 Nano Express
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Spin and ValleyDependent Electronic Structure in Silicene Under Periodic Potentials
Nanoscale Research Lettersvolume 13, Article number: 84 (2018)
Abstract
We study the spin and valleydependent energy band and transport property of silicene under a periodic potential, where both spin and valley degeneracies are lifted. It is found that the Dirac point, miniband, band gap, anisotropic velocity, and conductance strongly depend on the spin and valley indices. The extra Dirac points appear as the voltage potential increases, the critical values of which are different for electron with different spins and valleys. Interestingly, the velocity is greatly suppressed due to the electric field and exchange field, other than the gapless graphene. It is possible to achieve an excellent collimation effect for a specific spin near a specific valley. The spin and valleydependent band structure can be used to adjust the transport, and perfect transmissions are observed at Dirac points. Therefore, a remarkable spin and valley polarization is achieved which can be switched effectively by the structural parameters. Importantly, the spin and valley polarizations are greatly enhanced by the disorder of the periodic potential.
Background
Twodimensional (2D) Dirac materials with hexagonal lattice structures are being explored extensively since the discovery of graphene, such as silicene [1, 2], transition metal dichalcogenides [3, 4], and phosphorene [5]. Although graphene has many particular properties, its application is limited by the zero band gap and the weak spinorbit interaction (SOI). Recently, a silicon analog of graphene, silicene, has been fabricated via epitaxial growth [6–10], and its stability has been predicted by theoretical studies [11, 12]. Graphene and silicene have similar band structures around K and K^{′} valleys, and the low energy spectra of both are described by the relativistic Dirac equation [13]. Contrary to graphene, silicene has a strong intrinsic SOI and a buckled structure. The strong SOI could open a gap at Dirac points [13, 14] and lead to a coupling between the spin and valley degrees of freedom. The buckled structure allows us to control the band gap by an external electric field perpendicular to the silicene sheet [14–16]. Furthermore, silicene has the advantage that it is more compatible with existing siliconbased electronic technology. These characteristics make silicene an excellent material for the nextgeneration nanoelectronics. In particular, a silicene fieldeffect transistor at room temperature has been successfully fabricated by a growthtransferfabrication process in experiment [17].
The discovery of 2D Dirac materials provides new opportunities to explore quantum control of valley. The two inequivalent valleys K and K^{′} in the first Brillouin zone could be regarded as an additional degree of freedom besides charge and spin for quantum information and quantum computation [18–20]. For example, the valley degree of freedom can be incorporated to expand an electron spin qubit to a spinvalley qubit [18]. Therefore, valleytronics which aims to generate, detect, and manipulate the valley pseudospin has attracted considerable interest. In graphene, various schemes to achieve a valley polarization have been proposed by utilizing unique edge modes [21, 22], trigonal warping effect [23], topological line defects [24, 25], strain [26, 27], and electrostatic gates [28]. Compared to graphene, silicene has significant advantage in the study of valley pseudospin. It is found that silicene exhibits a rich variety of topological phases and Chern numbers under the modulation of different external fields [13, 16, 29, 30]. In the presence of electric field E_{ z } and exchange field h, Ezawa explored the phase diagram in the E_{ z }−h plane which is characterized by the spin and valley indices [16]. Further considering the Rashba SOI, a valleypolarized quantum anomalous Hall state is predicted in silicene owing to the topological phase transition [31]. Based on the state transition, a silicenebased spin filter with nearly 100% spin polarization is proposed which is robust against weak disorder [32]. Yokoyama studied the ballistic transport through a ferromagnetic (FM) silicene junction and demonstrated a controllable spin and valley polarized current [33]. In transition metal dichalcogenides with a broken inversion symmetry, the spin splitting of the valence bands arising from intrinsic SOI is opposite at the two valleys due to a timereversal symmetry [3, 34, 35]. The broken inversion symmetry could result in a valleydependent optical selection rule, which can be used to selectively excite carriers in the K or K^{′} valley via right or left circularly polarized light, respectively [3, 34]. In experiment, the signal of valley polarization has been probed by optical [36, 37] and transport [38, 39] measurements. A giant nonlocal valley Hall effect was observed in bilayer graphene subjected to a symmetrybreaking gate electric field, and the nonlocal signal persists up to room temperature [38]. A recent review of valleytronics in 2D Dirac materials is provided in Ref. [40].
