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Controllable Valley Polarization Using Silicene Double Line Defects Due to Rashba SpinOrbit Coupling
Nanoscale Research Letters volume 14, Article number: 350 (2019)
Abstract
We theoretically investigate the valley polarization in silicene with two parallel line defects due to Rashba spinorbit coupling (RSOC). It is found that as long as RSOC exceeds the intrinsic spinorbit coupling (SOC), the transmission coefficients of the two valleys oscillate with the same periodicity and intensity, which consists of wide transmission peaks and zerotransmission plateaus. However, in the presence of a perpendicular electric field, the oscillation periodicity of the first valley increases, whereas that of the second valley shortens, generating the corresponding wide peakzero plateau regions, where perfect valley polarization can be achieved. Moreover, the valley polarizability can be changed from 1 to −1 by controlling the strength of the electric field. Our findings establish a different route for generating valleypolarized current by purely electrical means and open the door for interesting applications of semiconductor valleytronics.
Introduction
Silicene, a lowbuckled monolayerhoneycomb lattice of silicon atoms, is a potentially attractive alternative to graphene for valleytronic applications. The lowbuckled structure gives rise to relatively large spinorbit coupling (SOC) in silicene, and a sizable energy gap of approximately 1.55 meV is estimated at the Dirac points K and K^{′}[1] Different from graphene, the low energy dispersion relation of silicene is parabolic rather than linear form. Facilitated by the buckling structure, the band structure of silicene can be controlled by applying an electric field, and even a topological phase transition from a quantum spin Hall insulator to a quantum Valley Hall insulator may occur[2, 3]. Silicene has been successfully synthesized on the surface of substrates such as Ag(111), Ir(111), and ZrB2(0001)[4–6], and its freestanding stable structure has also been predicted in several theoretical studies[7]. Most importantly, a roomtemperature silicene fieldeffect transistor (FET) has been successfully observed experimentally[8]. The electric field tunability and compatibility with existing siliconbased devices make silicene a potential twodimensional material for application in nextgeneration valleytronics.
In twodimensional (2D) materials such as graphene and transition metal dichalcogenides(MoS_{2}, etc.), grain boundaries between two domains of material with different crystallographic orientations are ideal choices to achieve the valley polarization and has attracted considerable attention[9–14]. Recently, the extended line defects (ELDs) in silicene have been extensively investigated according to firstprinciples calculations[15, 16], and the 558 ELD (abbreviated as "line defect” in the following) was found to be the most stable and most readily formed structure. The spin and valley polarization of the silicene line defect have been investigated theoretically[17–19]. The formation of a line defect can be visualized as the stitching of the zigzag edges of two Si grains by the adsorbed Si atoms, where either side of the line defect shows pseudoedgestatelike behavior and the grain boundaries of the zigzag edge act as the pseudoedge[16]. Obviously, such a lattice has mirror symmetry with respect to the line defect and the corresponding lattice vectors in the “left” and “right” domains separated by the defect are contrary[10, 11]. In such a line defect with inversion domain boundary, the A/B sublattices and valley indexes are exchanged upon crossing the defect. The line defect is semitransparent for the quasiparticles in graphene and a high valley polarization appears with a high angle of incidence. The valley polarization is q_{y} (the electron’s group velocity along the y direction) dependent across the line defect. For graphene, which has a linear dispersion and constant group velocity, the valley polarization can reach near 100% at large q_{y} (corresponding to high angle of incidence) while it decreases as q_{y} diminishes and vanishes as q_{y}∼0 [9, 14]. In contrast, silicene has two different transmission characteristics [17, 18]: firstly, the two valleys become indistinguishable as the Fermi energy is close to the band edge due to the parabolic dispersion relation, and secondly, the transmission is restrained because of the helical edge state flowing inversely on both sides of the line defect, as shown in Fig. 1c. Naturally, the system with SOC in a particular RSOC is a promising candidate for efficient spin FET. The RSOC generates an inplane effective magnetic field and induces the spin precession that is injected perpendicular to the plane of confinement. The spin polarization[20] and inversion[21] have been investigated in gated silicene nanoribbons. Theoretical calculations have shown that the energy band of silicene can be significantly modulated by RSOC [22, 23]. For instance, at a relative strong RSOC, the spindown (up) band at the K(K^{′}) valley shifts up while the other spin bands in the conduction band remain unchanged. In consideration of the peculiar transmission feature in the silicene line defect and the effect of RSOC in silicene, the practical allelectric schemes for generating valleypolarized carriers becomes feasible.
