Proximity exchange effects in MoSe$_2$ and WSe$_2$ heterostructures with CrI$_3$: twist angle, layer, and gate dependence

TL;DR

This study uses first-principles calculations to demonstrate how proximity exchange effects in MoSe2 and WSe2 heterostructures with CrI3 can be controlled by twist angle, layer stacking, and gating, revealing potential for magnetic property tuning in 2D materials.

Contribution

It reveals that proximity exchange in MoSe2 and WSe2 can be modulated by twist, gate, and layer configuration, and shows the effects are detectable via optical signatures, advancing 2D magnetic heterostructure design.

Findings

Proximity exchange can be tuned by twist angle and gating.

Antiferromagnetic CrI3 bilayers induce similar exchange effects as ferromagnetic layers.

Adding an hBN barrier reduces the proximity exchange significantly.

Abstract

Proximity effects in two-dimensional (2D) van der Waals heterostructures offer controllable ways to tailor the electronic band structure of adjacent materials. Exchange proximity in particular is important for making materials magnetic without hosting magnetic ions. Such synthetic magnets could be used for studying magnetotransport in high-mobility 2D materials, or magneto-optics in highly absorptive nominally nonmagnetic semiconductors. Using first-principles calculations, we show that the proximity exchange in monolayer MoSe$_2$ and WSe$_2$ due to ferromagnetic monolayer CrI$_3$ can be tuned (even qualitatively) by twisting and gating. Remarkably, the proximity exchange remains the same when using antiferromagnetic CrI$_3$ bilayer, paving the way for optical and electrical detection of layered antiferromagnets. Interestingly, the proximity exchange is opposite to the exchange of the adjacent antiferromagnetic layer. Finally, we show that the exchange proximity is confined to the layer adjacent to CrI$_3$, and that adding a separating hBN barrier drastically reduces the proximity effect. We complement our it ab initio results with tight-binding modeling and solve the Bethe-Salpeter equation to provide experimentally verifiable optical signatures (in the exciton spectra) of the proximity exchange effects.

Figures

Tables

Table 1. Table 1: Summary of fit parameters, calculated dipoles and distances for TMDC/CrI 3 systems. The orbital gap Δ Δ \Delta of the spectrum and the Fermi velocity v F subscript 𝑣 F v_{\textrm{F}} . The parameters λ c subscript 𝜆 c \lambda_{\textrm{c}} and λ v subscript 𝜆 v \lambda_{\textrm{v}} describe the SOC splittings, and B c subscript 𝐵 c B_{\textrm{c}} and B v subscript 𝐵 v B_{\textrm{v}} are the proximity exchange parameters for CB and VB. The dipole of the structures is given in debye and the distance d TMDC/CrI 3 TMDC/CrI 3 {}_{\textrm{TMDC/CrI${}_{3}$}} is defined in Fig. 1 (a). The twist angle 0 ∘ superscript 0 0^{\circ} ( 30 ∘ superscript 30 30^{\circ} ) corresponds to the small (large) supercell in Fig. 1 . The individual columns are for calculations of the bare TMDC (B) with the modified lattice constant from the heterostructure, the heterostructure without SOC (noSOC) and with SOC.

	MoSe2/CrI3			WSe2/CrI3			MoSe2/CrI3	WSe2/CrI3
dipole [Debye]	0.103			0.156			0.172	0.790
distance [Å]	3.506			3.497			3.517	3.529
twist angle [∘]	0			0			30	30
calculation	B	noSOC	SOC	B	noSOC	SOC	noSOC	noSOC
$\Delta$ [eV]	1.302	1.305	1.301	1.327	1.358	1.327	1.351	1.417
$v_{\text{F}}$ [${10}^5\text{m}/\text{s}$]	4.591	4.579	4.591	5.863	5.799	5.845	4.597	5.863
${\lambda }_{\text{c}}$ [meV]	-9.647	-	-9.678	13.90	-	13.81	-	-
${\lambda }_{\text{v}}$ [meV]	94.56	-	94.43	241.79	-	240.99	-	-
${\lambda }_{\text{R}}$ [meV]	-	-	-	-	-	-	-	-
$B_{\text{c}}$ [meV]	-	-2.081	-1.592	-	-2.223	-1.783	-1.641	-1.648
$B_{\text{v}}$ [meV]	-	-1.454	-1.426	-	-1.446	-1.583	0.502	1.896

