Spintronics,^[1,2] aiming to manipulate the charge and spin degrees of freedom of electrons in nanodevices, has received considerable interest from researchers due to its faster processing speed and lower energy consumption compared with traditional electronic devices.^[3,4] Among the various materials of spintronics, graphene is more attractive owing to its unique two dimensional hexagonal lattice structure,^[5] long spin relaxation time,^[6] high carrier mobility,^[7] and gate voltage controllable electron transport at room temperature.^[8] In order to realize future graphene spin devices, various kinds of current valves controlled by electrical methods are becoming increasingly important for their convenience in practice and the advantage that the electric field cannot break the spin coherence in devices.^[9,10] Because the bulk graphene is a gapless semiconductor, it is difficult to electrically modulate the carrier transport around the Dirac point for the Klein paradox,^[10] until the valley valve effect or band selective transport was found in the zigzag graphene nanoribbon (ZGR) junction.^[11]

The valley valve effect offers an efficient electrical way to manipulate the inter-valley electron transport in graphene by using so-called pseudo-parity conservation.^[11–17] For example, Rycerz et al.^[11] first proposed a gate tunable current valve based on a ZGR p/n junction. Niu et al.^[13] reported an even-chain ZGR junction with a parallel magnetization distribution can act as a bipolar spin filter due to the band selective transport. Wang et al.^[16] proposed an even-chain ZGR junction to produce and detect two 100% polarized spin-up and spin-down currents at two edges of the ribbon. In a recent work,^[17] a bipolar spin valve based on a three terminal ZGR junction with different even–odd chain connections was designed to separate and detect one species of spin current. Since the valley valve phenomenon mainly depends on the even–odd chain number of the ZGR, which is difficult to prepare in experiments, one needs to explore a new graphene-like material with a large spin–orbit coupling.

Silicene, a single layer of silicon atoms forming a graphene-like honeycomb lattice, shares almost all the electron transport properties of graphene.^[18–26] In particular, the valley valve effect was also predicted in the zigzag silicene nanoribbon (ZSR) p/n junctions^[20] and ferromagnetic junctions.^[21,22] For instance, Tsai et al.^[20] proposed a current valve very similar to the graphene one in Ref. [11] to separate spin and valley polarized currents in a ZSR p/n junction, and pointed out the unique band structure under the gate voltage is the main reason for the selective transport of the device. Subsequently, Zhou et al.^[21] reported a band selective spin filter in a ZSR ferromagnetic junction; they revealed that the valley valve effect mainly comes from the asymmetry edge of the ribbon, and predicted that the spin filter phenomenon exhibits the even–odd chain effect, which are very similar to those in the ZGR junction.

Moreover, silicene has a non-planar buckled structure and a large spin–orbit gap of about 1.55 meV,^[27] which provides new possibilities for manipulating a massive Dirac fermion around the energy gap electrically.^[28] A perpendicular electric field with the Kane–Mele-type spin–orbit interaction can break the structure inversion symmetry of silicene and induce the topological phase transition from the topological insulator to the band insulator,^[29] which can be used to filter spin-up and spin-down electrons at different energy valleys.^[30–33] For instance, Wang et al.^[31] reported a theoretical work that a bulk silicene with a spatially alternative electric field can generate a no dissipation spin-dependent valley current. By means of the Keldysh Green function method, Farokhnezhad et al.^[32] proposed a ZSR ferromagnetic junction with a perpendicular electric field as a perfect spin valve to generate a fully polarized spin current. Recently, the spin switching phenomena caused by electrical tunable edge modes of silicene were also confirmed by the DFT calculations^[34–36] and the low-energy electron diffraction experiments.^[37]

At present, the spin transport affected by the interplay between the valley valve effect and the controllable band structure of ZSR junction has not received enough attention.^[32,33] Thus, in this work, we design a ZSR ferromagnetic junction in Fig. 1 to study the band selective spin transport influenced by a tunable perpendicular electric field in the topological phase transition of silicene. By using the Landauer–Bütticker formula,^[38,39] we calculate the spin-resolved conductance spectrum of the system and find that the valley valve effect could exist in an even-chain ZSR junction when the perpendicular electric field is absent and the ZSR is in the topological insulator phase. The ON and OFF states of the spin conductance can be easily changed by the electrical methods. On the other hand, when the perpendicular electric field is nonzero and the ZSR is a band insulator, the spin valve phenomenon remains in the ZSR junction, which does not come from the valley valve effect, but from the energy gap opened by the perpendicular electric field. Furthermore, we also explain the two spin valve phenomena by means of the energy band theory.

