Spin-polarized Josephson current in superconductor/ferromagnet/superconductor junctions with inhomogeneous magnetization

We study numerically the properties of spin transport and charge transport in a current-biased nanoscale diffusive superconductor/ferromagnet/superconductor junction when the magnetization texture is nonuniform. Specifically, we incorporate the presence of a Bloch/Neel domain walls and conical ferromagnetism, including the role of spin-active interfaces. The superconducting leads are assumed to be of the conventional $s$-wave type. In particular, we investigate how the $0\text{−}\pi$ transition is influenced by the inhomogeneous magnetization texture and focus on the particular case where the charge current vanishes while the spin current is nonzero. In the case of a Bloch/Neel domain wall, the spin current can be seen only for one component of the spin polarization, whereas in the case of conical ferromagnetism the spin current has the three components. This is in contrast to a scenario with a homogeneous exchange field, where the spin current vanishes completely. We explain all of these results in terms of the interplay between the triplet anomalous Green’s function induced in the ferromagnetic region and the local direction of the magnetization vector in the ferromagnet. Interestingly, we find that the spin current exhibits discontinuous jumps at the $0\text{−}\pi$ transition points of the critical charge current. This is seen both in the presence of a domain wall and for conical ferromagnetism. We explain this result in terms of the different symmetry obeyed by the current-phase relation when comparing the charge and spin currents. Specifically, we find that whereas the charge current obeys the well-known relation $I_c\left(\varphi \right)=-I_c\left(-\varphi \right)$, the spin current satisfies $I_s\left(\varphi \right)=I_s\left(-\varphi \right)$, where $\varphi$ is the superconducting phase difference.

Figure 1

(Color online) The S/F/S junction (i) with Bloch domain wall of ferromagnet and (ii) with conical type of ferromagnet. The magnetization texture for the Neel wall is obtained by replacing the $x$ component of the magnetization with an $y$ component in case (i). The blue arrows in the central region show the magnetic moments in F layer. The magnetic moment for Bloch/Neel domain wall has two components and for conical type has three components.

Figure 2

(Color online) The variation in the normalized critical charge current versus the thickness of a homogeneous F layer. The inset panel zooms in on the behavior near the $0\text{−}\pi$ transition. As long as the exchange field is constant, we find that the spin current $I_s$ vanishes.

Figure 3

(Color online) The variations in normalized critical spin and charge currents vs increasing the thickness $d_F$ of F layer with a Bloch domain-wall structure.

Figure 4

(Color online) Plot of the normalized critical spin and charge currents vs the normalized thickness $d_F$ of the F layer for a conical type of magnetization texture. All three components of the spin polarization have a considerable magnitude in the entire range of $d_F$ considered here.

Figure 5

(Color online) Plot of the current-phase relation for the charge and spin currents. We are considering a conical magnetization texture, and the curves range from $d_F/{\xi }_S=0.7655$ to 0.7725 in steps of $1\times {10}^{-3}$ in the charge-current panel. For the spin-current panels, the variation in the current-phase relation upon changing $d_F$ is negligible and we give results only for $d_F/{\xi }_S=0.7725$. Note that the critical spin current nevertheless varies strongly with $d_F$ near the transition points as shown in Fig. 4, since the critical phase displays a strong dependence on $d_F$ in this regime.