Quantum limit of the triplet proximity effect in half-metal–superconductor junctions

We apply the scattering matrix approach to the triplet proximity effect in superconductor–half-metal structures. We find that for junctions that do not mix different orbital modes, the zero-bias Andreev conductance vanishes, while the zero-bias Josephson current is nonzero. We illustrate this finding on a ballistic half-metal–superconductor (HS) and superconductor–half-metal–superconductor (SHS) junctions with translation invariance along the interfaces and on HS and SHS systems where transport through the half-metallic region takes place through a single conducting channel. Our calculations for these physically single-mode setups—single-mode point contacts and chaotic quantum dots with single-mode contacts—illustrate the main strength of the scattering matrix approach. It allows for studying systems in the quantum mechanical limit, which is inaccessible for the quasiclassical Green’s function methods, the main theoretical tool in previous works on the triplet proximity effect.

Figure 1

(Color online) Composite HS junction consisting of a half-metallic contact (left), a superconducting contact (right), and a spin-active intermediate layer (center). In most of our considerations, the intermediate layer is taken to be ferromagnetic with a magnetization direction not collinear with the polarization of the half-metal. Transport through the HS junction is described by the scattering matrix $\mathcal{R}$, which is calculated in terms of the Andreev reflection matrix ${\mathcal{R}}_A$ of an ideal normal-metal–superconductor interface and the reflection and transmission matrices $r$, $r^{\prime }$, $t$, and $t^{\prime }$ of the nonsuperconducting region.

Figure 2

(Color online) Schematic drawing of an SHS junction. In the scattering approach, an SHS junction is seen as two opposing (composite) HS junctions, with scattering matrices ${\mathcal{R}}^{\prime }$ and $\mathcal{R}$, respectively. In the calculations of Sec. 4A, scattering phase shifts from the central half-metallic part are included into ${\mathcal{R}}^{\prime }$.

Figure 3

The subgap differential conductance $G$ versus the applied voltage $V$ for a ballistic single-mode HS quantum point contact. The small gray rectangle in the contact represents a region with a different magnetization than in the half-metallic part. Physically such a region can be present due to a misaligned magnetization at the half-metal surface (Ref. 14). In our calculations this corresponds to the ferromagnetic spacer layer. The curves correspond to different values of the phase angles in the ferromagnetic spacer, $\theta =0.8$ and $\rho =0.9$ (dashed curve), $\theta =1.4$ and $\rho =1.2$ (dashed-dotted curve), and $\theta =1.56$ and $\rho =1.53$ (dotted curve).

Figure 4

(Color online) Chaotic HS junction, consisting of a half-metallic contact (left), a half-metallic quantum dot (center), a ferromagnetic spacer layer, and a superconducting contact (right).

Figure 5

The ensemble-averaged subgap differential conductance $⟨G⟩$ versus the applied voltage $V$ for different phase angles $\theta$ and $\rho$ describing the ferromagnetic contact. The values of $\theta$ and $\rho$ are 0.8 and 0.9 (dashed curve), 1.4 and 1.2 (dashed-dotted curve), and 1.56 and 1.53 (dotted curve), respectively. The solid curve shows the special case corresponding to $\theta =\rho =\pi /2$.

Figure 6

(Color online) The contribution of a single transverse mode to the zero-temperature supercurrent $I$ of a short SHS junction, as a function of $\tilde{\varphi }$, for ferromagnetic phase angles $\theta ={\theta }^{\prime }=\rho ={\rho }^{\prime }=\pi /2$ (solid curve), $\theta ={\theta }^{\prime }=\rho ={\rho }^{\prime }=\pi /4$ (dotted-dashed curve), and $\theta ={\theta }^{\prime }=\pi /2,\rho ={\rho }^{\prime }=\pi /4$ (dashed curve). The supercurrent is shown in units of $I_{\text{short}}=e\Delta /\hslash$.

Figure 7

(Color online) The contribution of a single transverse mode to the nonoscillating component of the zero-temperature supercurrent $I$ of a long SHS junction, as a function of $\tilde{\varphi }$, for ferromagnetic phase angles $\theta ={\theta }^{\prime }=\rho ={\rho }^{\prime }=\pi /3$ (solid curve) and $\theta ={\theta }^{\prime }=\rho ={\rho }^{\prime }=\pi /4$ (dashed curve). The current is shown in units of $I_{\text{long}}=e{\hslash }^2{v}_{\mu }^3/\pi L^3{\Delta }^2$, where $v_{\mu }$ is the mode-dependent longitudinal velocity.

Figure 8

(Color online) Superconductor–half-metal-quantum-dot–superconductor junction. In the calculation, scattering from the half-metal quantum dot is included in the scattering matrix $\mathcal{R}$.

Figure 9

The ensemble-averaged Josephson current $⟨I⟩$ as a function of $\tilde{\varphi }$ for different phase angles $\theta$, $\rho$, ${\theta }^{\prime }$, and ${\rho }^{\prime }$ describing the ferromagnetic spacer layers. The values of $\theta$ and $\rho$ are $\theta =\rho =\pi /2$ (solid curve), $0.9\left(\pi /2\right)$ and $0.99\left(\pi /2\right)$ (dashed curve), and $0.5\left(\pi /2\right)$ and $0.6\left(\pi /2\right)$ (dotted curve), respectively. The values for the second contact are ${\theta }^{\prime }=1.05\left(\pi /2\right)$ and ${\rho }^{\prime }=0.95\left(\pi /2\right)$. The supercurrent is shown in units of $I_{\text{short}}=e\Delta /\hslash$.

Figure 10

Sketch of a possible experimental setup for testing the vanishing Andreev reflection at the Fermi level: a single-channel quantum point contact to an FS junction. The arrow in the quantum point contact indicates that the point contact transmits only one spin direction.