In this paper, we propose an efficient way to polarize the Dirac fermions of different valleys using the silicene double line defects, thus creating distinct valley polarization by utilizing the electric field in silicene. Our results show that when the Fermi energy is near the bottom of conduction band, the oscillation images of the transmission coefficients from two valleys, which comprise wide oscillating peaks and nadirs, coincide as long as RSOC exceeds the intrinsic SOC, while the presence of only a single line defect cannot disperse the valleydependent electrons. When two parallel line defects are involved, the oscillating nadirs evolve into zerotransmission plateaus, and effective modulation of valleydependent transport can be realized by changing the oscillation periodicity of the two Dirac valleys with a perpendicular electric field, where the oscillation periodicity of the two valleys increases and decreases and leads to the perfect valley polarization at the wide peakzero plateau corresponding regions. In experiment, one can detect such pure valley current by measuring the change of conductance with the electric field. This phenomenon provides a different route for effectively modulating the valley polarization in silicene devices by utilizing the RSOC and electric fields.
Methods
Let us start from the schematic of a twoterminal silicene line defect device, as shown in Fig. 1a, in which the spin precession is illustrated to generate the valleypolarized current due to the RSOC and electric field. It is supposed that RSOC exists on one side of the line defect with width W and WR in units of \(\sqrt {3}a\), where a=3.86 Å is the lattice constant of pristine silicene, as shown in Fig. 1a. When the Fermi energy is located at the bottom of conduction band, the states (K,↓)[ (K,↓) corresponds to a state in valley K with ↓(down) spin] and (K^{′},↑) are in the gap due to the manipulation of energy band from RSOC. The other two states, (K,↑) and (K^{′},↓), circulate along the pseudoedge because of the spinmomentum locking characteristic from SOC[24], as shown in Fig. 1a. For a definite spin state, it flows along the pseudoedge with opposite directions on both sides of the line defect which can act as a filter and restrain the transmission across the line defect, as depicted in Fig. 1c.
A lattice model in the tightbinding representation is used to describe the line defect system with RSOC as [17, 22]
where \(c_{i\alpha }^{\dag }\) and \(c_{i_{y}\alpha,\gamma /\delta }^{\dag }\) represent the electron creation operator with spin α at silicene site i and the line defect, respectively, and 〈〉/〈〈〉〉 runs over all nearest/nextnearestneighborhopping sites. The first three terms denote nearestneighbor hopping and the parameters t,τ_{1}, and τ_{2} denote various nearestneighbor hopping energies in the tightbinding model, as shown in Fig. 1b. The fourth term is the effective SOC with the hopping parameter t_{so}, and ν_{ij}=±1 for counterclockwise (clockwise) hopping between the nextnearestneighboring sites with respect to the positive zaxis. A theoretical investigation [16] has shown that the two nearest Si atoms in the defect region are relatively identical to those in the pristine region and that all Si atoms remain in the sp^{2}−sp^{3} hybridized state. Therefore, it is reasonable to set τ_{2}=τ_{1}=t. In the fifth term, Δ_{z} is the staggered sublattice potential that arises from an electric field perpendicular to the silicene sheet, and μ_{i}=±1 for the A(B) site. The last term represents the extrinsic RSOC term where t_{R} is the Rashba spinorbit hopping parameter. d_{ij} is the unit vector pointing from site j to i, and \(\vec {\sigma }=(\sigma ^{x},\sigma ^{y},\sigma ^{z})\) in Eq. 1 is the vector of real spin Pauli matrices. The RSOC arises from external potential applied by either an electric gate, metalatom adsorption or a substrates [20, 25] which can dramatically break the structure inversion symmetry of silicene. Notably, the extrinsic RSOC originating from the electric field is ignored because it is very weak.