Full text

transition-metal dichalcogenides, transition-metal trihalides, van der Waals heterostructures, proximity exchange effect, antiferromagnetism, intralayer exciton, valley polarization/splitting

I Introduction

Two-dimensional (2D) materials and their hybrids offer unforeseen opportunities, but also challenges, to electronics, spintronics, optics and magnetism Fiori et al. (2014); Ferrari et al. (2015); Choi et al. (2017); Schwierz et al. (2015); Žutić et al. (2004). Graphene, the prototypical 2D crystal, has excellent charge and spin transport properties Gurram et al. (2018); Castro Neto et al. (2009); Das Sarma et al. (2011); Han et al. (2014), but lacks an orbital band gap needed for digital transistor applications. Fortunately, we have now available a large class of air-stable 2D semiconductors—transition-metal dichalcogenides (TMDCs)—which have a band gap in the optical range Kormányos et al. (2014); Liu et al. (2015); Tonndorf et al. (2013); Tongay et al. (2012); Eda et al. (2011), and form a favorite platform for optical experiments including optical spin injection due to helicity-selective optical exitationsXiao et al. (2012): electrons in opposite valleys, but at the same energy, feel opposite spin-orbit fields, pointing out of the plane. This effect is called valley Zeeman coupling; unlike true Zeeman field, the valley coupling preserves time reversal symmetry, since it stems from spin-orbit coupling (SOC).

In fact, one can create a proximity structure from graphene and a TMDC, initially proposed by DFT calculations Gmitra and Fabian (2015) and confirmed experimentally Luo et al. (2017); Avsar et al. (2017) to facilitate transfer of the optically generated spin in TMDC into graphene. This is an example of a proximity effect Žutić et al. (2018) in van der Waals heterostructures. Proximity effects provide fascinating opportunities for band-structure engineering. Experiments and theory show that graphene can borrow different properties from a variety of substrates, be it SOC or magnetism Gmitra and Fabian (2015); Zollner et al. (2016); Qiao et al. (2014); Zhang et al. (2018); Averyanov et al. (2018); Khokhriakov et al. (2018); Hallal et al. (2017); Cardoso et al. (2018).

For all-2D spintronics, it is desirable to integrate 2D materials such as graphene and TMDCs with 2D magnets. Experimentalists have demonstrated magnetic order in 2D layered crystals, such as MnSe2 O’Hara et al. (2018); Kan et al. (2014), CrGeTe3 Li and Yang (2014); Carteaux et al. (1995); Gong et al. (2017); Siberchicot et al. (1996); Lin et al. (2017); Wang et al. (2018), and CrI3 Liu et al. (2016); Zhang et al. (2015); McGuire et al. (2015); Webster et al. (2018); Huang et al. (2017); Jiang et al. (2018a); Soriano et al. (2019); Huang et al. (2018); Jiang et al. (2018b); Wu et al. (2019), which are well suited for nanoelectronic devices Burch et al. (2018). Unlike thin films of conventional ferromagnets, which have magnetization typically in the plane (of the film), the 2D layered ferromagnets have magnetization pointing out of the plane, making them Ising-like.

This out-of-plane exchange interaction is a time-reversal breaking analog of the valley Zeeman splitting in TMDCs. The interplay of the two couplings, exchange and valley Zeeman, motivates explorations of stacked TMDCs and 2D ferromagnets. Certainly, one can introduce Zeeman coupling by applying an external magnetic field pointing out of the plane, but such fields produce modest valley splittings, about 0.1 - 0.2 meV per teslaSrivastava et al. (2015); Aivazian et al. (2015); Li et al. (2014); MacNeill et al. (2015). Proximity exchange fields can induce much stronger effects, perhaps up to hundreds of meVs, without significantly altering the band structure of TMDCs. Vitale et al. (2018); Langer et al. (2018); Schaibley et al. (2016); Ye et al. (2016); Zhong et al. (2017); Ji et al. (2018); Li et al. (2018); Qi et al. (2015); Zhang et al. (2016); Xu et al. (2018); Zhao et al. (2017); Ye et al. (2016); Zhong et al. (2017); Seyler et al. (2018); Peng et al. (2017). Conventional ferromagnetic substrates, such as EuO or MnO, were predicted to give 200 - 300 meV Qi et al. (2015); Zhang et al. (2016); Xu et al. (2018); experiments on EuS find only 2.5 meV Zhao et al. (2017), presumably due to uneven interfaces.

