Introduction

The intertwining of orbital and spin degrees of freedom underpin a wealth of phenomena, from the formation of topological insulators to the spin Hall effect of light1,2,3,4,5,6,7. In condensed matter systems, spin-orbit coupling (SOC) is a relativistic interaction due to the motion of electrons in the electric field of the crystal lattice, which can yield spin-dependent band structures and Berry-curvature effects that strongly influence the electrodynamics of quasiparticles8,9. Because the Berry curvature flux encodes global topological invariants (such as the Chern number for quantum anomalous Hall insulators), SOC is also a key mechanism behind quantised transport in topological phases of matter10,11.

Broken symmetries alter the spin-orbital character of electronic states12,13, and therefore provide pathways by which to realise novel spin phenomena. Among these, the emergence of spin textures in spin-orbit-coupled systems with broken spatial inversion symmetry has generated enormous excitement in the fields of spintronics and magnonics recently14,15. Owing to a close interplay of spin, lattice (pseudospin), and orbital degrees of freedom, SOC manifests both in real and momentum spaces—spin-momentum locking of spin-split Fermi surfaces16,17,18, magnetic skyrmions19,20,21, and persistent spin helices22,23,24 are prominent examples—and forms the basis of several transport effects of fundamental and practical interest. Chief among these is the current-driven spin polarisation that occurs in non-magnetic conductors with nontrivial spin textures, such as spin-momentum-locked Rashba interfaces and topological surfaces25,26,27. The ensuing net spin polarisations are often large (allowing current-induced magnetisation switching of ferromagnets28,29,30) and tend to lie perpendicularly to the applied electric field owing to the tangential nature of conventional Rashba-type spin textures. Moreover, recent studies have found that the net spin orientation can be tuned in chiral materials boasting more exotic spin textures due to fully broken reflection symmetries31,32,33,34, which has the potential to unlock unconventional spin-orbit torques35,36,37.

Likewise, the rich landscape of spin Hall effects (SHEs) reflects the symmetries underlying spin-orbit-coupled matter38. Of recent and growing interest is the SHE in vertical heterostructures built from graphene and two-dimensional (2D) semiconductors39. In these systems, the interfacial breaking of point-group symmetries leads to two main types of SOC that can be either induced or greatly enhanced via proximity effects: the sublattice-staggered SOC (underlying the valley-Zeeman effect) and the more familiar Rashba SOC40,41,42. Beyond featuring an exceptionally high degree of SOC tunability via strain and twisting effects43,44,45,46,47,48, proximitized 2D crystals support robust extrinsic SHEs due to scalar impurities, having no counterpart in other, non-Dirac 2D systems49,50 (for a recent review see ref. 42). Such symmetry-breaking effects are also of ubiquitous importance for 2D quantised transport51,52,53,54, as well as for metallic anomalous Hall and magnetic spin Hall phases55,56.

Despite this, most theoretical work so far has focused on translation invariant spin-orbit fields that reflect the periodicity of the underlying crystal structure, since this is the most conspicuous case. An interesting exception is the modulation of the strength of Rashba and Dresselhaus SOC induced in quantum-wire setups, previously explored in the context of spin-transistor devices57,58,59. Inspired by recent advances in the realisation of artificial Dirac band structures in graphene with one-dimensional (1D) superlattice potentials60,61,62, the purpose of this work is to show that the quantum geometry and electrodynamic response of 2D materials can be engineered via synthetic spin-orbit fields created by a metasurface. Our proposal, outlined in Fig. 1a, leverages proximity-induced effects between atomically thin crystals to engender effective spin-orbit fields with periodicity aS much greater than the lattice scale, which we call super-spin-orbit fields (SSOFs). We envision that the long-wavelength modulation of the spin-orbit field acting on charge carriers can be achieved by placing graphene on a patterned high-SOC substrate, akin to the patterning of electrostatic potentials in a lateral graphene superlattice60,61,62 (other possibilities are discussed below). As we shall see, the envisaged synthetic SSOFs not only lead to the formation of mini-bands but also impact their underlying quantum geometry, yielding a number useful effects. This includes the counterintuitive and exotic possibility of creating linearly dispersing spin-degenerate electronic states even for Rashba-type SSOFs where spatial inversion symmetry is strictly broken. Our proposal is, therefore, complementary to previous superlattice setups, where the presence of spatially uniform SOC components generally leads to spin-split energy bands with non-linear dispersion as well as energy gaps63,64,65. The SSOFs are also fundamentally distinct from superlattices arising from the periodic modulation of on-site staggered potentials66,67, which lack SOC effects. Another advantage of the SSOFs introduced here is that they intrinsically generate semimetallic phases without the need for periodic Zeeman fields68, which are difficult to implement. Furthermore, in analogy to spatially uniform SOC, SSOFs endow electronic states with spin Berry curvature, paving the way to SHEs with unique geometric features.

