Spin–orbit and exchange proximity couplings in graphene/1T-TaS2 heterostructure triggered by a charge density wave

Proximity-induced fine features and spin-textures of the electronic bands in graphene-based van der Waals heterostructures can be explored from the point of tailoring a twist angle. Here we study spin–orbit coupling and exchange coupling engineering of graphene states in the proximity of 1T-TaS2 not triggering the twist, but a charge density wave (CDW) in 1T-TaS2—a realistic low-temperature phase. Using density functional theory and effective model we found that the emergence of the CDW in 1T-TaS2 significantly enhances Rashba spin–orbit splitting in graphene and tilts the spin texture by a significant Rashba angle—in a very similar way as in the conventional twist-angle scenarios. Moreover, the partially filled Ta d-band in the CDW phase leads to the spontaneous emergence of the in-plane magnetic order that transgresses via proximity from 1T-TaS2 to graphene, hence, simultaneously superimposing along the spin–orbit also the exchange coupling proximity effect. To describe this intricate proximity landscape we have developed an effective model Hamiltonian and provided a minimal set of parameters that excellently reproduces all the spectral features predicted by the first-principles calculations. Conceptually, the CDW provides a highly interesting knob to control the fine features of electronic states and to tailor the superimposed proximity effects—a sort of twistronics without twist.

The CDW degree of freedom present in 1T-TaS 2 and its spatial topology provides a channel to control the proximity exchange coupling and spin-orbit coupling in graphene without any physical change in a twist of the heterostructure components, as the CDW phase can be manipulated locally, for example, by optical means [89,90,91], electrical pulses [78,79], or just temperature and in a fully reversible manner.Such a feature has been only achieved in optical lattice analog to bilayer graphene [92].Twisted graphene/TMDC heterostructures have been recently intensively studied [93,94,95,96,97] and control of the twist angle is of paramount importance for possible applications based on the design of spin-charge conversion utilizing the Rashba-Edelstein effect [8,98,91,99,100].Here we show that the same level of functionality can be expected also for graphene/CDW states where the "role of the twist" is played by the spatial topology of CDW.
In the paper, we study the van der Waals heterostructure of graphene and 1T-TaS 2 monolayer by means of first-principles calculations and propose an effective tight-binding model for graphene bands near the Dirac point extracting relevant model parameters.We found that the model parameters reflect induced spin-orbit coupling and exchange interaction proximity effects for transitions of the 1T-TaS 2 monolayer from the normal to CDW state, and then to the ferromagnetic ground state achieved by gradually decreasing temperature.

Effective Model Hamiltonian
Based on a given structural symmetry one can derive a specific effective model Hamiltonian for graphene π-bands [101,102].It has been noted [93,94] that the most general form of the induced Rashba spin-orbit coupling term in twisted graphene/TMDC heterostructures that obeys time reversal and threefold rotation C 3 symmetries can be written as U † H R U , where H R is the effective Rashba Hamiltonian, H R = λ R (κσ x s y − σ y s x ) [103,104] and U = e −iszφ R /2 , where σ x,y are the Pauli matrices acting in sublattice space, while s x,y,z are the Pauli spin matrices for the spin degree of freedom.The Rashba angle φ R originates from spin-orbit coupling matrix elements between d-orbitals of metal atoms in TMDC [93] and in general, represents a sum of the geometric angle and the quantum phase [105].The results provided below are given with respect to the ordered Bloch basis: |A(q), ↑ , |A(q), ↓ , |B(q), ↑ , |B(q), ↓ , where the states are sublattice (A, B) and spin (↑, ↓) resolved, and the crystal momentum q is measured from the Γ point.
The corresponding effective Hamiltonian in the studied cases has C 3 symmetry, and its low energy expansion close to the K (κ = 1) and K' (κ = −1) point is given as follows: The orbital and spin proximity effects result in the following parameters: E 0 corresponds to the Dirac point offset with respect to the Fermi energy, ∆ is the staggered potential discriminating electronic states located at sublattice A and B, λ A I and λ B I are the sublattice-resolved spin-conserving next-nearest neighbor spin-orbit couplings, λ R is the Rashba spin-orbit coupling parameter with phase angle φ R .Moreover, orbital nearest neighbor hopping t and the lattice constant a complete the set of parameters for the Hamiltonian as those can be subjected to CDW distortion.We should mention that the sublattice-resolved SOC parameters can be expressed as λ A I = λ I +δλ I and λ B I = λ I −δλ I , where λ I is conventionally called intrinsic, or Kane-Mele SOC.
The above-described Hamiltonian ( 2) was used to model graphene band structure in the vicinity of the K/K' points for normal and CDW phase of 1T-TaS 2 .The model parameters were determined to fit the density functional theory (DFT) calculations.A partially filled band in the vicinity of the Fermi level tends to spin split [53] at low temperatures and strong correlations of the Ta d-electrons play an important role in the positioning of the lower and upper Hubbard bands [106,47,56,107,87].This has an effect on the emergence of the in-plane magnetic polarization of the 1T-TaS 2 [54].In order to describe the spin-orbit coupling and also the exchange proximity coupling of the graphene bands around the Dirac points-we model the magnetic interaction between graphene and 1T-TaS 2 Hubbard band in terms of effective Zeeman-like interactions captured by the following Hamiltonian: The effective Hamiltonian H ex , independent of the valley, represents the coupling of a spin with the sublattice-resolved (subscripts A/B) in-plane magnetization that is parameterized by the amplitude ∆ A/B and the angle φ A/B between the x-axis and the magnetization direction.