Superlattice is an effective method of engineering the electronic structure in semiconductors and 2D materials [41]. Superlattice patterns with nanoscale can naturally arise in experiment when graphene or silicene is placed on top of metallic substrates [42, 43]. A superlattice in graphene could lead to renormalization of anisotropic Fermi velocity [44] and generation of new Dirac points in the spectrum [45–47] owing to the chiral nature, which have been experimentally observed [43, 48, 49]. In silicene superlattices with electric field E_{ z } and exchange field h, both spin and valley degeneracies are lifted. It is confirmed that miniband structure and minigaps caused by the superlattices depend on the spin and valley indices [50]. Furthermore, the spin and valley polarizations could be enhanced by the silicene superlattices [51]. Just like graphene, many novel electronic structures are expected in silicene superlattices. However, works on silicene superlattices are very few [50, 51]. In this paper, we discuss in detail a complementary aspect, namely, the spin and valleydependent band structure and transport property of silicene. We found that the spin and valley indices have different impacts on the extra Dirac points and anisotropic velocity which can be tuned by the structural parameters. The velocity is greatly suppressed due to the electric field and exchange field. A remarkable spin and valley polarization is achieved, which can be greatly enhanced by the disorder.
The paper is organized as follows. In the “Methods” section, we present the theoretical formalism and the dispersion relation. The numerical results on band structure, spin and valley polarized transmissions are shown in the “Results and discussions” section. Finally, we conclude with a summary in “Conclusions” section.
Methods
In the singleparticle approximation, the electronic structure of silicene in the vicinity of Dirac points obeys an effective Dirac Hamiltonian. The system under consideration is a onedimensional silicene superlattice formed by a series of local potential barriers U, exchange fields h, and perpendicular electric field E_{ z }. U, h, and E_{ z } are present only in the barrier regions with barrier width d_{ b }, whereas U=h=E_{ z }=0 in the well regions with well width d_{ w }, as shown in Fig. 1. The superlattice with a KronigPenney type varies only along x direction, and the length of one unit is d=d_{ b }+d_{ w }. Similar model has been discussed in Refs. [51, 52], which mainly focus on thermoelectric and electronic transport rather than the band structure and disorder effect studied in this work. Experimentally, U can be produced by the metallic gates and h can be produced by the magnetic proximity effect with FM insulators EuO [33], which are deposited periodically on top of the silicene layer (see Fig. 1). The electric field E_{ z } applied perpendicular to silicene can induce a staggered sublattice potential Δ_{ z }=ℓE_{ z }, with 2ℓ≈0.46Å the vertical separation of A and B sites of the two sublattices due to the buckled structure [16]. Hence, the electronic states can be described by the Hamiltonian,
Δ_{ η σ }=Δ_{ z }−ησλ_{ SO } describes the band gap for different spin and valley indices, which can be controlled by the staggered potential Δ_{ z } and the SOI λ_{ SO }. U_{ σ }=U−σh is the effective potential for different spin indices. η=±1 denotes the K and K^{′} valleys. σ=±1 denotes spinup and spindown states. v_{ F } is the Fermi velocity. In silicene, the intrinsic and extrinsic Rashba effects are very small and can be neglected [15].
Due to the translational invariance along the y direction, the transverse wave vector k_{ y } is conserved. The wave function for valley η and spin σ in each region has the form Ψ(x,y)=ψ(x)e^{iky}_{ y } with
In the barrier regions, ε_{ η σ }=ε_{ b }=(E−U_{ σ })+Δ_{ η σ } and the x component of the wave vector \( q_{\eta \sigma } = q_{b} = \sqrt {(E  U_{\sigma })^{2}  \Delta ^{2}_{\eta \sigma }  (\hbar v_{F} k_{y})^{2}} / \hbar v_{F} \). In the well regions, ε_{ η σ }=ε_{ w }=E−ησλ_{ SO } and \( q_{\eta \sigma } = q_{w} = \sqrt {E^{2}  \lambda _{SO}^{2}  (\hbar v_{F} k_{y})^{2}} / \hbar v_{F} \). k_{±}=q_{ η σ }±iηk_{ y }. The transmission probability T_{ η σ } can be calculated using the transfer matrix technique. The normalized conductance for a particular spin in a particular valley at zero temperature is given by
