Josephson $\pi$ State in a Ferromagnetic Insulator

We predict anomalous atomic-scale $0\mathrm{\text{−}}\pi$ transitions in a Josephson junction with a ferromagnetic-insulator (FI) barrier. The ground state of such junction alternates between 0 and $\pi$ states when thickness of FI is increasing by a single atomic layer. We find that the mechanism of the $0\mathrm{\text{−}}\pi$ transition can be attributed to thickness-dependent phase shifts between the wave numbers of electrons and holes in FI. Based on these results, we show that a stable $\pi$ state can be realized in junctions based on high-$T_c$ superconductors with a ${\mathrm{La}}_2{\mathrm{BaCuO}}_5$ barrier.

Figure 1

(a) The Josephson junction with a ferromagnetic-insulator (FI) barrier on the tight-binding lattice. The magnetic moment in FI is chosen along the $z$ axis in spin space. The band structure of a FI (b) and of standard band insulator (c) in the Bogoliubov–de Gennes picture. The dispersion for a hole with spin $\sigma$ is obtained as a mirror image of the dispersion for an electron with spin $\sigma$ with respect to Fermi energy.

Figure 2

(a) The Andreev levels ${\epsilon }_i\equiv {\epsilon }_{i,1}$ are plotted as functions of $\varphi$ for an one-dimensional S/FI/S junction with $W=1$. Left (right) panel shows the results for odd $L_F$ (even $L_F$) with $g=0.5t$. (b) Josephson critical current $I_C$ at $T=0.01T_c$ as a function of the thickness $L_F$ for $W=1$ and $g=0.5t$. The large red (small blue) circles indicate the $\pi (0)$ junction. In the inset, the $0\mathrm{\text{−}}\pi$ phase diagram on the $g\mathrm{\text{−}}{L}_F$ plane is shown for a two-dimensional S/FI/S junction with $W=10$ at $T=0.01T_c$, where the red and blue regimes correspond to the $\pi$ and 0 states, respectively.

Figure 3

(a) Schematic picture of a $c$-axis stack high-$T_c$ superconductor/FI/high-$T_c$ superconductor Josephson junction and (b) a $d$-wave ring to detect the $\pi$-junction behavior experimentally. (c) The Josephson critical current $I_C$ as a function of the FI thickness $L_F$ at $T=0.01T_c$ for a $c$-axis stack LSCO/LBCO/LSCO junction with $g/t=20$, ${\Delta }_d/t=0.6$, and $L_a=L_b=100$. The large red (small blue) circles indicate the $\pi (0)$ junction.