Electrically tunable exchange splitting in bilayer graphene on monolayer Cr$_2$X$_2$Te$_6$ with X=Ge, Si, and Sn

We investigate the electronic band structure and the proximity exchange effect in bilayer graphene on a family of ferromagnetic multilayers Cr$_2$X$_2$Te$_6$, X=Ge, Si, and Sn, with first principles methods. In each case the intrinsic electric field of the heterostructure induces an orbital gap on the order of 10 meV in the graphene bilayer. The proximity exchange is strongly band dependent. For example, in the case of Cr$_2$Ge$_2$Te$_6$, the low-energy valence band of bilayer graphene has exchange splitting of 8 meV, while the low energy conduction band's splitting is 30 times less (0.3 meV). This striking discrepancy stems from the layer-dependent hybridization with the ferromagnetic substrate. Remarkably, applying a vertical electric field of a few V/nm reverses the exchange, allowing us to effectively turn ON and OFF proximity magnetism in bilayer graphene. Such a field-effect should be generic for van der Waals bilayers on ferromagnetic insulators, opening new possibilities for spin-based devices.

For device operations, it is desirable to have electric control of the band structure. We already know that BLG on a substrate can open a gap (similarly to having a transverse electric field applied). Could we also control exchange coupling in a similar way, simply applying an electric field? One naturally expects that the proximity effect is strongest in the BLG layer adjacent to the ferromagnetic substrate. Orbitals from this layer form, say, the valence band, which will exhibit strong exchange splitting. The conduction band, on the other hand, would have no or only weak splitting. If an electric field is applied, the situation can be reversed, and now it would be the conduction band that has the largest exchange splitting. It has also been shown in model calculations, that BLG on a FMI can be a platform for field-effect magnetic or spin devices [49][50][51][52][53]. The combination of exchange and electric field leads to the control of the spin-dependent gap of the carriers and perfect switching of the spin polarization is predicted [51,52]. How realistic is electrical control of proximity exchange in BLG in real materials? We show here that the exchange splitting in BLG on monolayer Cr 2 Si 2 Te 6 , CGT, and Cr 2 Sn 2 Te 6 , does indeed show the band selectivity, and that the splitting in a given band can be switched on and off by a transverse electric field. This is similar to the field-effect proximity SOC in BLG on WSe 2 [54].
In this paper we investigate, by performing first principles calculations, the proximity induced exchange coupling in BLG on monolayer CGT, as well as on Cr 2 Si 2 Te 6 and Cr 2 Sn 2 Te 6 (which has been theoretically predicted [55] but not yet prepared experimentally). We find that the low energy bands of BLG show an indirect band gap of roughly 17 meV, which can be efficiently tuned by experimentally accessible electric fields of a few V/nm. The proximity induced exchange splitting of the valence band is giant, being around 8 meV, 30-times larger than the exchange splitting in the conduction band. This large difference arises from the fact, that the valence band is formed by non-dimer carbon atoms from the bottom graphene layer directly above CGT, where the proximity effect is strong, while the conduction band is formed by non-dimer carbon atoms from the top graphene layer, where the proximity effect is weak (in BLG with Bernal stacking, one pair of carbon atoms is vertically connected, which we call dimer, the other pair non-dimer). The most interesting result is the switching of the exchange splitting from the valence to the conduction band, via an external electric field that counters the built-in field of the BLG/CGT heterostructure. A proximity exchange valve is realized. We also include a brief discussion on possible device geometries based on the field-effect proximity exchange.

