Introduction

As signs continue to suggest that Moore’s Law has plateaued, researchers have begun to seek new routes to designing faster, smaller, more energy-efficient, and more versatile electronic devices. The key to realizing such devices will be discovering, characterizing, and designing novel nanoscale quantum electronic components whose many electronic degrees of freedom, including their electron spin1,2 and momenta3,4, can be manipulated to enable faster, more energy-efficient operations on denser data.

Along these lines, nanoscale spintronics have been hailed as extremely promising routes towards denser data storage and potentially faster and more efficient reading and writing. Unlike conventional electronics, which harness the charge of an electron, spintronic materials store information in electrons’ two possible spin states1, which can be manipulated more rapidly and with less energy than electrons’ charges5. Spintronic devices are also less volatile than conventional electronic devices because they can preserve their spin even in the absence of electric power1,6. Moreover, one of the primary advantages of spintronic devices is that they can be readily integrated into modern CMOS-based circuits6.

Since the birth of spintronics with the discovery of the giant magnetoresistive effect1, the world of spintronic device components has expanded to include various spintronic analogs to traditional resistors and transistors, as well as new components unique to controlling spin currents like spin filters and spin injectors7. Many of these components take advantage of the properties of magnetic materials, in which spins are already selectively ordered. For example, two spintronic analogs to traditional resistors, spin valves and magnetic tunnel junctions (MTJs), typically consist of two ferromagnetic layers separated by an insulating layer6,7. Varying the magnetic orientation of one of the magnetic layers and keeping the other fixed allows the resistance to spin currents to be changed by taking advantage of spin-selective quantum tunneling as in MTJs, or the giant magnetoresistive or spin-transfer torque effects as in spin valves. Perhaps the most fundamental spintronic device component, however, is that which enables the generation of spin current in the first place: the spin filter.

Spin filters are devices that generate spin-polarized currents from unpolarized electric currents by selectively transmitting electrons with a particular spin and blocking those with the opposite spin. In general, such devices have most often taken advantage of the inherent spin polarization in ferromagnetic or multiferroic materials8,9,10, spin-selective quantum tunneling using barrier materials with different spin-dependent transmission probabilities11, or spin-orbit coupling in Rashba-type spin filters to achieve this12,13. Spin filters can additionally be designed by placing ferromagnetic insulators in close proximity to superconducting junctions since the presence of an out-of-plane magnetic field can break time-reversal symmetry, causing the electrons to form spin-polarized currents through the spin-transfer torque effect14,15,16.

Recently, the concept of topological spin filters has been put forth as one promising option for improving the robustness of spin filters at higher temperatures by taking advantage of quantum anomalous Hall conductance, which is topologically protected from backscattering and could minimize dissipation as a result. By extension, edge conductance dominated by one spin channel amounts to nearly dissipationless conductance of one direction of quantum spin, also known as topological spin filtering, which can manifest as a chiral edge state along the edge of a partially exposed topologically trivial bilayer and nontrivial monolayer (see Fig. 1)17. The magnetic Weyl semimetal Co3Sn2S2 has recently been discussed as a potential avenue towards realizing such higher temperature chiral conducting edge states17,18, which could in principle also become spin-polarized. Along the same vein, a topological spin filter may be constructed by placing a ferromagnet near topologically nontrivial 2D materials such as 1\({T}^{{\prime} }\) transition metal dichalcogenides19, which could also give rise not only to spin-polarized currents, but spin-polarized helical edge modes that are topologically protected from backscattering.

Fig. 1: Schematics of the edge conductance in the materials studied in this manuscript.
figure 1

a Edge conductance around ML WTe2. b Proposed spin-polarized edge conductance/1D chiral edge state of the CrI3/WTe2 bilayer studied here.