Fig. 1.

Figure Option ViewDownloadNew Window

Fig. 1. (color online) Schematic plot of electrical controllable spin valves based on a ZSR ferromagnetic junction. The C region is an undoped ZSR, the left (L) and right (R) electrodes are two semi-infinite ZSRs with different local magnetization ML(R). The perpendicular electric field is applied on the C ZSR region, which can induce a staggered potential ηΔC and a weak disordered potential wi between the A (red dot, η = +1) and B (blue dot, η = −1) sublattices of silicene. The width and length of the C ZSR are described by N and LC, respectively.

The rest of the paper is arranged as follows. In Section 2, we give the detailed theoretical model of the ZSR junction, and derive the general formulae for the spin-dependent conductance of the tunneling junction. In Section 3, we present the numerical results of the spin transport affected by the interplay between the valley valve effect and the tunable energy band of the ZSR junction. We summarize our results in the last section.

We study a ZSR ferromagnetic junction as illustrated in Fig. 1, the whole device consists of three parts: the C nonmagnetic ZSR, the left (L) and right (R) semi-infinite ZSR ferromagnetic electrodes. The left and right ZSR leads have different local magnetization induced by two ferromagnetic insulator substrates. A perpendicular electric field is only applied on the C ZSR region.

The Hamiltonian of the system can be described by the next-nearest-neighbor tight-binding model^[18] 1 where the 1st term represents the nearest neighbor interaction with the transfer energy t, where 〈i, j〉 means all the nearest neighbor sites, and creates (annihilates) a quasi-particle with spin index α(β) = ↑↓ at lattice site i. The 2nd term describes the effective spin–orbit coupling with strength λ_so, where ν_ij = ±1 for the next-nearest neighbor hopping anti-clockwise (clockwise) around a hexagon with respect to the z axis, and σ = (σ^x, σ^y, σ^z) is the Pauli matrix in spin space. The 3rd term denotes the intrinsic Rashba spin–orbit interaction with λ_R, is a unit vector from the next-nearest neighbor site i to j. The notation σ⋅M in the 4th term depicts the local magnetization of the left and right ZSR leads. The last term means the perpendicular electric field applied on the C ZSR region, which may cause a staggered potential η_iΔ_C and a weak disordered potential w_i between the A (η_i = 1) and B (η_i = −1) sublattices due to the buckled structure of silicene.

Let us derive the linear conductance in the proposed ZSR device, the spin-dependent current I_Lα in the left lead can be obtained by applying the Heisenberg equation of motion^[40] and the non-equilibrium Green’s function method^[41,42] 2 where e is an electron’s charge and h = 2πℏ is the Planck constant. The notation Tr means trace over all the transverse lattices of the C ZSR region. and are retarded, advanced, lesser Green’s functions, and self-energy matrices, respectively. Using the Hermitian conjugation relations:^[38,39] , and , equation (2) can be expressed as 3 By employing the following Keldysh formula:^[43,44] , , and , equation (3) can be simplified as a two-terminal Landauer–Bütikker formula^[38,39] 4 where is the left (right) line-width function, which describes the coupling strength between the left (right) lead and the C region. f_L(R)(E) = 1/(e^{(E−μ_L(R))/kT} + 1) is the Fermi–Dirac distribution function with the chemical potential μ_L(R). The linear spin-resolved conductance at zero temperature limit can be defined as 5 where e²/h is the unit of the spin-dependent conductance. T_LRα is the transmission coefficient, and V = μ_L − μ_R is the external bias between the left and right ZSR electrodes. E_F is the Fermi level. and are (N × N)-dimensional block matrices of , and the retarded Green’s function can be solved by the matrix inversion directly 6 where H_C is the Hamiltonian of the C ZSR region, and I is a unit matrix. The influences of two semi-infinite ZSR leads are merged into the two self-energy terms , which can be calculated by the recursive Green’s function method.^[45,46] The above calculation of is only valid when the magnetization direction is along the z axis. If the right ZSR electrode has a magnetization angle θ against the z axis, should be rotated at angle θ in spin space.

