A first-principles study of bilayer 1T'-WTe2/CrI3: A candidate topological spin filter

The ability to manipulate electronic spin channels in 2D materials is crucial for realizing next-generation spintronics. Spin filters are spintronic components that polarize spins using external electromagnetic fields or intrinsic material properties like magnetism. Recently, topological protection from backscattering has emerged as an enticing feature through which the robustness of 2D spin filters might be enhanced. In this work, we propose and then characterize one of the first 2D topological spin filters: bilayer CrI3/1T'-WTe2. To do so, we use a combination of Density Functional Theory and maximally localized Wannier functions to demonstrate that the bilayer (BL) satisfies the principal criteria for being a topological spin filter; namely that it is gapless, exhibits charge transfer from WTe2 to CrI3 that renders the BL metallic despite the CrI3 retaining its monolayer ferromagnetism, and does not retain the topological character of monolayer 1T'-WTe2. In particular, we observe that the atomic magnetic moments on Cr from DFT are approximately 3.2 mB/Cr in the BL compared to 2.9 mB/Cr with small negative ferromagnetic (FM) moments induced on the W atoms in freestanding monolayer CrI3. Subtracting the charge and spin densities of the constituent monolayers from those of the BL further reveals spin-polarized charge transfer from WTe2 to CrI3. We find that the BL is topologically trivial by showing that its Chern number is zero. Altogether, this evidence indicates that BL 1T'-WTe2/CrI3 is gapless, magnetic, and topologically trivial, meaning that a terraced WTe2/CrI3 BL heterostructure in which only a portion of a WTe2 monolayer is topped with CrI3 is a promising candidate for a 2D topological spin filter. Our results further suggest that 1D chiral edge states may be realized by stacking strongly hybridized FM monolayers, like CrI3, atop 2D nonmagnetic Weyl semimetals like 1T'-WTe2.


