We investigate the effects of the circularly polarized light (CPL) and the electric field (EF) on the nonlocal transport in a silicene-based antiferromagnet/superconductor/ferromagnet (AF/S/F) asymmetrical junction. For case I (II), the CPL and the EF are applied simultaneously in the antiferromagnetic (ferromagnetic) region, whereas in the ferromagnetic (antiferromagnetic) region, only a constant EF is considered. The spin-valley-resolved conductance can be turned on or off by adjusting the CPL or the EF. The AF/S/F junction can be manipulated as a spin-locked valley filter for case I, while for case II, it can be used not only as a valley-locked spin filter but also as a nonlocal switch between two pure nonlocal processes. Such interesting nonlocal switch effect can be effectively controlled by reversing the direction of the incident energy axis, the handedness of the CPL, or the direction of the EF. These findings may open an avenue to the design and manufacture of the spintronic and valleytronic devices based on the asymmetrical silicene magnetic superconducting heterostructure.

## I. INTRODUCTION

As a counterpart of graphene for silicon, silicene consists of a monolayer of silicon atoms forming a two-dimensional hexagonal lattice and attracts great attention^{1,2} due to its unique physical properties and superior compatibility with modern Si-based electronics.^{3} Silicene and graphene have similar band structures, and their low-energy spectra obey the relativistic Dirac equation.^{4} Silicene possesses a stronger intrinsic spin–orbit interaction (SOI), which is about 1000 times greater than that of graphene.^{3,5} The SOI results not only in a gap in the low-energy band structure but also in a coupling between the spin-valley degrees of freedom.^{6,7} Moreover, silicene has a slight buckled structure because silicon atoms are close to bond in a mixed $s p 2$– $s p 3$ hybridization rather than an exclusive $s p 2$ hybridization.^{8} The silicene’s buckled structure can give rise to the modulation of its bandgap by external fields such as perpendicular electric field (PEF),^{9,10} ferromagnetic or antiferromagnetic exchange field,^{11–13} and off-resonant circularly polarized light (OCPL).^{14,15} Closing and reopening this bandgap leads to a transition between different topological phases, which exerts a crucial role in the modern condensed matter physics.^{16,17}

The conventional singlet superconductor (S) has been regarded as a natural source for entangled electrons^{18} because Cooper pairs are comprised of two electrons with momentum and spin entangled. The Cooper pair splitting (CPS) process belongs to the inverse crossed Andreev reflection (CAR).^{19} The CAR is a nonlocal process of transforming an electron or a hole from the one of leads to a hole or an electron in the other lead through an S.^{20} In the nonlocal transport, the CAR process often competes with the elastic co-tunneling (CT) process, thus many proposals have been devoted to enhance the CAR. A superconductor–graphene double quantum dot system has been studied experimentally by Tan *et al.*,^{21} and they observed 10% CPS efficiency. Li and Zhang^{22} have studied the quantum transport in the silicene–superconductor junction and found that the CPS efficiency is close to 50% for the middle (M) region covered by a single S and is about 100% for two edges of the M region covered by two Ss. The pure CAR and CT processes can be observed and be tuned by the Fermi level,^{23} the PEF,^{24,25} the ferromagnetic exchange field (FEF),^{12,26} and the handedness of the OCPL.^{27,28}

Antiferromagnet (AF) has attracted tremendous research interest due to some novel features such as zero net magnetization, negligible stray fields, insensitivity to disturbing magnetic fields, intrinsic high-frequency dynamics, and not emitting an external field to its surroundings.^{29,30} Zhai *et al.*^{31} have studied the giant thermal magnetoresistance effect in the ferromagnet/antiferromagnet (F/AF) junction and shown that the giant Seebeck magnetoresistance effect sensitively depends on the PEF. A transition from the spin to the valley filtering effect in the silicene-based AF/F junction has been reported by Niu.^{32} The coexistence of the AF with the S provides a significant potential for the application of spintronics. Based on the Bogoliubov–de Gennes (BdG) equation, the nonlocal transport through the electrically controlled AF/S/AF junction has been investigated theoretically in Refs. 29 and 33. It is demonstrated that the highly electrically controllable CPS efficiency can be achieved in the antiferromagnetic device.

In this paper, we theoretically investigate the nonlocal transport properties in an AF/S/F asymmetrical junction, which is modulated by the OCPL and the PEF. The comparative results of two cases have been discussed. It is revealed that a transition of the spin-valley indices occurs by tuning the handedness of the OCPL. Different from case I, in case II, the antiferromagnetic exchange field (AEF) can efficiently modulate the CAR and CT signals. Moreover, a nonlocal switch effect between two pure nonlocal processes can be observed in case II by reversing the direction of the incident energy axis, the handedness of the OCPL in the ferromagnetic region, or the direction of the PEF in the ferromagnetic region. A nonlocal switch, a spin-locked valley filter, and a valley-locked spin filter can be simultaneously achieved in the proposed system.

