Silicene is the silicon counterpart of graphene [1–3]. It consists of a monolayer of silicon atoms arranged in a honeycomb lattice with a low-buckled structure that can be described by the Dirac Hamiltonian in pseudospin space [4–6]. Because of its novel electronic properties such as a Dirac cone structure in its low-energy spectrum and the quantum spin and valley Hall effects [7–11], silicene has recently attracted much attention in condensed matter physics, not only for its fundamental scientific significance but also because of its potential applications in semiconductor spintronics [12–17] and valleytronics [18–22].

Compared to graphene, silicene has a much larger spin-orbit coupling strength and a buckled structure. These induce significant coupling between three spin-like degrees of freedom comprising the real, pseudo, and valley spins [11, 23, 24], the interplay of which leads to a rich transport behaviour [25–27]. The strong intrinsic spin-orbit coupling (SOC) in silicene can open a considerable bulk band gap at the Dirac points. Silicene is therefore a good candidate material for investigating the quantum spin Hall state [24, 28]. The coupling of the spin degrees with momentum (k) leads to a finite topological Berry curvature or Chern number [9, 29–31] and therefore, a finite Hall conductivity. This results in a robust topological transport that has been observed in, for example, topological insulators [32–34], skyrmions [35–39], and Weyl/Dirac semimetals [40–45]. In particular, because spin and valley Hall currents can coexist and flow without dissipation in silicene [46–48], it is possible to design devices with low power consumption.

In this work, we show how the quantum valley and spin Hall effects can be obtained and modulated in silicene by exploiting the interplay between the Rashba SOC and coupling to a ferroelectric layer (see Figure 1). We study how the Hall conductivities vary with the out-of-plane electric field, temperature, and Fermi energy. We show that a large band gap opening can be induced in the silicene system by varying the out-of-field electric field E _z to modulate the spin/valley Hall conductivities. Moreover, a sharp step-like change occurs in the valley (spin) Hall conductivities when the magnitude of the out-of-plane electric field exceeds (falls below) the Rashba SOC strength. The QSH, QVH, and quantum phase transitions are manifested by the relative strength of the applied electric field with respect to the SOC strength. This may potentially be utilized in a topological memory device to provide a robust read-out with distinct binary outputs that are insensitive to variations in the temperature and FE texture and other imperfections.

The low-energy Hamiltonian [9, 24] for silicene coupled to a ferroelectric (FE) layer is given by H k x , k y = ℏ v f σ x k x − η σ y k y − η s z Δ σ z + l E z σ z (1) where the σ _i s refer to pseudospin, and η = ±1 and s _z = ±1 are the valley and spin z indices, respectively. Δ is the k-independent effective SOC strength, and lE _Z the energy difference between the A and B sublattice sites under substrate effects or an applied out-of-plane electric field E _z [49] caused by the lattice buckling. The Hamiltonian can be written an effective field in pseudospin space as H = σ ⋅ b (η, s _z ) where the effective field is b (η, s _z ) = (ℏv _f k _x , − η _y ℏv _f k _y , − ηs _z Δ + lE _z ). The energy eigenvalues are then given by E ± , η , s z k = ± b = ± ℏ v f k 2 + Δ − η s z l E z 2 (2) where k = k x 2 + k y 2 and the corresponding eigenstates by | χ + , η , s z 〉 = cos θ η , s z 2 e i ϕ η , s z sin θ η , s z 2 (3) | χ − , η , s z 〉 = sin θ η , s z 2 e i ϕ η , s z − cos θ η , s z 2 (4) where θ ( η , s z ) = tan − 1 ( ℏ v f k η s z Δ − l E z ) and ϕ ( η , s z ) = tan − 1 ( η k y k x ) .

Because η and s _z in Eq. 1 can independently take the values of ± 1 and Eq. 1 is a two-by-two matrix, we introduce the collective index n = (λ, η, s _z ) for convenience to label the eight bands of the system where λ = ±1 denotes whether the band is a valence (λ = 1 with eigenenergy E ⁺) or conduction (λ = −1 with eigenergy E ⁻) band. There are therefore four bands at each of the two valleys.