INTRODUCTION
As signs continue to suggest that Moore's Law has plateaued after many decades, 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 spin 1,2 and momenta, 3,4 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 states, 1 which can be manipulated more rapidly and with less energy than electrons' charges. 5Spintronic devices are also less volatile than conventional electronic devices because they can preserve their spin even in the absence of electric power. 1,6Moreover, one of the primary advantages of spintronic devices is that they can be readily integrated into modern CMOS-based circuits. 6ince the birth of spintronics with the discovery of the giant magnetoresistive effect, 1 the world of spintronic device components has expanded to include various spintronic analogues a) Electronic mail: Authors to whom correspondence should be addressed: Brenda Rubenstein, brenda_rubenstein@brown.edu, Daniel Staros, daniel_staros@brown.edu, and Panchapakesan Ganesh, ganeshp@ornl.govto traditional resistors and transistors, as well as new components unique to controlling spin currents like spin filters and spin injectors. 7Many of these components take advantage of the properties of magnetic materials, in which spins are already selectively ordered.For example, two spintronic analogues to traditional resistors, spin valves and magnetic tunnel junctions (MTJs), typically consist of two ferromagnetic layers separated by an insulating layer. 6,7Varying 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 materials, 8-10 spin-selective quantum tunneling using barrier materials with different spin-dependent transmission probabilities, 11 or spin orbit coupling in Rashba-type spin filters to achieve this. 12,13][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 Figure 1). 17The magnetic Weyl semimetal Co 3 Sn 2 S 2 has recently been discussed as a potential avenue towards realizing such higher temperature chiral conducting edge states, 17,18 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 1T ′ transition metal dichalcogenides, 19 which could also give rise not only to spinpolarized currents, but spin-polarized helical edge modes that are topologically protected from backscattering.
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 devices. 209][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 metal. 32However, 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%. 2,33,349][40][41] 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 which consider the proximity effects of 2D magnets stacked atop a monolayer of 1T ′ -WTe 2 , which is the only MX 2 monolayer that exists in the 1T ′ phase in its ground state and the only such member which is topological as a freestanding monolayer. 42Until fairly recently, the only such example consisted of one layer of 1T ′ -WTe 2 interfaced with one layer of permalloy (Ni 80 Fe 20 ) to form a film with several-nm thickness that exhibited out-of-plane magnetic anistropy . 43,44More recently, proximity-induced magnetic order was observed in monolayer 1T ′ -WTe 2 placed onto antiferromagnetic trilayer CrI 3 , where edge conductance jumps were observed upon switching of CrI 3 's magnetic state. 45,46Most recently, proximityinduced half-metallicity and complete spin-polarization was predicted in bilayer 1T ′ -WTe 2 /CrBr 3 and attributed to strong orbital hybridization and charge transfer at the interface of the heterostructure. 47Nonetheless, to the best of our knowledge, no investigation of the topological properties of a 1T ′ -WTe 2 /CrX 3 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 1T ′ -WTe 2 /CrI 3 as a strong potential candidate for topological spin filtering which could leverage the perfect spin filtering of a 1T ′ -WTe 2 /CrX 3 hetrostructure in proximity to the dissipationless edge states of 1T ′ -WTe 2 .
Thus, in this manuscript, we use ab initio simulations to identify bilayer 1T ′ -WTe 2 /CrI 3 as a promising candidate for a 2D topological spin filter.1T ′ -WTe 2 is a nonmagnetic Weyl semimetal which exhibits topological edge conductance in its monolayer form, 19 while monolayer CrI 3 is a ferromagnetic Mott insulator. 48One can thus imagine that, by placing these two materials in proximity, the CrI 3 's magnetism could potentially polarize WTe 2 's edge conductance, forming a topological spin filter.To determine whether this is in fact the case, we predict the band structures, topological invariants, and interlayer charge and magnetization density transfer for BL CrI 3 /1T ′ -WTe 2 with and without spin-orbit coupling.In so doing, we unequivocally demonstrate that the proximity of CrI 3 to WTe 2 foremost results in strong interlayer coupling between the two layers, spin-polarization on the WTe 2 , and an overall trivial BL topology.These considerations, taken together with the metallic nature of the bilayer and previous evidence for spin-polarized helical edge modes in monolayer 1T ′ -WTe 2 , provide convincing evidence for the possibility of realizing chiral edge states at the interface of a terraced 1T ′ -WTe 2 /CrI 3 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 WTe 2 and exiting via spin-polarized edge conductance in chiral edge states around the monolayer WTe 2 portion; this terraced, strained 1T ′ -WTe 2 /CrI 3 bilayer is then likely a strong candidate for a highly robust, ultra-thin spin filter with 1D chiral edge states.

METHODS
In order to determine the ground electronic states of the constituent monolayers and 1T ′ -WTe 2 /CrI 3 bilayer, as well as the charge transfer present in the bilayer, we calculated ground electronic states and charge and spin densities using self-consistent Density Functional Theory.We then Wannierized the DFT orbitals into a form allowing for the calculation of Chern numbers, which determine whether a material is topologically nontrivial.As the different rhombohedral angles of monolayer R3 CrI 3 and 1T ′ -WTe 2 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 such that they could share a common cell.We therefore begin with a description of our methodology by first discussing the determination of the appropriate strain and interlayer distance of the bilayer before going into greater detail about the DFT calculations and computation of the Chern numbers.

Bilayer Supercell Construction
To construct our bilayer, we used a highly-accurate DMCoptimized monolayer CrI 3 structure containing 8 atoms in its unit cell and exhibiting triclinic (R3) symmetry. 49A monolayer 1T '-WTe 2 structure was obtained from the Materials Project website. 50During 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 WTe 2 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 CrI 3 structure is stretched by 2.5% relative to the monolayer, and WTe 2 (a = 3.505 Å) is stretched by 0.8% relative to the experimental bulk 1T ′ -WTe 2 value of 3.477 Å. 51 Additionally, our monolayer 1T ′ -WTe 2 lattice constant is close to the value of 3.502 Å previously obtained using DFT structural relaxation. 52

