Spin- and Valley-Dependent Electronic Structure in Silicene Under Periodic Potentials

Abstract

We study the spin- and valley-dependent 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 valley-dependent 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

Two-dimensional (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 spin-orbit 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 silicon-based electronic technology. These characteristics make silicene an excellent material for the next-generation nanoelectronics. In particular, a silicene field-effect transistor at room temperature has been successfully fabricated by a growth-transfer-fabrication 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 spin-valley 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 valley-polarized quantum anomalous Hall state is predicted in silicene owing to the topological phase transition [31]. Based on the state transition, a silicene-based 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 time-reversal symmetry [3, 34, 35]. The broken inversion symmetry could result in a valley-dependent 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 symmetry-breaking 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 valley-dependent 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 single-particle approximation, the electronic structure of silicene in the vicinity of Dirac points obeys an effective Dirac Hamiltonian. The system under consideration is a one-dimensional 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 Kronig-Penney 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,

$$\begin{array}{l}H=\hslash v_F(k_x{\tau }_x-\eta k_y{\tau }_y)+{\Delta }_{\mathrm{\eta \sigma }}{\tau }_z+U_{\sigma }.\end{array}$$ 1

Fig. 1.

Top: schematic of the silicene superlattices. The FM insulators, such as EuO and EuS, on the top of silicene induce the exchange fields in silicene, as proposed for graphene [53]. The metallic gates on the top of FM insulators control the Fermi level locally. Bottom: schematic of the energy spectrum in silicene with and without external fields

Δ_ησ=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 spin-up and spin-down 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

$$\begin{array}{l}\psi \left(x\right)=A\left(\begin{array}{c}1\\ \frac{\hslash v_F{k}_{-}}{{\epsilon }_{\mathrm{\eta \sigma }}}\end{array}\right)e^{i{q}_{\mathrm{\eta \sigma }}x}+B\left(\begin{array}{c}1\\ \frac{-\hslash v_F{k}_{+}}{{\epsilon }_{\mathrm{\eta \sigma }}}\end{array}\right)e^{-i{q}_{\mathrm{\eta \sigma }}x}.\end{array}$$ 2

In the barrier regions, ε_ησ=ε_b=(E−U_σ)+Δ_ησ and the x component of the wave vector $q_{\mathrm{\eta \sigma }}=q_b=\sqrt{{(E-U_{\sigma })}^2-{\Delta }_{\mathrm{\eta \sigma }}^2-{\left(\hslash v_F{k}_y\right)}^2}/\hslash v_F$. In the well regions, ε_ησ=ε_w=E−ησλ_SO and $q_{\mathrm{\eta \sigma }}=q_w=\sqrt{E^2-{\lambda }_{\mathit{\text{SO}}}^2-{\left(\hslash v_F{k}_y\right)}^2}/\hslash v_F$. k_±=q_ησ±iηk_y. The transmission probability T_ησ 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

$$\begin{array}{l}{G}_{\mathrm{\eta \sigma }}\left(E\right)=\frac{1}{2}\underset{-\pi /2}{\overset{\pi /2}{\int }}{T}_{\mathrm{\eta \sigma }}(E,Esin\theta )cos\mathrm{\theta d\theta },\end{array}$$ 3

where θ is the incident angle with respect to the x direction. The spin- and valley-resolved conductances are defined as $G_{\uparrow \left(\downarrow \right)}=\left(G_{\mathrm{K\uparrow }\left(\downarrow \right)}+G_{K^{\prime }\uparrow \left(\downarrow \right)}\right)/2$ and $G_{K\left(K^{\prime }\right)}=\left(G_{K\left(K^{\prime }\right)\uparrow }+G_{K\left(K^{\prime }\right)\downarrow }\right)/2$, respectively. Then, we introduce the spin polarization P_s and valley polarization P_v:

$$\begin{array}{l}{P}_s=(G_{\uparrow }-G_{\downarrow })/(G_{\uparrow }+G_{\downarrow }),\end{array}$$ 4

$$\begin{array}{l}{P}_v=(G_K-G_{K^{\prime }})/(G_K+G_{K^{\prime }}).\end{array}$$ 5