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While many such three-dimensional spin filters have been proposed, two-dimensional materials and their heterostructures possess a larger design space advantageous for engineering new spintronic devices20. Two-dimensional heterostructures can be designed to exhibit a wide array of emergent properties by mixing and matching the properties of their constituent monolayers21,22, twisting them23,24,25,26, straining them27, or placing them in proximity to electric and/or magnetic fields28,29,30,31. Despite this, researchers have only recently made significant strides towards truly 2D spin filters which promise to be smaller, more tunable, and ideally more efficient than their 3D counterparts. Graphene is one 2D material that originally garnered spintronic interest when it was predicted to exhibit nearly perfect spin filtering when interfaced with only a couple of layers of a ferromagnetic metal32. However, experimental attempts to realize such a graphene-based spin filter fell short, initially showing tunnel magnetoresistance ratios of 0.4% for graphene/NiFe, with additional attempts increasing this ratio to no more than 5%2,33,34. More recently, the pivotal discovery of giant magnetoresistance in bilayer CrI3 in 2018 has reinvigorated the search for better 2D spin filters based on atomically thin magnets35,36,37,38, and a slew of inspired studies have since been published that take advantage of their properties38,39,40,41,42. These examples suggest that, with the right combination of 2D monolayers, 2D spin filters, and even 2D topological spin filters, should also be within reach.

Notably, there are few studies that consider the proximity effects of 2D magnets stacked atop a monolayer of \(1{T}^{{\prime} }\)-WTe2, which is the only MX2 monolayer that exists in the \(1{T}^{{\prime} }\) phase in its ground state and the only such member that is topological as a freestanding monolayer43. Until fairly recently, the only such example consisted of one layer of \(1{T}^{{\prime} }\)-WTe2 interfaced with one layer of permalloy (Ni80Fe20) to form a film with a several-nm thickness that exhibited out-of-plane magnetic anistropy44,45. More recently, proximity-induced magnetic order was observed in monolayer \(1{T}^{{\prime} }\)-WTe2 placed onto antiferromagnetic trilayer CrI3, where edge conductance jumps were observed upon switching of CrI3’s magnetic state46,47. Most recently, proximity-induced half-metallicity and complete spin polarization were predicted in bilayer \(1{T}^{{\prime} }\)-WTe2/CrBr3 and attributed to strong orbital hybridization and charge transfer at the interface of the heterostructure48. Nonetheless, to the best of our knowledge, no investigation of the topological properties of a 1\({T}^{{\prime} }\)-WTe2/CrX3 bilayer has yet been performed, let alone with the goal of realizing a new type of topological spin filter. Taken together, these discoveries point towards bilayer 1\({T}^{{\prime} }\)-WTe2/CrI3 as a strong potential candidate for topological spin filtering, which could leverage the perfect spin filtering of a 1\({T}^{{\prime} }\)-WTe2/CrX3 hetrostructure in proximity to the dissipationless edge states of \(1{T}^{{\prime} }\)-WTe2.

Thus, in this manuscript, we use ab initio and quantum transport simulations to identify terraced bilayer 1\({T}^{{\prime} }\)-WTe2/CrI3 as a promising candidate for a 2D topological spin filter. 1\({T}^{{\prime} }\)-WTe2 is a nonmagnetic Weyl semimetal that exhibits topological edge conductance in its monolayer form19, while monolayer CrI3 is a ferromagnetic Mott insulator49. One can thus imagine that, by placing these two materials in proximity, the CrI3’s magnetism could potentially polarize WTe2’s edge conductance, forming a topological spin filter. To determine whether this is in fact the case, we predict the charge and magnetization density transfer, band structures, and topological invariants for BL CrI3/1\({T}^{{\prime} }\)-WTe2 with and without spin-orbit coupling. In so doing, we unequivocally demonstrate that the proximity of CrI3 to WTe2 foremost results in strong interlayer coupling between the two layers, spin polarization on the WTe2, and an overall trivial BL topology. To determine whether edge conductance is lost in a terraced bilayer, we also predict the conduction in a model terraced bilayer, showing that spin-polarized edge conductance is retained. These considerations, taken together with the metallic nature of the bilayer and previous evidence for spin-polarized helical edge modes in monolayer \(1{T}^{{\prime} }\)-WTe2, provide convincing evidence for the possibility of realizing chiral edge states at the interface of a terraced \(1{T}^{{\prime} }\)-WTe2/CrI3 bilayer. Specifically, our results imply that electric current injected into the metallic bilayer portion of the terraced heterostructure would become spin-polarized before transferring to WTe2 and exiting via spin-polarized edge conductance in chiral edge states around the monolayer WTe2 portion; this terraced, strained \(1{T}^{{\prime} }\)-WTe2/CrI3 bilayer is then likely a strong candidate for a highly robust, ultra-thin spin filter with 1D chiral edge states.