## II. MODEL AND FORMULAS

As shown in Fig. 1, we consider an AF/S/F asymmetrical junction based on an infinite silicene sheet occupying the $x$– $y$ plane, where the superconducting layer occupies the M region ( $0<x<d$), while the antiferromagnetic and ferromagnetic layers cover the left (L) ( $x<0$) and right (R) ( $x>d$) regions, respectively. A gate voltage $ U 0$ is applied to the M region to tune its chemical potential. The current flows along with the $x$-axis. To tune bandgap, both the OCPL and the PEF are applied in the L and R regions.

^{34}When the off-resonant condition

^{29}

^{,}$\u210f\Omega \u226b t 0$ ( $ t 0=\u210f v F/a$ is the nearest-neighbor hopping parameter with $ v F=5.5\xd7 10 5$ m/s and $a\u22483.86\xc5$ being the Fermi velocity

^{5}and the lattice constant

^{35}) is satisfied, we can reduce the influence of time-related electromagnetic potential to a static effective Hamiltonian by using the Floquet theory.

^{36,37}With the limit of $eA v F/\u210f\Omega \u226a1$ ( $e$ is the electron charge), the Dirac BdG equation can be expressed in the following form:

^{12,33,38,39}

^{9,40}$\eta =\xb11$ represents the valley index with $ \eta \xaf=\u2212\eta $, $\sigma =\xb11$ denotes the spin index with $ \sigma \xaf=\u2212\sigma $, $ \lambda S O$ is the intrinsic spin–orbit coupling strength, $ k ^ \alpha (\alpha =x,y,z)$ is the wave vector operator, $ \tau ^ \alpha (\alpha =x,y,z)$ is the $2\xd72$ Pauli matrix, $ \tau ^ 0$ is a $2\xd72$ unit matrix, and $u$ and $v$ are the electron-like and hole-like quasiparticle wavefunctions, respectively. $ \Delta Z(x)= \Delta Z j$ denotes the field-tunable on-site potential difference between two sublattices in the $j$th region with $j=L,M,$ and $R$, originating from the buckled structure. The chemical potential $\mu (x)$ is $ \mu M= \mu F+ U 0$ in the M region and $ \mu F$ in the L and R regions, and they can be tuned independently by the local gate voltage. Taking an abrupt approximation,

^{12}the FEF $h(x)$ is given by $h(x)= h 0\Theta (x\u2212d)$, where $ h 0$ denotes the ferromagnetic exchange energy and $\Theta (x)$ is the Heaviside function. The AEF $ \lambda A F(x)$ can be described as $ \lambda A F(x)= \lambda A F\Theta (\u2212x)$, with $ \lambda A F$ being the antiferromagnetic exchange energy. $ \lambda \Omega (x)= \lambda \Omega j$ is the effective energy to describe the OCPL, where $ \lambda \Omega j<0$ ( $>0$) corresponds to the L-handed (R-handed) OCPL. To ensure the validity of the mean-field approximation, the Fermi wavelength in the M region is much smaller than both the superconducting coherence length and the Fermi wavelength in the L and R regions so that the leakage of Cooper pairs is negligibly small.

^{9,12}In doing so, the superconducting pair potential $\Delta (x)$ can be described by a step function model, i.e., $\Delta (x)= \Delta 0 e i \varphi \Theta (x)\Theta (d\u2212x)$, where $ \Delta 0$ and $\varphi $ are the amplitude and phase of the pair potential.

^{41}According to Eqs. (5) and (6), by using the conservation of the probability current along the $x$ direction, the corresponding incident probability current of electron is $ J x i= 2 v F k x L e \tau L e +$, while the CAR and CT probability currents are $ J x C A R= 2 v F R e ( k x R h ) \tau R h \u2212| t \eta \sigma e h | 2$ and $ J x C T= 2 v F R e ( k x R e ) \tau R e +| t \eta \sigma e e | 2$, respectively. Thus, the CAR and CT probabilities can be, respectively, expressed as

^{42}the spin-valley-resolved conductances (SVRCs) of the CAR and CT processes at zero-temperature can be formulated as

^{12,27,43}

Based on the condition of the presence of the propagating incident modes, Fig. 2(a) shows the distribution for the four kinds of the spin-valley polarized states (SVPSs) within the subgap energy regime $|E|\u2264 \Delta 0$ in the ( $ \lambda \Omega L$, $ \Delta Z L$) parameter space, when $ \lambda A F=0.2 \Delta 0$ and $ \lambda S O=4.0 \Delta 0$. The distributions for the SVPSs of the CAR and CT processes with $|E|\u2264 \Delta 0$ in the ( $ \lambda \Omega R$, $ \Delta Z R$) parameter space are, respectively, plotted in Figs. 2(b) and 2(c), on account of the condition of the generation of the final modes. The colored areas indicate the regions for the four kinds of the SVPSs, while the white areas denote the regions where there are no propagating modes. It is seen from Eq. (2) that the PEF, the AEF, the OCPL, and the FEF can break the spin-valley degeneracies, resulting in the emergence of the SVPSs. The SVPSs are distributed in different areas and are crossed in pairs. The crossed areas represent the coexistence of the two kinds of the SVPSs. The specific SVPS can be obtained by tuning the OCPL and the PEF. In the ( $ \lambda \Omega L$, $ \Delta Z L$) parameter space, the widths of the areas for the four SVPSs are equal and are $ 2 \Delta 0$, whereas in the ( $ \lambda \Omega R$, $ \Delta Z R$) parameter space, this width is $ 2( \Delta 0+ h 0)$ for an incident electron with spin-up states and $ 2( \Delta 0\u2212 h 0)$ for an incident electron with spin-down states. We can also see from Figs. 2(b) and 2(c) that two pure nonlocal processes can be achieved and can also be modulated by the OCPL or the PEF. The reason for the emergence of the pure nonlocal processes is due to the impact of the FEF.