Fig. 1: Proposed experimental setup and predicted electronic structure.
figure 1

a 1D periodic modulation of the proximity-induced SOC. In this example, the SSOF is imprinted on graphene via the use of a dielectric metasurface decorated with 2D semiconductors (labelled TMD). b Energy dispersion of low-lying states around the K valley for a zero-mean square-wave profile with aS = 100 nm, λKM = 20 meV, and u = 0. c Same as b but with u = 15 meV. As a guide to the eye, the bare energy dispersion of graphene is shown in red (inner cones).

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To model the electronic properties of a graphene sheet subject to a proximity-induced SSOF, we employ a continuum low-energy description based on the Weyl-Dirac Hamiltonian69, supplemented with a 1D periodic perturbation comprising a scalar potential U(x)60 and SOC terms allowed by symmetry40,41,42. We focus exclusively on long-period perturbations, hence suppressing intervalley scattering60. The Hamiltonian in the valley-isotropic basis is

$${H}_{\tau }=v\,({{{\boldsymbol{\sigma }}}}\cdot {{{\bf{p}}}})\otimes {s}_{0}+U(x){\sigma }_{0}\otimes {s}_{0}+{H}_{{{{\rm{so}}}},\tau }(x),$$
(1)

where v is the bare Fermi velocity of 2D massless Dirac fermions (v = 106 m/s), σa and sa (a = x, y, z) are Pauli matrices acting on the pseudospin and spin subspaces, respectively, σ0 and s0 are 2 × 2 identity matrices, p = − iℏ∇ is the momentum operator, and τ = ± 1 is the valley index. For the broad class of Dirac Hamiltonians that are locally invariant under the C3v point group41,42, the SSOF term Hso,τ(x) receives up to 3 contributions, namely, a spin-flip Rashba term [HR(x) = λR Φ(x) (σx ⊗ sy − σy ⊗ sx)], a valley-Zeeman term due the broken sublattice symmetry [Hvz,τ(x) = τλvz Φ(x) σ0 ⊗ sz], and a Kane-Mele (KM) term [HKM(x) = λKM Φ(x) σz ⊗ sz]. Here, λR, λvz, and λKM respectively denote the nominal strength of the Rashba, valley-Zeeman, and intrinsic-like SOC induced by the application of the SSOF, while Φ(x) describes the spatial profile of the SOC modulation [Φ(x + m aS) = Φ(x) with m an integer]. We note that the presence of the periodic modulation yields a mini-band energy spectrum, εnk, where \(n\in {\mathbb{Z}}\backslash \{0\}\) is the mini-band index and k is the Bloch wavevector relative to a Dirac valley; see Methods for additional details.

An example of a graphene system subject to a SSOF with a square-wave profile is depicted in Fig. 1a. Recent measurements45,46,47 have shown that proximity-induced Rashba SOC in graphene/WSe2 attains giant values of up to 15 meV46, which is more than 350 times larger than graphene’s intrinsic SOC70, and makes group VI dichalcogenides ideal high-SOC substrates for our proposal. Lastly, we assume that the superlattice potential, when present, is designed to track the SSOF modulation (e.g. via a patterned bottom gate), and thus write U(x) = u Φ(x), where u is the scalar potential amplitude. The most striking scenario, on which we will focus our attention, concerns Rashba SSOFs with a zero-mean profile [that is, \(\langle \Phi (x)\rangle =(1/{a}_{{{{\rm{S}}}}})\int_{0}^{{a}_{{{{\rm{S}}}}}}dx\,\Phi (x)=0\)]. For example, this can be accomplished through the encapsulation of a graphene sheet between identical dielectric layers with a relative offset of aS/2. More exotic experimental routes, yet viable, include metal intercalation71, periodic folding of graphene72, proximity coupling to rippled group-VI dichalcogenides73,74 and deposition of graphene on stepped surfaces75. The low-energy physics in all these routes are captured by Eq. (1) (or simple generalisations thereof) with a suitable choice of parameters. Without loss of generality, we work within the K valley (τ = 1) with a valley-degeneracy factor of two properly accounted for in physical quantities like the spin Hall conductivity.