In this section, we mainly focus on the spin-dependent conductance G_LRα and the spin valve phenomena affected by the valley valve effect and the tunable band structure in a ZSR ferromagnetic junction. In the calculations, the zero temperature limit T = 0 K is considered. The local magnetizations of the left and right ZSR terminals are set to have the same amplitudes |M_L| = |M_R| = M = 0.05 eV, and different orientations θ_L = 0, θ_R = θ with respect to the positive z axis. The nearest transfer energy t = 1.6 eV, the Rashba spin–orbit coupling coefficient is λ_R = 0.7 meV and the effective spin–orbit interaction magnitude λ_so can be modulated by an external electric field. Because the valley valve effect occurs only in even-chain ZSR junctions and the disordered lattice potential is weak, in most cases unless otherwise stated, the width, the length, and the disordered potential strength of the C ZSR region are fixed N = 10, L_C = 20, and w = 0 eV, respectively.

In Fig. 2, we plot the spin-resolved conductance G_LRα versus Fermi level E_F for different system parameters when the perpendicular staggered potential Δ_C = 0 eV. Figures 2(a) and 2(b) describe G_LRα as the function of E_F with λ_so = 0 for the parallel and antiparallel magnetization distributions, respectively. As it is shown, when the zigzag-chain number is even and the magnetization direction is parallel in Fig. 2(a), the spin-up electron is blocked and G_LR↑ = 0 at the region E_F ∈ [0, M], the spin-down one meets G_LR↑(E_F) = G_LR↓(−E_F) and the prohibited region is [−M, 0]. Similar spin valve phenomena emerge in the antiparallel magnetization configuration as shown in Fig. 2(b), G_LR↑ = G_LR↓ = 0 at the region [−M, M], where both spin-up and spin-down electrons are forbidden to flow through the tunneling junction. Similar to that in a ZGR junction,^[12,13] the spin valve exhibits the even–odd chain effect, and it could not appear in the odd-chain ZSR junction in Fig. 2(c). In addition, when the disordered lattice potential w ≠ 0, a conductance peak (valley) appears in the vicinity of E_F = 0 for the even (odd)-chain ZSR junction, which means that the weak w cannot break the valley valve effect, but it can affect the ON/OFF states of spin transport around the Fermi zero point. The spin switching effect influenced by an electrical tunable spin–orbit coupling λ_so in the antiparallel magnetized ZSR junction is presented in Fig. 2(d), the spin valve phenomenon is still preserved when a weak λ_so is applied, the blocked region of the spin transport (G_LR↑ = 0) is divided into two parts [−M + λ_so, − λ_so] and [λ_so, M − λ_so]. The width of each forbidden region is reduced to M − 2λ_so, therefore, it is possible to allow and prohibit the spin transport by electrically modulating the spin–orbit coupling λ_so.

Fig. 2.

Figure Option ViewDownloadNew Window

Fig. 2. (color online) Spin-dependent conductance GLRα versus Fermi level EF in the ZSR ferromagnetic junction for (a) parallel magnetization θ = 0, and (b) antiparallel magnetization θ = π. (c) and (d) GLR↓ versus EF for different even–odd chain number N, disordered lattice potential strength w, and effective spin–orbit coupling λso in the antiparallel magnetized ZSR junction, respectively. The perpendicular staggered potential ΔC = 0 eV.

We use the energy band theory and pseudo-parity conservation to explain the spin valve phenomena in Fig. 2. The band structures of the left, C, and right ZSRs are plotted in Fig. 3. As it is shown, the ZSRs are gapless semi-metals when the perpendicular electric field Δ_C and the spin–orbit coupling λ_so are both absent in Fig. 3(a). The spin-up (-down) subbands shift a Zeeman splitting energy ±M from the Fermi energy zero point, which may cause the spin-up (-down) electrons on the left (right) and C regions to have opposite pseudo-parities within the energy range [0, M]([−M, 0]) in the parallel magnetized ZSR junction, therefore the valley valve effect appears and the corresponding conductance G_LR↑(G_LR↓) = 0 at [0, M]([−M, 0]), as shown in Fig. 2(a). When the spin–orbit coupling λ_so ≠ 0 and the magnetization orientation is antiparallel in Fig. 3(b), the spin-up and spin-down subbands are exchanged, the energy ranges of the asymmetric pseudo-parities for spin-up and spin-down electrons are both extended to [−M, M]. On the other hand, the ZSR is a topological insulator under a weak spin–orbit coupling, its topological edge states (the bands crossing the gap in Fig. 3(b)) [−λ_so, +λ_so] and [±M∓λ_so, ±M±λ_so] are allowed for the spin transport, due to the interplay between the valley valve effect and the edge state, the spin-up and spin-down conductances are both suppressed at two regions [−M + λ_so, −λ_so] and [λ_so, M − λ_so] in Fig. 2(d). For the zigzag-chain number is odd in Fig. 2(c), the valley valve effect of the system disappears, the electron transport no longer satisfies the pseudo-parity conservation, thus the spin-resolved conductances are nonzero in the whole energy region.

