Quantum transport in ballistic $s_{\pm }$-wave superconductors with interband coupling: Conductance spectra, crossed Andreev reflection, and Josephson current

We study quantum transport in ballistic $s_{\pm }$-wave superconductors where coupling between the two bands is included and apply our model to three possible probes for detecting the internal phase shift of such a pairing state: tunneling spectroscopy in a $\text{N}\mid s_{\pm }$-wave junction, crossed Andreev reflection in a two-lead $\text{N}\mid s_{\pm }\text{-wave}\mid \text{N}$ system, and Josephson current in a $s\text{-wave}\left|\text{I}\right|s_{\pm }$-wave Josephson junction. Whereas the first two probes are insensitive to the superconducting phase in the absence of interband coupling, the Josephson effect is intrinsically phase dependent and is moreover shown to be relatively insensitive to the strength of the interband coupling. Focusing on the Josephson current, we find a $0\text{−}\pi$ transition as a function of the ratio of effective barrier transparency for the two bands, as well as a similar phase-shift effect as a function of temperature. An essential feature of this $s_{\pm }$-wave model is nonsinusoidality of the current-phase relation and we compute the dependence of the critical current on an external magnetic field, showing how this feature may be experimentally observable for this system. We also comment on the possible experimental detection of the phase-shift effects in $s_{\pm }$-wave superconductors.

Figure 1

(Color online) Schematic drawing of the systems under consideration in this work: (a) the model of $\text{N}\mid s_{\pm }$-wave junction for tunneling spectroscopy as studied in Sec. 3A, (b) the model of the two-lead $\text{N}\mid s_{\pm }\text{-wave}\mid \text{N}$ junction for the study of crossed Andreev reflection in Sec. 3B, and (c) the model of the $s\text{-wave}\left|\text{I}\right|s_{\pm }$-wave Josephson junction considered in Sec. 3C. For system (b), we have illustrated how an electron in the left-hand lead is converted to a hole in the right-hand lead by the formation (together with a electron from the right-hand lead) of a Cooper pair in the superconducting interlayer.

Figure 2

(Color online) Comparison of the probabilities of the reflection processes in a $\text{N}\mid s_{\pm }$-wave junction and a two-band $\text{N}\mid s$-wave junction as described in the text. We have chosen zero barrier strength, $Z=0$, and gap ratio $r_{\Delta }=1.5$. We have used a value $\tilde{\alpha }=1$ for the interband coupling (for the $s$-wave and $s_{\pm }$-wave cases) while for the decoupled case we have $\tilde{\alpha }=0$.

Figure 3

(Color online) (a) Conductance for a $\text{N}\mid s_{\pm }$-wave junction and (b) a two-band $\text{N}\mid s$-wave junction for various strengths of interband coupling $\tilde{\alpha }$ normalized on its normal-state value, where we have set $Z=1$ and $r_{\Delta }=1.5$.

Figure 4

(Color online) Nonlocal conductance through a $\text{N}\mid s_{\pm }\text{-wave}\mid \text{N}$ junction for a relatively thin superconducting interlayer, $L=2\times {10}^4{k}_{\text{F}}^{-1}$, and for $r_{\Delta }=1.5$. The upper panels show the probability measure for crossed Andreev reflection while the lower panels show for elastic cotunneling. We have used barrier strengths $Z=0$ (left) and $Z=4$ (right), and a number of values for the interband coupling $\tilde{\alpha }$.

Figure 5

(Color online) Nonlocal conductance through a $\text{N}\mid s_{\pm }\text{-wave}\mid \text{N}$ junction for a relatively thick superconducting interlayer, $L=8\times {10}^4{k}_{\text{F}}^{-1}$, and for $r_{\Delta }=1.5$. The upper panels show the probability for crossed Andreev reflection while the lower panels show the probability for elastic cotunneling. We have used barrier strengths $Z=0$ (left) and $Z=4$ (right), and a number of values for the interband coupling $\tilde{\alpha }$.

Figure 6

(Color online) Critical current for a $s\text{-wave}\left|\text{I}\right|s_{\pm }$-wave Josephson junction as a function of the ratio $r_Z$ between the effective barrier strength of band 2 and band 1. There is no interband coupling $\left(\tilde{\alpha }=0\right)$ and we have set $Z_1=6$ and $T=0$. Also shown is the critical current in the case that the right-hand side is a (two-band) $s$-wave superconductor.

Figure 7

(Color online) Current-phase relation for a $s\text{-wave}\left|\text{I}\right|s_{\pm }$-wave Josephson junction with zero interband coupling $\left(\tilde{\alpha }=0\right)$, with $Z=6$ and $T=0$.

Figure 8

(Color online) Josephson coupling for a $s\text{-wave}\left|\text{I}\right|s_{\pm }$-wave Josephson junction for various values of interband coupling strength $\tilde{\alpha }$, with (a) dispersion of the two Andreev bound states with $E<0$ shown to the left and (b) critical current as a function of barrier strength ratio $r_Z$ shown to the right. Both results are given for intermediate barrier strength $Z=4$.

Figure 9

(Color online) Fraunhofer-type magnetic diffraction pattern for a $s\text{-wave}\left|\text{I}\right|s_{\pm }$-wave junction in an external magnetic field for various values of barrier strength ratio. We have used the parameter values $Z=6$ and $\tilde{\alpha }=0$.

Figure 10

(Color online) Temperature dependence of the critical current for the parameters $r_s=0.5$ and $r_{\Delta }=0.3$ with $Z=6$ and $r_Z=1$. Both the results for a two-gap $s$-wave and a $s_{\pm }$-wave superconductor are shown. Inset: temperature dependence of gap magnitudes for the same parameter set.

Figure 11

(Color online) Energy of Andreev bound states $E_{\lambda }^{-}$ for two temperatures close to the point where $\left|{\Delta }_2\left(T\right)\right|={\Delta }_s\left(T\right)$ for $r_s=0.5$ and $r_{\Delta }=0.3$. Other parameters are $Z=6$ and $r_Z=1$. Shown with dotted lines are the relevant gap edges which the energy states track.

Figure 12

(Color online) Current-phase relation for a system slightly below $\left(T/T_c=0.4\right)$ and slightly above $\left(T/T_c=0.7\right)$ the thermally induced phase shift appearing in the Josephson junction with gap ratios $r_s=0.5$ and $r_{\Delta }=0.3$ with $Z=6$ and $r_Z=1$. The arrows indicate the phase difference supporting the critical current $\left(I>0\right)$ for the two temperatures.

Figure 13

(Color online) The upper panel shows the temperature dependence of the critical current for the parameters $r_s=0.5$, $r_{\Delta }=0.3$, and $Z=6$, similarly as in Fig. 10 but for various values of barrier strength ratios $r_Z$. The lower panel shows the phase difference ${\phi }^{\ast }$ supporting the critical current as a function of temperature for the same parameter set as above, illustrating the effect of the (discontinuous) phase shift from ${\phi }^{\ast }>0$ to ${\phi }^{\ast }<0$.

Figure 14

(Color online) Illustration of the physical interpretation of the barrier strength ratio $r_Z$. The outer circle represents the Fermi surface of band 1 of the $s_{\pm }$-wave superconductor and the inner circle represents band 2. The two dashed circles in between represent the Fermi surface of the $s$-wave superconductor on the left-hand side of the junction for two situations: largest FWVM with respect to band 1 $\left(Z_1>Z_2\right)$ and largest FWVM with respect to band 2 $\left(Z_1<Z_2\right)$.