Results

Interlayer charge transfer and magnetic induction

As a first step toward understanding the physics of our bilayer, we began by examining how the layers influence each other’s electronic structure. To do so, we analyzed the difference between the bilayer and individual monolayers’ charge and spin densities. If proximity effects are truly at play, we would expect to see significant differences in their bilayer charge and spin densities relative to the separate monolayer densities. That said, when a monolayer of CrI3 is stacked on a monolayer of 1\({T}^{{\prime} }\)-WTe2, the density functional theory (DFT)-predicted charge density difference between the bilayer and monolayers, ρBL − ρCrI3 − ρWTe2, clearly shows charge accumulating near the CrI3/WTe2 interface as the charge is drawn downwards (see Fig. 2). This suggests that CrI3 is a charge acceptor and WTe2 is a charge donor in the bilayer. We see this charge transfer effect both with (noncollinear calculations) and without (collinear calculations) including spin-orbit coupling (SOC) in our DFT calculations (see Supplementary Fig. 3), suggesting that it is a robust feature of the bilayer.

Fig. 2: Charge transfer upon placing monolayer CrI3 onto monolayer WTe2.
figure 2

Left: Collinear PBE charge density of monolayer WTe2 in the absence of CrI3 with an isosurface level value of 0.8. Right: Collinear PBE+U charge density difference between the bilayer and individual monolayers, ρBL − ρCrI3 − ρWTe2, with an isosurface level value of 0.0025. Charge transfer from WTe2 to CrI3 is evident. Yellow indicates a positive charge density and light blue indicates a negative charge density. Chromium atoms are colored dark blue and iodine atoms are purple, while tungsten atoms are gray and tellurine atoms are beige.

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Interestingly, charge also accumulates in between and around the bilayer with charge accumulating near the CrI3 within the vdW interface, and withdrawn from the portion of CrI3, which is facing away from the interface. The existence of significant charge density within the bilayer gap confirms the strong hybridization between the iodine and tellurium atoms, which is also reflected in the significant atomic overlaps of all four atomic species in the partial densities of states or PDOS (see Fig. 3). In addition, the metallicity of the bilayer is reflected in the PDOS occupations of all four atomic species at and near the Fermi level, indicating that this hybridization causes CrI3 to lose its Mott insulating nature when it is interfaced with WTe2. The Lowdin charges of the constituent monolayers and bilayer are tabulated in Supplementary Tables I and III to quantify the extent of charge transfer in this bilayer. Summing the individual atomic charges of the monolayers and the bilayer yields an electron transfer of approximately 0.06 e per primitive bilayer cell.

Fig. 3: Band structures of bilayer 1\({\rm{T}}^{{\prime}}\)-WTe2/CrI3.
figure 3

Top: Collinear PBE+U band structure (black) and partial density of states (PDOS) of bilayer 1\({T}^{{\prime} }\)-WTe2/CrI3 with the interpolated MLWF band structures overlain (light blue). Bottom: Noncollinear PBE+U band structure with spin-orbit coupling.