The ELDs of silicene are shown in Fig. 1a, which extends immensely along the y direction. The translational symmetry of the lattice structure along the y direction indicates that k_{y} is a conserved quantity and that the creation (annihilation) operators can be rewritten as follows, according to the Fourier transformation (the spin index is ignored)[17]:
Then, the Hamiltonian matrix in Eq. 1 is decoupled into \(H=\sum _{k_{y}}H_{k_{y}}\), where \(H_{k_{y}}\) can be described in the following form:
where \(\varphi _{i,l}^{\dag }=\left [ c_{{{k}_{y}},i,l,A\uparrow }^{\dag }, c_{{{k}_{y}},i,l,A\downarrow }^{\dag }, c_{{{k}_{y}},i,l,B\uparrow }^{\dag },c_{{{k}_{y}},i,l,B\downarrow }^{\dag }\right ]\), i in the set of index (i,l) represents the position of a supercell \((\bar {i}=i)\), and l=1 or 2 denotes different zigzag chains in a supercell, as shown in the dashed rectangle in Fig. 1b. \(\hat {T_{ll'}}\) represents the Hamiltonian matrix of each zigzag chain (l=l^{′}) in a supercell or the interplay between different zigzag chains (l≠l^{′}).
It is noted that the two valleys K and K^{′} are now cast at [0,±π/3a] due to the insertion of the line defect. The transmission matrix of the η(η=K/K^{′}) valley is calculated using the generalized Landauer formula[26, 27],
where
and
Here, \(Im\Sigma _{L,R}=\left (\Sigma _{L,R}^{r}\Sigma _{L,R}^{a}\right)/ 2i\) are positive semidefinite matrices with a welldefined matrix square root, where \(\Sigma _{L,R}^{a}=\left [\Sigma _{L,R}^{r}\right ]^{\dag }\) are the retarded/advanced selfenergy of the left/right lead. The 16×16 submatrix G^{r} is the retarded Green’s function, which connects the first and last supercells along the x direction and can be calculated using the recursive Green’s function method. The total transmission coefficients of the η valley are \(T_{\eta }=T^{\uparrow \uparrow }_{\eta }+T^{\uparrow \downarrow }_{\eta }+ T^{\downarrow \uparrow }_{\eta }+T^{\downarrow \downarrow }_{\eta }\), and the spin polarization P_{s} and valley polarization P_{η} can be given by
Results and Discussion
In the calculations of the spindependent transmission coefficients, we set τ_{2}=τ_{1}=t=1 as the energy unit, the SOC strength t_{so}=0.005t, and the Fermi energy E_{f}=1.001t_{so}, which is situated at the bottom of the conduction band. The width of the scattering region is W=1000 for the single line defect and an additional width WR=1000 is also taken into account for the two parallel line defects, as shown in Fig. 1a.
Figure 2 depicts the spinconserved/spinflip transmission coefficients of valley \(\eta, T^{sc}_{\eta }/T^{sf}_{\eta }\), as a function of the incident angles α (a) and of the RSOC strength t_{R} (b–d). Figure 2a–c correspond to the case of the single line defect, and (d) is for the case of the two parallel line defects. It is shown that at a definite t_{R} (for instance, t_{R}=5t_{so} as in Fig. 2a), the spindependent transmission coefficients \(T^{sc}_{K}/T^{sf}_{K}\) are constant and independent of the incident angles due to the parabolic dispersion relation, as shown in Fig. 2a. Therefore, in the following calculations, we can use the incident angle α=0 as an example. For a weak t_{R}, an oscillating phenomenon similar to that in a twodimensional electron gas [26, 27] appears due to the Rashba splitting, as shown in the inset of Fig. 2b. As t_{R} increases (t_{R}>t_{so}), \(T_{K}^{\uparrow \uparrow }\) and \(T_{K}^{\uparrow \downarrow }\) have the same oscillating periodicity and nearly the same magnitudes as t_{R} which consists of some oscillation peaks and nadirs, while \(T_{K}^{\downarrow \downarrow }/T_{K}^{\downarrow \uparrow }\) tends to zero because the Fermi energy lies in its gap, as shown in Fig. 2b. Thus, the total transmission coefficient of K valley is mainly contributed by the spin up state. In fact, the oscillation images of the two valleys, K and K^{′}, coincide while the transmission coefficients of K^{′} valley is mainly contributed by the spindown electrons.