Based on the Bloch’s theorem and the continuity condition of wave functions, the dispersion relation E(k_x) for spin-up and spin-down electrons near the K and K^′ valleys can be calculated,

$$\begin{array}{cc}cos\left(k_{x}d\right)& =cos\left(q_w{d}_w\right)cos\left(q_b{d}_b\right)\\ -\frac{{\left({\epsilon }_b{q}_w\right)}^2+{\left({\epsilon }_w{q}_b\right)}^2+{({\epsilon }_b-{\epsilon }_w)}^2{k}_y^2}{2{\epsilon }_w{\epsilon }_b{q}_w{q}_b}sin\left(q_w{d}_w\right)sin\left(q_b{d}_b\right),\end{array}$$ 6

and k_x is the Bloch wave number. In order to simplify the calculation, the dimensionless units are introduced: $E\to \mathit{\text{Ed}}/\hslash v_F$, $U\to \mathit{\text{Ud}}/\hslash v_F$, ${\lambda }_{\mathit{\text{SO}}}\to {\lambda }_{\mathit{\text{SO}}}d/\hslash v_F$, ${\Delta }_z\to {\Delta }_{z}d/\hslash v_F$, $h\to \mathit{\text{hd}}/\hslash v_F$, k_y→k_yd, k_x→k_xd, 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 Valley-Dependent 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 spin-up electron near K valley (K↑ electron) and spin-down electron near K^′ valley (K^′↓ electron) are gapless (see Fig. 2 (a1, a4)), while the spin-down electron near K valley (K↓ electron) and spin-up electron near K^′ valley (K^′↑ electron) have a large gap (see Fig. 2 (a2, a3)). The minibands of spin-up (or spin-down) 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

$$\begin{array}{l}{E}_0=\frac{{(U-\mathrm{\sigma h})}^2-{\Delta }_z^2+2\mathrm{\eta \sigma }{\Delta }_z{\lambda }_{\mathit{\text{SO}}}}{2(U-\mathrm{\sigma h})}.\end{array}$$ 7

Fig. 2.

Energy spectrum versus k_y for several different values of potential U. (a1–a4) U=0; (b1–b4) U=40.66 meV; (c1–c4) U=100.63 meV; (d1–d4) U=135.0 meV. The values of parameters are h=8.0 meV, Δ_z=7.8 meV, and k_x=0

For K↑ and K^′↓ electrons with ησ=1, when Δ_z=2λ_SO, Eq. (7) can reduce to

$$\begin{array}{l}{E}_0=\frac{U-\mathrm{\sigma h}}{2}.\end{array}$$ 8

Correspondingly, Eq. (6) turns into

$$\stackrel{2}{cos}\left(q_w{d}_w\right)-\frac{\left({\epsilon }_b^2+{\epsilon }_w^2\right)q_w^2+{({\epsilon }_b-{\epsilon }_w)}^2{k}_y^2}{2{\epsilon }_w{\epsilon }_b{q}_w^2}\stackrel{2}{sin}\left(q_w{d}_w\right)=1,$$ 9

which is satisfied when $\left({\epsilon }_b^2+{\epsilon }_w^2\right)q_w^2+{({\epsilon }_b-{\epsilon }_w)}^2{k}_y^2=-2{\epsilon }_w{\epsilon }_b{q}_w^2$ or q_wd=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_wd=2nπ have the form

$$\begin{array}{l}{k}_{y0}=\pm \frac{1}{d}\sqrt{\frac{\left(E_0^2-{\lambda }_{\mathit{\text{SO}}}^2\right)d^2}{{\left(\hslash v_F\right)}^2}-{\left(2\mathrm{n\pi }\right)}^2}.\end{array}$$ 10

When $\sqrt{E_0^2-{\lambda }_{\mathit{\text{SO}}}^2}d/2\mathrm{\pi \hslash }{v}_F\ge n$, k_y0 is real, and the new Dirac points will arise which are exactly located at (E₀,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,

$$\begin{array}{l}{N}_D=2\left[\frac{\sqrt{E_0^2-{\lambda }_{\mathit{\text{SO}}}^2}d}{2\mathrm{\pi \hslash }{v}_F}\right]+1\end{array}$$ 11

for K↑ and K^′↓ electrons, while

$$\begin{array}{l}{N}_D=2\left[\frac{\sqrt{E_0^2-{\lambda }_{\mathit{\text{SO}}}^2}d}{2\mathrm{\pi \hslash }{v}_F}\right]\end{array}$$ 12

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 spin-down (or spin-up) 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.