where θ is the incident angle with respect to the x direction. The spin and valleyresolved conductances are defined as \(G_{\uparrow (\downarrow)} = \left (G_{K \uparrow (\downarrow)} + G_{K^{\prime } \uparrow (\downarrow)} \right) / 2 \) and \(G_{K (K^{\prime })} = \left (G_{K (K^{\prime }) \uparrow } + G_{K (K^{\prime }) \downarrow } \right) / 2\), respectively. Then, we introduce the spin polarization P_{ s } and valley polarization P_{ v }:
Based on the Bloch’s theorem and the continuity condition of wave functions, the dispersion relation E(k_{ x }) for spinup and spindown electrons near the K and K^{′} valleys can be calculated,
and k_{ x } is the Bloch wave number. In order to simplify the calculation, the dimensionless units are introduced: \(E \rightarrow E d / \hbar v_{F}\), \(U \rightarrow U d / \hbar v_{F}\), \(\lambda _{SO} \rightarrow \lambda _{SO} d / \hbar v_{F}\), \(\Delta _{z} \rightarrow \Delta _{z} d / \hbar v_{F}\), \(h \rightarrow h d / \hbar v_{F}\), k_{ y }→k_{ y }d, k_{ x }→k_{ x }d, d_{ w }→d_{ w }/d, and d_{ b }→d_{ b }/d. Note that at Δ_{ z }=λ_{ SO }=h=0, Eq. (6) is reduced to the one found for gapless graphene in a periodic potential, where both spin and valley are degenerated [44–47]. From Eq. (6), we can see that exchange field h alone could induce the split of spin, while the valley keeps degeneracy. However, the valley degeneracy can be lifted by the electric field E_{ z } with the help of the SOI λ_{ SO }. Thus, a combination of the exchange field and the electric field could lift the spin and valley degeneracies [16, 31–33], as shown in Fig. 1. In the proposed system, electrons with different spins near different valleys would present various band structures and transport features.
Results and discussions
In this section, we would use the above equations to calculate the band structures and transport properties for different spin and valley indices in silicene superlattices. The widths of barriers and wells are assumed to be the same in what follows. The results for the case with unequal well and barrier widths (d_{ b }≠d_{ w }) are similar to those in gapless graphene [47]. Some parameters are set as d_{ b }=d_{ w }=50 nm and λ_{ SO }=3.9 meV in silicene, unless otherwise stated. We shall concentrate on the first two minibands (the lowest valence and conduction minibands) near the Fermi level.
Spin and ValleyDependent Band Structure
First, the effect of potential U on minibands is depicted in Fig. 2. In order to discuss the gapped case and gapless case of energy bands simultaneously, we set Δ_{ z }=7.8 meV=2λ_{ SO }. In the absence of potential (U=0), the spinup electron near K valley (K↑ electron) and spindown electron near K^{′} valley (K^{′}↓ electron) are gapless (see Fig. 2 (a1, a4)), while the spindown electron near K valley (K↓ electron) and spinup electron near K^{′} valley (K^{′}↑ electron) have a large gap (see Fig. 2 (a2, a3)). The minibands of spinup (or spindown) electron shift to the negative (or positive) energy range from E=0 by h, due to the effective potential U_{ σ }=U−σh. The band structures of K↑ (or K↓) electron and K^{′}↓ (or K^{′}↑) electron present mirror symmetry with respect to E=0, consistent with Eq. (6). However, this mirror symmetry is destroyed in the presence of U. Observably, as U increases, extra Dirac points appear, the number of which increases in the meantime. The extra Dirac points can be demonstrated by the chirality of the wave functions in their vicinity [46]. The features of Dirac points in silicene system rely heavily on the spin and valley degrees of freedom, as shown in Fig. 2. For example, at U=135 meV in Fig. 2 (d1–d4) for K↑, K↓, K^{′}↑ and K^{′}↓ electrons, the numbers of Dirac points are 5, 6, 4, and 7, respectively. For specific values of U, such as U=40.66 meV for K↓ electron (see Fig. 2 (b2)) and U=100.63 meV for K^{′}↑ electron (see Fig. 2 (c3)), a new Dirac point can be generated at k_{ y }=0, and it will split into a pair which move in opposite directions away from the k_{ y }=0 point but always keeping k_{ x }=0, as U further increases. In consequence, the band gaps for K↓ and K^{′}↑ electrons are closed (see Fig. 2 (b2, c3)), and the gapped system becomes gapless. In order to find the critical value of U, we set d_{ b }=d_{ w } and k_{ x }=0. Analogous to the rule in gapless graphene [47], taking into account the implicit function theorem, one can conclude that the longitudinal wavevectors at the new Dirac points satisfy q_{ b }=q_{ w } when