II. COMPUTATIONAL DETAILS AND GEOMETRY
The heterostructure of BLG on CGT is shown in Fig. 1, where a 5 × 5 supercell of BLG is placed on a √ 3 × √ 3 CGT supercell. The considered heterostructure model contains 130 atoms in the unit cell. We keep the lattice constant of graphene unchanged at a = 2.46Å and stretch the lattice constant of CGT by roughly 4% from 6.8275Å [34] to 7.1014Å. Theoretical calculations predict that the tensile strain leaves the ferromagnetic ground state unchanged, but enhances the band gap and the Curie temperature of CGT [41,44]. In Fig. 1(a) the side view of the structure shows the Bernal stacking of BLG, with an average distance relaxed to 3.266Å between the graphene layers, in good agreement with experiment [56]. The average distance between the lower graphene layer and CGT was relaxed to 3.516Å, consistent with literature [26].
Previous calculations show that the relative alignment of graphene on CGT does not influence the electronic bands much, see Ref. 26, and we will only consider the supercell with stacking as in Fig. 1. We will also not consider SOC in the calculation, since it was shown that proximity induced exchange in graphene, caused by CGT, is one order of magnitude larger than proximity SOC [26]. The electronic structure calculations and structural relaxation of BLG on CGT were per- formed by means of density functional theory (DFT) [57] within Quantum ESPRESSO [58]. Self-consistent calculations were performed with the k-point sampling of 18 × 18 × 1 to get converged results for the proximity exchange splittings. We have performed open shell calculations that provide the spin polarized ground state with a collinear magnetization. Theory and experiment predict that CGT is a FMI with magnetic anisotropy favoring a magnetization perpendicular to the CGT-plane [33-38, 48, 55]. A Hubbard parameter of U = 1 eV was used for Cr d-orbitals, being in the range of proposed U values especially for this compound [37]. The value results from comparison of DFT and experiment on the magnetic ground state of bulk CGT. Other theoretical calculations report, that their results are qualitatively independent from the used U values [43,44]. We used an energy cutoff for charge density of 500 Ry, and the kinetic energy cutoff for wavefunctions was 60 Ry for the scalar relativistic pseudopotential with the projector augmented wave method [59] with the Perdew-Burke-Ernzerhof exchange correlation functional [60]. For the relaxation of the heterostructures, we added van-der-Waals corrections [61,62] and used quasi-newton algorithm based on trust radius procedure. In order to simulate quasi-2D systems the vacuum of 20Å was used to avoid interactions between periodic images in our slab geometry. Dipole corrections [63] were also included to get correct band offsets and internal electric fields. To determine the interlayer distances, the atoms of BLG were allowed to relax only in their z positions (transverse to the layers), and the atoms of CGT were allowed to move in all directions, until all components of all forces were reduced below 10 −3 [Ry/a 0 ], where a 0 is the Bohr radius.
The electronic structure of bare BLG contains four parabolic bands near the Fermi energy [64,65]. Two low energy bands close to the charge neutrality point are formed by (p z ) orbitals of non-dimer atoms B 1 and A 2 , see Fig. 1. In addition, there are two higher lying bands formed by the orbitals of dimer atoms A 1 and B 2 , see Fig. 1. Since the dimer atoms are connected by direct interlayer hopping, the bands are shifted roughly 400 meV from the Fermi level, such that they can be ignored for transport.
In Fig. 2(a) we show the calculated electronic band structure of BLG on monolayer CGT along high symmetry lines. The bands near the Fermi level resemble closely the bands from bare BLG [64,65], even though the high energy conduction band of BLG is located within the conduction bands of CGT. Very important for transport is that the low energy bands of BLG are located within the band gap of CGT. The offset between the conduction band of BLG at K and the bottom of conduction band minimum of CGT is roughly 100 meV. However, there are two important differences compared to bare BLG. First, the heterostructure possesses an intrinsic dipole (transverse to the sheets) and thus the two graphene layers are at a different potential en-ergy resulting in a small indirect orbital gap of roughly 17 meV, see Fig. 2(b). The dipole of 1.505 Debye points from CGT towards BLG and therefore B 1 (A 2 ) electrons have lower (higher) energy forming the valence (conduction) states. Second, the ferromagnetic CGT substrate interacts mainly with the lower graphene layer and, by proximity exchange, splits the low energy bands originating from this graphene layer by 8 meV, see Fig. 2 The upper graphene layer is far away and experiences almost no proximity effect, resulting in a comparatively small spin splitting of the corresponding bands. The calculated magnetic moment of the Cr-atoms is positive and roughly 3.2 µ B , but the Te-atoms and consequently the C-atoms are polarized with a small negative magnetic moment. Thus the BLG low energy valence band is spin split, with spin down states being lower in energy, see Fig. 2(b).