Here, we treat Δ as a fixed material parameter with the value of 3.9 meV [50] and assume that lE _z is a freely adjustable parameter, which can be varied in an experiment by varying the substrate material or applying a gate voltage [49]. It is instructive to examine how the energy distributions of the four bands change with the variation of lE _z . As we shall show later, the sign of the Hall conductivity of a completely filled valence band is opposite that of the σ _z expectation of the states in the band. Eq. 1 therefore implies that the sign of the Hall conductivity of the completely filled valence band is given by that of η(lE _z − ηs _z Δ). Depending on the relative magnitudes of lE _z and Δ, the sign of (lE _z − ηs _z Δ) follows that of lE _z for the two valence bands corresponding to the two values of ± 1 for s _z in each of the two valleys if |lE _z | > |Δ|, and has opposite signs in the two valleys for the same value of s _z if |lE _z | < |Δ|. The change in the relative signs of (lE _z − ηs _z Δ) between the two spin polarizations in a given valley at the critical value of lE _z = ±Δ changes whether the Hall conductivities of the four valence bands in the two valleys add up constructively or cancel out in the calculation of the Hall and valley conductivities. Moreover, Eq. 2 implies that the band gap between the conduction and valence bands is 2‖lE _z | − |Δ‖, and that the size of the band gap can be changed by modulating the value of E _z . The critical value of lE _z = ±Δ at which the band gap vanishes corresponds to a topological phase transition.

The energy dispersion relation is plotted as a function of ℏv _f k _x in Figure 2. The eigenenergies of the bands in one valley with a given value of s _z = ±1 have the same values as those of the corresponding bands in the other valley with the opposite value of s _z . As mentioned earlier, when |lE _z | < |Δ| (Figure 2A), the two valence band states within the same valley have opposite signs of ⟨σ _z ⟩ (Figure 2A) and when |lE _z | > |Δ|, the band gap reopens, but the two valence states in each valley would now have the same sign of ⟨σ _z ⟩ (Figure 2B). The system becomes an insulator if the Fermi energy is located within the gap [8].

The coupling of the silicene monolayer to the FE layer breaks the time-reversal symmetry of the system (as evident from the lE _z σ _z term in Eq. 1, which flips sign under time reversal), and results in a finite Berry curvature Ω _n (k), which is defined as Ω n k = i 〈 ∂ k x χ n k | ∂ k y χ n k 〉 − 〈 ∂ k y χ n k | ∂ k x χ n k 〉 . (5)

For a two-by-two Hamiltonian with the form of H = b ⋅ σ , Ω _n can be calculated using Ω λ , η , s z k = − λ 1 2 b 3 b η , s z ⋅ ∂ k x b η , s z × ∂ k y b η , s z (6) and is explicitly given by Ω λ , η , s z k = ℏ v f 2 2 λ η l E z − s z Δ η ℏ v f 2 + l E z − s z Δ η 2 3 / 2 , (7) which, as mentioned earlier, is proportional to λ⟨ k ; λ, η, s _z |σ _z | k ; λ, η, s _z ⟩. Sundaram et al. [51] showed that when an electric field is applied in the x direction, the Berry curvature gives rise to an anomalous velocity in the y direction perpendicular to the applied electric field v n y = E x Ω n . This anomalous velocity gives rise to a Hall conductivity σ x y η , s z = e 2 π 2 1 ℏ ∑ λ ∫ Ω λ , η , s z k f E λ , η , s z k d k = σ 0 π ∫ η l E z − s z Δ η ℏ v f k 2 + l E z − s z Δ η 2 3 / 2 f E + , η , s z k − f E − , η , s z k d k , (8) where f(E) is the Fermi-Dirac distribution f(E) = 1/(1 + exp ((E − E _f )/k _B T)), and σ 0 = e 2 2 h is half the quantum conductance.

We then define the spin and valley Hall conductivities σ x y Spin and σ x y Val as σ x y Spin = σ x y 1,1 + σ x y − 1,1 − σ x y 1 , − 1 + σ x y − 1 , − 1 , (9) σ x y Val = σ x y 1,1 + σ x y 1 , − 1 − σ x y − 1,1 + σ x y − 1 , − 1 . (10)

For convenience, we define I k , λ , η , s z ≡ ∫ σ 0 π λ η l E z − s z Δ η ℏ v f k 2 + l E z − s z Δ η 2 3 / 2 d k (11) = 2 σ 0 ∫ λ η k l E z − s z Δ η ℏ v f k 2 + l E z − s z Δ η 2 3 / 2 d k (12) = − σ 0 η λ l E z − s z Δ η l E z − s z Δ η 2 + ℏ v f k 2 . (13)

We first consider the scenario where E _f = 0, i.e., the Fermi level lies within the bulk band gap. In this case, f ( E − 1 , η , s z ) = 1 for all combinations of η and s _z , and f ( E + 1 , η , s z ) = 0 . Noting that I ∞ , λ , η , s z = 0 (14) I 0 , − 1 , η , s z = σ 0 η sgn l E z − s z Δ η , (15)

We have σ x y η , s z = I ∞ , − 1 , η , s z − I 0 , − 1 , η , s z = η sgn l E z − s z Δ η . (16)

Here, we note from Eq. 1 that sgn (lE _z − s _z Δη) = − sgn ⟨σ _z ⟩ for a valence band, as mentioned earlier.

