Switch effect and 0-$\pi$ transition in Ising superconductor Josephson junctions

We theoretically study the Josephson current in Ising superconductor–half-metal–Ising superconductor junctions. By solving the Bogoliubov–de Gennes equations, the Josephson currents contributed by the discrete Andreev levels and the continuous spectrum are obtained. For very short junctions, because the direct tunneling of the Cooper pair dominates the Josephson current, the current-phase difference relation is independent of the magnetization direction, which is the same as the conventional superconductor-ferromagnet-superconductor junctions. On the other hand, when the length of the half-metal is similar to or greater than the superconducting coherence length, the spin-triplet Josephson effect occurs and dominates the Josephson current. In this case, the current-phase difference relations show the strong magnetoanisotropic behaviors with the period $\pi$. When the magnetization direction points to the $\pm z$ directions, the current is zero regardless of the phase difference. However, the current has a large value when the magnetization direction is parallel to the junction plane, which leads to a perfect switch effect of the Josephson current. Furthermore, we find that the long junctions can host both the 0 state and $\pi$ state, and the $0\text{−}\pi$ transitions can be achieved with the change of the magnetization direction. The physical origins of the switch effect and $0\text{−}\pi$ transitions are interpreted from the perspectives of the spin-triplet Andreev reflection, the Ising pairing order parameter and the Ginzburg-Landau type of free energy. In addition, the influences of the chemical potential, the magnetization magnitude, and the strength of the Ising spin-orbit coupling on the switch effect and $0\text{−}\pi$ transitions are also investigated. Furthermore, the two-dimensional Josephson junctions are also investigated and we show that the spin-triplet Josephson effect can exist always. These results provide a convenient way to control the Josephson critical current and to adjust the junctions between the 0 state and $\pi$ state by only rotating one magnetization.

Figure 1

(a) Schematic illustration of the ISC-HM-ISC junction. The junction is in the $xy$ plane. The interface is located at $x=0$ and $x=L$. The direction of magnetization $\mathit{M}$ is depicted by the polar angle ${\theta }_m$ and the azimuthal angle ${\phi }_m$. (b) The energy bands near $\mathit{K}$ and $-\mathit{K}$ valleys for the normal phase of ISC. The black solid lines and dashed lines indicate the Fermi energy for the double-band case and the single-band case, respectively. The red and blue arrows represent two electrons with the opposite spin and opposite wave vector from different valleys, which combine to form a Cooper pair. (c) The energy bands for ferromagnet. Here the Fermi energy (black solid lines) is across the energy band with spin antiparallel to $\mathit{M}$ only.

Figure 2

(a)–(d) The discrete Andreev levels $E_n^{+}$ for ${\theta }_m=0$, $0.1\pi$, $0.2\pi$, and $0.5\pi$, respectively. (e) The discrete Josephson current $I_d$ and (f) the continuum Josephson current $I_c$ versus the phase difference $\varphi$ for the different ${\theta }_m$. The related parameters are $k_{F}L=100$, ${\mu }_s=1.3$, ${\beta }_s=1.1$, ${\mu }_f=1.0$, and $M=1.2$.

Figure 3

The total current $I$ as a function of the phase difference $\varphi$ with ${\theta }_m=0,0.1\pi$, $0.2\pi$, $0.3\pi$, $0.4\pi$, and $0.5\pi$ for (a) $k_{F}L=0.01$, (b) $k_{F}L=5$, (c) $k_{F}L=100$, and (d) $k_{F}L=300$. Other parameters have the same values as those in Fig. 2.

Figure 4

The total current $I$ with $\varphi =0.5\pi$ as a function of the polar angle ${\theta }_m$ for $k_{F}L=0.01$, 5, 10, 100, and 300. The currents for $k_{F}L=0.01$ (the black solid line) and $k_{F}L=5$ (the red dashed line) have been taken as $1/100$ and $1/10$ of their real values. Other parameters have the same values as those in Fig. 2.

Figure 5

The total current $I$ as a function of the phase difference $\varphi$ with ${\theta }_m=0$, $0.1\pi$, $0.2\pi$, $0.3\pi$, $0.4\pi$, and $0.5\pi$ for (a) $k_{F}L=0.01$, (b) $k_{F}L=5$, (c) $k_{F}L=100$, and (d) $k_{F}L=300$. The related parameters are ${\mu }_s=1.0$, ${\beta }_s=1.1$, ${\mu }_f=1.0$, and $M=1.2$.

Figure 6

The total current $I$ with $\varphi =0.5\pi$ as a function of the polar angle ${\theta }_m$ for $k_{F}L=0.01$, 5, 10, 100, and 300. The current for $k_{F}L=0.01$ (the black solid line) has been taken as $1/20$ of its real value. Other parameters have the same values as those in Fig. 5.

Figure 7

(a) The Andreev levels $E_n^{+}$ (the blue dashed curves) and $E_n^{-}$ (the red dotted curves) as functions of ${\theta }_m$ for $k_{F}L=5$ and $\varphi =0.5\pi$. The black solid line represents $E_F=0$. The symbol A denotes the intersection point between the Andreev levels and $E_F=0$. (b) The enlarged figure in the vicinity of the point A in (a) with $\varphi =0.5\pi$ and $0.501\pi$. (c) The Andreev levels versus the phase difference $\varphi$ for ${\theta }_m=45.7^{\circ }<{\theta }_m^c$, $45.{758}^{\circ }={\theta }_m^c$, and $45.8^{\circ }>{\theta }_m^c$. Other parameters have the same values as those in Fig. 6.

Figure 8

The total current $I$ with $\varphi =0.5\pi$ as a function of ${\theta }_m$ for various values of ${\mu }_s$. The parameters are $k_{F}L=300$, ${\beta }_s=1.1$, ${\mu }_f=1.0$, and $M=1.2$.

Figure 9

The total current $I$ with $\varphi =0.5\pi$ as a function of ${\theta }_m$ for (a) ${\mu }_s=1.3$, ${\beta }_s=1.1$ and various values of $M$, (b) ${\mu }_s=1.3$, $M=1.2$ and various values of ${\beta }_s$, (c) ${\mu }_s=1.0$, ${\beta }_s=1.1$ and various values of $M$, and (d) ${\mu }_s=1.0$, $M=1.2$ and various values of ${\beta }_s$. Other parameters are $k_{F}L=300$ and ${\mu }_f=1.0$.

Figure 10

The Josephson current $I\left(k_y\right)$ with different $k_y$ as a function of ${\theta }_m$ for (a) $k_{F}L=5$, (b) $k_{F}L=10$, (c) $k_{F}L=20$, and (d) $k_{F}L=100$. Other parameters are ${\mu }_s=1.0$, ${\beta }_s=1.1$, ${\mu }_f=1.0$, and $M=1.2$.

Figure 11

The Josephson current for the two-dimensional junctions along the $x$ axis as a function of ${\theta }_m$ for $k_{F}L=5$, 10, 20, and 100. Other parameters have the same values as those in Fig. 10.