IV. PROXIMITY EXCHANGE VALVE
In order to design devices for spintronics it is desirable if one can electrically control both spin and orbital properties. Such a control should be highly efficient in our heterostructures. In Fig. 3 we show the low energy band structures of BLG, in the presence of a transverse electric field. At zero field strength, see Fig. 3(d), the situation is as explained above; the hole band formed by p z -orbitals of B 1 is strongly split by proximity exchange, while the electron band, formed by p z -orbitals of A 2 is much less split. The reason is simply because the atoms B 1 are closer to the ferromagnetic substrate experiencing a stronger proximity effect than atoms A 2 . The ordering of the bands is determined by the built-in electric field, pointing from CGT towards BLG, and thus B 1 electrons are at lower energy. Now, if we apply a positive external electric field, see Fig. 3(e), it adds to the internal field, opening the orbital gap further, keeping the banddependent exchange splitting unchanged. But when the applied field is negative and strong enough to compensate the built-in electric field, the orbital gap closes, see Fig. 3(c). This compensation happens at roughly −0.25 V/nm and the states from atoms B 1 and A 2 are almost at the same potential energy, closing the gap. Still, it is the valence band that experiences the giant proximity spin splitting.
A further increase of the negative field leads to the reopening of the band gap, see Fig. 3(b), but with a switched character of the bands. Now, the conduction band is formed by atoms B 1 , which still experience the stronger proximity effect, while the valence band is formed by atoms A 2 , which are far from the ferromagnetic substrate. The switching of the band character happens, because the total (external and built-in) field is pointing now from BLG towards CGT, opposite to the zero-field case. Thus, atoms B 1 experience a higher potential than atoms A 2 . We get a proximity exchange valve. Are the relevant BLG low-energy bands still within the orbital band gap of CGT? In Fig. 4(a), we see that the calculated dipole depends linearly on the applied electric field. The built-in dipole is roughly compensated by an external field of −0.4 V/nm; the amplitude of the intrinsic field is 0.406 V/nm. Figure 4(b) shows that the energy offset between the BLG conduction band at K and the bottom of the conduction band minimum of CGT also changes with electric field. However, it is important to note that the low energy states of BLG are located within the band gap of CGT for electric fields that allow to observe the proximity exchange switching of the states. For fields larger than 0.5 V/nm, the conduction band of BLG at K is shifted above the bottom of the conduction band minimum of CGT and BLG gets hole doped.
In order to see by how much the switching of the band character affects the spin splitting of the low energy bands, we compare the splittings of the conduction and valence bands for the electric field strengths of 0 V/nm and −0.5 V/nm, see Fig. 5. For zero field we see that the valence band is strongly split, around 8 meV near the K point, while the conduction band splitting is negative and around −0.3 meV, see Figs. 5(a,b). Applying a negative electric field of −0.5 V/nm, the band character is switched and also the splittings change. Now the conduction band is strongly split, while the valence band splitting is small, see Figs. 5(c,d). We have effectively switched the spin splitting by a factor of 30.

V. BLG ON Cr2Si2Te6 AND Cr2Sn2Te6
We predict similar scenarios for X=Si and Sn. In Fig. 6  (Fig. 7) we show the band structure of BLG on Cr 2 Si 2 Te 6 (Cr 2 Sn 2 Te 6 ). Very similar to the bands of BLG on CGT, we see proximity spin splitting in the low energy valence band, much larger than in the conduction band. Here we would like to mention that the compounds with Si and Ge have been identified in experiments [33,34], whereas the compound with Sn atoms has been only predicted theoretically [55] so far.
In the Si (Sn) case, proximity induced exchange is smaller (larger), as the valence band spin splitting is about 3 meV (12 meV) at the K point and the indirect gap is roughly 13 meV (17 meV). The energy offset between the conduction band of BLG at K and the bottom of conduction band minimum of Cr 2 Si 2 Te 6 (Cr 2 Sn 2 Te 6 ) is roughly 44 meV (211 meV). The intrinsic dipole is weaker (stronger), about 1.336 (2.013) Debye, which in the end leads to slightly different electric fields to observe the switching of the band character, as shown for BLG on CGT. The average distance between the lower graphene layer and Cr 2 Si 2 Te 6 (Cr 2 Sn 2 Te 6 ) was relaxed to 3.563 (3.531)Å. From the minimal difference in distance, we conclude that the spin splitting strongly depends on the material itself, whereas in Ref. 26, it was already shown that proximity induced exchange splitting strongly depends on the distance between graphene and the CGT substrate.
For model transport calculations it is useful to have realistic parameters in order to estimate the conductance through BLG in this heterostructures. In Tab. I, we summarize some important structural information, as well as values for the exchange splitting and the orbital gap. For this we define the orbital gap ∆, as the average of the spin up (∆ ↑ ) and spin down (∆ ↓ ) orbital gaps, ∆ = (∆ ↑ + ∆ ↓ )/2 at the K point. The exchange splitting is defined as the energy difference between spin up and spin down band, λ ex = E ↑ − E ↓ , at the K point for conduction band (CB) and valence band (VB) respectively. The orbital gap, which is a measure for the strength of the intrinsic dipole across the junction, is consistent with our dipole values. The exchange splitting of the valence band is always roughly 30 times larger than the splitting of the conduction band, independent of the material. Note that λ CB ex has the opposite sign of λ VB ex , see Tab. I, meaning that the order of the spin bands for conduction valence band is different at the K point.