There already are experiments demonstrating proximity exchange in TMDCs. Recent measurements in TMDC/CrI3 heterostructures Seyler et al. (2018); Zhong et al. (2017) show a few meV of proximity exchange. CrI3 is especially interesting, because the monolayer is a ferromagnet (FM) Jiang et al. (2019a); Huang et al. (2017), while bilayer CrI3 shows antiferromagnetic (AFM) coupling Huang et al. (2017); Jiang et al. (2018c); Song et al. (2018a), in contradiction to the existing theory Jiang et al. (2019a); Soriano et al. (2019); Sivadas et al. (2018) which predicts a FM state for the low temperature phase. Remarkably, the magnetization of the CrI3 can be tuned optically Seyler et al. (2018), thereby influencing proximity exchange and the photoluminescence (PL) spectrum of the TMDC. In addition, the magnetism in few layer CrI3 can be controlled by gating and external magnetic fields Jiang et al. (2018b); Huang et al. (2018), opening a new path for gate controlled devices, such as spin-filter tunnel junctions Song et al. (2018a); Klein et al. (2018); Ghazaryan et al. (2018), and AFM spintronics Baltz et al. (2018); Jungwirth et al. (2016).

Here, we provide a systematic theoretical analysis of the proximity exchange coupling in TMDC/CrI3 heterostructures (with MoSe2 and WSe2 as TMDC) from first-principles. First, we confirm that the magnetic insulator substrate couples weakly to the TMDC by van der Waals forces, preserving the characteristic electronic band structure of the TMDC. The proximity exchange coupling splits the conduction (CB) and valence band (VB) of the TMDC by roughly 1–5 meV, and combined with the intrinsic (valley Zeeman) SOC of the TMDC lifts the valley degeneracy. We introduce a minimal model Hamiltonian to describe the proximity effects in TMDC due to CrI3, extracting realistic proximity exchange parameters which should be useful for modeling transport and optics.

Next, and this is the main result of the paper, we find wide tunability of proximity exchange effects with respect of twisting and gating, and the absence of effects coming from additional layers (both TMDC and CrI3). In particular, we find that proximity exchange splittings depend on the twist angle between the TMDC and the CrI3. We investigated 0 and 30 degrees structures, and observed that not only the magnitudes of the exchange differ, but, remarkably, the direction of the exchange field for holes changes sign. The exchange parameters can be tuned by a few meVs by gating, using accessible electric fields of a few V/nm.

It is rather fascinating that adding another layer of CrI3 does not affect the proximity exchange in TMDCs (we used MoSe2), given that the two magnetic layers are antiferromagnetically coupled and they have zero net magnetic moment. The proximity effect can then be used to detect, optically or electrically, the magnetic moment of the adjacent (to the TMDC) CrI3 layer even in the antiferromagnetic state. We also explicitly prove the short-range nature of the proximity effects by investigating bilayer-MoSe2/CrI3 and MoSe2/hBN/CrI3. In the former, the proximity affects the adjacent TMDC layer, while in the latter the insulating hBN layer drastically reduces the proximity exchange. This message is important since experimentally it may be desirable to cover CrI3 with hBN first, to improve its stability under ambient conditions. For proximity effects, using hBN barriers would be detrimental.

Finally, we give specific predictions for optical signatures of the proximity exchange effects, by calculating the excitonic absorption spectra employing the Bethe-Salpeter equation. The twist-angle and gate-bias dependence of the proximity exchange is mapped into the valley splitting of the first intralayer exciton peak. This is a valuable and experimentally testable fingerprint of our results.

