Spin-triplet supercurrent through inhomogeneous ferromagnetic trilayers

Motivated by a recent experiment [J. W. A. Robinson, J. D. S. Witt, and M. G. Blamire, Science 329, 5987 (2010)], we report here the possibility of establishing a long-range spin-triplet supercurrent through an inhomogeneous ferromagnetic region consisting of a $\text{Ho}\mid \text{Co}\mid \text{Ho}$ trilayer sandwiched between conventional $s$-wave superconducting leads. We utilize a full numerical solution in the diffusive regime of transport and study the behavior of the supercurrent for various experimentally realistic configurations of the ferromagnetic trilayer. We obtain qualitatively very good agreement with experimental data regarding the behavior of the supercurrent as a function of the width of the Co layer, $L_{\text{Co}}$. Moreover, we find a synthesis of $0\text{−}\pi$ oscillations with superimposed rapid oscillations when varying the width of the Ho layers symmetrically, which pertain specifically to the spiral magnetization texture in Ho. Although we are not able to reproduce the anomalous sharp peaks in the supercurrent vs Ho-layer thickness observed experimentally in this regime, the results obtained are quite sensitive to the exact magnetization profile in the Ho layers. This might be the reason for the discrepancy between our results and the experimental reported data for this particular aspect. We also investigate the supercurrent in a system where the intrinsically inhomogeneous Ho ferromagnets are replaced with domain-wall ferromagnets and find similar behavior as in the $\text{Ho}\mid \text{Co}\mid \text{Ho}$ case. Furthermore, we propose magnetic Josephson junctions including only a domain-wall ferromagnet and a homogeneous ferromagnetic layer. The hybrid structure not only is simple regarding the magnetization profile but also offers a tunable long-range spin-triplet supercurrent. Finally, we discuss some experimental aspects of our findings.

Figure 1

(Color online) (A) The schematic setup of a $\text{Ho}\mid \text{Co}\mid \text{Ho}$ trilayer. The spiral curves show the trajectory of the magnetization vector in the Ho layer which rotates along a conical profile in the $\widehat{x}$ direction. (i) shows the configuration in which the magnetization patterns of the two Ho layers are completely identical, whereas in (ii) the magnetization patterns follow a continuous spiral magnetization in the two Ho layers. (B) Schematic model of the experimental setup of our proposed ferromagnetic Josephson junction including a domain-wall and homogeneous ferromagnetic layer which enables a controllable triplet-supercurrent. The angle $\theta$ represents the orientation of the homogeneous magnetization in the F1 layer with respect to the $\widehat{z}$ direction.

Figure 2

(Color online) The normalized flowing critical charge current through three types of magnetization textures. Top, middle, and bottom frames show critical supercurrent through sandwiched uniform, domain-wall, and spiral ferromagnetic layers, respectively.

Figure 3

(Color online) The normalized critical supercurrent through a magnetic $\text{Ho}\mid \text{Co}\mid \text{Ho}$ trilayer vs the length of the Co layer for three values of Ho-layer lengths, $L_{\text{Ho}}=2$, 5, and 8 nm.

Figure 4

(Color online) The normalized critical supercurrent through a ferromagnetic trilayer of $\text{Ho}\mid \text{Co}\mid \text{Ho}$ vs the length of the Ho layer for three values of Co-layer lengths, $L_{\text{Co}}=2$, 5, and 8 nm.

Figure 5

(Color online) The normalized critical supercurrent through a trilayer of $\text{Ho}\mid \text{Co}\mid \text{Ho}$ vs the length of the Ho layer for two values of Co-layer lengths, $L_{\text{Co}}=2$ and 8 nm. In this case, we assume that the magnetization vector in Ho layers follow a continuous spiral pattern without being interrupted by the Co layer.

Figure 6

(Color online) The normalized critical supercurrent through sandwiched $\text{Bloch-wall}\mid \text{Co}\mid \text{Bloch-wall}$ trilayer vs length of Bloch-wall layers for three different values of Co-layer lengths, $L_{\text{Co}}=2$, 5, and 8 nm. Here magnetization direction of Co layer is along the $\widehat{z}$ axis.

Figure 7

(Color online) The normalized flowing critical supercurrent through our proposed $\text{Bloch-wall}\mid \text{homogeneous}$ ferromagnetic bilayer vs the length of homogeneous ferromagnetic layer $L_{F1}$ for four values of magnetic orientation angle of homogeneous ferromagnet with respect to $\widehat{z}$ direction, $\theta =0$, $\pi /6$, $\pi /3$, and $\pi /2$.