$0\mathrm{\text{−}}\pi$ Transitions in a Superconductor/Chiral Ferromagnet/Superconductor Junction Induced by a Homogeneous Cycloidal Spiral

We study the $\pi$ phase in a superconductor-ferromagnet-superconductor Josephson junction, with a ferromagnet showing a cycloidal spiral spin modulation with in-plane propagation vector. Our results reveal a high sensitivity of the junction to the spiral order and indicate the presence of $0\mathrm{\text{−}}\pi$ quantum phase transitions as function of the spiral wave vector. We find that the chiral magnetic order introduces chiral superconducting triplet pairs that strongly influence the physics in such Josephson junctions, with potential applications in nanoelectronics and spintronics.

Figure 1

S-CM-S Josephson junction where CM is a chiral ferromagnet with a cycloidal spiral spin modulation; i.e., the spins are confined to a plane (the $x\mathrm{\text{−}}y$ plane) parallel to the spiral propagation direction (the $y$ axis).

Figure 2

Josephson critical current $I_c$ vs the spiral wave vector $Q$ for a few thicknesses of the ferromagnet: curves from top to bottom $d_f/{\xi }_f=0.1$, 1, 3, 5. Here $T=0.1T_c$ and $J=20T_c$. The inset shows the flow of the real and imaginary parts of the eigenvalues $k_{\pm 1}$ with varying $\eta =(Q{\xi }_J{)}^2/4$ for ${\epsilon }_n=\pi T_c$.

Figure 3

(a) Critical current vs spiral wave vector for a few temperatures, in the region $0\le Q{\xi }_J\le 2$, where $0\mathrm{\text{−}}\pi$ transitions are possible. Here $d_f=2.7{\xi }_f$ and $J=20T_c$. (b) The $0\mathrm{\text{−}}\pi$ transition is observable as function of temperature for a certain thickness (here $d_f=0.45{\xi }_f$) by tuning the spiral wave vector. The other model parameters are the same as in (a).

Figure 4

(a) ($T-Q$) phase diagram for a Josephson junction with a chiral magnet ($J=20T_c$) between two singlet superconductors. The transition from a $\pi$ junction at smaller chiral wave vector $Q$ to a 0 junction at larger $Q$ is indicated for several thicknesses of the ferromagnetic layer. (b) Corresponding low-$T$ ($d_f-Q$) phase diagram ($T=0.1T_c$).