II Band structure, geometry, and twist effects

To study proximity exchange effects, we set up a common unit cell for the TMDC/CrI3 heterostructures 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results.. We consider a $2\times 2$ supercell for the TMDCs (MoSe2 and WSe2) above a $1\times 1$ cell of CrI3, as well as a larger $7\times 7$ supercell of the TMDC on top of a ($2\sqrt{3}\times 2\sqrt{3}$)R$30^{\circ}$ supercell of CrI3. In Figs. 1(a,b), we show the geometry of the MoSe2/CrI3 heterostructure, as a typical structure of the small supercell with $0^{\circ}$ twist angle. In Fig. 1(f), we show the top view of the larger supercell which is twisted by $30^{\circ}$ relative to the underlying CrI3.

In Figs. 1(c-e) we show the calculated band structure with a fit to our model Hamiltonian 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results. for the small MoSe2/CrI3 supercell structure without SOC (to extract the exchange coupling). We find that the bands of the TMDC are nicely preserved but are marked with a proximity exchange. The spin up CBs, originating from the CrI3, are located within the band gap of the TMDC, see Fig. 1(c). In experiments it was already shown that the spin polarized in-gap states from the CrI3 quench the PL spectrum for one light helicity only Seyler et al. (2018); Zhong et al. (2017), due to additional nonradiative relaxation processes. The energy gap of the full heterostructure is $\Delta E\approx 400$ meV, as defined in Fig. 1(c). The band edges of the TMDC can be almost perfectly described by our model Hamiltonian, as shown in Figs. 1(d,e). Due to proximity exchange, the bands are spin split by about $5$ meV.

In Figs. 1(g-i) we present the calculated band structure with a fit to our model Hamiltonian for the larger MoSe2/CrI3 supercell structure with $30^{\circ}$ twist angle without SOC. As there are much more atoms in the supercell, more in-gap states from the CrI3 are located within the band gap of the TMDC. The proximity exchange is clearly visible. In comparison to the smaller, $0^{\circ}$ supercell, the VB splitting is opposite in sign. Unfortunately, studying other, especially small twist angles is beyond our DFT approach. But the different direction of the exchange field in VB seen for $0^{\circ}$ and $30^{\circ}$ structures shows that twisting can be an effective tool to modify the proximity effect. Similar effect was predicted in SOC proximity effect, by placing graphene on a topological insulator. Two different twist angles produced qualitatively different spin-orbit fields in graphene Song et al. (2018b).

To investigate the optical signatures of the proximity exchange due to the twist angle, we calculate the absorption spectra of the intralayer excitons, i.e., electron-hole pairs created directly at the TMDC layer that are experimentally accessible in PL spectraZhong et al. (2017); Seyler et al. (2018). For these calculations we apply the effective Bethe-Salpeter equationRohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019) using the model Hamiltonian fitted to the first-principles band structure, see Figs. 1(d,e,h,i). For these calculations we also consider the effects of SOC by combining the parameters summarized in Tab. 1 (see Supplemental Material 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results. for details of the excitonic calculations and the model Hamiltonian). Focusing on the lowest energy excitonic levels, we show in Figs. 1(j,k) the first absorption peak at K ($\sigma_{-}$ polarization) and K’ ($\sigma_{+}$ polarization) valleys for $0^{\circ}$ and $30^{\circ}$ twist angles. We find that the valley splitting (the energy separation between the two absorption peaks) shows a substantial 3-fold increase by changing the twist angle, from $\sim$1.13 meV for $0^{\circ}$ to $\sim$3.89 meV for $30^{\circ}$. Interestingly, these energy splittings calculated within the single-particle picture ($\sim$1.25 meV for $0^{\circ}$ and $\sim$4.29 meV for $30^{\circ}$) are in reasonable agreement with the excitonic calculations. We also performed a similar investigation for WSe2/CrI3 heterostructures and found an excitonic (single-particle) valley splitting of $\sim$1.43 (1.61) meV for $0^{\circ}$ and $\sim$6.35 (7.09) meV for $30^{\circ}$, thus providing a dramatic $\sim$4.4-fold increase due to twist angle. In recent experiments by Zhong et al.Zhong et al. (2017), the measured valley splitting in WSe2/CrI3 is $\sim$3.5 meV (equivalent to $\sim$13 T external magnetic field in bare WSe2), and thus it is reasonable to expect that the structure (or part of it, depending on the quality) was twisted. Undoubtedly, the WSe2/CrI3 heterostructure demands further experimental investigations of proximity exchange, especially with respect to different values of the twist angle. A similar twist angle dependence of proximity SOC has been reported for graphene/TMDC heterostructures Li and Koshino (2019).

