Collective modes in two-band superconductors in the dirty limit

We analyze collective modes in two-band superconductors in the dirty limit. It is shown that these modes exist at all temperatures $T$ below $T_c$ provided the frequency of the modes is higher than the inelastic scattering rate and lower than the energy gaps ${\Delta }_{a,b}$. At low temperatures, these modes are related to counterphase oscillations of the condensate currents in each band. The spectrum of the collective oscillations is similar to the spectrum of the Josephson “plasma” modes in a tunnel Josephson junction but the velocity of the mode propagation in the case under consideration is much lower. At higher temperatures $({\Delta }_b<T<T_c)$, the spectrum consists of two branches. One of them is gapless (soundlike) and the second one has a threshhold that depends on coupling between the bands. We formulate the conditions under which both types of collective modes can exist. The spectrum of the collective modes can be determined by measuring the $I\text{−}V$ characteristics of a Josephson junction in a way as it was done by Carlson and Goldman [Phys. Rev. Lett. 34, 11 (1975)].

Figure 1

(a) Calculated CM spectrum for the case of low temperatures $(T⪡{\Delta }_{a,b})$. There is a gap ${\Omega }_0$ in the CM spectrum. $(k_o={\Omega }_0∕v_{\mathrm{CM}})$. (b) The CM spectrum for high temperatures $(T⪢{\Delta }_{a,b})$. There are two branches of the CMs: the soundlike mode and the mode with a gap (analogous to the Leggett mode). The soundlike mode exists for frequencies greater than the inverse energy relaxation time: $\Omega >\gamma$.

Figure 2

CM spectrum at intermediate temperatures $({\Delta }_b<T<{\Delta }_a)$ for two cases: (a) $\Omega ⪢{\Omega }_{a,b}$ and (b) $\Omega ⪡{\Omega }_{a,b}$. There are two branches of the CM spectrum in both limiting cases: the soundlike mode and the mode with a gap. The frequency and wave vector of CMs are normalized to ${\Omega }_{*}$ and $k_{*}$ equal to (a) ${\Omega }_{*}=E_1(T=8{\Delta }_b)$, $k_{*}=E_1∕v_1$ and (b) ${\Omega }_{*}=E_1(T=2{\Delta }_b)$, $k_{*}=E_1∕v_1$. With increasing temperature, the two modes are getting closer to each other.

Figure 3

Corrections to the $I\text{−}V$ characteristics of a Josephson tunnel junction due to CMs in the two-band superconductor. The Josephson junction consists of one-band and two-band superconductors. At voltages $V$ satisfying the equation ${\Omega }_V^2={\Omega }_0^2+{\left(K_H{v}_{\mathit{CM}}\right)}^2$, a spike arises on the $I\text{−}V$ characteristics. The position of the spike depends on an applied magnetic field $H$. The current is normalized to the value $j_0=\gamma (j_{Ja}-j_{Jb})(E_{Ja}^2∕{\overline{\nu }}_a-E_{Jb}^2∕{\overline{\nu }}_b)$.