Density Functional Simulations
All simulations of structural and electronic properties were performed using DFT as implemented within the Quantum ESPRESSO package. 53,54The PBE and PBE+U functionals 55 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 CrI 3 . 49,56ur calculations used norm-conserving, scalar-relativistic Cr and relativistic I pseudopotentials and recently developed spin-orbit relativistic effective W and Te pseudopotentials. 57e 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 WTe 2 and the bilayer heterostructure, subsets of DFT singleparticle Bloch functions were bijectively rotated onto sets of maximally-localized Wannier functions (MLWF's) 58 starting from selected columns of the density matrix from DFT via the SCDM-k method. 59This mapping was performed for the isolated set of 31 monolayer 1T ′ -WTe 2 bands ranging from -10 eV below to 0.6 eV above the monolayer Fermi level, and for entangled sets of bilayer 1T ′ -WTe 2 /CrI 3 bands as detailed in the Supplementary Information.All of the obtained MLWF's 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 firstprinciples calculation. 60,61Thus, 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 WTe 2 's spin-orbitinduced gap opening 19 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 the Z2Pack software. 60,61

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 CrI 3 is stacked on a monolayer of 1T ′ -WTe 2 , the DFT-predicted charge density difference between the bilayer and monolayers, ρ BL − ρ CrI3 − ρ W Te2 , clearly shows charge accumulating near the CrI 3 /WTe 2 interface as ithe charge is drawn downwards (see Figure 3).This suggests that CrI 3 is a charge acceptor and WTe 2 is a charge donor in the bilayer.We see this charge transfer effect both with (non-collinear calculations) and without (collinear calculations) including SOC in our DFT calculations (see Figure S3), suggesting that it is a robust feature of the bilayer.
Interestingly, charge also accumulates in between and around the bilayer with charge accumulating near the CrI 3 within the vdW interface, and withdrawn from the portion of CrI 3 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 Figure 5).Additionally, 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 CrI 3 to lose its Mott insulating nature when it is interfaced with WTe 2 .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.
Lastly, we consider the spin polarization that accompanies these charge transfer effects by evaluating the collinear spin density difference s BL − s CrI3 − s W Te2 and noncollinear magnetization density difference m BL − m CrI3 − m W Te2 between the bilayer and individual monolayers.Both quantities point to spin-polarized charge transfer, resulting in induced magnetization on the WTe 2 as exhibited by the positive magnetization density (yellow) on WTe 2 in Figure 2. As can be observed from Figure 2, spin-orbit coupling enhances this polarization over the collinear case because spin-orbit coupling increases the time-reversal symmetry-breaking in CrI 3 , which in turn influences the WTe 2 .As before, we tabulate the DFTpredicted atomic magnetic moments of ML CrI 3 and bilayer 1T ′ -WTe 2 /CrI 3 in Supplementary Tables SII and SIV to quantify the spin-polarized charge transfer.The atomic magnetic moments on the Cr atoms are enhanced by WTe 2 , with net negative atomic moments of -0.1 µ B induced on the W atoms, which are of the same order as the iodine moments in the freestanding monolayer.The induced spin polarization is thus stronger than the electron transfer.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 1T ′ -WTe 2 possesses a band crossing below the Fermi level, which gives rise to its nontrivial topology. 42As a first step, we thus determined the band structures of our strained WTe 2 monolayer with and without spin-orbit coupling (SOC) to verify that our slight distortion does not change the bands significantly.
Indeed, as shown in Figure 4, a band crossing occurs midway between the Γ and X high-symmetry points as is also observed in simulations of pristine WTe 2 without spin-orbit coupling.Additionally, 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 5d xz and 5d z 2 orbitals. 42These 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 MLWFs up to the band circled in Figure 4 yields a Chern number of 0, meaning the topology is trivial as expected, since spin-orbit coupling was turned off.The same analysis of the noncollinear WTe 2 monolayer when spin-orbit coupling is included (bottom of Figure 4) exhibits a band gap within the 5d xz and 5d z 2 bands about halfway between the Γ and X high-symmetry points.Integration of the Berry curvature over the fiber bundle of these bands yields a Chern number of 1, verifying that this monolayer is topologically nontrivial. 42ext, we calculated the band structure, partial density of states, and Chern number for the bilayer composed of ML 1T ′ -WTe 2 and ML CrI 3 (Figure 5) with and without SOC.This bilayer loses the Mott insulating behavior of CrI 3 , with the bilayer exhibiting a metallic band structure and finite density of states at and around the Fermi level.The bilayer also maintains monolayer 1T ′ -WTe 2 's band crossing below the Fermi level, which opens when spin-orbit coupling is introduced as is visible in the upper panel of Figure 5.The integra- tion of the Berry curvature over the fiber bundle of MLWF's up to and including the W 5d xz and 5d 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-orbitcoupled MLWF's also yields a Chern number of 0, indicating that this bilayer loses the topological character of monolayer 1T ′ -WTe 2 when CrI 3 is stacked on top of it.Since the constituent 1T ′ -WTe 2 layer is itself topological, this observation lends itself to the possibility of realizing a chiral edge state in a terraced 1T ′ -WTe 2 /CrI 3 bilayer due to the change in the Chern number from 0 in the bilayer to 1 as the CrI 3 layer ends but the topological 1T ′ -WTe 2 layer continues.The step acts as the boundary where the chiral edge state should exist due to the bulk-boundary correspondence.In addition, the strong hybridization of the WTe 2 with the ferromagnetic CrI 3 should break the degeneracy of the two chiral edge states of the ML WTe 2 , causing the step edge of CrI 3 to host only a spin-polarized chiral edge state.