As a magnetic field or proximity exchange breaks time-reversal symmetry, we show the calculated band structure for MoSe2/CrI3 with SOC in Fig. 2, for the $0^{\circ}$ twist angle supercell. We find a very good agreement between the model Hamiltonian and the calculated bands around K and K’ valley. The valley degeneracy is now broken, especially when comparing the CB edges at K and K’ valley, see Figs. 2(b,c). Therefore, a TMDC/CrI3 heterostructure shows valley polarization of the TMDC, in agreement with recent measurements Seyler et al. (2018); Zhong et al. (2017), and other first-principles calculations Lin et al. (2019). Inclusion of SOC effects for the large supercell structure is beyond our computational possibilites and therefore not included here.

In order to qualitatively extract the influence from the FM substrate, we calculate the band structure of the TMDC/CrI3 heterostructures, for MoSe2 and WSe2 in the small supercell geometry, and fit it to our low energy Hamiltonian 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results.. To obtain reasonable fit parameters we consider three situations. First, we calculate the band structure for the bare TMDC, removing the CrI3, and check the SOC parameters for this situation, with the modified lattice constant used in the heterostructure geometry. This is crucial, because an increase in the lattice constant diminishes for example the band gap of the TMDC Fang et al. (2018); Lloyd et al. (2016); Frisenda et al. (2017); Plechinger et al. (2015). Second, we calculate the band structure for the TMDC/CrI3 heterostructures without SOC in order to obtain the proximity exchange splitting parameters. Finally, we calculate the band structure for the TMDC/CrI3 heterostructures with SOC. In Tab. 1 we summarize the fit results for the three mentioned calculations, for $0^{\circ}$ twist angle. Additionally, we give the fit results for the $30^{\circ}$ twist angle case, corresponding to Figs. 1(f-i) without SOC.

Especially interesting are the proximity exchange parameters, being roughly $2$ meV in magnitude, translating into about 10 T exchange field Srivastava et al. (2015); Aivazian et al. (2015); Li et al. (2014); MacNeill et al. (2015), in agreement with recent experiments Zhong et al. (2017); Seyler et al. (2018). The calculation of the atomic magnetic moments reveals, that the magnetization direction of the TMDC is the same as in the I atoms, opposite to the Cr atoms, therefore giving negative proximity exchange parameters for a net CrI3 magnetization pointing along positive $z$-direction towards the TMDC. As there are many atoms in the unit cells, we calculate averaged magnetic moments for the different atomic species. The averaged magnetic moments for the small non-twisted supercell are: Cr ($+3.53~{}\mu_{B}$), I ($-0.19~{}\mu_{B}$), Mo ($-0.0039~{}\mu_{B}$), and Se ($-0.0046~{}\mu_{B}$). We averaged only over Se atoms closer to the CrI3 substrate, as they mediate the proximity exchange to the Mo atoms. The averaged magnetic moments for the large twisted supercell are: Cr ($+3.46~{}\mu_{B}$), I ($-0.18~{}\mu_{B}$), Mo ($-0.0029~{}\mu_{B}$), and Se ($-0.0031~{}\mu_{B}$).

Including SOC in the heterostructure calculations, we find the parameters to be barely different than those from the individual calculations for the bare TMDCs or for the heterostructures without SOC. Therefore, combining the model Hamiltonian, the SOC parameters from the bare TMDC monolayer, and the exchange parameters from the heterostructure calculation without SOC, already suffices to get a reasonable low energy band structure. Our fit shows that the Rashba parameter is not necessary to capture the essentials of the band structure for the TMDC/CrI3 stacks, because there is no in-plane component of the spin expectation value present around the band edges.

From the experimental point of view, when materials are mechanically exfoliated and stacked, one can expect various local interface (stacking) configurations between the TMDC and CrI3, different to what is shown in Fig. 1(b). However, our calculations show, that proximity exchange splittings are marginally affected by different interface configurations, see Supplemental Material 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results..

