron-phonon interaction. The model is applied to calculate the electronic contribution to the specific heat, superconducting gaps, and electron-phonon coupling constants of the multiband layered superconductors MgB${}_2$ and Ba(Fe${}_{0.925}$Co${}_{0.075}$)${}_2$As${}_2$. For both compounds, the energy gaps and electron-phonon coupling constants are determined, and the temperature behavior of the specific heat is obtained that describes the experimental data well. It is shown that the well-known two-band α model that assumes independent bands is formally valid only in the case of relatively strong interband scattering in the band with a smaller gap.

Figure 1

Dependence of reduced gaps ${\delta }_2$($t$) = Δ${}_2$($t$)/Δ${}_2$(0) on the reduced temperature $t$ = $T$/$T_c$ calculated with $\theta$(0) = 0.275, ${\alpha }_1$ = 1.78 for two superconductors with different interband couplings, (a) ${\tilde{\lambda }}_{12}$ = 0.05 and ${\tilde{\lambda }}_{21}$ = 0.1 (filled symbols) and (b) ${\tilde{\lambda }}_{12}$ = 0.22 and ${\tilde{\lambda }}_{21}$ = 0.8 (open symbols). Dashed line shows the ${\delta }_0$($t$) dependence calculated according to the two-band α model that does not consider the interband interaction. [In the two cases, the reduced gaps ${\delta }_1$($t$) practically coincide with ${\delta }_0$($t$) and are not shown.]

Figure 2

Temperature dependence of normalized electronic specific heat of the band with smaller gap calculated with $\theta$(0) = 0.275, ${\alpha }_1$ = 1.78 for two superconductors with different interband coupling, (a) ${\tilde{\lambda }}_{12}$ = 0.05 and ${\tilde{\lambda }}_{21}$ = 0.1 (filled symbols) and (b) ${\tilde{\lambda }}_{12}$ = 0.22 and ${\tilde{\lambda }}_{21}$ = 0.8 (open symbols). Dashed line shows calculation according to the two-band α model that does not consider the interband interaction.

Figure 3

Temperature dependence ($t$ = $T$/$T_c$)of normalized electronic specific heat calculated with $\theta$(0) = 0.275, ${\alpha }_1$ = 1.78 for two superconductors with different interband couplings, (a) ${\tilde{\lambda }}_{12}$ = 0.05 and ${\tilde{\lambda }}_{21}$ = 0.1 (filled symbols) and (b) ${\tilde{\lambda }}_{12}$ = 0.22 and ${\tilde{\lambda }}_{21}$ = 0.8 (open symbols). Dashed line corresponds to calculations according to the two-band α model with ${\gamma }_1$/${\gamma }_n$ = 0.55 and ${\gamma }_2$/${\gamma }_n$ = 0.45.

Figure 4

Temperature dependence of the gap 2Δ${}_{\pi }$ (filled symbols) and of the Leggett mode energy Ω${}_L$ (empty symbols) of Mg${}_{1-x}$Al${}_x$B${}_2$ with $T_c$ = 32 K (Ref. 13). Thick and thin solid lines are calculations within the α${}^{\prime }$ approximation of the gap and Leggett mode energies, respectively. Lines with filled and empty square symbols are calculations according to the α approximation of the gap and Leggett mode energies, respectively. Grey line shows calculation according to α approximation.

Figure 5

Temperature dependence of normalized superconducting condensate density of Ba(Fe${}_{0.9}$Co${}_{0.1}$)${}_2$As${}_2$ (Ref. 20) (symbols). Lines correspond to calculations according to the α${}^{\prime }$ approximation with ${\tilde{\lambda }}_{12}\sim {\tilde{\lambda }}_{21}\sim 0.1$ (thick), α approximation (thin) and two-band α model (grey).

Figure 6

Temperature dependence of electronic specific heat of Mg${}^{11}$B${}_2$ (Ref. 2) (filled symbols). Grey line shows calculations according to the two-band α model; open symbols represent calculation according to the α${}^{\prime }$ approximation with parameters given in Table .

Figure 7

Temperature dependence of electronic specific heat of Ba(Fe${}_{0.925}$Co${}_{0.075}$)${}_2$As${}_2$ (Ref. 3) (filled symbols). Grey line shows calculations according to the two-band α model; open symbols represent calculations according to the α${}^{\prime }$ approximation with parameters given in Table .