Results and Discussion

Kane-Mele SSOF case

To build intuition, we first consider an SSOF with zero spatial average (〈Φ(x)〉 = 0) that locally preserves all the spatial symmetries of the honeycomb lattice, i.e. with a single term (HKM). The energy spectrum is two-fold spin degenerate in this case and exhibits the typical mini-band structure caused by a periodic perturbation. In Fig. 1b, we show numerically exact results for a long-period square-wave modulation of type KM; details are provided in the Methods. The most striking feature of the low-energy spectrum is the band touching at zero energy, i.e. the SOC spatial modulation precludes the opening of a topological gap3. (Higher-energy mini-bands are located at energies  ≈ ± 2πvℏ/aS ≈ ± 40 meV and thus lie outside the energy range of Fig. 1.) Importantly, the linearly dispersing zero energy states in our system cannot be gapped out without breaking the global average condition of the periodic perturbation. In other words, the Dirac point degeneracy survives SSOFs with 〈Φ(x)〉 = 0 (the reader is referred to Supplementary Notes 1–4 for numerical and analytical evidence supporting the generality of this statement). What is more, the emergent 2D Dirac fermions unveiled here remain massless even for SSOFs that locally break one or more spatial symmetries, such as a spatially modulated Rashba SOC. The robustness of the crossing point between the electron and hole mini-bands hints at significant quantum geometry effects, which will be discussed shortly.

Next, we observe that the KM-SSOF renormalises the group velocity along the modulation direction, \(\hat{x}\), while it produces no change perpendicularly to it. This is the opposite behaviour of graphene under a periodic (scalar) potential60, and provides a simple mechanism to fine tune charge carrier propagation. This possibility is highlighted in Fig. 1c, showing that the combined action of a SSOF and a periodic potential squeezes the Dirac cones along both parallel and perpendicular directions to the reciprocal superlattice vector. Perturbation theory provides further insights, as detailed in the Supplementary Notes 2 and 3. In the limit u, λKM ≪ ℏv/aS, the component of the group velocity parallel to the wavevector k of the low-lying Dirac states may be obtained directly from Supplementary Equation (2) of Supplementary Note 2. We find

$${v}_{\hat{{{{\bf{k}}}}}}\cong v\left[1-\xi \,\frac{{u}^{2}{\sin }^{2}{\theta }_{{{{\bf{k}}}}}+{\lambda }_{{{{\rm{KM}}}}}^{2}{\cos }^{2}{\theta }_{{{{\bf{k}}}}}}{{\hslash }^{2}{v}^{2}{{G}_{1}}^{2}}\right],$$
(2)

where θk is the wavevector angle, ξ is a geometric factor (ξ ≈ 1.645 for a square-wave SSOF), and G1 = 2π/aS. Equation (2) shows that the periodic perturbation can be tuned to yield an isotropic group velocity. Indeed, setting u = ± λKM results in isotropic Dirac cones, thus mimicking the low-energy physics of bare graphene without SOC. The situation becomes richer when considering realistic systems with broken spatial symmetries as shown below. For example, Dirac fermions with isotropic behaviour can be realised by means of a pure Rashba SSOF, bypassing the need for a scalar periodic potential.

