Quasiparticle interference in multiband superconductors with strong coupling

We develop a theory of the quasiparticle interference (QPI) in multiband superconductors based on the strong-coupling Eliashberg approach within the Born approximation. In the framework of this theory, we study dependencies of the QPI response function in the multiband superconductors with the nodeless $s$-wave superconductive order parameter. We pay special attention to the difference in the quasiparticle scattering between the bands having the same and opposite signs of the order parameter. We show that at the momentum values close to the momentum transfer between two bands, the energy dependence of the quasiparticle interference response function has three singularities. Two of these correspond to the values of the gap functions and the third one depends on both the gaps and the transfer momentum. We argue that only the singularity near the smallest band gap may be used as a universal tool to distinguish between the $s_{++}$ and $s_{\pm }$ order parameters. The robustness of the sign of the response function peak near the smaller gap value, irrespective of the change in parameters, in both the symmetry cases is a promising feature that can be harnessed experimentally.

Figure 1

Density of states for the bands $a$ and $b$ calculated in strong coupling at various temperatures. The coupling constants are ${\lambda }_{bb}=0.5$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.1$, ${\lambda }_{aa}=3$. The superconducting critical temperature is $T_c=28{\text{cm}}^{-1}$. The DOS is normalized with respect to the normal state and is set equal to 1 for each band.

Figure 2

The response function $I(\omega$) for the $s_{++}$ and $s_{\pm }$ case with the strong-coupling $\lambda$ matrix defined as (${\lambda }_{aa}=3$, ${\lambda }_{ab}=\pm 0.2$, ${\lambda }_{ba}=\pm 0.1$, ${\lambda }_{bb}=0.5$) and $T_c=28{\text{cm}}^{-1}$. Here “+” is for the $s_{++}$ case and “−” for the $s_{+-}$ case. Below we will put only the absolute values for the coupling constants.

Figure 3

3D plots of the response function vs temperature at fixed energy $\omega$ for $s_{++}$ (upper) and $s_{\pm }$ (lower) cases, respectively. The coupling parameters are the same as used above.

Figure 4

The QPI response function for the $s_{++}$ and $s_{\pm }$ case at very strong couplings $\tilde{\lambda }$, i.e., ${\lambda }_{aa}=6$, ${\lambda }_{bb}=1$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.4$ with the transition temperature $T_c=46{\text{cm}}^{-1}$.

Figure 5

2D plots of the QPI response function $I(\omega$) vs $\omega$ and the momentum $\mathbf{q}$ for the strong-coupling case with $\epsilon =\delta \mu =0$ at temperature $T=1{\mathrm{cm}}^{-1}$. The values of the coupling constants are ${\lambda }_{aa}=3$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.1$, ${\lambda }_{bb}=0.5$.

Figure 6

3D plots of the QPI response function $I(\omega$) vs $\omega$ and momentum $\mathbf{q}$ with the zero ellipticity $\epsilon =0$ and zero shifted Fermi surface energy $\delta \mu =0$ for the strong-coupling case at temperature $T=1{\mathrm{cm}}^{-1}$. The values of the coupling constants are ${\lambda }_{aa}=3$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.1$, ${\lambda }_{bb}=0.5$.

Figure 7

2D plots of the QPI response function $I(\omega$) vs $\omega$ and momentum $\mathbf{q}$ for the strong-coupling case, with ${\mathrm{\Delta }}_a={\mathrm{\Delta }}_b$, and $\epsilon =\delta \mu =0$ at temperature $T=1{\mathrm{cm}}^{-1}$, and with the coupling constants given as ${\lambda }_{aa}=3$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.1$, ${\lambda }_{bb}=0.5$.

Figure 8

2D plots of the QPI response function $I(\omega$) vs $\omega$, momentum $\mathbf{q}$ and the shifted Fermi surface energy $\delta \mu$ for the strong-coupling case at temperature $T=1{\mathrm{cm}}^{-1}$. In the inset, the dependence of $\left|I\right(\mathbf{q}-\mathbf{Q}\left)\right|$ is shown at fixed energy close to the smaller gap, i.e., $\omega \approx 18{\text{cm}}^{-1}$. The values of the coupling constants are ${\lambda }_{aa}=3$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.1$, ${\lambda }_{bb}=0.5$.

Figure 9

2D plots of the QPI response function $I(\omega$) vs $\omega$ and the shifted Fermi surface energy $\delta \mu$ for the strong-coupling case at temperature $T=1{\mathrm{cm}}^{-1}$ at ellipticity $\epsilon$ and the momentum $\tilde{q}=0$. The values of the coupling constants are ${\lambda }_{aa}=3$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.1$, ${\lambda }_{bb}=0.5$.

Figure 10

3D plots of the QPI response function $I(\omega ,\delta \mu$) vs $\omega$ and the nonzero Fermi surface energy $\delta \mu$ at zero ellipticity for the strong-coupling case at temperature $T=1{\mathrm{cm}}^{-1}$ for $s_{++}$ and $s_{\pm }$. There is a large amplitude for the response function in the region $\delta \mu =[100,200]$ for the latter case. The values of the coupling constants are ${\lambda }_{aa}=3$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.1$, ${\lambda }_{bb}=0.5$.

Figure 11

2D plots of the QPI response function $I(\omega$) vs $\omega$ and the ellipticity $\epsilon$ for the strong-coupling case with value of shifted Fermi surface energy $\delta \mu$ and the momentum $\tilde{q}=0$, at temperature $T=1{\mathrm{cm}}^{-1}$. The values of the coupling constants are ${\lambda }_{aa}=3$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.1$, ${\lambda }_{bb}=0.5$.

Figure 12

3D plots of the QPI response function $I(\omega$) vs $\omega$ and the ellipticity $\epsilon$ for the strong-coupling case with value of shifted Fermi surface energy $\delta \mu$ and the momentum $\tilde{q}=0$, at temperature $T=1{\mathrm{cm}}^{-1}$. The values of the coupling constants are ${\lambda }_{aa}=3$, ${\lambda }_{ab}=0.2$, ${\lambda }_{ba}=0.1$, ${\lambda }_{bb}=0.5$.