We thus obtain σ x y Spin = 2 σ 0 sgn l E z + Δ − sgn l E z − Δ , (17) σ x y Val = − 2 σ 0 sgn l E z − Δ + sgn l E z + Δ . (18)

One consequence of Eq. 16 is that when |Δ| > |lE _z | in the quantum spin Hall effect (QSHE) regime, the Berry curvatures for the spin up states (s _z = 1) in the two valleys have the same sign, and this is opposite to that of their respective spin down (s _z = −1) states. This results in the spin up and spin down states being driven along opposite directions perpendicular to the applied electric field by their anomalous velocity [52], giving rise to a spin Hall current (Eq. 17). In contrast, when |Δ| < |E _z | in the quantum valley Hall effect (QVHE) regime, the states in the two spin polarizations in each of the two valleys are driven along the same direction opposite that of the other valley, giving rise to a valley Hall current (Eq. 18). Experimentally, the spin and valley Hall currents may be detectable using electrical means with Hall bar geometries, which has been achieved for superlattices of the related two-dimensional material graphene [53, 54]. Special precautions may need to be taken to protect the silicene sample from exposure to air in the experiment [55]. The results of Eqs. 17, 18 at E _f = 0 meV are summarized in the lE _z − Δ phase diagram in Figure 3.

We next consider a finite value of the Fermi energy. In an experiment, the Fermi energy can be set through applying gate voltages [55, 56] or by doping [57, 58]. Owing to the anti-symmetries of the Berry curvature about E = 0 as a result of the factor of λ in Eq. 7 and the relation f (E − E _f ) = 1 − f (E _f − E) obeyed by the Fermi-Dirac distribution (Figure 4), the Hall conductivities at E = ±E _f are equal to each other. Using this equality and evaluating Eq. 8 for a finite E _F at zero temperature, we have σ x , y η , s z = σ 0 η l E z − s z Δ η | E f | , (19) regardless of whether E _f is positive or negative. Note that this expression is consistent with Eq. 16 because |E _f | = |lE _z − s _z Δη| at the minimum of the (λ = +1, η, s _z ) band and the maximum of the (λ = −1, η, s _z ) band. The corresponding spin and valley conductivities when E _f is positive and lies above the minima of all the λ = +1 bands, or when E _f is negative and lies below the band maxima of all the λ = −1 bands, is then given by σ x y Spin = σ 0 4 Δ | E f | , (20) σ x y Val = − σ 0 4 l E z | E f | . (21)

Notice that because |E _f | assumes its minimum value at the band minima or maxima, the spin and valley Hall conductivities assume their largest values when E _f is located within the band gap. This is because when E _f lies within the band gap, the valence band will be completely filled and thus all the valence band states will contribute. Because all of these contributions have the same sign, they sum up cumulatively. However, when E _f increases above the band minima of the conduction bands, there will be contributions from some of the conduction band states, which lie below E _f . These contributions have opposite signs from those of the valence bands and thus cancel the contributions from those valence band states, resulting in a lowering of the spin and valley condutivities.

We now investigate the effects of a finite temperature. Figures 4B,C show the spin and valley Hall conductivities at 1 K and 10 K at E _f = 0 meV and E _f = 8 meV, respectively. At the lower temperature of 1 K, the Fermi-Dirac distribution does not deviate very much from the step function profile at 0 K. The step-like switchover between the quantum valley Hall and qunatum spin Hall regimes at lE _z = ±Δ with the variation of lE _z at E _f = 0 meV implied by Eqs. 17, 18, and Figure 3 is evident in the 1 K curves in Figure 4B. The changes in the Hall conductivities are slightly rounded at lE _z = ±Δ because of the finite slope of the Fermi-Dirac distribution with respect to E _f at a finite temperature. In comparison, the variation of the Hall conductivities with lE _z are smoother at 10 K because of the larger thermal broadening. In addition, the magnitudes of the peak Hall conductivities tends to be smaller because some of the valence band states near the band maxima are unoccupied and do not contribute, while some of the conduction band states near the band minima are occupied, and contribute with an opposite sign.

