Andreev reflection at half-metal/superconductor interfaces with nonuniform magnetization

Andreev reflection at the interface between a half-metallic ferromagnet and a spin-singlet superconductor is possible only if it is accompanied by a spin flip. Here we calculate the Andreev reflection amplitudes for the case where the spin flip originates from a spatially nonuniform magnetization direction in the half-metal. We calculate both the microscopic Andreev reflection amplitude for a single reflection event and an effective Andreev reflection amplitude describing the effect of multiple Andreev reflections in a ballistic thin film geometry. It is shown that the angle and energy dependence of the Andreev reflection amplitude strongly depends on the orientation of the gradient of the magnetization with respect to the interface. We calculate the resulting effects on the subgap conductance as well as Josephson current for a few exemplary cases. Establishing a connection between the scattering approach employed here and earlier work that employs the quasiclassical formalism, we connect the symmetry properties of the Andreev reflection amplitudes to the symmetry properties of the anomalous Green function in the half-metal.

Figure 1

(a) Lateral contact between a thin half-metallic film (H) and a superconductor (S). Andreev reflection at the HS interface is possible at those interface positions where the magnetization direction (indicated by thin black arrows) is not uniform, such as a domain wall. The situation shown in the figure is generic, since domain walls at the HS interface are bound to occur if the HS contact is larger than the domain size in the half-metallic film. In this geometry, there are two possible ways to define the Andreev reflection amplitude: The amplitude $r_{\mathrm{he}}$ for a single Andreev reflection (filled red arrows), appropriate for the interface between a semi-infinite half-metal and a superconductor (b), and the effective amplitude $r_{\mathrm{he}}^{\mathrm{eff}}$ representing the combined effect of multiple Andreev reflections in the same region of nonuniform magnetization for quasiparticles moving in the half-metallic film (c). In (b) and (c), the black arrows in the top panel show the spatial variation of the magnetization direction, whereas the filled red arrows in the bottom panel show an example of a trajectory for an incoming electron (solid) and an Andreev reflected hole (dashed).

Figure 2

Illustration of the scattering setup. Coordinates are chosen, such that the half-metal–superconductor interface is the plane $z=0$. Electron-like quasiparticles incident on the HS interface are either reflected as an electron (normal reflection), or as a hole (Andreev reflection). At a translationally invariant interface, the projection ${\mathbf{k}}_{\parallel }$ of the wave vector on the interface is conserved upon reflection.

Figure 3

Illustration of a magnetization gradient perpendicular to the HS interface (a) and parallel to the HS interface (b).

Figure 4

Schematic of an SHS junction with a magnetization gradient perpendicular to the HS interfaces (a) or parallel to the HS interface (b). The magnetization angle ${\theta }_{\mathrm{m}}$ is chosen to be a symmetric function of $z$.

Figure 5

Illustration of the thin-film geometry. The coordinates are chosen such that the magnetization direction does not change in the $y$ direction (perpendicular to the plane of the drawing).

Figure 6

Illustration of a lateral HS contact with a magnetization gradient that is perpendicular to the HS interface (left) or perpendicular to the HS interface (right).

Figure 7

Illustration of a lateral SHS junction with a spatially nonuniform magnetization in the half metal. Two magnetization profiles are considered in the text: the gradient $\nabla {\theta }_{\mathrm{m}}$ of the magnetization angle perpendicular to the HS interface (top) and parallel to the HS interface (bottom).