For K↑ and K^{′}↓ electrons with ησ=1, when Δ_{ z }=2λ_{ SO }, Eq. (7) can reduce to
Correspondingly, Eq. (6) turns into
which is satisfied when \(\left (\epsilon _{b}^{2} + \epsilon _{w}^{2}\right) q_{w}^{2} + (\epsilon _{b}  \epsilon _{w})^{2} k^{2}_{y} = 2 \epsilon _{w} \epsilon _{b} q_{w}^{2}\) or q_{ w }d=2nπ (n is a positive integer). Based on Eq. (8), we have ε_{ b }=−ε_{ w }, and so the former equality is fulfilled only if k_{y0}=0 for K↑ and K^{′}↓ electrons at Δ_{ z }=2λ_{ SO }, corresponding to the original Dirac point. The solutions of q_{ w }d=2nπ have the form
When \(\sqrt {E_{0}^{2}\lambda _{SO}^{2}}d / 2\pi \hbar v_{F} \geq n\), k_{y0} is real, and the new Dirac points will arise which are exactly located at (E_{0},k_{y0}). At low values of U, k_{y0} is imaginary, and there is no solution for n, which means no extra Dirac points. The Dirac points appear only after a critical value of U, such as U=40.66 meV for K↓ electrons in Fig. 2 (b2), corresponding to n=1. According to Eq. (10), The number of Dirac points N_{ D } can be obtained. When Δ_{ z }=2λ_{ SO },
for K↑ and K^{′}↓ electrons, while
for K↓ and K^{′}↑ electrons, where [...] denotes an integer part. Note that at the critical value of U, such as U=40.66 meV and 100.63 meV, the number of Dirac points is N_{ D }=2n−1 for K↓ and K^{′}↑ electrons (see Fig. 2 (b2, c3)).
Equations (7) and (10) manifest that the positions and the numbers of Dirac points could be adjusted by the electric field E_{ z } and exchange field h. Figure 3 exhibits the number of Dirac points N_{ D } as a function of U for different values of E_{ z } and h. When Δ_{ z }=7.8 meV in Fig. 3a, with increasing U, N_{ D } for K↑ and K^{′}↓ electrons increases in the form of odd number, consistent with Eq. (11). N_{ D } for K↓ and K^{′}↑ electrons increases in the form of even number, consistent with Eq. (12), except for N_{ D } at the critical value. Comparison between Fig. 3a and b indicates that as h increases, the critical value for spindown (or spinup) electron decreases (or increases) gradually. When Δ_{ z }=15 meV≠2λ_{ SO } in Fig. 3c, N_{ D } for all electrons increases in the form of even number, except for N_{ D } at the critical value. Distinctly, the critical values of U are different for electron with different spins and valleys. The Dirac points could be controlled by a joint modulation of the parameters U, E_{ z }, and h.
The potential U and barrier width d_{ b } could be used to regulate the band gap, as illustrated in Fig. 4. The gaps for K↑ and K^{′}↓ electrons are small, while the gaps for K↓ and K^{′}↑ electrons are large due to Δ_{ η σ }=Δ_{ z }−ησλ_{ SO }. As U increases, all the minibands gradually move toward high energy region (see Fig. 4a), and all the band gaps display damped oscillation with U (see Fig. 4b). When U=σh, the effective potential is zero, and the gap reaches maximum value. The gap is closed at the critical value of U, on account of the emergence of new Dirac points. Figure 4c, d depicts the dependence of minibands and band gaps on barrier width d_{ b } at U=0. In the absence of external field (d_{ b }=0), the minibands keep degenerate, and the gap at Fermi level is 2λ_{ SO }. With the appearance of d_{ b }, the miniband is split, where valley and spin become nondegenerate. The minibands of K↑ (or K↓) and K^{′}↓ (or K^{′}↑) electrons keep mirror symmetry about E=0 (see Fig. 4c). As d_{ b } increases, the gaps of K↓ and K^{′}↑ electrons are broaden gradually. The gaps of K↑ and K^{′}↓ electrons decrease to zero when d_{ b } satisfies d_{ b }/d_{ w }=λ_{ SO }/Δ_{ z }, and thereafter increase with d_{ b } (see Fig. 4d). The gap widths approach to saturation with the further increase of d_{ b }. Furthermore, the width of miniband is narrowed as d_{ b } increases (not shown in the figure), due to the less coupling of eigenstates. The effect of electric field on band gap is analogous to that in previous study [50].