## III. RESULTS AND DISCUSSION

In this section, we present the numerical results and compare the nonlocal transport properties of the AF/S/F asymmetrical junction for two different situations. For case I, both the OCPL and the PEF are applied simultaneously in the L region (i.e., $ \lambda \Omega L\u22600$ and $ \Delta Z L\u22600$), while in the R region, the OCPL is not considered and the PEF is finite (i.e., $ \lambda \Omega R=0$ and $ \Delta Z R\u22600$). For case II, only the PEF is exerted to the L region (i.e., $ \lambda \Omega L=0$ and $ \Delta Z L\u22600$), while in the R region both the OCPL and the PEF are presented (i.e., $ \lambda \Omega R\u22600$ and $ \Delta Z R\u22600$). In the numerical calculation, we choose parameters $ \Delta 0=1.0$ meV, $ \lambda S O=4.0 \Delta 0$, $ \mu F=0$, $ h 0=0.2 \Delta 0$, $ \lambda \Omega M=0$, and $ \Delta Z M=0$. The width $d$ of the M region is in units of the superconducting coherence length $\xi =\u210f v F/ \Delta 0$. The phase $\varphi $ of the pair potential is set as zero, and its value does not affect our results.^{44} The relevant parameters are set as follows unless otherwise indicated: $ \lambda A F=0.2 \Delta 0$, $ U 0=50 \Delta 0$, and $d=0.7\xi $.

In Figs. 4 and 5, we present the SVRCs $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ as functions of the incident energy $E$. Figure 4 corresponds to the case I for different (a) and (b) $d$, (c) and (d) $ \lambda A F$, and (e) and (f) $ \Delta Z L$, where the related parameters are $ \lambda \Omega L=2.0 \Delta 0$, $ \Delta Z L=2.0 \Delta 0$, and $ \Delta Z R=4.0 \Delta 0$, unless otherwise specified in the figure. Figure 5 corresponds to the case II for different (a) and (b) $d$, (c) and (d) $ \lambda A F$, and (e) and (f) $ \Delta Z R$, where the related parameters are $ \Delta Z L=4.0 \Delta 0$, $ \lambda \Omega R=2.0 \Delta 0$, and $ \Delta Z R=2.0 \Delta 0$, unless otherwise specified in the figure. The considered region of $E$ only exists the $K\u2193$ and $ K \u2032\u2191$ hole states and the $K\u2191$ and $ K \u2032\u2193$ electron states, while the other hole and electron states do not contribute to the nonlocal transport. This feature can be explained by the band structures, which is shown in Fig. 3. For case I with $\u22126.0 \Delta 0\u2264E\u22646.0 \Delta 0$, the incident electron may come from the $K\u2191$, $ K \u2032\u2191$, or $ K \u2032\u2193$ states, but only the $K\u2193$ and $ K \u2032\u2191$ ( $K\u2191$ and $ K \u2032\u2193$) hole (electron) bands in the R region appear, as shown in Figs. 3(a)–3(c). For case II with $\u22126.0 \Delta 0\u2264E\u22646.0 \Delta 0$, the incident electron may come from the $K\u2191$ or $ K \u2032\u2193$ states, while the transmitted hole (electron) with the $K\u2191$, $K\u2193$, and $ K \u2032\u2191$ ( $K\u2191$, $ K \u2032\u2191$, and $ K \u2032\u2193$) states appear. In the absence of the spin- and valley-flip scattering, the $K\u2191$ hole and $ K \u2032\u2191$ electron bands in the R region do not contribute to the nonlocal transport. Moreover, there exists zero $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ energy regions for cases I and II; this happens due to the occurrence of the evanescent modes in the L and R regions. Interestingly, the widths of the zero $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ energy regions are different under different parameters. This phenomenon can be best understood from the band structures. As indicated by the band structures plotted in Figs. 3(a) and 3(d), the transmission gap of the incident electron with spin $\sigma $ and valley $\eta $ satisfies the condition $\u2212| m \eta \sigma L e|<E<| m \eta \sigma L e|$, which means that there are no transport channels and such transmission gap can be effectively tuned by the PEF, the AEF, and the OCPL of the L region. For case I, the zero $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ energy regions are determined by the transmission gap of the incident electron due to the gapless and coincidence of the electron and hole bands in the R region, where the transmission gap $2| m \eta \sigma L e|$ for an incident electron with $K\u2191$ states is located in the range $E\u2208[\u22120.2 \Delta 0,0.2 \Delta 0]$ and that for an incident electron with the $ K \u2032\u2191$ or $ K \u2032\u2193$ states is located in the range $E\u2208[\u22124.2 \Delta 0,4.2 \Delta 0]$, as shown in Figs. 3(a)–3(c). Different from case I, only the $ K \u2032\u2191$ transmitted hole and $K\u2191$ transmitted electron bands are gapless for case II, thus the energy regions of the zero $ G K \u2032 \u2191 C A R$ and $ G K \u2191 C T$ are determined by the transmission gap of the incident electron [see Figs. 3(d)–3(f)], whereas the $K\u2193$ transmitted hole and $ K \u2032\u2193$ transmitted electron bandgaps occur in the regions of $\u2212| m \eta \sigma R h|\u2212\sigma h 0<E<| m \eta \sigma R h|\u2212\sigma h 0$ ( $\u22124.2 \Delta 0<E<3.8 \Delta 0$) and $\u2212| m \eta \sigma R e|\u2212\sigma h 0<E<| m \eta \sigma R e|\u2212\sigma h 0$ ( $\u22123.8 \Delta 0<E<4.2 \Delta 0$). The transmitted hole and electron bandgaps can be tuned by the PEF, the OCPL, and the FEF of the R region and are robust against the width of the M region and the AEF. We can also see from Figs. 4 and 5 that the maximum value of $ G K \u2032 \u2191 C A R$ ( $ G K \u2193 C A R$) is smaller than that of $ G K \u2193 C A R$ ( $ G K \u2032 \u2191 C A R$) in case I (II). This indicates that the $K\u2193$ and $ K \u2032\u2191$ states dominate the hole transport in cases I and II, respectively. The maximum value of $ G K \u2032 \u2193 C T$ is smaller compared with $ G K \u2191 C T$ in cases I and II, which means that the $K\u2191$ states dominate the electron transport in cases I and II.

