We theoretically study the valley- and spin-resolved scattering through magnetic barrier in a one layer thick silicene, using the mode-matching method for the Dirac equation. We show that the spin-valley filtering effect can be achieved and can also be tuned completely through both a top and bottom gate. Moreover, when reversing the sign of the staggered potential, we find the direction of the valley polarization is switched while the direction of spin polarization is unchanged. These results can provide some meaningful information to design valley valve residing on silicene.

We investigate the Cooper pair splitting based on crossed Andreev reflection in a two-dimensional narrow superconductor coupled to two leads, where the two leads are van der Waals (vdW) spin valves formed by inserting bilayer graphene (BLG) between two antiparallel insulating ferromagnetic layers. In this BLG-based $n$-type vdW spin valve/superconductor/$p$-type vdW spin valve junction with edge-contact geometry, when controlling the Fermi energy and the interlayer electric field in BLG independently by top- and bottom-gate voltages in the vdW spin valve region, we first demonstrate that the two electrons of perfect Cooper pair splitting can remain nonlocal valley entanglement. This effect can be explained as follows: When both the two antiparallel insulating ferromagnetic layers and the interlayer electric field are taken into account, BLG exhibits a spin-splitting but valley-degenerate band gap near the Dirac point. Thus, our setups allow the Cooper pair in the BCS scenario to serve as a natural source of valley entanglement. We further predict large oscillations of cross conductance as a function of the doping of the superconductor, the junction length, and the bias voltage. Our results may provide a tool for realizing fast nonlocal switching of the valley-entanglement state.

Based on the Bogoliubov–de Gennes equations, we provide an exact numerical solution for the critical current of Josephson junctions with a composite ferromagnetic bilayer. We demonstrate that for the antiparallel orientation of the magnetic moments of the bilayer, the presence of a potential barrier at the bilayer interface results in large oscillations of the critical current as a function of ferromagnet thickness and/or exchange field. Because of this, and remarkably, in the range of small exchange field and thicknesses, the magnetism leads to the increase of the critical current. This effect is well pronounced at low temperature but disappears near $T_c$. If the potential barrier is replaced by a spin-active barrier at the bilayer interface the conventional 0-$\pi$ transition, similar to the case of an uniform ferromagnetic Josephson junction, is observed. Strikingly, for a parallel orientation of the magnetic moments of the bilayer, the presence of the spin-active barrier restores the anomalous behavior—potential barrier in the antiparallel case. These behaviors result from the resonant tunneling of Cooper pairs across the composite barrier—an effect related to the spin-dependent Fermi vector in the presence of the ferromagnets' exchange field.

We demonstrate that a Josephson junction with a half-metallic weak link integrated into the superconducting loop enables the pumping of magnetic flux piercing the loop. In such a junction, the ground state phase ψ is determined by the mutual orientation of magnetic moments in two ferromagnets surrounding the half-metal. Thus, the precession of magnetic moment in one of the two ferromagnets controlled, e.g., by the microwave radiation, results in the accumulation of the phase ψ and subsequent switching between the states with different vorticities. The proposed flux pumping mechanism does not require the application of voltage or an external magnetic field, which enables the design of electrically decoupled memory cells in superconducting spintronics.

Based on the Bogoliubov–de Gennes equations, we investigate the transport of the Josephson current in a $\mathrm{S}/f_L-{\mathrm{F}}_1-f_C-{\mathrm{F}}_2-f_R/\mathrm{S}$ junction, where S and ${\mathrm{F}}_{1,2}$ are superconductors and ferromagnets, and $f_{L,C,R}$ are the left, central, and right spin-active interfaces. These interfaces have noncollinear magnetizations, and the azimuthal angles of the magnetizations at the $f_{L,C,R}$ interfaces are ${\chi }_{L,C,R}$. We demonstrate that, if both the ferromagnets have antiparallel magnetizations, the critical current oscillates as a function of the exchange field and the thickness of the ferromagnets for particular ${\chi }_L$ or ${\chi }_R$. By contrast, when the magnetization at the $f_C$ interface is perpendicular to that at the $f_L$ and $f_R$ interfaces, the critical current reaches a larger value and is hardly affected by the exchange field and the thickness. Interestingly, if both the ferromagnets are converted to antiparallel half-metals, the critical current maintains a constant value and rarely changes with the ferromagnetic thicknesses and the azimuthal angles. At this time, an anomalous supercurrent can appear in the system, in which case the Josephson current still exists even if the superconducting phase difference $\varphi$ is zero. This supercurrent satisfies the current-phase relation $I=I_{c}sin(\varphi +{\varphi }_0)$ with $I_c$ being the critical current and ${\varphi }_0=2{\chi }_C-{\chi }_L-{\chi }_R$. We deduce that the additional phase ${\varphi }_0$ arises from phase superposition, where the phase is captured by the spin-triplet pairs when they pass through each spin-active interface. In addition, when both the ferromagnets are transformed into parallel half-metals, the $f_C$ interface never contributes any phase to the supercurrent and ${\varphi }_0={\chi }_R-{\chi }_L+\pi$. In such a case, the current-phase relation is similar to that in a $\mathrm{S}/f_L-\mathrm{F}-f_R/\mathrm{S}$ junction.

We study the distinctive features of the phase-pumping effect in Josephson transport through a three-layered ferromagnet ${\mathrm{F}}_1/\mathrm{F}/{\mathrm{F}}_2$ with noncoplanar magnetization. Using Gor'kov and Bogoliubov–de Gennes formalisms we go beyond the quasiclassical approximation and analyze the dependence of the spontaneous Josephson phase $\psi$ on the exchange field $h$ in the F layer and details of magnetization profile. The pumping of the Josephson phase can be generated by the mutual rotation of magnetizations in ${\text{F}}_1$ and ${\text{F}}_2$ layers resulting in the nontrivial phase gain at the rotation period (Berry phase). The increase in $h$ is shown to cause changes in the topology of the phase evolution: the gain of the Josephson phase at the pumping period switches from 0 to $2\pi$. We study the scenario of these switchings originating from the interplay between several competing local minima of the free energy of the junction versus the superconducting phase difference. Our analysis provides the basis for the search of experimental setup realizing the phase-pumping phenomenon.