Electrically Tunable Valley Polarization in Silicene via Rashba Spin‑Orbit Coupling and Double Line Defects

Introduction

Silicene—a low‑buckled monolayer of silicon—offers a platform that rivals graphene for valleytronic devices. Its buckling induces a sizable intrinsic spin‑orbit coupling, opening a ~1.55 meV band gap at the Dirac points K and K′, and allows the band structure to be tuned by an out‑of‑plane electric field. This field can even trigger a quantum spin Hall to quantum valley Hall phase transition. Silicene has been fabricated on Ag(111), Ir(111), and ZrB₂(0001) and predicted to be stable in free‑standing form; a room‑temperature field‑effect transistor has already been demonstrated. Consequently, silicene is a compelling candidate for next‑generation valleytronic devices that leverage existing silicon technology.

Grain boundaries in two‑dimensional crystals are natural valley filters. In silicene, extended line defects (ELDs) formed by stitching zigzag edges of two grains have been studied by first‑principles calculations. The 5‑5‑8 ELD is the most stable and readily formed. The defect acts as an inversion‑domain boundary that exchanges sublattice A/B and valley indices across the line. In graphene, a single line defect is semi‑transparent, yielding a valley polarization that depends on the transverse momentum qy. For silicene, however, the parabolic dispersion renders the two valleys indistinguishable near the band edge, and the helical edge states circulating on opposite sides of the defect suppress transmission. RSOC, generated by a perpendicular electric field, metal adsorption or substrate interaction, introduces an in‑plane effective magnetic field that precesses spins injected perpendicular to the plane. It can therefore be harnessed to manipulate valley‑dependent transport.

In this work we show that a pair of parallel line defects in silicene, combined with RSOC and a controllable electric field, can produce a robust, fully electric valley polarization. We present a tight‑binding model that captures the defect geometry and SOC terms, calculate transmission using the Landauer approach, and demonstrate that the valley polarization can be tuned from +1 to –1 by varying the field. We also discuss how the valley‑polarized current can be detected experimentally via conductance measurements.

Methods

We consider a two‑terminal silicene device with two parallel line defects, as illustrated in Fig. 1a. RSOC is present on one side of each defect, with widths W and WR (in units of √3 a, where a = 3.86 Å). The Fermi energy is set just above the conduction band minimum, so that the states (K,↓) and (K′,↑) lie in the SOC‑induced gap, while (K,↑) and (K′,↓) propagate along pseudo‑edges due to spin‑momentum locking.

The system is described by the tight‑binding Hamiltonian

H = t∑⟨i,j⟩α cᵢα† cⱼα + τ₂∑⟨γδ⟩α cᵢyα,γ† cᵢyα,δ + τ₁∑⟨i,γ⟩α cᵢα† cᵢyα,γ + i t_so/(3√3)∑⟨⟨i,j⟩⟩αβ ν_ij cᵢα† σ^z_{αβ} cⱼβ + Δ_z∑ᵢα μᵢ cᵢα† cᵢα + i t_R∑⟨i,j⟩αβ cᵢα† (σ × d_ij)^z_{αβ} cⱼβ + h.c.

Here t is the nearest‑neighbor hopping, τ₁ and τ₂ are hopping parameters within the defect, t_so is the intrinsic SOC, Δ_z is the staggered sublattice potential induced by a perpendicular electric field, and t_R is the Rashba SOC strength. The lattice symmetry allows a Fourier transform along the y‑direction, yielding a decoupled set of H_k_y Hamiltonians (Eq. 2–3). Transmission is computed via the recursive Green’s function method and the generalized Landauer formula (Eqs. 4–6). Spin and valley polarizations are defined as P_s = (T_K^{↑↑}+T_K^{↑↓}−T_K^{↓↓}−T_K^{↓↑}+T_K′^{↑↑}+T_K′^{↑↓}−T_K′^{↓↓}−T_K′^{↓↑})/(T_K+T_K′) and P_η = (T_K−T_K′)/(T_K+T_K′), respectively.

Results and Discussion

With τ₂=τ₁=t=1 (energy unit), intrinsic SOC t_so=0.005 t, and Fermi energy E_f=1.001 t_so, we first examine a single line defect (W = 1000). Figure 2a shows that the spin‑conserved transmission T_K^{↑↑} is independent of the incident angle, a consequence of the parabolic band. As the Rashba strength t_R increases beyond t_so, T_K^{↑↑} and T_K^{↑↓} develop identical oscillations with peaks and valleys, while T_K^{↓↓} and T_K^{↓↑} are suppressed by the SOC gap. The two valleys K and K′ exhibit mirror‑image oscillations, but a single defect cannot fully suppress transmission of one valley.

Introducing a second, parallel line defect (additional width WR = 1000) creates zero‑transmission plateaus at the minima of the oscillations (Fig. 2d). The peak width narrows while the trough widens, and the period between neighboring peaks remains constant (3.25 t_so). Applying a perpendicular electric field Δ_z lifts valley degeneracy: the K‑valley period elongates, the K′ period shortens (Fig. 3a,b). By tuning Δ_z, the peak‑gap regions can be shifted, producing perfect valley polarization (P_η = ±1) in wide plateau intervals (Fig. 3c,d). Simultaneously, high spin polarization emerges when P_η reaches ±1.

While the Rashba strength is difficult to control precisely, the electric field is readily tunable. Fig. 4a demonstrates that for a fixed t_R = 7.2 t_so, varying Δ_z produces oscillatory T_K/T_K′ with alternating peaks and gaps, leading to perfect valley polarization near the peak maxima. The effect persists for Fermi energies up to 1.5 t_so and for realistic on‑site energies (E = 0.15 t). Conductance, proportional to the total transmission, thus displays a pronounced peak–dip structure from which the valley‑polarized current can be inferred experimentally. Estimated conductances (~0.17 (e²/h) for E = 0.15 t) are within measurable limits. The observable energy window (~0.5 t_so) can be widened by proximity‑induced SOC (e.g., Bi(111) bilayer), which raises the intrinsic gap to ~44 meV. Similar mechanisms apply to other buckled 2D materials such as germanene, stanene, and MoS₂, which possess larger band gaps and SOC strengths.

Conclusions

We have demonstrated that two parallel line defects in silicene, together with Rashba spin‑orbit coupling and an externally applied perpendicular electric field, enable a fully electrical control of valley polarization. The transmission spectra of the K and K′ valleys oscillate with identical periodicity when RSOC dominates, and the field breaks this symmetry, creating distinct peak‑gap regions that host perfect valley‑polarized currents. This approach offers a practical, tunable route for generating and detecting valley currents in silicene and related 2D materials, paving the way for future valley‑based electronics.

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:

Two‑dimensional

ELD:

Extended line defect

FET:

Field‑effect transistor

RSOC:

Rashba spin‑orbit coupling

SOC:

Intrinsic spin‑orbit coupling