Spin-flip scattering and critical currents in ballistic half-metallic $d$-wave Josephson junctions

We analyze the dc Josephson effect in a ballistic superconductor/half-metal/superconductor junction by means of the Bogoliubov–de Gennes equations. We study the role of spin-active interfaces and compare how different superconductor symmetries, including $d$-wave pairing, affect the Josephson current. We analyze the critical current as a function of junction width, temperature, and spin-flip strength and direction. In particular, we demonstrate that the temperature dependence of the supercurrent in the $d_{xy}$ symmetry case differs qualitatively from the $s$ and $d_{x^2-y^2}$ symmetries. Moreover, we have derived a general analytical expression for the Andreev bound-state energies that shows how one can either induce $0-\pi$ transitions or continuously change the ground-state phase of the junction by controlling the magnetic misalignment at the interfaces.

Figure 1

Schematic illustration of our system. A half metal of width $L$ is sandwiched between two superconductors, which are considered as reservoirs. The superconductors may have either an $s$-wave or $d$-wave symmetry. The interface regions are assumed to be spin active due to, e.g., magnetic disorder or misaligned local magnetic moments.

Figure 2

The barrier magnetic moment at the interface and its misalignment angles $\Psi$ and $\phi$. The bulk magnetization in the ferromagnet is assumed to be aligned with the $z$ axis.

Figure 3

Critical current as a function of junction width for $\rho =0$ (dashed lines) and $\rho =0.5$ (solid lines) with misalignment angles $\Psi =\pi /2$ and $\phi =\pi /2$. Blue, green, and red lines indicate $s$-, $d_{x^2-y^2}$-, and $d_{xy}$-wave pairing symmetry, respectively.

Figure 4

Critical current as a function of the misalignment angle $\phi$. Solid, dashed, and dash-dotted lines correspond to junction widths $kL=10,100$, and 1000, respectively. Panels (a)–(c) show $s$-, $d_{x^2-y^2}$-, and $d_{xy}$-wave pairing symmetry, respectively.

Figure 5

Current-phase relation for a junction with $d_{xy}$ superconductors on both sides. The black solid line shows parallel misalignment at both interfaces, while the red dashed line shows antiparallel misalignment.

Figure 6

Current-phase relation for a junction with $d_{xy}$ superconductors on both sides. The misalignment field in the $xy$ plane is rotated from parallel to perpendicular, i.e., ${\Psi }_{\mathrm{R}}$ changes from 0 to $\pi /2$ in steps of $\pi /8$ (left to right). ${\Psi }_{\mathrm{L}}=0$ and ${\rho }_{\mathrm{L}}={\rho }_{\mathrm{R}}=0.5$. Here, ${\phi }_L={\phi }_R=\pi /2$.

Figure 7

Critical current as a function of temperature for all three superconducting symmetries at junction width $kL=100$. Notice that the $d_{xy}$-symmetry decreases monotonically, while $s$- and $d_{x^2-y^2}$-symmetries have a non-monotonic behavior. The legends show the misalignment angle $\phi$ in units of $\pi$.