In the presence of a perpendicular electric field, the valley degeneracy is lifted, and the oscillating behaviors of the two valleys differs: the oscillating periodicity of the K valley increases, while that of the K^{′} valley decreases, as shown in Fig. 2c. However, it seems infeasible to filter one conical valley state with only a single line defect because the oscillating nadirs have a definite magnitude. Naturally, one may consider the oscillating phenomenon with two parallel line defects to further restrain the transmission, as shown in Fig. 2d. Comparing Fig. 2b with d reveals that the oscillation peak becomes narrow and acute, while the oscillation nadir broadens and weakens, which forms the zerotransmission platform. The space between two neighboring oscillation peaks is fixed at 3.25t_{so}, as characterized by the two dashed lines in Fig. 2d.
To achieve a better valley filter effect, we concentrate our attention on the effect of the perpendicular electric field. The results of this effect are shown in Fig. 3. As discussed above, the oscillating periodicity of the two valleys change in an opposite manner, and the original overlapping oscillation peaks in Fig. 2d are relieved. Meanwhile, the zerotransmission plateau broadens and narrows for T_{K} and \(T_{K^{\prime }}\), respectively, as shown in Fig. 3a and b. At Δ_{z}=0.15t_{so}, the space between the two neighboring oscillation peaks develops into 3.6t_{so} for T_{K}, while it is reduced to 3.1t_{so} for \(T_{K^{\prime }}\), as indicated by the two blue and red dashed lines shown in Fig. 3a. As the electric field strengthens, the space between the two neighboring oscillation peaks continues to increase/decrease for T_{K}/\(T_{K^{\prime }}\), which is 5.4t_{so}/2.8t_{so} at Δ_{z}=0.3t_{so}, as shown in Fig. 3b. The change in the oscillation periodicity will lead to the corresponding regions of wide peakzero plateau, where perfect valley polarization with P_{η}=±1 plateaus can be realized, as shown in Fig. 3c and d. Simultaneously, it is shown that high spin polarization P_{s} also arises when P_{η}=±1.
However, due to the uncontrollability of RSOC, it is still difficult to detect such pure valley currents experimentally, even though the RSOC induced in the line defect can be greater than the intrinsic SOC. To conveniently probe the pure valley current experimentally, we also investigate the transmission coefficients and valley polarization as a function of electric field, which can be continuously controlled during an experiment. It is shown that the perfect valley polarization with P_{η}=±1 can emerge in a certain range of Δ_{z} and that it can change from P_{η}=1 to P_{η}=−1 as the electric field increases, as shown in Fig. 4a. For a definite t_{R} (for instance t_{R}=7.2t_{so}, as indicated with a dashed line in Fig. 4a), the transmission coefficients \(T_{K}/T_{K^{\prime }}\) oscillate with Δ_{z}, where the wide transmission peaks of the K(K^{′}) valley correspond to the zerotransmission plateaus of the K^{′}(K) valley. The total transmission coefficients are basically contributed by one valley as the electric field varies, and perfect valley polarization can always occur around the maximal value of \(T_{K}/T_{K^{\prime }}\), as shown in Fig. 4b. As the Fermi energy departs from the band edge, the perfect valley polarization can still survive even at E_{f}=1.5t_{so}, where the plateau relation can be well maintained, as shown in Fig. 4c. During an experiment, one can analyze the valleypolarized electrical currents from the left to right lead with an experimentally measurable quantity such as the conductance, which is proportional to the total transmission coefficient. The maximal conductance between two minimum values (sometimes, they are zero) should be from one valley. We can estimate the magnitude of the conductance according to the formula \(G=\frac {e^{2}}{h}\int _{k_{F}}^{k_{F}}T\frac {dk_{y}}{2\pi /L_{y}}=\frac {e^{2}}{h}\frac {Ly\sqrt {E^{2}t^{2}_{so}}}{2\pi \hbar v_{F}}2T\) [28], where L_{y}=2a≈7.72Å is the width of silicene line defect, v_{F}=5.5×10^{5}m/s is the Fermi velocity, \(\hbar =h/2\pi \) is the reduced Planck constant with \(\phantom {\dot {i}\!