Realistic SSOFs and quantum geometry effects

Now, we turn to the class of SSOFs that admix valley-Zeeman (Hvz) and Rashba (HR) terms due to the breaking of spatial symmetries. Unlike the KM-type SOC in the example above, both λR and λvz can reach experimentally relevant energy scales, which is ideal for our proposal. We primarily focus on pure Rashba SSOFs which can be realised via twist-angle engineering in graphene-on-transition metal dichalcogenide (TMD) heterostructures43,44,45,46. We neglect the KM-type SOC, which due to its smallness70 is unimportant. The idea is to tune the twist angle, so that the effective SOC of charge carriers on A and B sublattices coincide, yielding a vanishing valley-Zeeman effect, λvz = (λA − λB)/2 = 0. The resulting SOC is thus of Rashba type (allowed by the broken z → −z symmetry)43,44. This intriguing possibility has been confirmed experimentally via quasiparticle interference imaging46, showing that λvz ≈ 0 and λR ≈ 15 meV for 30∘ twist-angle graphene-on-WSe2 systems. Armed with this important insight, we start by investigating the electronic structure induced by a square-wave Rashba SSOF. The energy dispersion of charge carriers in the three lowest-lying bands, above and below the charge neutrality point, is shown in Fig. 2a. The calculated spectrum contains several genuine fingerprints of the SSOFs proposed in this work. Similar to the case above, the zero energy modes exhibit linear dispersion (i.e. the Dirac point degeneracy is protected). Furthermore, the behaviour is isotropic. Thus, with regards to energy dispersion, this system emulates pristine graphene with a strongly renormalised Fermi velocity (see below). The massless nature of low-energy excitations is a robust feature of the 2D van der Waals metamaterials underpinning the SSOFs. In fact, only perturbations breaking the zero-average condition (〈Φ〉 = 0) can gap out the massless Dirac states (see Supplementary Note 5). As such, the zero-energy modes can be mode as robust as desired in a realistic setup, by ensuring that the fabrication method preserves the global average of the periodic perturbation. This confers protection against local SOC fluctuations that are unavoidable in realistic systems.

Fig. 2: Energy dispersion and spin Berry curvature due to a square-wave Rashba SSOF.
figure 2

a Dispersion of low-lying mini-bands along a cut with kx = 0. b Spin Berry curvature along the same k-path. Mini-bands are labelled by integers next to curves [positive (negative) n labels conduction (valence) bands]. Other parameters: aS = 100 nm and λR = 20 meV.

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Next, we ask whether the SSOFs can endow 2D massless Dirac fermions with quantum geometric properties. We start by noting that the mini-bands due to a square-wave Rashba SSOF [see Fig. 2a] are two-fold spin degenerate, thus lacking a spin texture of their own. This is intriguing because the Rashba SOC breaks the spatial inversion symmetry and thus can lead to spin splittings. To explain this counter-intuitive result, we analytically compute the dispersion of the low-lying Dirac states using perturbation theory. While a standard second-order expansion in λR predicts a spin-degenerate spectrum, a cumbersome third-order calculation yields

$${\varepsilon }_{n = \pm 1,{{{\bf{k}}}}s}^{(3)}\approx \pm \left(\hslash {v}_{{{{\rm{ren}}}}}| {{{\bf{k}}}}| \,+s\tilde{\xi }\frac{{\lambda }_{{{{\rm{R}}}}}^{3}}{{\hslash }^{2}{v}^{2}{G}_{1}^{4}}| {{{\bf{k}}}}{| }^{2}\right),$$
(3)

where s = ± 1 for spin-up (spin-down) low-energy branch, \({v}_{{{{\rm{ren}}}}}=v[1-\xi {({\lambda }_{{{{\rm{R}}}}}/\hslash v{G}_{1})}^{2}]\) and \(\tilde{\xi }\) is a geometric factor that equals zero for sine- and square-wave modulations, but is otherwise non-zero (e.g, for Kronig-Penney-type modulations, \(| \tilde{\xi }|\) attains values close to 0.26; see Supplementary Note 3 for additional details. Hence, Rashba SSOFs with more general profiles can lift the spin degeneracy (as intuition would suggest), but only perturbatively. While the resulting spin splittings are typically small, a sizeable effect can be achieved by combining SSOFs with a periodic potential, providing a rich phenomenology for future exploration.