Fig. 3.

Number of Dirac points N_D versus potential U. (a) h=8.0 meV and Δ_z = 7.8 meV; (b) h = 20.0 meV and Δ_z = 7.8 meV; (c) h = 8.0 meV and Δ_z=15.0 meV

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_ησ=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].

Fig. 4.

(a) Minibands near Fermi level and (b) their band gaps at original Dirac point versus potential U, at d_b=d_w=50 nm. (c) Minibands near Fermi level and (d) their band gaps at original Dirac point versus d_b, at U=0 and d_w=50 nm. The values of other parameters are h=8.0 meV, Δ_z=15.0 meV, and k_x=k_y=0

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

$$\begin{array}{l}{v}_x/v_F=\mathrm{\partial E}/\partial k_x,v_y/v_F=\mathrm{\partial E}/\partial k_y.\end{array}$$ 13

Fig. 5.

(a–d) Velocity versus potential U, and the parameters are set as h=20.0 meV and Δ_z=7.8 meV. The black, red, blue, and green solid curves are the velocities v_0x, v_1x, v_2x, and v_3x, respectively. The black, red, blue, and green dashed curves are the velocities v_0y, v_1y, v_2y, and v_3y, respectively

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_ησ 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 spin-up electron is always larger (or less) than the one for spin-down 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.

Fig. 6.

Velocities v_0x and v_0y versus (a) Δ_z and (b) h, for K↑ electron. (a) h=20.0 meV and λ_SO=Δ_z/2. (b) Δ_z=7.8 meV

Spin- and Valley-Polarized Transport

The spin- and valley-dependent 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 valley-polarized transport through a finite silicene superlattice. Figure 7 shows the transmission probability T_ησ 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_ησ 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_ησ=1 occur precisely at the positions of the Dirac points, suggesting an application of the system as a spin and valley filter.

Fig. 7.

Contour plot of the transmission T_ησ(E,k_y) for (a), (c) K↑ electron and (b), (d) K↓ electron. The values of parameters are the same as these in Fig. 2 (d1–d4), and n=10

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_ησ, 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 spin-up and spin-down electrons could be alternative distribution by adjusting h properly. Consequently, $G_{K\left(K^{\prime }\right)\uparrow }$ and $G_{K\left(K^{\prime }\right)\downarrow }$ present alternative distribution as well, i.e., $G_{K\left(K^{\prime }\right)\uparrow }$ nearly vanishes for those regions where $G_{K\left(K^{\prime }\right)\downarrow }$ 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 valley-polarized 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.

Fig. 8.

Minibands, conductances G_ησ, spin polarization P_s, and valley polarization P_v versus potential U. (a) Δ_z=4.0 meV. (b) Δ_z=12.0 meV. Other parameters are set as h=7.0 meV, E=6.0 meV, d_b=d_w=120 nm, and n=10

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.

Fig. 9.

Contour plot of (a) spin polarization P_s(U,h) and (b) valley polarization P_v(U,h), at Δ_z=10.0 meV. The values of other parameters are the same as these in Fig. 8

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〉=h₀, respectively. The fluctuations are given by

$$\begin{array}{l}{\Delta }_z{|}_i={\Delta }_{z0}(1+\delta {\zeta }_i),h{|}_i=h_0(1+\delta {\zeta }_i),\end{array}$$ 14

Fig. 10.

Conductances (a) G_↑ and (b) G_↓ versus potential U, when the electric field presents disorder, at n=50 and Δ_z0=20.0 meV. The solid, dashed, dotted, and dash-dotted curves correspond to the disorder strength δ=0.0, 0.1, 0.3, and 0.6, respectively. The values of other parameters are the same as these in Fig. 8

Fig. 11.

Polarizations P_s and P_v versus potential U when a the electric field or b the exchange field presents disorder. Δ_z0=20.0 meV and h=7.0 meV in (a). Δ_z=20.0 meV and h₀=7.0 meV in (b). The values of other parameters are the same as these in Fig. 10

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 valley-dependent 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

Two-dimensional

FM

Ferromagnetic

SOI

Spin-orbit interaction

Authors’ contributions

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.

Competing interests

The authors declare that they have no competing interests.

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