With increasing $d$, both $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ exhibit significant oscillations and the number of oscillation peaks increases, arising from the quantum interference effects of quasiparticle in the M region.^{45} The peak values of $ G K \u2193 C A R$ for case I and $ G K \u2032 \u2191 C A R$ for case II increase with $d$ and approach to their maximum values at $d\u2248\xi $, which can be seen in Figs. 4(a) and 5(a).^{12} As $ \lambda A F$ increases, the widths of the energy regions for zero $ G K \u2193 C A R$ in Fig. 4(c) and zero $ G K \u2032 \u2191 C A R$ in Fig. 5(c) are broaden gradually. The feature of zero $ G K \u2191 C T$ energy region in Figs. 4(d) and 5(d) is consistent with the results of the zero $ G K \u2193 C A R$ and zero $ G K \u2032 \u2191 C A R$ energy regions referred to above, whereas the width of the energy region for zero $ G K \u2032 \u2193 C T$ in Fig. 5(d) is robust against the change of $ \lambda A F$, which is different from Fig. 4(d). We can also see from Fig. 4(c) that the peak value of $ G K \u2193 C A R$ decreases with $ \lambda A F$ in the positive energy region, while it increases slightly in the negative energy region. In Fig. 5(c), the peak value of $ G K \u2032 \u2191 C A R$ increases significantly with $ \lambda A F$, which is much larger than that of $ G K \u2193 C A R$ in Fig. 4(c). This means that a larger CAR conductance can be easily obtained in case II by changing $ \lambda A F$. Different from Figs. 4(e) and 4(f), the widths of the zero $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ energy regions in Figs. 5(e) and 5(f) become narrow with the increase in $ \Delta Z R$, and a nonlocal switch effect between two pure nonlocal processes can be observed in case II by reversing the direction of the incident energy axis.