First-principles calculations details
We consider a heterostructure with a structural model formed by the supercell of √ 13× √ 13 for 1T-TaS 2 and 5×5 for graphene, see Fig. 1.We study a slab geometry with a vacuum of about 15 Å in the direction perpendicular to the planes in order to separate periodic images.The graphene lattice parameter was compressed by approximately 1.4% to appropriately fit the supercell, capturing the commensurate CDW in 1T-TaS 2 , using the experimental lattice constant of 12.1323 Å [108].The commensurate CDW can be described by a wavector q CDW whose orientation stems from the emerging of √ 13 × √ 13 reconstruction and rotation angle of 13.9 • [108, 109], see Fig. 1(b).
All the electronic structure calculations were performed using the Quantum Espresso suite [110,111], implementing a plane wave basis for DFT calculations [112].The Perdew-Burke-Ernzerhof exchange-correlation functional was utilized [113] for the projector augmented wave method [114].For non-collinear DFT calculations including spin-orbit coupling, fully relativistic pseudopotentials were applied [115].Small Methfessel-Paxton energy level smearing [116] of 1 mRy was used along with the kinetic energy cut-off of 53 Ry.The semiempirical van der Waals corrections were included [117,118] and the dipole correction [119] was applied to properly determine Dirac point energy offset due to dipole electric field effects.The relaxation of atomic positions within the spin-unpolarized DFT approach was allowed with a threshold force of 10 −4 Ry per Bohr radius.For spin-polarized DFT+U calculations [120], we used on-site Hubbard U = 3 eV for Ta d-orbitals, inspired by the calculations reported in Refs.[21,88], with 9 × 9 mesh of k-points used for sampling the first Brillouin zone.

