Positive magnetoresistance in anapole superconductor junctions

This paper presents a method to detect time-reversal symmetry breaking in noncentrosymmetric superconductors using only transport measurements. Specifically, if time-reversal symmetry is broken via a phase difference between singlet and triplet correlations, as in anapole superconductors, the differential conductance in superconductor-ferromagnet-normal metal junctions is enhanced by increasing the exchange field strength in the ferromagnet. This is in sharp contrast with the negative magnetoresistance when using superconductors in which time-reversal symmetry is preserved. Moreover, results show a large quadrupolar component of the magnetoresistance which is qualitatively different from the bipolar giant magnetoresistance in strong ferromagnets.

Figure 1

A schematic of the SFN junction. The superconductor (SC) is characterized by the phase difference ${\chi }_t$ between the singlet and triplet components and by the $d$ vector of the triplet component. The ferromagnet (F) is characterized by its exchange field. Both the SC and the normal metal (N) serve as electrodes.

Figure 2

The magnetic field dependence of the differential conductance of $is$ $+$ helical $p$-wave junctions for ${\chi }_t=1$ for parallel (a) and perpendicular (b) orientations when $r=0.5$. The sharp peak that without magnetic field is at zero bias splits into two peaks at $\left|eV\right|=h$ for small $h$, disappearing completely for large $h$. For voltages just below ${\mathrm{\Delta }}_0$, the differential conductance is enhanced by the exchange field. This effect is stronger for the perpendicular orientation than for the parallel orientation.

Figure 3

The magnetic-field dependence of the differential conductance of $is$ $+$ helical $p$-wave junctions for ${\chi }_t=1$ and $r=2$ for parallel (a) and perpendicular (b) orientations of the exchange field with respect to the $d$ vector. The sharp peak that without magnetic field is at zero bias splits into two peaks at $\left|eV\right|=h$ for small $h$, disappearing for large $h$. For voltages just below ${\mathrm{\Delta }}_0$, the differential conductance is enhanced by the exchange field. This effect is stronger for the perpendicular orientation than for the parallel orientation.

Figure 4

The quadrupolar dependence of the differential conductance of $is$ $+$ helical $p$-wave junctions at $eV/{\mathrm{\Delta }}_0\approx 0.75$ on the angle $\alpha$ between the exchange field and $d$ vector of the superconductor for $r=0.5$ (a) and $r=2$ (b). The exchange field strength is $h/{\mathrm{\Delta }}_0=0.2$. For $s$ $+$ helical $p$-wave superconductors, the differential conductance is maximized for the parallel orientation. On the other hand, for $is$ $+$ helical $p$-wave superconductors, the differential conductance is minimized in the parallel orientation and the angular dependence is much larger. There is also a small bipolar component, $\sigma (\alpha =\pi )\ne \sigma (\alpha =0)$, however, the quadrupolar component dominates.

Figure 5

The magnetic-field dependence of the differential conductance of $is$ $+$ helical $p$-wave junctions for ${\chi }_t=1$ for $r=0.9$ (a) and $r=1.1$ (b) when the field is perpendicular to the $d$ vector. The sharp peak that without magnetic field is at zero bias splits into two peaks at $\left|eV\right|=h$ for small $h$, disappearing completely for large $h$. The increase in the differential conductance by applying a magnetic field is larger than for $r=0.5$ or $r=2$, confirming that it is an effect for which the coexistence of even and odd-parity contributions to the pair potential.

Figure 6

The differential conductance in $is$ $+$ chiral $p$-wave junctions in the presence of parallel (a) and perpendicular (b) exchange fields. The presence of a small exchange field can increase the differential conductance in the junction. The increase is less visible then for $is$ $+$ helical $p$-wave junctions because the phase is mode dependent. Therefore, only the differential conductance of a selection of modes is enhanced, while that of the others is suppressed. The superconductor is $s$-wave dominant, with $r=0.5$.

Figure 7

The differential conductance in $is$ $+$ chiral $p$-wave junctions in the presence of parallel (a) and perpendicular (b) exchange fields. The superconductor is $p$-wave dominant, with $r=2$. The sharp zero-bias conductance peak splits upon application of a parallel exchange field, but hardly changes by application of a perpendicular field. This happens because the ZBCP is due to the oblique modes, for which the phase difference between the modes is almost 0.

Figure 8

Schematic showing the difference between helical and chiral ($i$)$s$ $+$ p-wave superconductors. The phase difference between singlet and triplet superconductors ${\varphi }_{sp}\left(\varphi \right)$ is illustrated by color. For ($i$)$s$ $+$ helical p-wave superconductors, this phase difference is independent of the direction of momentum, and hence all modes contribute to an enhancement of the differential conductance. On the other hand, for the ($i$)$s$ $+$ chiral $p$-wave superconductor, the phase difference between the triplet and singlet components depends on the angle of momentum, and hence for some modes the differential conductance is enhanced while for others it is suppressed.

Figure 9

The differential conductance of the $is$ $+$ helical $p$-wave junction with $r=2$ for different exchange fields if the boundary resistance is lowered such that ${\gamma }_{BS}=0.5$, with fields in the (a) parallel or (b) perpendicular orientation. In both orientations, there is an enhancement of conductance by an exchange field, which is larger in the perpendicular compared to the parallel orientation. This quadrupolar dependence can be used to distinguish the $is$ $+$ helical $p$-wave enhancement from the peak at $eV=h$. We recommend using $h>{\mathrm{\Delta }}_0$ to avoid the latter effect.

Figure 10

The differential conductance of the $is$ $+$ helical $p$-wave junction with $r=2$ for different exchange fields if the boundary resistance is enhanced such that ${\gamma }_{BS}=5$, with fields in the (a) parallel or (b) perpendicular orientation. In the perpendicular orientation, there is a clear enhancement of conductance by an exchange field, while in the parallel orientation this effect is almost absent. The peak at $eV=h$ is small and hence does not play a role in this parameter regime.

Figure 11

The differential conductance of the $is$ $+$ helical $p$-wave junction with $r=2$ for different exchange fields if the length is increased, so $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.01$, with fields in the (a) parallel or (b) perpendicular orientation. In the perpendicular orientation, there is a clear enhancement of conductance by an exchange field; in the parallel orientation the enhancement is small. The peak at $eV=h$ is very sharp in this parameter regime, so the quadrupolar enhancement can be easily distinguished from this effect.

Figure 12

The differential conductance of the $is$ $+$ helical $p$-wave junction with $r=2$ for different exchange fields if the length is decreased, so $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.04$, with fields in the (a) parallel or (b) perpendicular orientation. In both orientations, there is a clear enhancement. The peak at $eV=h$ is relatively broad. For this reason, we recommend using the regime $h>{\mathrm{\Delta }}_0$ to disentangle the two effects.

Figure 13

The differential conductance of the $is$ $+$ helical $p$-wave junction with $r=2$ for different exchange fields if the transparency is increased, so $z=0.5$, with fields in the (a) parallel or (b) perpendicular orientation. In both orientations, there is a clear enhancement of conductance by an exchange field. It is harder to distinguish between the parallel and perpendicular orientations due to the similar contribution of modes with normal and oblique incidence.

Figure 14

The differential conductance of the $is$ $+$ helical $p$-wave junction with $r=2$ for different exchange fields if the transparency is decreased, so $z=2$, with fields in the (a) parallel or (b) perpendicular orientation. There is a clear difference between the perpendicular orientation and parallel orientation, with a large enhancement of conductance if $\mathit{h}$ and $\langle \mathit{d}\rangle$ are perpendicular that is almost absent if these two vectors are parallel.