Spin-orbit coupling assisted magnetoanisotropic Josephson effect in ferromagnetic graphene Josephson junctions

The spin-flipping effect can be induced by the Rashba spin-orbit coupling (RSOC), leading to triplet equal-spin pairs in a superconducting hybrid structure. Herein, by combining the Dirac–Bogoliubov–de Gennes equation and the Furusaki-Tsukada formalism at a finite temperature, we theoretically investigate the Josephson effect in graphene-based superconductor-ferromagnet-R-superconductor junctions, where R refers to a region with the RSOC. It is demonstrated that as a result of the RSOC, one $0\text{−}\pi$ transition can be attained by tuning the orientation of the exchange field $\vec{h}$, which is determined by its magnitude $h$ that could be periodically taken, and the periodical $0\text{−}\pi$ transitions are also caused by manipulating $h$ during a considerable scope of the orientation. More interestingly, although varying the RSOC strength $\lambda$ cannot give rise to the $0\text{−}\pi$ transition in itself, not only is it a necessary condition for the $0\text{−}\pi$ transition induced by modulating the orientation of $\vec{h}$ but also can produce the shift of the crossover point. Furthermore, two different kinds of anomalous Josephson current effect are exhibited by controlling the orientation of $\vec{h}$. Particularly, the out-of- and in-plane magnetoanisotropic Josephson currents always exist, varying monotonically with $\lambda$ while nonmonotonically with $h$. The characteristics may provide more insights into the proximity-induced RSOC and pave the way to a new class of tunable superconducting spintronic devices based on large-scale graphene.

Figure 1

(a) Schematic of the graphene-based S/F/R/S junction, with F, R, and S denoting the ferromagnetic, RSOC, and superconducting regions, respectively. (b) The orientation of $\vec{h}$ presented by the polar angle $\theta$ and azimuthal angle $\phi$.

Figure 2

The current-phase relation $I\left(\varphi \right)$ at different $\theta$ in the $x\text{−}z$ plane ($\phi =0$), where $E_F=15{\mathrm{\Delta }}_0$, $L_1=0.2\xi$, $L_2=0.1\xi$, $\lambda =4{\mathrm{\Delta }}_0$, $h=4{\mathrm{\Delta }}_0$ (a) and $15{\mathrm{\Delta }}_0$ (b).

Figure 3

$I\left(\varphi \right)$ (a) for different $\theta$ in the $y\text{−}z$ plane ($\phi =-\pi /2$) and (b) for different $\phi$ in the $x\text{−}y$ plane ($\theta =\pi /2$), where $h=4{\mathrm{\Delta }}_0$ and the other parameters are the same as those in Fig. 2.

Figure 4

Spin-splitting dispersion of ABS $\pm {\varepsilon }_{\sigma }$ for different $h$ and $\theta$ of $\vec{h}$ in the $x\text{−}z$ plane ($\phi =0$) at the incident angle $\alpha =0$. Here, $\theta =0$ for (a) and (d), $\theta =0.2\pi$ for (b) and (e), $\theta =0.5\pi$ for (c) and (f), $h=4{\mathrm{\Delta }}_0$ for the top row, and $15{\mathrm{\Delta }}_0$ for the bottom row, and the other parameters are the same as those in Fig. 2.

Figure 5

Free energy $F\left(\varphi \right)$ for different $\theta$ in the $x\text{−}z$ plane ($\phi =0$), where (a) $h=4{\mathrm{\Delta }}_0$, (b) $h=15{\mathrm{\Delta }}_0$, and the other parameters are the same as those in Fig. 2.

Figure 6

The critical supercurrent $I_c^{\pm }$ (a) as a function of $\theta$ and $h$ at $L_1=0.2\xi$ and (b) a function of $\theta$ and $L_1$ at $h=4{\mathrm{\Delta }}_0$, where the other parameters are unchanged.

Figure 7

$I_c$ as a function of $\theta$ for $h=4{\mathrm{\Delta }}_0$ with the other parameters unchanged.

Figure 8

(a) The in-plane MAJC as a function of $\phi$ and (b) out-of-plane MAJC as a function of $\theta$ for different $h$. The other parameters are unchanged.

Figure 9

The in-plane MAJC as a function of $\phi$ for $h/E_F=1.5$ (a) and 1 (c), and the out-of-plane MAJC as a function of $\theta$ for the same $h$ (b) and (d) as in (a) and (c), respectively. Here, various $\lambda$ are marked in the figure and the other parameters are unchanged.