In order to study the influence of the OCPL on the nonlocal transport, the SVRCs $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ of cases I and II are plotted in Figs. 6 and 7 as functions of the OCPL $ \lambda \Omega L$ ( $ \lambda \Omega R$) for different (a) and (b) $d$, (c) and (d) $ \lambda A F$, and (e) and (f) $ \Delta Z L$ ( $ \Delta Z R$). The incident energy is fixed at $E=1.0 \Delta 0$. The other parameters in Figs. 6 and 7 are the same as in Figs. 4 and 5, respectively. For considered region of the OCPL, there only exists the $K\u2193$ and $ K \u2032\u2191$ hole and $K\u2191$ and $ K \u2032\u2193$ electron states, and these two SVPSs occur in the separated region of the OCPL which means that the OCPL plays the role of a spin-valley valve that is off and on depending on its value. $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ exhibit a conversion of the spin-valley indices by modulating the handedness of the OCPL. Figure 6 shows that only $ G K \u2193 C A R$ and $ G K \u2191 C T$ exist when the R-handed OCPL is irradiated, while when the L-handed OCPL is irradiated they are suppressed completely and $ G K \u2032 \u2191 C A R$ and $ G K \u2032 \u2193 C T$ are nonzero. This is different from Fig. 7, where $ G K \u2193 C A R$ and $ G K \u2032 \u2193 C T$ are vanishing and $ G K \u2032 \u2191 C A R$ and $ G K \u2191 C T$ are finite for the R-handed OCPL, while for the L-handed OCPL there only exists $ G K \u2193 C A R$ and $ G K \u2032 \u2193 C T$. This can be clearly seen from the band structures of Figs. 3 that for case I with the R-handed OCPL and the subgap energy, the incident electron comes from the $K\u2191$ states, and the $K\u2193$ and $ K \u2032\u2191$ hole and $K\u2191$ and $ K \u2032\u2193$ electron bands in the R region are present in the full energy regime. Due to the absence of the spin-flip process, the spin-up hole and spin-down electron bands do not contribute to the subgap transport. Thus, the CAR and CT processes are presented by the $K\u2193$ and $K\u2191$ states, respectively [see Fig. 6], whereas for case II with the R-handed OCPL and the subgap energy, the incident electron may come from the $K\u2191$ or $ K \u2032\u2193$ states and the transmitted hole (electron) with $ K \u2032\u2191$ ( $K\u2191$) states appears. So the CAR and CT processes are presented by the $ K \u2032\u2191$ and $K\u2191$ states, respectively [see Fig. 7]. The situation of the L-handed OCPL can be analyzed in the same way. Besides, in Fig. 6 both the CAR and CT processes appear in the OCPL region $\u2212E\u2212 \Delta Z L+ \lambda S O\u2212\sigma \lambda A F\u2264\eta \lambda \Omega L\u2264E\u2212 \Delta Z L+ \lambda S O\u2212\sigma \lambda A F$, so their region widths depend on just the incident energy and are $2E$, whereas the OCPL regions of the CAR and CT processes in Fig. 7 satisfy $\u2212E+ \Delta Z R\u2212 \lambda S O\u2212\sigma h 0\u2264\eta \lambda \Omega R\u2264E+ \Delta Z R\u2212 \lambda S O+\sigma h 0$ and $\u2212E\u2212 \Delta Z R+ \lambda S O\u2212\sigma h 0\u2264\eta \lambda \Omega R\u2264E\u2212 \Delta Z R+ \lambda S O+\sigma h 0$, respectively. The widths of the OCPL regions depend not only on the incident energy but also on the FEF, which can be expressed as $2(E+\sigma h 0)$ with an incident electron with spin $\sigma $. Such a difference for the width of the OCPL region between the CAR and CT processes implies that unlike case I, for case II the pure CAR and CT processes can be, respectively, achieved in the L- and R-handed OCPL regions without changing the other parameters.

In Figs. 6 and 7, the peak value of $ G \eta \sigma C A R$ increases gradually as $d$ increases, while the value of $ G \eta \sigma C T$ decreases continually. As $ \Delta Z L$ or $ \Delta Z R$ increases, the peak values of $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ are almost invariant and their positions shift toward the weaker OCPL. It is worth noting that with increasing $ \lambda A F$, the peak values of $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ in Fig. 6 remain almost unchanged, but their positions move to the L-handed OCPL direction. In Fig. 7, the peak values of $ G K \u2193 C A R$ and $ G K \u2191 C T$ decrease and of $ G K \u2032 \u2191 C A R$ and $ G K \u2032 \u2193 C T$ increase with $ \lambda A F$, whereas the location of the peak is robust against the change of parameter $ \lambda A F$. The results indicate that the intensity of the CAR signal in case II can be easily tuned by the AEF, which is consistent with the results of Fig. 5.

Next, the SVRCs $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ are plotted as functions of the gate voltage $ U 0$ for different (a) and (b) $d$, (c) and (d) $ \lambda A F$, and (e) and (f) $ \Delta Z L$ or $ \Delta Z R$. The results are depicted in Figs. 8 and 9 with cases I and II, respectively. The incident energy is $E=1.0 \Delta 0$. The other parameters in Figs. 8 and 9 are the same as in Figs. 4 and 5, respectively. In the considered region of $ U 0$, $ G \eta \sigma C A R$ is contributed by the $K\u2193$ states in case I and the $ K \u2032\u2191$ states in case II, while $ G \eta \sigma C T$ is contributed by the $K\u2191$ states for cases I and II. This result is determined by the choice of values for the OCPL and the PEF, which could be best understood in Figs. 2 and 3. Moreover, the related explanation is also given by the analytical results in Figs. 6 and 7. $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ show a periodic oscillatory behavior with $ U 0$, attributing to the quantum interference effects inside the M region. With increasing $d$, the number of the oscillations peaks for $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ increases and the value of $ G K \u2191 C T$ decreases. In Figs. 8(c)–8(f), the values of $ G K \u2193 C A R$ and $ G K \u2191 C T$ decrease and their peak-to-valley ratios also slightly decrease or remain unchanged by increasing $ \lambda A F$ or $ \Delta Z L$. The important difference from Figs. 8(c) and 8(d) is that in Figs. 9(c) and 9(d), the peak value and the peak-to-valley ratio of $ G K \u2032 \u2191 C A R$ increase significantly with increasing $ \lambda A F$, while $ G K \u2191 C T$ is reduced and its peak-to-valley ratio is enhanced. Such a character means that by tuning $ \lambda A F$, the AF/S/F asymmetrical junction for case I suppresses the CAR and CT signals, while for case II, the CAR signal is enhanced and the CT signal is suppressed. Figures 9(e) and 9(f) show that $ G K \u2032 \u2191 C A R$ and $ G K \u2191 C T$ are weakly dependent on $ \Delta Z R$, which is comparable with Figs. 8(e) and 8(f).