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Lastly, we consider the spin polarization that accompanies these charge transfer effects by evaluating the collinear spin density difference sBL − sCrI3 − sWTe2 and the z-component of the noncollinear magnetization density difference mBL − mCrI3 − mWTe2 between the bilayer and individual monolayers. We find that charge transfer in the absence of SOC is minimally spin polarized, though the collinear spin density difference indicates that the Cr moments become slightly larger, which can be attributed to charge transfer with a slightly larger spin-up character (see Fig. 4). Strikingly, the charge transfer we observe is significantly more spin-polarized when SOC is considered. The charge transfer is mostly spin-up as in the collinear case, but to such a large extent that the W atoms in WTe2 become polarized in the opposite, spin-down direction. We attribute this to the SOC-enhanced splitting of WTe2’s majority up and down spin bands, which yields majority down spin WTe2 bands that are lower in energy than its up spin bands similar to that observed in bilayer 1\({T}^{{\prime} }\)-WTe2/CrBr348. This change is additionally accompanied by an increase in the magnitude of the bilayer Cr magnetic moments relative to those in monolayer CrI3 with SOC (see Supplementary Table VI), which we attribute to the effect of strong Te/I hybridization on the anisotropic exchange interactions that stabilize monolayer CrI3’s Ising-like ferromagnetism50. It is clear from the partial density of states in Fig. 3 that the (eg) conduction bands of the bilayer with SOC have nearly equal Te and I p-orbital character, indicating strong hybridization. As the iodine SOC is instrumental in mediating the anisotropic exchange interactions which stabilize out-of-plane ferromagnetic order in monolayer CrI350, this hybridization must enhance the exchange interactions in a way that ultimately increases the magnitude of the z-component of the magnetic moments on the Cr atoms. Strong interlayer eg-eg interactions in bilayer CrI3 favor interlayer antiferromagnetic (AFM) coupling51, so the observed AFM interlayer coupling along with strong hybridization between the Te and I p-orbitals with eg character suggests a similar mechanism in this heterostructure.

Fig. 4: Bilayer spin polarization is enhanced by spin-orbit coupling.
figure 4

Left: Collinear PBE+U spin density difference (yellow = positive) sBL − sCrI3 − sWTe2 with an isosurface level value of 0.0025 shows slightly enhanced positive magnetization on the Cr atoms due to spin-polarized charge transfer. Right: Noncollinear PBE+U magnetization density z-component difference (yellow = positive) mBL − mCrI3 − mWTe2 with an isosurface level value of 0.0025 shows more drastic Cr magnetization enhancement, and additionally magnetic induction in WTe2.

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Topological properties of ML 1\({T}^{{\prime} }\)-WTe2 and BL WTe2/CrI3

To assess the potential for topology that can give rise to chiral edge states in our bilayer, we first examined the band structures of the individual monolayers and combined bilayer structure. Previous modeling has shown that 1\({T}^{{\prime} }\)-WTe2 possesses a band crossing below the Fermi level, which gives rise to its nontrivial topology43. As a first step, we thus determined the band structures of our strained WTe2 monolayer with and without SOC to verify that our slight distortion does not change the bands significantly.

Indeed, as shown in Fig. 5, a band crossing occurs mid-way between the Γ and X high-symmetry points as is also observed in simulations of pristine WTe2 without SOC. In addition, the density of states in this region has slightly more W d-orbital character than Te p-orbital character and is consistent with the previous assignment of these bands to the hybridized W 5dxz and 5\({d}_{{z}^{2}}\) orbitals43. These band structures confirm that the strain applied to the monolayer did not alter its topology. Integration of the Berry curvature over the fiber bundle of maximally localized Wannier functions (MLWFs) up to the band circled in Fig. 5 yields a Chern number of 0, meaning the topology is trivial as expected, since SOC was turned off. The same analysis of the noncollinear WTe2 monolayer when SOC is included (bottom of Fig. 5) exhibits a band gap within the 5dxz and 5\({d}_{{z}^{2}}\) bands about halfway between the Γ and X high-symmetry points. Integration of the Berry curvature over the fiber bundle of these MLWFs yields a Chern number of 1, verifying that this monolayer is topologically nontrivial43.

Fig. 5: Band structures of monolayer 1\({T}^{{\prime}}\)-WTe2.
figure 5

Top: Collinear PBE band structure (black) and partial density of states (PDOS) of 0.8%-strained 1\({T}^{{\prime} }\)-WTe2 (relative to bulk) with the interpolated MLWF band structure overlain in light blue. Bottom: Noncollinear PBE band structure with spin-orbit coupling.