To examine the quantum geometry of SSOF-induced mini-bands, we map out the momentum-space distribution of the spin Berry (SB) curvature8

$${\Omega }_{yx,n}^{z}({{{\bf{k}}}})=-2{\hslash }^{2}\,{{{\rm{Im}}}}\, {\sum}_{m\ne n}\frac{\langle n{{{\bf{k}}}}| {v}_{y}^{z}| m{{{\bf{k}}}}\rangle \langle m{{{\bf{k}}}}| {v}_{x}| n{{{\bf{k}}}}\rangle }{{({\varepsilon }_{n{{{\bf{k}}}}}-{\varepsilon }_{m{{{\bf{k}}}}})}^{2}},$$
(4)

where vi = v σi ⊗ s0 and \({v}_{i}^{z}=v\,{\sigma }_{i}\otimes {s}_{z}\) (here, i = x, y) are the charge and spin velocity operators, respectively. This quantity governs the spin Hall transport of electron wavepackets and therefore is the geometric analogue of the Berry curvature in the anomalous Hall effect56,76. The SB curvature around the Dirac points is shown in Fig. 2b. We see that the linearly dispersing zero energy modes (labelled n = ± 1) are endowed with significant SB curvature, despite their massless nature. This is evidently at variance with other 2D gapless Dirac systems, which have vanishing SB and (charge) Berry curvature10. Thus, the emergent 2D Dirac cones reported here are not only robust against perturbations sharing the global average of the SSOFs but also display quantum geometric effects. To explore this further, Fig. 3a, b show 3D plots of the SB curvature of mini-bands with n = ± 1, ± 2 in the mini-Brillouin zone. Two features are of note. First, the central peaks in the SB curvature of the massive Dirac mini-bands n = ± 2, ± 3 discussed earlier are seen to arise from local hot spots of SB curvature [see the local maximum of \({\Omega }_{yx,\pm 2}^{z}\) at k = 0 in Fig. 3b]. The SB curvature also displays pronounced local minima near the zone edges, where Ωyx,±2 attains large negative values. Second, the massless mini-bands (n = ± 1) have a giant SB curvature at the edges of the mini-Brillouin zone (kx = ± π/aS) [see Fig. 3a], about twice as large as the Dirac-point hot spot of the massive mini-bands. We attribute this curious feature to the emergence of large pseudo-gaps along the SSOF direction; see Supplementary Figs. 1 and 2. Finally, we note that the general behaviour is highly anisotropic, except in the immediate vicinity of the Dirac point.

Fig. 3: Spin Berry curvature and spin Hall conductivity.
figure 3

a Momentum-space distribution of the SB curvature of the massless Dirac mini-bands (n = ± 1). b Same as in a for the mini-bands n = ± 2. SSOF parameters as in Fig. 2. c Spin Hall conductivity \({\sigma }_{yx}^{z}\) as a function of the chemical potential for kBT = 25 meV and selected twisting-induced SOC modulations, namely: pure Rashba SSOF (solid lines), admixed Rashba-valley-Zeeman SSOFs with λvz = λR/2 (dashed lines) and λvz = λR (dot-dashed lines). Here, green (blue) curves correspond to λR = 10 meV (20 meV).