In order to study the combined influence of the OCPL and the PEF on the nonlocal transport, we show the contour plots of the SVRCs $ G \eta \sigma C A R$ (first row) and $ G \eta \sigma C T$ (second row) for cases I and II in Figs. 10 and 11, respectively. The first, second, third, and fourth columns of Fig. 10 correspond to $ \Delta Z R=3.0 \Delta 0$, $5.0 \Delta 0$, $\u22123.0 \Delta 0$, and $\u22125.0 \Delta 0$, respectively; while the first, second, third, and fourth columns of Fig. 11 correspond to $ \Delta Z L=3.0 \Delta 0$, $5.0 \Delta 0$, $\u22123.0 \Delta 0$, and $\u22125.0 \Delta 0$, respectively. The incident energy is $E=1.0 \Delta 0$. The other parameters in Figs. 10 and 11 are taken the same values as those in Figs. 4 and 5, respectively. In Fig. 10, $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ can be observed simultaneously in the same area. The reason is that the propagating condition of the CAR and CT processes is equal and satisfies $E\u2265| \Delta Z R\u2212\eta \sigma \lambda S O|\u2212\sigma h 0$. The area width of $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ is robust against the SVPS and the PEF of the R region. From Figs. 10(a) and 10(b), we can find that for the cases of $ \Delta Z R=3.0 \Delta 0$ and $5.0 \Delta 0$, $ G K \u2193 C A R$ and $ G K \u2191 C T$ exactly occur in the area of the $K\u2191$ state for Fig. 2(a), and other SVPSs are filtered. The scenarios for the cases of $ \Delta Z R=\u22123.0 \Delta 0$ and $\u22125.0 \Delta 0$ have similar results, where $ G K \u2032 \u2193 C A R$ and $ G K \u2032 \u2191 C T$ exactly happen in the area of the $ K \u2032\u2191$ state for Fig. 2(a) [see Figs. 10(c) and 10(d)]. This indicates that by reversing the direction of $ \Delta Z R$, there exists a spin-locked valley filter, where $ G K \u2193 C A R$ and $ G K \u2191 C T$ exist for $ \Delta Z R>0$ and $ G K \u2032 \u2193 C A R$ and $ G K \u2191 C T$ exist for $ \Delta Z R<0$, while the nonlocal transport in other states is completely suppressed. Comparing Figs. 10(a) and 10(b), a strong CAR signal can occur in a wider range of the ( $ \lambda \Omega L$, $ \Delta Z L$) plane when $ \Delta Z R$ is low. Changing the direction of $ \Delta Z R$, similar results can also be obtained, as discussed in Figs. 10(c) and 10(d).

As shown in Fig. 11, the area occupied by $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ is in agreement with the region of the SVPSs of Figs. 2(b) and 2(c). The area of $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ for an incident electron with spin-up states is wider than that for an incident electron with spin-down states due to the effect of the FEF. $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ with the same SVPS are crossed because the propagating conditions of the CAR and CT processes are $E\u2265| m \eta \sigma R h|\u2212\sigma h 0$ and $E\u2265| m \eta \sigma R e|\u2212\sigma h 0$, respectively. These phenomena are consistent with those in Fig. 2. From Fig. 11(a), we find that for $ \Delta Z L=3.0 \Delta 0$, $K\u2191$ electron enters the system, where the CAR and CT conductances appear at the areas of the $K\u2193$ and $K\u2191$ states in Figs. 2(b) and 2(c), respectively. Accordingly, the CAR conductance is the $K\u2193$ polarized while the CT conductance is the $K\u2191$ polarized. In the cross area of the $K\u2191$ electron and $K\u2193$ hole states, the coexistence of the CAR and CT processes. While in the uncrossed area, the pure CAR and CT processes can be observed in the appropriate parameters and be switched by reversing the handedness of the OCPL in the R region or the direction of the PEF in the R region. The scenarios for other values of $ \Delta Z L$ have similar results. A comparison of Figs. 11(a) with 11(c) or Figs. 11(b) with 11(d) shows that a valley-locked spin filter can be obtained by changing the direction of the PEF in the L region, where the one valley one (other) spin state devotes to the nonlocal transport for $ \Delta Z L>0$ ( $<0$) and the other valley has no contribution. Moreover, there exists a transition of the spin-valley indices, i.e., one spin and one valley state switches to other spin and other valley state, when the magnitude of $ \Delta Z L$ changes and its direction unchanged [see Figs. 11(a) and 11(b) or Figs. 11(c) and 11(d)]. This demonstrates that a perfect spin-valley filtering effect can be achieved and be modulated by the PEF in the L region. A strong CAR signal happens in a wider range of the ( $ \lambda \Omega R$, $ \Delta Z R$) plane when $ \Delta Z L$ is high, which is opposite of the results in Fig. 10. The CAR and CT signals for Fig. 11 are enhanced significantly comparing to Fig. 10, which means that the AF/S/F asymmetrical junction for case II can be as a promising candidate to obtain a strong CAR signal.