Electronic band structures and results of fitting
We study two stacking configurations of graphene on 1T-TaS 2 , particularly, the top position where the carbon atom is placed over the Ta atom of the David star center and the hollow position where the center of graphene ring and the David star are placed on top of each other, see insets in Fig. 3.The CDW formation lowers the heterostructure energy per supercell by 152 meV for the top and by 136 meV for the hollow stacking, respectively.We note that those energies reasonably correspond to the transition of 1T-TaS 2 to the commensurate CDW phase [32].
In Figure 2, we show calculated band structures of the supercell heterostructure for the top stacking along the high symmetry lines in the first Brillouin zone for the case where 1T-TaS 2 is in the normal and in the CDW state (including the magnetic state, which we discuss in Section 5).The fat bands shown by the dark gray circles correspond to the states originating from the carbon atoms.The graphene gets hole-doped and its Dirac cone is shifted about 0.25 eV and 0.30 eV above the Fermi level for the normal and CDW phase, respectively.The π-bands of graphene hybridize with the 1T-TaS 2 states in the vicinity of the Fermi level.We note that in the case of the CDW phase, the bands of 1T-TaS 2 significantly change and the Dirac states hybridize near the Fermi level with the partially filled Mott band originating predominantly from the 5d orbitals of Ta atoms.The change in hybridization has a direct effect on proximity-induced spinorbit coupling in graphene.The band structure for the hollow stacking (not shown) possesses on the energy scale as shown in Fig. 2 almost the same bands' dispersion as the top stacked configuration.The bands differ noticeably in the vicinity of the Dirac point for two stackings in the CDW phase as will be seen from discussion of Fig. 3.
The Figure 3 shows details of the band structure topology of graphene in the vicinity of the Dirac point and corresponding s z , s y , and s x spin expectation values owing to proximity effects for the normal and CDW phase of 1T-TaS 2 .The calculated DFT results near the K point (k = 0 corresponds to the K point) are shown by circles while the solid lines are tight-binding model fits.Four different colours of these lines mark distinct bands in all the panels of the figure.In each case, both the electron and hole bands are spin split.For the normal phase at the top and hollow stacking, the band dispersions are electron-hole-like symmetric with significant band inversion and avoided crossings due to proximity-induced Rashba spin-orbit coupling.A similar band topology has been observed in the case of graphene on 1H-WSe 2 leading to quantum spin Hall effect and pseudohelical states predicted in graphene nanoribbons [121,122].We note that in the normal phase, the s x and s y spin expectation values are identical for pairs of particle-hole symmetric bands, so that they overlap in the plots.Also, s x and s y spin expectation values vanish exactly at the K point for all cases considered in Fig. 3 due to the inherent C 3 symmetry.The emergence of CDW in the underlying 1T-TaS 2 monolayer leads to the strong band topology modification due to a change in the spin-orbit coupling proximity-induced effects.
The complex features of the proximity-induced effects on graphene electronic structure can be studied by analyzing an effective tight-binding model.We used the Hamiltonian given by Eq. ( 2) and fit its parameters to the DFT data.The parameter values are summarized in Table 1 and provide a very good reproduction of the electronic band structure close to the Dirac point as well as the corresponding spin expectation values; compare DFT data (circles) and the tight-binding model (solid lines) in Fig. 3.The effect of the CDW phase in the case of top stacking leads to the band topologies mostly controlled by the sign change of the staggered potential ∆ and significant spin expectation values variation.In the case of hollow stacking with CDW, the bands' dispersions are similar to the case of bare graphene in a strong transverse electric field which induces |λ R | |λ I | [123] and enforces the degeneracy of a pair of bands at the K point.However, in this case, the bands are nearly degenerate and the physical mechanism is more complex due to the presence of significant staggered potential and the sublattice-asymmetry of the intrinsic spin-orbit coupling (see the parameter values in Table 1).The most striking effect of the CDW phase is the enhancement of the Rashba angle φ R in both stackings.This notably cants the in-plane spin components towards the radial direction, see Fig. 4 showing the behavior of the in-plane spin components along the circular contour in wavevector space at the distance of 10 −3 Å−1 from the K point.It is noteworthy that each arrow color corresponds to an individual energy band as in Fig. 3.A somehow similar effect regarding the presence of a radial component of the Rashba field was predicted in twisted graphene/TMDC structures [94,97,105].In the context of the enhancement of the Rashba angle by the emergence of CDW, let us emphasize the fundamental change in the band structure in the vicinity of Dirac point focusing on the bands originating from TaS 2 (compare Fig. 2(a) and (b)).In the spirit of the model presented in Ref. [93] (discussing the case of a twisted heterostructure of graphene with TMDC), the change of the Rashba angle is a consequence of a pronounced difference between the TMDC energy band structure in the normal phase and the CDW phase.