VI. BILAYER GRAPHENE DEVICE
The electrical switching of the exchange splitting is a nice platform to realize novel spintronics devices, particularly since high spin-polarization can be achieved. Indeed, without any applied field, the chemical potential can be tuned close to the maximum of the valence band where states are 100% spin polarized. The same would hold for the conduction band in an applied negative field. Based on theoretical model calculationson magnetoresistance in BLG [66] and spin transport in graphene [28,67], both being subject to proximity exchange -model transport calculations in FMI/BLG were performed [49,50]. It was found that the proximity induced spin splitting allows BLG to act as a spin filter and spin rotator, being electrically controllable.
In Fig. 8  TABLE I. Summary of parameters for bilayer graphene on monolayer Cr2X2Te6 for zero field and X = Si, Ge, and Sn. The parameter ∆ describes the average orbital gap in the spectrum at K, λex are the exchange splittings for valence band (VB) and conduction band (CB) at K. The strength of the calculated intrinsic dipole across the heterostructure is given in debye, and d is the average distance between the lower graphene layer and the substrate. The lattice constant a0 is the one from the ferromagnetic substrate. In order to match the 5 × 5 graphene unit cell on the FMI we had to stretch its lattice constant to 7.1014Å. The strain gives, by how much we stretch the substrate. Finally Tc is the Curie temperature, either from experiment (exp.) on bulk samples or from Monte-Carlo simulations (theo.) on monolayers.
is placed and gate voltages V G1 and V G2 can be applied. The magnetization directions M of the two FMIs are opposite. Suppose the gate voltage V G1 is negative, then the low energy conduction bands of BLG below that first FMI are formed by the upper graphene layer and are strongly spin split, see Fig. 8(a). A spin unpolarized current is injected from the left and enters the first heterostructure. If the chemical potential µ is between the two spin split electron bands, one spin component is filtered. In the central part we have just bare BLG where, due to small intrinsic SOC, the spin keeps its direction aligned until it enters the second heterostructure. Depending on the gate voltage V G2 , we can have a ON or an OFF state. Since the magnetization of the second FMI is opposite to the first one the electrons cannot travel on in the upper layer if V G2 is also negative, because only the opposite spin channel is open, see Fig. 8  A different approach is based on the spin rotation, see Fig. 9. Similar to the spin filter, we polarize first a spin unpolarized current injected from the left, which is then analyzed in the right heterostructure. The difference is that there is an additional FMI able to control the carrier spins via gate voltage V G3 . The magnetization directions of the FMIs from left to right are −z, y, z and spin rotation happens below the central FMI. Depending on the gate voltage V G3 , carriers will flip their spin in the central region and the channel on the right is open, see Fig.  9(a), or closed, see Fig. 9(b). The origin of spin rotation and the efficiency of this device has been discussed in Refs. [49,50], and is based on wave function localization and short ranged proximity exchange.
Above are but a few examples of spin devices based on the field-effect exchange coupling concepts. BLG on ferromagnetic insulating substrates could be employed to build bipolar spintronics elements, such as spin diodes and spin transistors [68] or spin logic devices [69], which require exchange coupling (but non necessarily magnetic moments) to make their electronic bands spin dependent.

VII. SUMMARY AND CONCLUSION
In conclusion, we have studied from first principles the electronic structure of bilayer graphene on Cr 2 Ge 2 Te 6 . We have shown that we can efficiently, and most impor-tant fully electrically, switch the exchange interaction by one order of magnitude, of electrons and holes. At low enough energy the electronic states can be 100% spin polarized, which can lead to interesting new device concepts. We also expect that our DFT-derived parameters will be useful in model transport calculations involving exchange proximitized BLG.