III Short-rangeness of proximity effects

III.1 Bilayer TMDC on monolayer CrI3

In experiment, when the TMDC is exfoliated from bulk crystals, it may also happen that a bilayer TMDC is transferred onto the magnetic CrI3 substrate. Does the second TMDC layer also experience proximity exchange, or only the closest layer? To answer this question, we consider 2H bilayer MoSe2 on top of monolayer CrI3. We first allow for relaxation of the whole geometry, similar to the monolayer MoSe2 case, to get reasonable interlayer distances.

In Fig. 3(a) we show the calculated band structure of 2H bilayer MoSe2 on top of monolayer CrI3 without considering SOC effects. The bilayer TMDC has an indirect band gap, consistent with literature Kormányos et al. (2018); Bussolotti et al. (2018). For the CB edge, see Fig. 3(b), we can identify the bands originating from the first and second layer of the bilayer MoSe2. The bands from the first layer are spin-split, similar to what we observe for the monolayer MoSe2 case, while the bands originating from the second layer experience negligible splitting. The VB edge around the K point, see Fig. 3(c), is formed by two non-degenerate bands, a spin up and a spin down one, which are split in energy. The magnitude of the splitting is again similar to the monolayer case and a result of proximity exchange. However, we find that each band is formed by orbitals from both TMDC layers, which does not allow us to trace back the splitting to proximity exchange in a specific layer. For further insights, we show the calculated layer- and spin-resolved density of states, see Supplemental Material 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results.. In additon, the calculated magnetic moments for the second layer ($-0.0001~{}\mu_{B}$) are much smaller than for the first layer ($-0.004~{}\mu_{B}$). Thus we conclude that the proximity exchange is mainly induced in the TMDC layer closest to the CrI3 substrate.

III.2 TMDC on antiferromagnetic bilayer CrI3

Another feature which was especially observed in bilayer CrI3 is the switching from FM to AFM coupled layers by gating Huang et al. (2017, 2018). For the AFM state, there are two energetically degenerate states of the bilayer (first layer $\uparrow$, second layer $\downarrow$, or vice versa). Therefore, we calculate the band structure, with and without SOC, for MoSe2 on top of bilayer CrI3, which is stacked in the low temperature rhombohedral phase Jiang et al. (2019a); Soriano et al. (2019); Sivadas et al. (2018). We find a total energy difference of $E_{\textrm{AFM}}-E_{\textrm{FM}}\approx 24$ meV between the FM and the AFM state calculated without SOC of the whole TMDC/bilayer-CrI3 stack, which contains four Cr atoms in our supercell. In agreement with previous DFT calculations Soriano et al. (2019); Jiang et al. (2019a); Sivadas et al. (2018), we find that the FM state of the bilayer CrI3 is energetically favorable compared to the AFM state in contrast to experiments Huang et al. (2017); Jiang et al. (2018c); Song et al. (2018a).

In Fig. 3(f) we show the calculated band structure of MoSe2 on top of bilayer CrI3 without SOC, when the bilayer CrI3 is in the AFM ($\uparrow\downarrow$) configuration (the magnetization of the CrI3 layer directly below the TMDC is pointing $\uparrow$). In the Supplemental Material 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results., we show the band structures including SOC for both cases, the FM and AFM configuration. The fit to the low energy bands, see Figs. 3(g,h) are similar to what is shown in Fig. 1 for MoSe2 on top of monolayer CrI3. The naive expectation is that, depending on the total magnetization of the bilayer CrI3, we can enhance or reduce proximity exchange in the TMDC, compared to the monolayer CrI3 case. However, we find that the FM ($\uparrow\uparrow$) or AFM ($\uparrow\downarrow$) coupled bilayer give almost no difference in the fit parameters compared to the monolayer case 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results..