We next consider the case where E _f has a finite positive value of 2 meV (Figure 4C), which is smaller than Δ. In contrast to the E _f = 0 meV scenario considered above where the Fermi energy is always within the bulk gap between the conduction and valence bands except at lE _z = ±Δ, the location of E _f relative to the band minima and maxima of the conduction and valence bands now varies with lE _z because the band gap is dependent on lE _z . When E _f has a finite positive value that is smaller than |Δ|, E _f lies in between the band minima of the two conduction bands in each valley when ‖Δ| − |E _f ‖ < lE _z < ‖Δ| + |E _f ‖, and within the bulk band otherwise. When E _f lies between the band minima of the two conduction bands, the spin and valley Hall conductivities both show a linear variation with lE _z as shown in the lE _z values between Δ ± E _f in Figure 4C. This is because from Eq. 13 with k = k _F for the occupied bands and k = 0 for the unoccupied ones, the net spin and valley Hall conductivities is a sum of terms that are linear in both Δ and lE _z . The size of this linear region increases with |E _f |. When |lE _z | falls outside this range of values, the Fermi energy is located within the bulk band gap, and the qualitative behavior of the spin and valley currents are similar to those of the E _f = 0 case considered in Figure 4B where the system is in the quantum spin Hall regime for |lE _z | < |Δ − |E _f ‖ and the quantum valley Hall regime for |lE _z | > |Δ + |E _f ‖. Similarly, at a higher temperature, there is a smoother variation of the Hall conductivities with lE _z .

Finally, we consider the case where E _f has a finite positive value of 8 meV (Figure 4D), which is larger than the positive value of Δ considered here. Unlike the |E _f | < |Δ| case considered in the previous paragraph in which E _f is always below the band minima of at least one of the conduction bands, E _f can now lie above the band minima of both conduction bands when |lE _z | < E _f − Δ. The independence of the spin Hall conductivity with respect to lE _z in Eq. 20 and the linear variation of the valley Hall conductivity with respect to lE _z in Eq. 21 in this range of lE _z are evident from the 1 K curves. (Note the noticeably smaller magnitude of the spin Hall conductivity and more obvious slope of the valley Hall conductivity in Figure 4D compared to those in Figure 4C in the small |lE _z | region. The small slope of the valley Hall conductivity in Figure 4C in the small |lE _z | region is due to thermal broadening.) When E _f − Δ < |lE _z | < E _f + Δ, the Fermi energy lies between the band minimum of the two bands and both the spin and valley Hall conductivities show a linear variation with lE _z . When |lE _z | > E _f + Δ, the Fermi energy lies below the minima of both conduction bands in both valleys, i.e., within the band gap. In this case, because |lE _z | is also larger than Δ, the system is in the quantum valley Hall regime in which the valley Hall conductivity approaches the constant value of ± 4σ ₀ and the spin Hall conductivity approaches the constant value of 0. Similar to the E _f = 0 case considered just now, a higher temperature results in a smoother variation of the Hall conductivities with lE _z and smaller peak values of the Hall conductivities.

In this work, we study the quantum spin Hall (QSH) and quantum valley Hall (QVH) conductivities in a silicene-ferroelectric coupled system, and analyzed the effects of the ferroelectric coupling strength (which in turn affects the energy band gap), temperature, and Fermi energy on these conductivities. Our results show that the QSH and QVH conductivities in the silicene-ferroelectric system can be readily controlled by tuning the electric field arising from the ferroelectric coupling. The coupling could be utilized to ensure a large gap opening (which is proportional to 2‖Δ| − |lE _z ‖), which would make it easier in practice to align the Fermi energy within the band gap. When this alignment is achieved, the quantum Hall conductivities would attain their maximum quantized values. Additionally, the ferroelectric coupling can be modulated to make the electric field strength equal to the SOC coupling Δ, at which point the QSH and QVH conductivity values exhibit sharp step-like jumps. These sharp transitions between quantized plateau values would be useful for memory applications where either Hall conductivity can serve as a read-out for the binary states of the stored data [59, 60]. Lastly, the QSH and QVH outputs are linked to a topological invariant, i.e., the Berry phase of the system, which confers protection against imperfections and perturbations and robustness against thermal broadening. Hence, utilizing the QSH and QVH effects in the silicene-ferroelectric coupled system as memory outputs would open a new avenue for topological spintronic and valleytronic devices that can be modulated by electrical means.