Since the above discussion is the case of the undoped silicene sheet, it is necessary to further investigate the effect of the chemical potential $ \mu F$ on the nonlocal transport properties. In Figs. 12 and 13, the SVRCs $ G \eta \sigma C A R$ and $ G \eta \sigma C T$ of cases I and II are, respectively, shown as functions of $ \mu F$ for different (a) and (b) $d$, (c) and (d) $ \lambda A F$, and (e) and (f) $ \Delta Z L$ or $ \Delta Z R$. The incident energy is $E=1.0 \Delta 0$. The other parameters in Figs. 12 and 13 are same as these in Figs. 4 and 5, respectively. The considered region of $ \mu F$, only the $K\u2193$ and $ K \u2032\u2191$ hole and $K\u2191$ and $ K \u2032\u2193$ electron states, contributes to the nonlocal transport. Consistent with the result of the undoped silicene sheet, the $K\u2193$ ( $K\u2191$) states dominate the hole (electron) transport in case I, while in case II the $ K \u2032\u2191$ ( $K\u2191$) states dominate the hole (electron) transport. The number of oscillations peaks of the SVRCs for the dominated CAR and CT processes increases with increasing $d$. As shown in Fig. 12, $ G K \u2193 C A R$ decreases almost linearly with $ \mu F$, approaches to zero at $ \mu F=E+ h 0$, and then begins to increase in an oscillatory manner. For undoped silicene $ \mu F=0$, $ G K \u2193 C A R$ is suppressed by increasing $ \lambda A F$ or $ \Delta Z L$, while for doped silicene $ \mu F=6.0 \Delta 0$, it is enhanced by $ \lambda A F$ and suppressed by $ \Delta Z L$. For other cases of the doped silicene, $ G K \u2193 C A R$ is almost independent of the AEF and the PEF of the L region. $ G K \u2191 C T$ oscillates irregularly with $ \mu F$ and becomes gentler for higher $ \mu F$. $ G K \u2191 C T$ decreases initially by increasing $ \lambda A F$ or $ \Delta Z L$ and then does not show any sensitivity for higher $ \mu F$. In the region of $0< \mu F<| m \eta \sigma L e|\u2212E$, since $ \mu F$ lies in the electron bandgap with $ K \u2032\u2193$ states, there is no transport channels, resulting in zero $ G K \u2032 \u2191 C A R$ and $ G K \u2032 \u2193 C T$. This electron bandgap becomes wider with increasing $ \lambda A F$ or decreasing $ \Delta Z L$, which means that the zero $ G K \u2032 \u2191 C A R$ and $ G K \u2032 \u2193 C T$ regions can be effectively tuned by the AEF and the PEF of the L region, while they are robust against change of the value of $d$. Outside the region of $0< \mu F<| m \eta \sigma L e|\u2212E$, $ G K \u2032 \u2191 C A R$ shows the oscillating behavior near zero, whereas $ G K \u2032 \u2193 C T$ increases with $ \mu F$ initially and then gradually saturates. For higher value of $ \mu F$, $ G K \u2032 \u2191 C A R$ and $ G K \u2032 \u2193 C T$ do not depend on the AEF and the PEF of the L region.

In Fig. 13, $ G K \u2032 \u2191 C A R$ decreases almost linearly with $ \mu F$, reaching a minimum in the quasi-hole bandgap, and then begins to show an oscillatory behavior, approaching zero for higher $ \mu F$. This quasi-hole bandgap becomes narrow and gradually disappears with increasing $ \Delta Z R$, and it satisfies $\u2212| m \eta \sigma R h|+E+\sigma h 0< \mu F<| m \eta \sigma R h|+E+\sigma h 0$, which means that such quasi-hole bandgap can be modulated by changing the PEF and the OCPL of the right region. The oscillating behavior of $ G K \u2191 C T$ for case II is analogous to that for case I. Different from Fig. 12, $ G K \u2193 C A R$ and $ G K \u2032 \u2193 C T$ of Fig. 13 are, respectively, suppressed completely in the regions of $0< \mu F<| m \eta \sigma R h|+E+\sigma h 0$ and $0< \mu F<| m \eta \sigma R e|\u2212E\u2212\sigma h 0$ since $ \mu F$ lies in the $K\u2193$ hole and $ K \u2032\u2193$ electron bandgaps of the R region. The electron and hole bandgaps become narrow with the increase in $ \Delta Z R$, but they are robust against the variations in $d$ and $ \lambda A F$. Outside the hole bandgap, $ G K \u2193 C A R$ is gradually enhanced as $d$, $ \lambda A F$, or $ \Delta Z R$ increases. Outside the electron bandgap, $ G K \u2032 \u2193 C T$ increases monotonously and then tends to saturate with increasing $ \mu F$, while it is suppressed with increasing $d$ and is nearly independent of $ \lambda A F$ and $ \Delta Z R$.