Fig. 3.

Figure Option ViewDownloadNew Window

Fig. 3. (color online) Energy–momentum relations of the left (ferromagnetic), C (nonmagnetic), and right (ferromagnetic) ZSRs for (a) λso = 0 eV, θ = 0 and (b) λso = 5 meV, θ = π. The “+ (−)” represents the even (odd) pseudo-parity (symmetry) of the quasi-particle, and it could be different for the conduction bands and the valence bands. The arrows show the electron transport owing to the band selection rule. The areas between two solid (dashed) lines depict the forbidden transport regions of the spin-up (-down) electrons.

We draw the conductance G_LRα as a function of magnetization M for different spin–orbit coupling λ_so in the parallel magnetized ZSR junction in Fig. 4(a). As it is shown, there exists a critical magnetization M_C = E_F for filtering one species of spin. When λ_so = 0 and M < M_C, both spin-up and spin-down electrons are permitted to flow through the ZSR junction. When M > M_C, only a spin-up electron can transport through the tunneling junction. When λ_so ≠ 0, the topological edge state should be considered, G_LR↓ is neither equal to 0 nor equal to 1 in the region [M_C − λ_so, M_C + λ_so] as also shown in Fig. 4(a). Therefore, the device can be designed as an electrical controllable spin valve by modulating λ_so. We also calculate G_LRα versus magnetization angle θ for different λ_so around E_F = M_C in Fig. 4(b). When λ_so = 0, the spin-up electron is allowed to pass through the junction and the spin-down one is completely blocked, i.e., G_LR↑ ≠ 0 and G_LR↓ = 0. The nonzero conductance G_LR↓ is a cosine-like curve of magnetization angle θ, when θ = 2kπ with k is an integer, G_LR↓ reaches its maximum (minimum) value for the parallel (antiparallel) magnetization configuration. Thus, the parallel magnetized ZSR junction can act as a perfect spin valve to block the spin-down electron when λ_so is tuned to 0. When λ_so ≠ 0, the topological edge effect takes effect, both G_LR↑ ≠ 0 and G_LR↓ ≠ 0 no matter how θ changes. Furthermore, two spin conductances are oscillating out of phase with the magnetization angle θ, as also shown in Fig. 4(b).

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (color online) (a) GLRα versus magnetization strength M for different spin–orbit coupling λso, (b) GLRα versus magnetization angle θ for different λso in the parallel magnetized ZSR junction. The other parameters are set as follows: EF = 0.05 eV in panel (a), EF = 0.045 eV in panel (b).

In Fig. 5(a), we give the numerical results of G_LRα versus E_F influenced by a nonzero perpendicular staggered potential in the parallel magnetized ZSR junction. When the perpendicular staggered potential Δ_C ≠ 0 and spin–orbit coupling λ_so ≠ 0, the spin valve phenomena (G_LRα = 0) still appear in the ZSR junction. Different from the case of Δ_C = 0, both spin-up and spin-down conductances are equal to 0 at the same region [−Δ_C + λ_so, Δ_C − λ_so] which cannot be explained by the pseudo-parity conservation of the valley valve effect. To further study the effect of perpendicular staggered potential on the spin transport, we also plot G_LRα of the antiparallel magnetized ZSR junction in Fig. 5(b). The spin-up and spin-down conductances are almost degenerate, and the transport blocked regions are the same as those in the parallel magnetized ZSR junction. Although this current valve can also be tuned by the staggered potential Δ_C and the spin–orbit interaction λ_so, it can only be used to switch charge current. One cannot further separate and detect spin-up and spin-down currents from the charge current in the above ZSR junction when a nonzero Δ_C is applied.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. (color online) (a) GLRα versus EF for (a) parallel magnetization and (b) antiparallel magnetization under a nonzero perpendicular staggered potential ΔC = 0.02 eV. The other parameter is λso = 5 meV.