The group velocity depends strongly on the spin and valley indices, as shown in Fig. 5. The components (v_{ x },v_{ y }) of velocity can be defined as
Figure 5 presents the velocity components v_{ mx } and v_{ my } in units of v_{ F } at original Dirac point (m=0) and new Dirac points (m=1,2,3). One can see that as U increases, v_{0y} oscillates in a decayed way and v_{0x}≈v_{ F } is almost unaffected (see Fig. 5a, d). At the critical value of U where the new Dirac points emerge, v_{ mx }≈v_{ F } but v_{0y}=v_{ my }=0, indicating a collimation behavior along the k_{ x } direction for specific spins and valleys. When U exceeds the critical value and further increases, v_{ my } increase to v_{ F } but v_{ mx } decrease to zero gradually. The effect of the periodic potential is highly anisotropic, as a result of the chiral nature. The features of anisotropic velocity are various for different spins and valleys owing to the gap Δ_{ η σ } and the potential U_{ σ }, which can be commanded by employing U. Taking U=20 meV for example, v_{0y}=v_{ F } for K↑ electron is much greater than v_{0y}=0.16v_{ F } for K^{′}↓ electron, and no v_{0y} for K↓ and K^{′}↑ electrons due to the band gap. v_{ mx } (or v_{ my }) for spinup electron is always larger (or less) than the one for spindown electron in the same valley. Notably, Fig. 5 also implies that for small value of U, v_{0x}, v_{0y}, and v_{ mx } are less than v_{ F } due to Δ_{ z } and h, other than the gapless system [44]. For instance, v_{1x}=0.98v_{ F }, 0.89v_{ F }, 0.89v_{ F }, and 0.98v_{ F } for K↑, K↓, K^{′}↑ and K^{′}↓ electrons, respectively, when the Dirac point appears. In order to illuminate the influence of Δ_{ z } and h on the group velocity, Fig. 6 shows the velocities (v_{0x},v_{0y}) as a function of (a) Δ_{ z } and (b) h for K↑ electron. From Fig. 6a we can clearly see that v_{0x} is monotonically decreasing with Δ_{ z } while v_{0y} is insensitive to the change of Δ_{ z }. On the contrary, v_{0x} is desensitized to h, while v_{0y} increases to maximum value v_{0y}=v_{ F } at h=σU and then decreases with h. The results indicate that the group velocity can be suppressed by Δ_{ z } and h in silicene.
Spin and ValleyPolarized Transport
The spin and valleydependent band structure is reflected in transport property and provides a guide in controlling the transport. In this section, we discuss the properties of spin and valleypolarized transport through a finite silicene superlattice. Figure 7 shows the transmission probability T_{ η σ } for (a, c) K↑ and (b, d) K↓ electrons, and the period number n=10. The red dashed curves are the minibands, which are also the borders for different electronic states deciding the transmission. We can see that the transmission is restricted in the miniband region and no transmission in the band gap region (see Fig. 7a, b). The distribution of transmission is symmetric around k_{ y }=0 due to the symmetric minibands. The resonant characteristic of transmission arises from the resonant states. It should be noted that the transmission still exists in the gap region near k_{ y }=0 due to the tunnel effect of eigenstates. T_{ η σ } at Fermi level for K↑ and K↓ electrons are shown in Fig. 7c, d), respectively. One can clearly see that many thin resonant peaks with T_{ η σ }=1 occur precisely at the positions of the Dirac points, suggesting an application of the system as a spin and valley filter.
The strong dependence of band structure on the spin and valley indices is beneficial to the realization of high spin and valley polarizations. Figure 8 presents the minibands, conductances G_{ η σ }, spin polarization P_{ s }, and valley polarization P_{ v } as a function of potential U. It can be found that the distribution of conductance is completely in agreement with the band structure, that is, the conductance (or conductance gap) corresponds to the miniband (or band gap). The minibands for spinup and spindown electrons could be alternative distribution by adjusting h properly. Consequently, \(G_{K(K^{\prime })\uparrow }\) and \(G_{K(K^{\prime })\downarrow }\) present alternative distribution as well, i.e., \(G_{K(K^{\prime })\uparrow }\) nearly vanishes for those regions where \(G_{K(K^{\prime })\downarrow }\) is in resonance and vice versa. This result directly leads to a remarkable spin polarization, proposing a switching effect of spin polarization (see Fig. 8a). By changing Δ_{ z }, the minibands and conductances for electrons near K and K^{′} valleys could be controlled, leading to a fully valleypolarized current (see Fig. 8b). Compared with spin polarization, the valley polarization is not perfect enough. However, this drawback could be remedied via the disorder structure of the system, as discussed in the following.