III.3 Effects of hBN barrier

Finally, when introducing a hBN buffer layer between the MoSe2 and CrI3, proximity exchange splittings of the MoSe2 bands are drastically reduced to about $100~{}\mu$eV 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results.. The calculated averaged magnetic moments for the case of MoSe2/hBN/CrI3 are: Cr ($+3.50~{}\mu_{B}$), I ($-0.19~{}\mu_{B}$), N ($-0.0016~{}\mu_{B}$), B ($0.0~{}\mu_{B}$), Mo ($-0.0001~{}\mu_{B}$), and Se ($-0.0001~{}\mu_{B}$). The magnetic moments in the TMDC are reduced by one order of magnitude, compared to the case without the hBN layer. By looking at the proximity exchange in the hBN layer, we find that the bands originating from hBN are strongly hybridized with the CrI3 bands, see Supplemental Material 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results.. This can be helpful for interpreting tunneling experiments of such heterostructures. All these results indicate that proximity exchange is truly a short range effect, and can be used to create and detect the magnetic order in the layered AFM, bilayer CrI3 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results..

IV Gate tunable proximity exchange and exciton splitting

Motivated by recent experiments Jiang et al. (2018b); Huang et al. (2018, 2017), showing the electric field control of magnetism in few layer CrI3, and the optical tuning of proximity exchange in TMDC/CrI3 heterostructures Seyler et al. (2018), we perform additional calculations for our heterostructures, where we apply a transverse electric field across the geometry consisting of one monolayer of TMDC and one of CrI3.

As calculations without SOC already give reasonable proximity exchange parameters, we neglect SOC for the electric field study. In Fig. 4 we show the fit parameters as a function of a transverse electric field for TMDC/CrI3 heterostructures calculated without SOC, for the $0^{\circ}$ twist angle structures. We find that the gap parameter $\Delta$, as well as the Fermi velocity $v_{\textrm{F}}$ are barely affected by external electric fields. The dipole of the heterostructure depends linearly on the electric field. By applying an electric field, the band offsets can be changed. The band gap $\Delta$E of the heterostructure, defined in Fig. 1(c), shrinks linearly with applied electric field. This tunability of the band offsets could be very important for other effects. Imagine electrons located in the CrI3 layer coupled to holes in the TMDC layer. As we apply an electric field, we tune the band gap $\Delta$E, possibly affecting the lifetime of interlayer excitons. Especially interesting is the fact that the CB states, originating from the CrI3, are spin polarized, see Fig. 1(c), which then gives additional valley control, depending on the magnetization of the CrI3, due to spin-valley coupling in the TMDC. Most important, the two exchange parameters $B_{\textrm{c}}$ and $B_{\textrm{v}}$ can be tuned by the external electric field. In general, the proximity exchange increases, when the electric field is tuned from negative to positive values, which enables the gate control of proximity exchange.

Let us now look at the tunability of proximity exchange reflected in the valley splitting of the TMDC intralayer excitons. In Fig. 4(g) we sketch the energy levels for the top VB and first two CBs at K and K’ valleys for MoSe2 and WSe2. We set the top VB to zero, which simplifies the analysis by just looking at the allowed optical transitions that satisfy the spin-valley locking. The effective signature of proximity is the interplay of SOC and exchange parameters in the optically allowed transitions from the VB to the first (second) CB in MoSe2 (WSe2). The resulting valley splittings are shown in Fig. 4(h) for the first exciton absorption peak (dashed lines with points) and the single-particle of the optically active bands (solid lines). Unlike the strong nonlinear behavior observed in the exchange parameters $B_{\text{c}}$ and $B_{\text{v}}$ under applied electric field, see Figs. 4(d-e), the optical valley splitting due the proximity exchange shows a rather linear behavior, with the single-particle results following closely the excitonic calculations. Specifically for MoSe2, the valley splitting changes sign at an electric field of about -2.5 V/nm. This trend might also happen for WSe2 for further negative values of electric field. Therefore, in addition to the control of the twist angle, see Figs. 1(j,k), the application of external electric fields can modify the value of the valley splitting. In real samples it is reasonable to expect an interplay of both effects, the twist angle and the electric field. Regarding the parameters used in the calculations shown in Fig. 4(h), we used the model Hamiltonian with the values of $\Delta$, $v_{\text{F}}$, $\lambda_{\text{c}}$, $\lambda_{\text{v}}$ given in Tab. 1 with SOC and $B_{\text{c}}$, $B_{\text{v}}$ extracted from the data presented in Figs. 4(d,e). See Supplemental Material 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results. for details and the calculated absorption spectra used to extract the exciton valley splittings.