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Next, we calculated the band structure, partial density of states, and Chern number for the bilayer composed of ML 1\({T}^{{\prime} }\)-WTe2 and ML CrI3 (Fig. 3) with and without SOC. This bilayer loses the Mott insulating behavior of CrI3, with the bilayer exhibiting a metallic band structure and finite density of states at and around the Fermi level. The bilayer also maintains monolayer 1\({T}^{{\prime} }\)-WTe2’s band crossing below the Fermi level, which opens when SOC is introduced as is visible in the upper panel of Fig. 3. The integration of the Berry curvature over the fiber bundle of MLWF’s up to and including the W 5dxz and 5\({d}_{{z}^{2}}\) orbitals yields a Chern number of 0, when spin-orbit coupling is not included in the DFT calculation, while the same process for the spin-orbit-coupled MLWFs also yields a Chern number of 0, indicating that this bilayer loses the topological character of monolayer 1\({T}^{{\prime} }\)-WTe2 when CrI3 is stacked on top of it.

Conductance in a model terraced 1\({T}^{{\prime} }\)-WTe2/CrI3 heterostructure

In principle, the bulk-boundary correspondence guarantees that a conducting edge must exist at the interface between topologically trivial and nontrivial regions of a system. In the spirit of completeness, we nevertheless performed a direct calculation of the conductance in a simplified model heterostructure, which suggests that even when interfacial scattering is considered, edge conductance persists at the boundary between monolayer WTe2 and bilayer 1\({T}^{{\prime} }\)-WTe2/CrI3.

The schematic in Fig. 6 depicts the hopping sites in our model heterostructure, as well as the calculated current, which is overlain onto a transparent version of the same grid. The red portion of the grid is a semi-infinite lead representing the conducting bilayer, from and to which current flows. This portion is represented by a four-band model, which contains the four bands involved in the topological crossing of WTe252 split by a magnetic field-induced potential that emulates the magnetic field produced by the proximate ferromagnetic monolayer CrI3 (see Supplementary Information). The blue region is the scattering region of pristine monolayer 1\({T}^{{\prime} }\)-WTe2, and it is represented by a four-band model which is the same as that of the lead except that it does not have the additional magnetic field-induced potential. Note that we do not include terms describing interlayer hybridization due to the fact that the primary effect of hybridization is to remove the topology of WTe2, which was already established via our calculation of the Chern number in the bilayer portion of the terraced heterostructure. The length scale of our tight-binding model is set by the spacing of the grid points, which is equal to the lattice parameters of pristine ML 1\({T}^{{\prime} }\)-WTe2 in units of Å.

Fig. 6: Edge conductance persists on the exposed 1T′-WTe2 monolayer of a model terraced heterostructure.
figure 6

Left: Top view of the sites in a simplified heterostructure in which the red region represents the lead corresponding to WTe2 subject to the magnetic field produced by CrI3 and the blue region is the scattering region of pristine ML WTe2. Right: The calculated current (in varying shades of red) overlain on a transparent version of the model heterostructure is the largest around the edge of the pristine WTe2.

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Finally, we used the Kwant software package53 to solve the scattering problem for the scattering matrix and eigenfunctions of our model heterostructure. We then constructed the current operator in terms of the scattering matrix at the energy at which the topological crossing occurs in the Kwant-computed band structure (see Supplementary Fig. 4), and calculated the conductance by applying this operator to the scattering eigenfunctions at that same energy and summing their contributions, obtaining a conductance of 7.0 e2/h. When we plot the corresponding current in the right panel of Fig. 6, it is evident that the current mostly flows around the edge, consistent with previous conductance experiments on pristine monolayer 1\({T}^{{\prime} }\)-WTe243,54. In addition, this current is down-spin (polarized) since the proximity of CrI3 lowers the energy of the down-spin bands in ML WTe2.