Figure 9 shows the (a) spin polarization P_{ s } and (b) valley polarization P_{ v } in (U,h) space. Interestingly, both P_{ s } and P_{ v } present periodical changes in the considered region, which is not observed in the ferromagnetic silicene junction [33]. Both distributions of P_{ s } and P_{ v } are antisymmetric with respect to h→−h. It is possible to achieve independently a full spin and valley polarization by a proper tuning of the fields U and h. For example, when h=6 meV and U=42 meV, P_{ s }≈1 and P_{ v }≈1, meaning that the current is mainly contributed by K↑ electrons. When h=6 meV and U=44 meV, P_{ s }≈1 and P_{ v }≈−1 while P_{ s }≈−1 and P_{ v }≈−1 at h=6 meV and U=46 meV. The results demonstrate that a spin and valley polarization can be switched effectively.
In experiment, the structural imperfection of the model is unavoidable due to the limitations of the experimental techniques. Therefore, it is necessary to discuss the effect of the disorder on transmission. When the electric field or exchange field presents disorder, the conductance, spin polarization, and valley polarization are shown in Figs. 10 and 11. We set disorder situations of Δ_{ z } and h fluctuate around their mean values, given by 〈Δ_{ z }〉=Δ_{z0} and 〈h〉=h_{0}, respectively. The fluctuations are given by
where {ζ_{ i }} is a set of uncorrelated random variables or white noise, − 1<ζ_{ i }<1, δ is the disorder strength, and i is the site index. Note that the disorder only takes place in the x direction, and the system is always homogeneous in y direction. Thus, k_{ y } still keeps conservation. Figure 10 exhibits the effect of the disorder of the electric field on the conductances (a) G_{ ↑ } and (b) G_{ ↓ }. With the presence and increase of the disorder strength δ, both G_{ ↑ } and G_{ ↓ } are suppressed gradually, and each resonant peak splits into many small peaks. One may find that the conductance range is narrowed while the conductance gap range is broadened. Hence, the allowable (or forbidden) ranges of G_{ ↑ } completely fall into the forbidden (or allowable) ranges of G_{ ↓ }, giving rise to an excellent spin polarization (see Fig. 11). Furthermore, the positions of conductances and conductance gaps are nearly invariable as δ changes, suggesting that the miniband and band gap are insensitive to the disorder. Note that the disorder effect of the electric field on G_{ K } and G_{ K }^{′} is similar to that observed in Fig. 10. Figure 11 presents the disorder effects of (a) the electric field and (b) the exchange field on polarizations P_{ s } and P_{ v }. Obviously, with the increase of δ, P_{ s } and P_{ v } increase greatly, and the polarization platform is broadened. Thus, a full spin and valley polarization is realized. Comparison between Fig. 11a and b indicates that the disorder effect of exchange field is more prominent. The results demonstrate that the disorder could enhance the spin and valley polarizations compared with the order case, which is an advantage in realistic application.
Conclusions
In summary, we demonstrated detailedly that band structure and transport property of silicene under a periodic field strongly depend on the spin and valley degrees of freedom. The numerical results indicate that electrons with different spins and valleys have various characteristics in Dirac point, bang gap, and group velocity. In particular, owing to the electric field and exchange field, the anisotropic velocity is restrained, which displays a collimation behavior for specific spins and valleys. Therefore, the transmission presents strong spin and valleydependent feature, consistent with the band structure, resulting in a significant spin and valley polarizations. In addition, the disorder could greatly enhance the spin and valley polarizations. Finally, we hope these results can be conducive to the potential applications of the spin and valley indices.
Abbreviations
 2D:

Twodimensional
 FM:

Ferromagnetic
 SOI:

Spinorbit interaction
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Acknowledgements
We thank the reviewers for their valuable comments.
Funding
This work was supported by the Natural Science Foundation of Shandong Province (Grants No. ZR2017JL007 and No. JQ201602) and the NSFC (Grants No. 11404157 and No. 51302126).
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WTL carried out the calculations, analyzed the results, and wrote the manuscript. YFL and HYT participated in the result analysis and manuscript preparation. All authors reviewed the manuscript. All authors read and approved the final manuscript.
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Correspondence to WeiTao Lu or HongYu Tian.
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WTL and YFL are the professors in Linyi University. HYT is a lecturer in Linyi University.
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Keywords
 Silicene
 Energy band
 Valley polarization
 Spin polarization