Finally, we want to discuss several experimental uncertainties, one has to consider, before directly comparing them with our results. For example, the twist angle and a resulting moirè pattern between the TMDC and CrI3 in micrometer size samples, can influence proximity exchange. We have seen, that the twist angle can lead to a giant enhancement of the valley splitting. However, when twist angle is not an issue and can be precisely controlled in experiment, there are still several different local interface configurations. We have studied this as different interface geometries of the small supercell in the Supplemental Material 111See Supplemental Material at [URL will be inserted by publisher], including Refs. Bahn and Jacobsen (2002); Jiang et al. (2018a); McGuire et al. (2015); Schutte et al. (1987); James and Lavik (1963); Song et al. (2018b); Hohenberg and Kohn (1964); Giannozzi et al. (2009); Kresse and Joubert (1999); Perdew et al. (1996); Liechtenstein et al. (1995); Grimme (2006); Barone et al. (2009); Bengtsson (1999); Qi et al. (2015); Kormányos et al. (2014); Rohlfing and Louie (2000); Scharf et al. (2017); Tedeschi et al. (2019); Faria Junior et al. (2019); Rytova (1967); Keldysh (1979); Cudazzo et al. (2011); Berkelbach et al. (2013); Huang et al. (2017); Berkelbach et al. (2013); Scharf et al. (2017); Haug and Koch (2009); Scharf et al. (2019); Zhang et al. (2019); Seyler et al. (2018); Catellani et al. (1987); Huang et al. (2018); Jiang et al. (2018c, b); Dirnberger et al. (2019); Zhong et al. (2017); McGuire (2017); Webster and Yan (2018); Zhang et al. (2015); Handy and Gregory (1952); Wakabayashi et al. (1975); Webster et al. (2018); Liu et al. (2016); Lado and Fernández-Rossier (2017); Jiang et al. (2019b); Kumar Gudelli and Guo (2019); Behera et al. (2019), for details about the used supercell geometries, the computational methods, model Hamiltonian, and futher calculation results.. Local variations in the magnitude of the proximity exchange and valley splitting can occur. One can even speculate about a vanishing global proximity exchange in the TMDC, when thinking of magnetic domains in the CrI3 substrate. We have also seen, that proximiy exchange depends on the actual electric dipole field across the sample. In contrast to our approach of mono- and bilayer CrI3 as substrate, experiments may utilize thicker CrI3 samples (few layers) affecting the overall electrostatics and the band alignment in the heterostructure. In this context, one must also consider the effect of an additional SiO2 or hBN substrate/capping layer to protect the system from the environment. In additon, recent first-principles calculations have shown that the optical and magneto-optical properties of CrI3 are also dominated by strongly bound excitons Wu et al. (2019). When studying the absorption spectra of TMDC/CrI3 heterostructures, one has to be aware that quasiparticles can in principle be created in both layers simultaneously. We conclude that experimentally (and also theoretically) one has to be very careful in preparing the heterostructures and analyzing the data, in order to make qualitative statements about the proximity exchange effects. Even though our presented analysis is very systematic, we can at most give predictions for idealized structures.

V Summary

By combining DFT calculations with a low energy model Hamiltonian of exchange proximitized TMDCs, we have shown that a CrI3 substrate causes sizable proximity exchange in the TMDCs MoSe2 and WSe2. Crucial for the magnitude of the induced valley splitting, is the twist angle between the TMDC and CrI3, as we find from the $0^{\circ}$ and $30^{\circ}$ twist angle cases, by calculating optical absorption spectra. By applying experimentally accessible electric fields transverse to the heterostructure, we can tune band offsets, proximity exchange, and consequently the valley splitting in the TMDCs. Finally, we have seen that proximity exchange originates only from the FM CrI3-layer closest to the TMDC by investigating TMDC/bilayer-CrI3 heterostructures. The observed twist angle dependence, electric field tunability, and short-rangeness of proximity exchange are experimentally testable fingerprints of our results, and should be generally valid for other 2D van der Waals heterostructures.

Acknowledgements.

This work was supported by DFG SPP 1666, SFB 1277 (B05, B07), the European Unions Horizon 2020 research and innovation program under Grant No. 785219, the Alexander von Humboldt Foundation and Capes (grant No. 99999.000420/2016-06).