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The enhanced SB curvature of the SSOF-induced mini-bands indicates that the 2D van der Waals metamaterials proposed here support large spin Hall responses. To confirm this, we compute the intrinsic spin Hall conductivity (\({\sigma }_{ij}^{z}\)) from the flux of SB curvature using standard methods8 (see also Methods). According to linear response theory, the z-polarised spin current density generated by an external electric field is \({{{{\bf{j}}}}}_{s}^{z}={\sum }_{i,j = x,y}{\sigma }_{ij}^{z}{E}_{j}{{{{\bf{e}}}}}_{j}\), where Ej are the field components and ei is the unit vector along the i-axis. As shown in Fig. 3c, the spin Hall response has a strong energy dependence and can reach sizeable values on the order of e/4π for typical values of proximity-induced SOC at room temperature. This behaviour is robust to imperfections in the SSOF even when the vanishing global average condition, 〈Φ(x)〉 = 0, is not exactly met. We verified this with different types of SOC and SSOF spatial patterns Φ(x). For example, the spin Hall conductivity presented in Fig. 3c is found to vary by less than 10% in the presence of a spatially uniform Rashba-like SOC as large as 50% of the SSOF magnitude itself; see Supplementary Note 5 for additional details. Moreover, at variance with 2D conductors subject to the usual uniform Rashba effect49,77, the spatial dependence of the Rashba SSOF protects our quantum geometry-driven spin Hall effect from exact cancellations due to impurity-scattering corrections. In fact, a semiclassical conservation law for expectation values involving the spin current can be derived in the vein of ref. 49 yielding \(\langle {H}_{{{{\rm{so}}}}}(x){v}_{i}^{z}\rangle =0\), with 〈. . . 〉 denoting a quantum and disorder average. For a uniform Rashba field, this relation (which holds in the presence of arbitrary non-magnetic impurity potentials) implies \(\langle {v}_{i}^{z}\rangle =0\) in steady-state conditions and thus \({{{{\bf{j}}}}}_{s}^{z}=0\). However, in our system, the condition \(\langle {v}_{i}^{z}\rangle =0\) is circumvented due to the oscillatory nature of Hso(x). The SSOF-driven SHE thus appears to be more robust than its counterpart in standard Rashba-coupled graphene.

We now briefly address the case of 2D metamaterials with concurrent Rashba-type and valley-Zeeman SSOFs. Here, the condition 〈Φ(x)〉 = 0 could be achieved by alternating the relative rotation angle of consecutive TMD layers, exploiting the anti-periodicity of the valley-Zeeman effect, λvz(θ) = −λvz(θ ± π/3)43,44. The ensuing SSOF, in this case, strongly renormalises the group velocity of wavepackets that propagate parallel to the SSOF direction. The leading correction to Eq. (3) is given by \(\delta {\varepsilon }_{(n = \pm 1),{{{\bf{k}}}}s}=\pm \xi {\Lambda }_{{{{\rm{vz}}}}}\,{\sin }^{2}{\theta }_{{{{\bf{k}}}}}\), with \({\Lambda }_{{{{\rm{vz}}}}}={\lambda }_{{{{\rm{vz}}}}}^{2}/(\hslash v{G}_{1}^{2})\), yielding an anisotropic dispersion and SB curvature at low energies (see Supplementary Note 2). The valley-Zeeman SSOF leads to an overall decrease in the SB curvature magnitude, which is reflected in the spin Hall conductivity [Fig. 3c]. This is also at odds with the expected behaviour in (standard) proximitised graphene, where the spin Hall conductivity has a non-monotonic behaviour with λvz, with λvz ≠ 049,50 being essential to observe the SHE.

In closing, we have shown that the spatial patterning of symmetry-breaking spin-orbit fields gives rise to rich physics beyond that of conventional superlattices, in particular, the emergence of 2D massless Dirac fermions with anomalous electrodynamic responses. The proposed periodic modulation of interface-induced SOC is within reach of current nano-fabrication methods and is likely to have broad applications beyond those described in this work.

Methods

Numerical implementation

Here we present the numerical framework used to study the electronic structure of 2D metamaterials described by the Hamiltonian in Eq. (1). The wavefunctions in valley τ = ± 1 are 4-component spinors of the form \({\Psi }_{\tau }({{{\bf{r}}}})={({\psi }_{\tau \uparrow }^{A}({{{\bf{r}}}}),{\psi }_{\tau \downarrow }^{A}({{{\bf{r}}}}),{\psi }_{\tau \uparrow }^{B}({{{\bf{r}}}}),{\psi }_{\tau \downarrow }^{B}({{{\bf{r}}}}))}^{{{{\rm{t}}}}}\) for τ = 1 and \({\Psi }_{\tau }({{{\bf{r}}}})={(-{\psi }_{\tau \uparrow }^{B}({{{\bf{r}}}}),-{\psi }_{\tau \downarrow }^{B}({{{\bf{r}}}}),{\psi }_{\tau \uparrow }^{A}({{{\bf{r}}}}),{\psi }_{\tau \downarrow }^{A}({{{\bf{r}}}}))}^{{{{\rm{t}}}}}\) for τ = −1. Moving to reciprocal space, the eigenproblem formally reduces to solving an infinite set of coupled equations for the plane-wave amplitudes \(\{{\psi }_{{{{\bf{k}}}}s}^{\sigma }\}\) for each valley:

$$\begin{array}{ll}\hslash v| {{{\bf{k}}}}| {e}^{-i\sigma {\theta }_{{{{\bf{k}}}}}}{\psi }_{{{{\bf{k}}}}s}^{-\sigma }+&\sum_{p\in {\mathbb{Z}}}\left[\left(u+s\sigma {\lambda }_{{{{\rm{KM}}}}}+s\tau {\lambda }_{{{{\rm{vz}}}}}\right){\Phi }_{{G}_{p}}{\psi }_{{{{\bf{k}}}}-{{{{\bf{G}}}}}_{p},s}^{\sigma }\right.\\ &\left.+i(s-\sigma ){\lambda }_{{{{\rm{R}}}}}{\Phi }_{{G}_{p}}{\psi }_{{{{\bf{k}}}}-{{{{\bf{G}}}}}_{p},-s}^{-\sigma }\right]=E{\psi }_{{{{\bf{k}}}}s}^{\sigma }\,,\end{array}$$
(5)

where s = ± (  ≡ ↑, ↓) and σ = ±  ( ≡ A, B) are the spin and pseudospin indices, respectively; the valley index is omitted for brevity. Furthermore, k is the Bloch wavevector from the Dirac point, \({\theta }_{{{{\bf{k}}}}}=\angle ({{{\bf{k}}}},\hat{x})\), \({{{{\bf{G}}}}}_{p}={G}_{p}\hat{x}\) with Gp = 2πp/aS (\(p\in {\mathbb{Z}}\)), and \({\Phi }_{{G}_{p}}\) are the Fourier coefficients of the periodic modulation.

The summation over Fourier components in Eq. (5) is truncated to a finite but large number of terms, i.e. ∣p∣≤N. The resulting system of equations is solved numerically, yielding d = 4(2N + 1) bands {εnk} and associated 4-component eigenvectors {Ψnk}, where 1≤∣n∣≤d/2. In practice, we restrict kx to the first Brillouin zone (kx ∈ ] − ks, ks] with ks = π/aS) and choose the ky interval such that the energy ranges along kx and ky directions are similar, i.e., ky ∈ ] − (2N + 1)ks, (2N + 1)ks]. The k-space intervals are discretised uniformly with Nk equally spaced points used to cover the interval] 0, ks], yielding a total of (2Nk + 1) discrete kxs and 2(2N + 1)Nk + 1 discrete kys. Good accuracy is typically achieved with N = 3 and Nk = 40, corresponding to 28 energy bands and a grid of 81 by 561 discrete points. Numerical convergence with respect to the number of Fourier components and k-space grid points was established in all calculations.

Spin Hall conductivity

The linear-response intrinsic spin Hall conductivity is calculated from the SB curvature [Eq. (4)] according to

$${\sigma }_{yx}^{z}=(e/2)\mathop{\sum}_{{{{\bf{k}}}}}\mathop{\sum}_{n}f({\varepsilon }_{n{{{\bf{k}}}}})\,{\Omega }_{yx,n}^{z}({{{\bf{k}}}}),$$
(6)

where f(ε) is the Fermi-Dirac distribution function8.

Spatial profile of SSOFs

In the main text, we focus on Kronig-Penney (KP) and sinusoidal perturbations with zero spatial average. The KP profile is

$$\Phi (x)=2\Phi \mathop{\sum }_{m=-\infty }^{\infty }\left(R(x+m{a}_{{{{\rm{S}}}}})-r\right),\,\,\,\,\,\,\,\,,0 \, < \, r \, < 1,$$
(7)

where Φ is the amplitude, R(x) = Θ(x + ℓ/2)Θ(ℓ/2 − x), Θ is the Heaviside step function, aS is the lattice width, ℓ is the barrier width (ℓ < aS), and r = ℓ/aS (for a square wave r = 0.5). For pure sinusoidal modulations, we use \(\Phi (x)=\Phi \cos ({G}_{1}x)\).