As far as the realistic realization of our proposed junction is concerned, it should be possible to fabricate such a geometry with the currently available experimental techniques. Silicene has been fabricated via epitaxial growth on Ag (111),^{46}^{,} $ ZrB 2$ (0001),^{47} Ir (111),^{48} and $ MoS 2$^{49} substrates. Due to the interaction between the silicene and the different substrates, the various surface reconstructions have been revealed by the theoretical calculations and the experiments, including $3\xd73$, $ 7\xd7 7$, and $ 3\xd7 3$ superstructures with respect to $1\xd71$ silicene lattice.^{50} These superstructures may break the inversion symmetry of silicene.^{51} However, the $ 3\xd7 3$ superstructure is the most stable and robust, which can be observed at Ag, $ ZrB 2$, and Ir substrates.^{48,52} The observation of quasiparticle interference patterns in the case of a $ 3\xd7 3$ superstructure is in agreement with the analysis based on the theoretical band structure of silicene, which is extra proof that the underlying atomic structure is graphene-like with only some buckling that does not change the basic electronic structure.^{52,53} Moreover, the influence of the interaction between the silicene and the substrate on the AR is almost negligible due to the strong interaction between the silicon atoms.^{48} The superconductivity of silicene can be induced by proximity-coupled to a conventional s-wave S like Al or $ NbSe 2$,^{54,55} and this approach has been successfully employed in graphene-based devices.^{56,57} The proximity-induced ferromagnetism and antiferromagnetism can be attained by depositing the magnetic insulators, such as EuO, EuS, and YIG,^{58,59} on top of the silicene sheet. Once such proximity effect is realized in silicene, the fabrication of our proposed junction is feasible. By the first-principles calculation, the intrinsic spin–orbit coupling strength of silicene is computed to $ \lambda S O\u223c3.9$ meV at the critical electric field strength $ E c\u224817$ meV/Å.^{5} The buckling parameter of silicene is $l\u22480.23$ Å. In the experiments, the proximity-induced superconducting energy gap may be of order $ \Delta 0\u223c1.1\xb10.1$ meV.^{60} For such proximity-induced superconducting energy gap, the chemical potential in the L and R regions is $ \mu F\u223c0\u221210$ meV. Moreover, in order to satisfy the mean-field requirement, the chemical potential in the M region should be much larger than the superconducting energy gap, i.e., $ \mu S\u226b \Delta 0$.

## IV. CONCLUSION

We studied the nonlocal transport properties in the AF/S/F asymmetrical junction controlled by the OCPL and the PEF. On the basis of the Dirac BdG equation, the SVRCs of the CAR and CT processes for two cases have been calculated. For case I (II), the OCPL and the PEF are applied simultaneously in the L (R) region, while in the R (L) region, only a constant PEF is taken into account. It is revealed that the OCPL or the PEF could act as a switch to turn on or off the SVRCs of the CAR and CT processes. The spin-valley indices can be transformed by modulating the handedness of the OCPL. In case I, the $K\u2193$ states dominate the hole transport and the $K\u2191$ states dominate the electron transport, whereas, in case II, the $ K \u2032\u2191$ states dominate the hole transport and the $K\u2191$ states dominate the electron transport. The width of the OCPL region for the CAR and CT processes is dependent on just the incident energy in case I, while in case II, it is related not only to the incident energy but also to the FEF. By increasing the AEF, both the CAR and CT signals are suppressed in case I, but in case II, the CAR signal is enhanced and the CT signal is suppressed. Different from case I, the nonlocal switch effect between two pure nonlocal processes can be observed in case II by reversing the direction of the incident energy axis, the handedness of the OCPL in the R region, or the direction of the PEF in the R region. The proposed junction can be simultaneously used as a nonlocal switch, a spin-locked valley filter, and a valley-locked spin filter. All of these findings would be favorable for designing and manufacturing the spintronic and valleytronic devices based on the asymmetrical silicene magnetic superconducting heterostructure.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (NNSFC) (Grant No. 12274305), the Natural Science Foundation of Hebei Province (Grant Nos. A2024205022 and A2019205053), and the Key Program funded by Science and Technology Project of Hebei Education Department (Grant No. ZD2019040).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Shuo Ma:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). **Hongmei Zhang:** Conceptualization (equal); Investigation (equal). **Jianjun Liu:** Conceptualization (equal); Formal analysis (equal); Methodology (equal). **De Liu:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.