}h=4.13566743\times 10^{15}eV\cdot s, T=T_{K}+T_{K'}\) is the total transmission coefficient and E is the onsite energy of the incident electrons. Then, the conductance is about \(G\approx \left [0.7T\sqrt {E^{2}t^{2}_{so}}/eV\right ]\frac {e^{2}}{h}\). It is also found that as the onsite energy in the incident side is raised to E=0.15t(t=1.6eV), the transmission coefficients of the two valleys change only a little compared with Fig. 4c due to spin and momentum conservation and the transmission peakzero plateau relation maintains still, as shown in Fig. 4d. In this case, the conductance is about \(G\approx 0.17T\frac {e^{2}}{h}\) which is sizable and can be detectable in experiment. The energy window to observe this phenomenon is about 0.5t_{so}(t_{so}<E<1.5t_{so}) which is proportional to t_{so}. In experiment, it is not difficult to control the Fermi energy near the band edge and the SOC gap can even be radically increased to 44 meV by proximity with Bi(111) bilayer[29] which can greatly improve the energy region to detect the pure valley current. Moreover, the computational model can also be applicable to other lowbuckled counterparts of graphene, germanene[30],stanene and MoS_{2}[31–36],which have even larger band gaps[37, 38] as well as the SOC strengths(SOC strength can reach 0.1eV for stanene[38, 39]). In a real experiment, it is easy to realize a strong RSOC which can exceed the intrinsic SOC by breaking the inplane mirror symmetry with the special substrate[40]. Therefore, this scheme can be completely feasible in experiment.
Conclusions
We have proposed an electrical method for generating a valleypolarized current in silicene line defects. In sharp contrast to the conventional electrical approaches that are used to produce valleypolarized current, we explore the RSOC, which is considered to tune the widely used spin polarization in spinpolarized FETs. It is found that the transmission coefficients of the two valleys oscillate with the same periodicity and intensity, which is composed of transmission peaks and zerotransmission plateaus. The valleypolarized current can be generated by tuning the oscillating periodicity of the two valleys with an electric field, which can destroy the symmetry of the valley states and bring about the corresponding transmission peakzero plateau regions. Moreover, we also provide a scheme to detect the pure valley current in experiment and the results may shed light on the manipulation of valleypolarized currents by electrical means.
Availability of Data and Materials
The datasets generated during and/or analyzed during the current study are available from the corresponding authors on reasonable request.
Abbreviations
 2D:

Twodimensional
 ELD:

Extended line defect
 FET:

Fieldeffect transistor
 RSOC:

Rashba spinorbit coupling
 SOC:

Intrinsic spinorbit coupling
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Funding
This work was supported by the NNSF of China (nos. 11974153, 11704165, and 11864047), the Natural Science Foundation of Shandong Province (nos. ZR2017JL007, ZR2019MA030, and ZR2016AL09), China Scholarship Council (no. 201908320001), the Science Foundation of Guizhou Provincial Education Department (no. QJHKYZ[2016]092), the Major Research Project for Innovative Group of Education Department of Guizhou Province (no. KY[2018]028) and the Scientific Research Fund of Hunan Provincial Education Department (no. 17A193).
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HYT and CDR together carried out the physical idea and were major contributors in writing the manuscript. BHZ derived the algorithm to calculate the transmission coefficients with RSOC, HYT, and CDR carried out the numerical results of calculations. WTL provided guidance in improving the quality of the manuscript. YFL, JL, and SYZ 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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Tian, H., Ren, C., Zhou, B. et al. Controllable Valley Polarization Using Silicene Double Line Defects Due to Rashba SpinOrbit Coupling. Nanoscale Res Lett 14, 350 (2019). https://doi.org/10.1186/s1167101931963
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Keywords
 Silicene
 Line defect
 Rashba spin orbit coupling
 Valley polarization