To explain the current valve phenomenon under a nonzero staggered sublattice potential in Fig. 5 that is different from the valley valve effect, we draw the energy band of the device in Fig. 6. The C ZSR is a band insulator when a nonzero perpendicular electric field is applied. The staggered sublattice potential induced by perpendicular electric field can open an energy gap with an approximate width ±Δ_C around E_F = 0 in the C region, the width of the energy gap can be electrically modulated by the spin–orbit coupling λ_so. The quasi-particles in the gap cannot produce any current and the conductance is zero. For the parallel magnetized ZSR junction in Fig. 5(a), the gaps of spin-up and spin-down subbands of C ZSR are both opened at [−Δ_C+λ_so, Δ_C − λ_so], therefore G_LR↑ = G_LR↓ = 0 at the same energy regions. Because the width of the energy gap can only be tuned by Δ_C and λ_so, which is independent of the magnetization distribution, the regions of G_LR↑↓ = 0 in the antiparallel magnetized junction in Fig. 5(b) are very similar to those in the parallel one.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. (color online) Energy band structure of the parallel magnetized ZSR junction under ΔC = 0.01 eV and λso = 5 meV. The areas between two solid lines [−ΔC+λso, ΔC − λso] are two transport prohibited regions for both spin-up and spin-down electrons. The other parameters and symbols are the same as those in Fig. 3.

Finally, we also discuss the spin valve affected by the controllable staggered potential Δ_C and the spin–orbit coupling λ_so of the antiparallel magnetized ZSR junction in Fig. 7(a). As it is shown, for a given Fermi level E_F, when Δ_C is weak, that is, Δ_C < E_F − λ_so, both spin-up and spin-down electrons can pass through the ferromagnetic junction and G_LR↑↓ ≈ 1, that is because the energy gap opened by Δ_C is too small to block the transmission of spin electrons at the Fermi energy. G_LR↑ ≈ 1 and G_LR↓ ≈ 0 in the range [E_F − λ_so, E_F + λ_so], which can be used to filter the spin-down electron by tuning λ_so electrically. When Δ_C increases to more than E_F + λ_so, the gap is big enough and the Fermi level is located within it, therefore, both spin-up and spin-down conductances decrease to 0.

Fig. 7.

	Figure Option ViewDownloadNew Window
	Fig. 7. (color online) (a) GLRα versus staggered potential ΔC for different spin–orbit interaction strength λso with EF = 0.05 eV, (b) GLRα versus λso with EF = 0.01 eV and ΔC = 0.02 eV in the antiparallel magnetized ZSR junction. The spin polarization degree χ as a function of λso is presented in the inner of Fig. 7(b).

We also plot G_LRα as a function of λ_so in the antiparallel magnetized ZSR junction in Fig. 7(b), when λ_so < E_F < Δ_C, both spin-up and spin-down electrons are forbidden to transport and G_LR↑↓ ≈ 0, which can be explained that the transmission electron at the Fermi level is within the gap, λ_so is so small that it cannot help electrons pass through the tunneling junction. When λ_so increases and E_F < λ_so < Δ_C, both spin-up and spin-down conductances increase rapidly with the increasing of λ_so. When λ_so > Δ_C, the spin-dependent conductance decreases with the increasing of λ_so, that is because the gap is closed by λ_so, no more electrons can participate in the transport. We also draw the spin-polarized degree χ ≡ G_LR↑ − G_LR↓/G_LR↑ + G_LR↓ versus λ_so in the inner of Fig. 7(b), one can manipulate the spin-polarized degree of the electron from about −20% to 50% by using the spin–orbit coupling λ_so electrically.

To conclude, we have investigated the spin valve effect in a ZSR ferromagnetic junction with a perpendicular electric field applied on the C ZSR region. By calculating the spin-resolved conductance of the device, we found that the band selective transport could be preserved in the ZSR junction when the perpendicular electric field is zero and the ZSR is under a topological insulator phase. The valley valve phenomenon in the system emerges as an even–odd chain effect and can be electrically modulated by the Fermi level, the spin–orbit coupling, and the local magnetization. When a nonzero perpendicular electric field is considered, the spin valve phenomenon still exists in the device, but it is mainly determined by the energy gap of the C ZSR, which can be modulated by the interplay between the staggered sublattice potential and the spin–orbit coupling. The proposed device may serve as electrical controllable spin valves or spin filters.