Magnetism in graphene on 1T-TaS 2 monolayer
The partially filled band at the Fermi level at the CDW phase of the 1T-TaS 2 monolayer is suspected to indicate non-vanishing magnetic polarization according to recent firstprinciples studies [27,54].In order to study the presence of spontaneous magnetic polarization in graphene/1T-TaS 2 monolayer, we have performed spin-polarized DFT+U calculations considering the CDW phase for 1T-TaS 2 .The stability of the magnetically ordered state can be evaluated by computing the energy difference between the non-  magnetic and the magnetic ground state.Our DFT+U calculations yielded values (per supercell) equal to 34.7 meV and 33.1 meV for top and hollow stacking, respectively.On the other hand, analogous calculations performed for 1T-TaS 2 monolayers with the graphene layer removed resulted in values of 29.8 meV and 28.2 meV, respectively.The results support the view that the creation of a heterostructure increases slightly the stability of the magnetic state in 1T-TaS 2 .For the calculations we have considered a preferential in-plane magnetization along the x-direction, as we found that the magnetic anisotropy energy, i.e., the in-plane minus out-of-plane configuration energy (per supercell) amounts to 0.56 meV and 0.76 meV for top and hollow stacking, respectively, supporting the easy-plane anisotropy.The same type of anisotropy is predicted to emerge for a bare 1T-TaS 2 monolayer with magnetic moments centered on David stars [54].
The band structure resulting from DFT+U calculations with CDW phase in 1T-TaS 2 is shown in Fig. 2(c), where the fat bands originating from Ta atoms are marked with circles with color scale representing the degree of spin-polarization, linear color scale from red color for spin-up polarization, through gray to blue color for spin-down polarization.The main feature distinguishing this case from unpolarized CDW phase shown in Fig. 2(b) is pronounced spin splitting of the Hubbard flat Ta band located close to the Fermi level to a pair of strongly polarized bands.The remaining bands for the plotted energy range exhibit significantly lower degree of polarization.
The effect of the in-plane magnetization on the graphene bands in the vicinity of the K/K' point is well described by the effective Zeeman-like Hamiltonian H ex , given by Eq. ( 3), representing the coupling of a spin with the sublattice-dependent in-plane magnetization, having the interaction strength ∆ A/B and an angle φ A/B with respect to the considered x-direction.Using the Hamiltonian H κ + H ex we were able to fit relativistic non-collinear DFT data.The obtained parameters are given in Table 2, while the comparison between the DFT and fitted data in Fig. 5 shows that the addition of the H ex is able to capture the dominant behavior of the graphene bands in the vicinity of the K/K' point when the magnetic polarization emerges.In both the top and the hollow stacking, the sublattice magnetization direction φ A/B mostly points towards the x-direction.It resembles the situation found in monolayer 1T-TaS 2 , in which the Mott-like band that lies on the Fermi level strongly contributes to the x-component of magnetization [54].The in-plane spin pattern, shown in Fig. 6, differs significantly from the one depicted in Fig. 4, for the non-magnetic case with CDW.The tilt angle shows pronounced angular dependence and, also, differs significantly for all four bands involved.
Our fitting shows that the presence of magnetism influences the graphene bands not only qualitatively through the appearance of the effective Zeeman-like Hamiltonian, but also quantitatively through the change of the nonmagnetic parameters of graphene Hamiltonian (Eq.( 2)).This is not a surprise, since the exchange parameters ∆ A/B of the order of 1 meV correspond to the giant built-in magnetic field of 17 T (see a comparison between the nonmagnetic data in Table 1 and magnetic data in Table 2), thus justifying the observed interference between the proximity-induced spinorbit coupling and proximity-induced-magnetism.The effective model Hamiltonian we developed can be used in future studies probing interplays of several proximity-induced interactions, for instance, investigating spin-to-charge inter-conversion phenomena [124] or novel anisotropic Rashba-Edelstein effects that can be tuned by a twist [99] or by the engineering of CDW.The temperature in such cases can be used as an easy knob for switching between different phases and for turning on/off spin-orbit and exchange interactions.

