Josephson effect between a two-band superconductor with $s++$ or $s\pm$ pairing symmetry and a conventional $s$-wave superconductor

In this work, we investigate the Josephson effect between a two-band superconductor either with the $s++$ (two energy gaps have the same sign and are fully gapped) pairing symmetry or $s\pm$ (two energy gaps have $\pi$ phase difference and are fully gapped) pairing symmetry and a conventional $s$-wave superconductor. The ground state, critical current, plasma modes, flux-flow dynamics, response to external ac electric field, and possible soliton solutions are investigated. For junctions with the charge neutrality breaking, we find a new plasma mode for junctions, which gives rise to new resonance peaks in the Josephson flux-flow region. Because of the frustrated interaction in junctions with $s\pm$ pairing symmetry, time-reversal symmetry (TRS) can be broken if the frustration is optimized. In the TRS broken (TRSB) state, there is a nontrivial phase difference between the two Josephson tunneling channels, which results in a nontrivial interference. Furthermore, we find a massless plasma mode at the TRSB transition for junctions with the charge neutrality breaking. In the TRSB state, a spontaneous magnetic flux appears where there is a spatial inhomogeneity in the Josephson coupling, thus providing a possible smoking-gun evidence for the underlying pairing symmetry.

Figure 1

(a) Josephson junctions between two-band superconductors with the $s++$ pairing symmetry and $s$-wave single-band superconductors. The interaction between condensates is attractive and they have the same phase in the ground state as shown in (b). (c) Josephson junctions between two-band superconductors with the $s\pm$ pairing symmetry and $s$-wave single-band superconductors. The interaction between condensates ${\theta }_1$ and ${\theta }_2$ is repulsive. Under appropriate conditions, the system is strongly frustrated, resulting in TRSB, and the ground state is depicted in (d).

Figure 2

(a) Phase diagram of Josephson junctions with $s\pm$ pairing symmetry. Here, $J_{s2}=J_{s1}=1$. For $J_{12}<-0.5$, TRSB occurs and the transition is continuous. There are two degenerate ground states $\widehat{\phi }=({\phi }_{s1},{\phi }_{s2})$ and $-\widehat{\phi }$. Only one configuration of phases in the ground state with TRSB is shown in the figure. (b) Energy landscape in the TRSB state with $J_{12}=-1$ and $J_{s2}=1$. There are many degenerate and separated energy minima (blue region), $\widehat{\phi }$, $-\widehat{\phi }$ and other minima obtained by shifting phases by $2n\pi$ with an integer $n$. To contrast the minima, we plot $\ln (E+1.6)$.

Figure 3

Dependence of critical current on magnetic fields in three different regimes: $J_{12}>0$ with ${\phi }_{s1}={\phi }_{s2}$, $-J_{12}\gg J_{si}$ with ${\phi }_{s1}={\phi }_{s2}+\pi$, and the TRSB state.

Figure 4

Dispersion of two Josephson plasma modes according to Eqs. (45) and (46). We use $\zeta =1$, $\alpha =0.05$, and ${\epsilon }_b=1$ in the plot. Inset is the energy gap ${\omega }_1(k=0)$ and ${\omega }_2(k=0)$ as a function of $J_{12}$. The branch ${\omega }_2\left(k\right)$ becomes massless at the TRSB transition.

Figure 5

(a) $IV$ curve in the flux-flow region. The resonance peaks shown in the figure are caused by the excitation of cavity modes (Fiske resonance). Here, we only plot the resonance mode $k_m=\pi /L$ with $L=0.2$. We use $\zeta =1$, $\alpha =0.05$, ${\epsilon }_b=1$, and $B_a=\pi$. The Ohmic contribution in the figure is ${\beta }_d\omega$ with ${\beta }_d=0.01$. (b) Eck resonance in the flux-flow region. We use $\zeta =1$, $\alpha =0.05$, ${\epsilon }_b=1$, $B_a=\pi$, and ${\beta }_d=0.05$.

Figure 6

(a) Profile of phases and magnetic field in (a) a conventional $2\pi$ solition, where ${\phi }_{s1}$ changes by $2\pi$, (b) soliton between two TRSB pair states. Here, ${\zeta }_1=10$, ${\zeta }_2=24$, ${\zeta }_s=20$, $J_{s1}=1.0$, $J_{s2}=1.5$. $J_{12}=1.0$ in (a) and $J_{12}=-1.0$ in (b).

Figure 7

Profile of the phases and magnetic field with a step modulation of the Josephson coupling in a junction with $s\pm$ pairing symmetry. Here, ${\phi }_{s2}={\phi }_{s1}+\pi$ and ${\lambda }_e=J^{\prime }=1$.

Figure 8

(a) Experimental proposal to detect the pairing symmetry in multiband superconductors. (b) Grain-boundary Josephson junctions in multiband superconductors.

Figure 9

Profile of the phases and magnetic field with a defect in the junction with $s\pm$ pairing symmetry obtained from Eqs. (88, 89, 90). The defect is modeled as inhomogeneous Josephson coupling in the junction. (a) and (b) correspond to two TRSB pair states. Inset is the phase configuration in the ground state.

Figure 10

Profile of the phases and magnetic field with a Gaussian-type defect $J_{s2}\left(x\right)=1-0.4\exp [-{(x-L/2)}^2]$ in the Josephson coupling in a junction with $s\pm$ pairing symmetry obtained by numerical calculations. (a) and (b) correspond to two TRSB pair states. Inset is the phase configuration in the ground state. For clarity, the magnetic field is amplified by 20 times.