CONCLUSIONS
In conclusion, we have used a combination of Density Functional Theory and maximally-localized Wannier function-based tight-binding models to demonstrate that bilayer 1T ′ -WTe 2 /CrI 3 is a topologically nontrivial metallic material which exhibits enhanced Cr magnetic moments and spin-polarized charge transfer from WTe 2 to CrI 3 .Most notably, the topologically nontrivial monolayer WTe 2 becomes trivial when a monolayer of CrI 3 is placed on top of it, and this is reflected in a Chern number of 1 for monolayer 1T ′ -WTe 2 and 0 for the bilayer.When taken in conjunction with the spin-polarized and metallic nature of this bilayer, our findings suggest that it be a possible candidate for realizing a chiral conducting edge state via the construction of a terraced bilayer composed of a sheet of monolayer 1T ′ -WTe 2 that is partly covered by a monolayer of CrI 3 .Additionally, 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.
In this section, we provide additional details about the selection of energy windows and bands used in the SCDMk method 1 for calculating the maximally localized Wannier functions (MLWF's) 2 from which Chern invariants were calculated in Z2Pack. 3,4The Hilbert subspace used for calculation of MLWF's in monolayer 1T ′ -WTe 2 is illustrated in Supplementary Figure 1.The subspace is well-isolated from the other bands and the original subset of bands which were Wannierized is spanned by the obtained MLWF's (light blue).a) Electronic mail: Authors to whom correspondence should be addressed: Brenda Rubenstein, brenda_rubenstein@brown.edu and Daniel Staros, daniel_staros@brown.edu Additionally, the Hilbert subspace used for calculation of MLWF's in the bilayer required more careful evaluation.A fairly-small subspace for the collinear calculation resulted in MLWF's which were well-localized to within the dimensions of the cell, and their shape reliably reproduces that of the original bands over the relevant subspace (Supplementary Figure 2).The bottom of the total subspace deviates slightly from the original bands due to the large entanglement of bands, but this does not affect calculation of the Chern number, which is determined solely by the bands in proximity to the Fermi level and not by the core bands, which are all topologically trivial.Additionally, both CrI 3 and WTe 2 in the absence of spin-orbit coupling are known already to be topologivally trivial.For the noncollinear MLWF calculations, a much larger band subspace was required to obtain robust MLWF's; these were localized to within the dimensions of the cell and reproduced the shape of the original bands over the entire Hilbert subspace despite being entangled at the bottom.Here, we further quantify the charge-transfer and magnetic induction in bilayer 1T ′ -WTe 2 .Firstly, the charge-transfer from WTe 2 to CrI 3 is visible when spin-orbit coupling is included in the DFT calculations.As in the collinear case in the main manuscript, significant areas of negative charge density (blue) exist in the area between the two layers (Figure 3).Additionally, we now tabulate the atomic charges and magnetic moments obtained from our DFT calculations respectively.The atomic charges and moments we report are those calculated automatically by Quantum Espresso at the end of self-consistent field calculations using a semi-empirical weighted integration of electron or magnetization density around individual atomic spheres. 5,6hese data are separated into the collinear case (Tables I  and II) and the noncollinear case (Tables III and IV).As ev-idenced by changes only in the fourth significant digit of the atomic charges between collinear and noncollinear monolayers and bilayers, the inclusion of spin-orbit coupling changes atomic charges by very little.In the noncollinear bilayer, it is apparent that the Cr atoms gather a total of approximately 0.03 e − of charge and the I atoms gather a total of approximately 0.03 e − while the W atoms lose a total of approximately 0.06 e − compared to their constituent monolayers.
As the noncollinear calculation is more representative of the true physics of this system, we focus our discussion on the details of the noncollinear bilayer magnetic induction.Quantum Espresso reports the Cartesian components of magnetic moment m x atom , m y atom , and m z atom .Table IV contains the m z atom values for individual atoms, and m x atom /m y atom values were comparatively negligibly small, or in other words the Ising-like nature of CrI 3 's magnetism persists in both the noncollinear monolayer and the CrI 3 -containing bilayer.
When CrI 3 is placed on top of WTe 2 , the magnetic moments on the Cr atoms are enhanced by about 0.3 µ B from 2.9 to 3.2, the iodine magnetic moments are also enhanced by almost 0.1 µ B , and the W atoms gain atomic magnetic moments of about -0.1 µ B which were not present at all in monolayer WTe 2 while the Te atoms gain negligible moments.These changes are an order of magnitude larger than the sum of changes in atomic charges, leading us to conclude that the magnetic induction effect is stronger in this bilayer than charge transfer, although both are present.

