Leggett modes and multiband superconductivity in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ is a prototypical multiband superconductor with three bands crossing the Fermi level. These bands exhibit distinct dimensional characteristics, with one quasi-2D $\gamma$ band and two quasi-1D $\alpha$ and $\beta$ bands. This leads to the expectation that the superconductivity on the $\gamma$ band may be only weakly Josephson-coupled to that on the other two bands. Based on an explicit microscopic weak coupling calculation appropriate for ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, we study the collective Leggett modes associated with the relative phase oscillations between the bands and show that a relatively soft Leggett mode exists due to the comparatively weaker interband Josephson coupling. These calculations also provide insight into why the superconducting gap magnitudes may be comparable on all three bands, despite the noticeable differences between the $\gamma$ and $\alpha /\beta$ bands. The analyses can be readily applied to other multiband superconductors.

Figure 1

The three Fermi surfaces of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ in the $k_z=0$ plane. The $c$-axis dispersion is small and is ignored in our calculations.

Figure 2

Typical low-frequency electronic Raman response in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ in the $A_{1g}$ channel in the cases of two different gap ratios between the 1D and 2D bands: (a) $|{\mathrm{\Delta }}_{0\gamma }|\simeq 2.8|{\mathrm{\Delta }}_{0\alpha ,\beta }|$ and (b) $|{\mathrm{\Delta }}_{0\gamma }|\simeq |{\mathrm{\Delta }}_{0\alpha ,\beta }|$. The continuum contribution (black dashed), exhibiting nonvanishing intensity below $\omega =2{\mathrm{\Delta }}_0$, is evaluated using an anisotropic three-band chiral $p$-wave model. The gap anisotropy resembles the one obtained in Ref. [15]. A small imaginary part $\tau =0.004{\mathrm{\Delta }}_0$ is used in the analytic continuation of the Raman susceptibility for regularization. The Leggett mode contribution is shown in red (solid). Here we have simplified the calculation by using a broadening of ${\tau }_L=0.5{\mathrm{\Delta }}_0$ to model the damping due to the coupling with the low-lying quasiparticle states. This does not alter the Leggett mode peak position [62].