Final remarks
In the paper, we report a study of the electronic properties of van der Waals graphene/1T-TaS 2 heterostructure focusing on the proximity effects in the vicinity of the graphene Dirac point.The underlying band structures were computed by DFT calculations and then modeled analytically employing an effective tight-binding Hamiltonian derived from the symmetry considerations.The model parameters were determined by fitting the tight-binding to the DFT data.A crucial finding is the fundamental influence of CDW formation on the proximity effect, resulting in a profound change in the parameters, spin textures, and fine-band topologies.As a consequence, controlling the CDW presence in a heterostructure consisting of graphene on 1T-TaS 2 provides a promising tool to manipulate the electronic properties of graphene near the Dirac point, in particular, a possibility to tailor the radial Rashba SOC, which cants the in-plane spin projections.In addition to SOC proximity, the in-plane magnetic polarization of the CDW phase superimposes the SOC, and exchange proximity effects offering to control the proximity effects by temperature.
Our findings demonstrate that the emergence of CDW in 1T-TaS 2 profoundly modifies the proximity effects in the studied heterostructure with graphene.The effect bears resemblance to the physical situation in twisted heterostructures, where modification of the twist angle allows control over the proximity effects.It should be strongly emphasized that in our system this effect takes place without physical modification of the relative twist angle of 13.9 • between the graphene layer and 1T-TaS 2 layer, as enforced by the lattice commensurability for √ 13 × √ 13 reconstruction.Therefore, a specific sort of spintronics/twistronics without a twist may be realized based on CDW control in our heterostructure.
The proximity-induced SOC controlled with the twist angle in van der Waals graphene/TMDC heterostructure might find application in spin-to-charge interconversion [125,99].In particular, the modification of the in-plane spin texture with the twist angle enables tunability of the Rashba-Edelstein effect [99].In this context, we might mention that the controllability of the proximity effects in our heterostructure offers a route to master also the spin-to-charge inter-conversion but without the need to tune physically the twist angle and with full reversibility.

Figure 1 .
Figure 1.Structure details of graphene/1T-TaS 2 heterostructure.(a) side view of the heterostructure.(b) first Brillouin zone of considered heterostructure (gray) with charge density wave vector q CDW and the Brillouin zone of the undistorted 1T-TaS 2 (yellow).Sketch of the graphene Brillouin zone with the high-symmetry path used in band structure calculations.(c) top view of the supercell with charge density wave and graphene in top stacking with carbon atom centered in the middle of a David star (grayish background).

Figure 2 .
Figure 2. Calculated electronic band structure of graphene/1T-TaS 2 heterostructure in the top configuration for normal state (a), CDW state (b) and CDW magnetic state (c) of the 1T-TaS 2 monolayer.The fat bands for graphene states are shown with filled dark gray circles.The fat bands from Ta atoms are shown in (c) with filled circles, where the color indicates the degree of spin polarization (ranging within a linear scale from red for fully polarized spin-up through gray to blue for spin-down states).

Figure 3 .
Figure 3. Calculated electronic band structures of graphene on 1T-TaS 2 as well as s z , s y , and s x spin expectation values close to the Dirac point in normal and CDW phase for top and hollow stacking.The solid lines are a tight-binding model fit to the DFT data shown by circles.Each of four line colors corresponds to a distinct band throughout the figure.The wavevectors point towards Γ and K, as Γ ← K → M; wavevector k = 0 corresponds to the K point.

Figure 4 .
Figure 4. In-plane spin expectation values along the circular contour in wavevector space, centered at the Dirac point K in normal and CDW phase for top and hollow stacking, calculated with DFT.The four arrow colors correspond to the four individual bands as in Fig. 3 (the top row presents data for two conduction-like bands and bottom one for two valence-like bands).

Figure 5 .
Figure 5. Electronic band structures and spin expectation values of s z , s y and s x close to the Dirac point in CDW phase for top and hollow stacking, calculated with DFT+U.The solid lines are tight-binding model fit to the DFT+U data shown by solid circles.Each of four line colors corresponds to a distinct band throughout the figure.The k vectors are centered around the K point, k = 0, heading towards Γ ← K → M.

Figure 6 .
Figure 6.In-plane spin expectation values along the circular contour in wavevector space, centered at the Dirac point K for top and hollow stacking, calculated with DFT+U.The four arrow colors correspond to the four individual bands as in Fig. 5 (top row presents data for two conduction-like bands and bottom one for two valencelike bands).

Table 1 .
Tight-binding model parameters for graphene on 1T-TaS 2 determined by fitting the model given by Hamiltonian in Eq. (2) to DFT data.

Table 2 .
Tight-binding model parameters for graphene on 1T-TaS 2 in CDW phase and in-plane magnetization determined by fitting the model given by Eq. (2) and (3) to DFT data.