FIG. 2 .
FIG. 2. (Left) Collinear PBE+U spin density difference (yellow = positive) s BL − s CrI3 − s W Te2 with an isosurface level value of 0.0025 shows positive magnetization on the Cr atoms.The WTe 2 layer is magnetized, but the isosurface level does not resolve it.(Right) Noncollinear PBE+U magnetization density difference (yellow = positive) m BL − m CrI3 − m W Te2 with an isosurface level value of 0.0025 shows enhanced magnetization relative to the collinear non-SOC bilayer.

FIG. 3 .
FIG. 3. (Left) Collinear PBE charge density of monolayer WTe 2 in the absence of CrI 3 with an isosurface level value of 0.8.(Right) Collinear PBE+U charge density difference between the bilayer and individual monolayers, ρ BL − ρ CrI3 − ρ W Te2 , with an isosurface level value of 0.0025.Charge transfer from WTe 2 to CrI 3 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 grey and tellurine atoms are beige.

FIG. 5
FIG. 5. (Top) Collinear PBE+U band structure (black) and partial density of states (PDOS) of bilayer 1T ′ -WTe 2 /CrI 3 with the interpolated MLWF band structures overlain (light blue).The PDOS of the W d-orbitals, Te p-orbitals and d-orbitals, and I p-orbitals are shown respectively in grey, red, dark blue, and purple.(Bottom) Noncollinear PBE+U band structure with spin-orbit coupling.

FIG. 2 .
FIG. 2. (Top) Zoomed out collinear PBE+U band structure (black) and partial density of states of bilayer 1T ′ -WTe 2 /CrI 3 with the interpolated MLWF band structures overlain (light blue).(Bottom) Zoomed out noncollinear PBE+U band structure and partial density of states of bilayer 1T ′ -WTe 2 /CrI 3 with the interpolated spinor MLWF band structure overlain.The MLWF's reproduce the chosen Hilbert subspace well around the Fermi level.

FIG. 3 .
FIG. 3. Noncollinear PBE+U charge density difference between the bilayer and individual monolayers, ρ BL − ρ CrI3 − ρ W Te2 with an isosurface level value of 0.0025.Charge transfer from WTe 2 to CrI 3 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 grey and tellurine atoms are beige.