Discussion

We have used a combination of density functional theory and MLWF-based tight-binding models to demonstrate that bilayer 1\({T}^{{\prime} }\)-WTe2/CrI3 is a topologically nontrivial metallic material that exhibits enhanced Cr magnetic moments and spin-polarized charge transfer from WTe2 to CrI3. Most notably, the topologically nontrivial monolayer WTe2 becomes trivial when a monolayer of CrI3 is placed on top of it, and this is reflected in a Chern number of 1 for monolayer 1\({T}^{{\prime} }\)-WTe2 and of 0 for the bilayer.

Since the constituent 1\({T}^{{\prime} }\)-WTe2 layer is itself topological, this observation lends itself to the possibility of realizing a chiral edge state in a terraced 1\({T}^{{\prime} }\)-WTe2/CrI3 bilayer. According to the bulk-boundary correspondence14,55, a chiral edge state should exist at the boundary between topologically trivial and topologically nontrivial materials. If we place half of a layer of CrI3 atop a layer of WTe2 to form a terraced bilayer, the Chern number should change from 0 in the bilayer to 1 as soon as the CrI3 layer ends. This suggests that a chiral edge state should exist at this boundary. In addition, the magnetic field of the ferromagnetic CrI3 should break the degeneracy of the two chiral edge states of the ML WTe2, causing the step edge of CrI3 to host a spin-polarized chiral edge state.

We further simulated the conduction of a simplified model of the terraced bilayer, finding that the edge conductance of WTe2 is retained in such a heterostructure, and is spin polarized by the CrI3. Taken together, our findings suggest that terraced bilayer 1\({T}^{{\prime} }\)-WTe2/CrI3 in which a monolayer of 1\({T}^{{\prime} }\)-WTe2 is partly covered by a monolayer of CrI3 may exhibit a chiral conducting edge state and is thus a candidate for being a 2D topological spin filter. This is the first evidence for this type of behavior in a system composed of a nonmagnetic Weyl semimetal placed next to an atomically thin magnet, thus expanding the concept of such terraced chiral edge states beyond magnetic Weyl semimetal materials.

Methods

In order to determine the ground electronic states of the constituent monolayers and 1\({T}^{{\prime} }\)-WTe2/CrI3 bilayer, as well as the charge transfer present in the bilayer, we calculated ground electronic states and charge and spin densities using self-consistent DFT. We then Wannierized the DFT orbitals we obtained into a form allowing for the calculation of Chern numbers, which determine whether a material is topologically nontrivial. Finally, we characterized the conduction observed in our heterostructure by extending a previously parameterized k ⋅ p model of 1\({T}^{{\prime} }\)-WTe2 to a simplified version of our terraced heterostructure. As the different rhombohedral angles of monolayer R3 CrI3 and 1\({T}^{{\prime} }\)-WTe2 do not lend themselves to the simple construction of a commensurate supercell without the introduction of different strain to both layers, the layers were strained slightly by hand so that they could share a common cell. We therefore begin this methodology section by discussing the determination of the appropriate strain and interlayer distance of the bilayer before going into greater detail about the DFT calculations, the computation of the Chern numbers, and the details of our edge conductance simulations.

Bilayer supercell construction

To construct our bilayer, we used a highly accurate diffusion Monte Carlo (DMC)-optimized monolayer CrI3 structure containing 8 atoms in its unit cell and exhibiting triclinic (R3) symmetry56. A monolayer 1T’-WTe2 structure was obtained from the Materials Project website57. During the simulation of these monolayers, more than 20 Å of vacuum was added to both structures to prevent spurious self-interactions.

In stacking the layers, special consideration was given to how to align them since their monolayer structures are incommensurate. In particular, the WTe2 cell was rotated such that the original lattice constants for the monolayer cells were strained as little as possible. The result of this process was a bilayer with a lattice constant of 7.01 Å, which means that our DMC-optimized ML CrI3 structure is stretched by 2.5% relative to the monolayer, and WTe2 (a = 3.505 Å) is stretched by 0.8% relative to the experimental bulk 1\({T}^{{\prime} }\)-WTe2 value of 3.477 Å58. In addition, our monolayer 1\({T}^{{\prime} }\)-WTe2 lattice constant is close to the value of 3.502 Å previously obtained using DFT structural relaxation59.

Density functional simulations

All simulations of structural and electronic properties were performed using DFT as implemented within the Quantum ESPRESSO package60,61. The PBE and PBE+U functionals62 with U = 2 eV on the chromium atoms were selected to model these materials because previous studies demonstrated that a trial wavefunction utilizing a Hubbard U value of 2–3 eV minimizes the fixed-node error in DMC calculations of CrI356,63. Our calculations used norm-conserving, scalar-relativistic Cr and relativistic I pseudopotentials and recently developed spin-orbit relativistic effective W and Te pseudopotentials64. We employed a Monkhorst-Pack k-point mesh with dimensions 10 × 10 × 1 and a plane wave energy cutoff of 300 Ry.

Calculating topological invariants

To calculate the Chern numbers for monolayer WTe2 and the bilayer heterostructure, subsets of DFT single-particle Bloch functions were bijectively rotated onto sets of MLWFs65 starting from selected columns of the density matrix from DFT via the SCDM-k method66. This mapping was performed for the isolated set of 31 monolayer 1\({T}^{{\prime} }\)-WTe2 bands ranging from –10 eV below to 0.6 eV above the monolayer Fermi level, and for entangled sets of bilayer 1\({T}^{{\prime} }\)-WTe2/CrI3 bands as detailed in the Supplementary Information. All of the obtained MLWFs were well-localized and replicated the DFT band structure well over the span of bands involved in calculating topological invariants. Next, the hopping terms and correction terms for the lattice vectors of the hopping terms output by Wannier90 were used as input to the tight-binding model for the Z2Pack software for calculating topological invariants. Z2Pack is capable of calculating the evolution of hybrid Wannier charge centers across the surface defined by an explicit Hamiltonian H(k), a tight-binding model, or an explicit first-principles calculation67,68. Thus, with our tight-binding model as input, we used Z2Pack to calculate the hybrid Wannier center evolution of the MLWF’s corresponding to the bands up to and including the two orbitals involved in WTe2’s spin-orbit-induced gap opening19 on a small k-space sphere with a radius of 0.001 centered at the Γ-point of the first Brillouin zone. All of the Z2Pack calculations passed the line and surface convergence checks to within the default tolerances of Z2Pack67,68.

Simulating edge conductance

Lastly, in order to obtain more direct evidence that our heterostructure behaves like a topological spin filter, we modeled the edge conductance of ML 1\({T}^{{\prime} }\)-WTe2 by constructing a four-band k ⋅ p model, solving the corresponding scattering problem, and calculating the conductance of this model using the quantum transport software Kwant53.

Our model consists of two regions: a semi-infinite conducting lead representative of BL 1\({T}^{{\prime} }\)-WTe2/CrI3 and a scattering region representative of ML 1\({T}^{{\prime} }\)-WTe2, which are connected to one another. To mimic the magnetic field that would be induced by the CrI3 in the bilayer portion, we subject half of the ML 1\({T}^{{\prime} }\)-WTe2 to a magnetic field. The bilayer portion of the Hamiltonian ignores hybridization between WTe2 and CrI3 (see the Results section for discussion). We provide the forms of the continuous Hamiltonians for the lead and scattering regions in the Supplementary Information.

Next, we discretized this model onto a rectangular grid with site spacing commensurate with ML WTe2’s lattice parameters: a = 3.50 Å and b = 6.34 Å. Both regions used previously reported k ⋅ p model parameters for pristine 1\({T}^{{\prime} }\)-WTe252, and the bilayer region contained an additional energy offset for the magnetic field-induced splitting of the WTe2 bands. We then solved the scattering problem according to the Landauer-Büttiker formalism53, and applied current operators to the resulting eigenfunctions to calculate the conductance.