Theory of proximity-induced exchange coupling in graphene on hBN/(Co, Ni)

Graphene, being essentially a surface, can borrow some properties of an insulating substrate (such as exchange or spin-orbit couplings) while still preserving a great degree of autonomy of its electronic structure. Such derived properties are commonly labeled as proximity. Here we perform systematic first-principles calculations of the proximity exchange coupling, induced by cobalt (Co) and nickel (Ni) in graphene, via a few (up to three) layers of hexagonal boron nitride (hBN). We find that the induced spin splitting of the graphene bands is of the order of 10 meV for a monolayer of hBN, decreasing in magnitude but alternating in sign by adding each new insulating layer. We find that the proximity exchange can be giant if there is a resonant $d$ level of the transition metal close to the Dirac point. Our calculations suggest that this effect could be present in Co heterostructures, in which a $d$ level strongly hybridizes with the valence-band orbitals of graphene. Since this hybridization is spin dependent, the proximity spin splitting is unusually large, about 10 meV even for two layers of hBN. An external electric field can change the offset of the graphene and transition-metal orbitals and can lead to a reversal of the sign of the exchange parameter. This we predict to happen for the case of two monolayers of hBN, enabling electrical control of proximity spin polarization (but also spin injection) in graphene/hBN/Co structures. Nickel-based heterostructures show weaker proximity effects than cobalt heterostructures. We introduce two phenomenological models to describe the first-principles data. The minimal model comprises the graphene (effective) $p_z$ orbitals and can be used to study transport in graphene with proximity exchange, while the $p_z\text{−}d$ model also includes hybridization with $d$ orbitals, which is important to capture the giant proximity exchange. Crucial to both models is the pseudospin-dependent exchange coupling, needed to describe the different spin splittings of the valence and conduction bands.

Figure 1

Structure of the graphene/hBN/Co system, with labels for the different atoms. (a) Top view of the structure, with one unit cell emphasized by the dashed line. (b) Side view with stacking configuration: ${\mathrm{C}}_{\mathrm{B}}$ is over boron, and ${\mathrm{C}}_{\mathrm{A}}$ is over hBN hexagon. Nitrogen is at the top site above Co, and boron is above the fcc site of Co. The distances indicated are measured between graphene/Co and the nitrogen atom of hBN since the hBN layer is slightly corrugated by $\mathrm{\Delta }z=0.113\AA$. The boron atom is closer to the Co surface. Numbers in parentheses indicate the Co layer.

Figure 2

Band structure of the graphene/hBN/Co system. (a) Calculated spin-resolved band structure along the high-symmetry path $\mathrm{M}\text{−}\mathrm{K}\text{−}\mathrm{\Gamma }$ using DFT. Spin-up (-down) states are shown in solid red (blue). Thicker bands at energies smaller (larger) than $-4$ eV (2 eV) correspond to the highest- (lowest-) lying valence (conduction) band of hBN which is spin split. (b) Hamiltonian ${\mathcal{H}}_0$ gives the linear dispersion of graphene with $v_{\mathrm{F}}$ defining the slope. (c) ${\mathcal{H}}_0+{\mathcal{H}}_{\mathrm{\Delta }}$ describes gapped graphene with a gap of $2\mathrm{\Delta }$. (d) ${\mathcal{H}}_0+{\mathcal{H}}_{\mathrm{\Delta }}+{\mathcal{H}}_{\mathrm{ex}}$ describes gapped graphene in which the spin degeneracy of the conduction (valence) band gets lifted with a splitting of $2{\lambda }_{\mathrm{ex}}^{\mathrm{A}}$ $\left(2{\lambda }_{\mathrm{ex}}^{\mathrm{B}}\right)$. (e) Close-up of the band structure from (a) around the $\mathrm{K}$ point with labels for the different orbital and sublattice contributions of graphene; that is, the upper (lower) two bands are formed by $p_z$ orbitals of sublattice A (B).

Figure 3

Calculated spin-polarized band structure of the graphene/hBN/Co heterostructure for one layer of hBN. (a) Band structure in the vicinity of the Dirac point with labels for the main orbital contributions from which the individual bands are formed; for example, $d_{z^2}\left(3\right)$ corresponds to the $d_{z^2}$ orbital of Co atom (3) from Fig. 1. Labels $E_j,j=\mathrm{u},\mathrm{v},\mathrm{w}$, are the energy bands, which correspond to the Co $d$ states used to fit the $p_z\text{−}d$ model Hamiltonian in Eq. (5). The energies $E_j$, in Eq. (5), are measured with respect to the energy $E_0$. (b) The fit to the $p_z\text{−}d$ model with a side view of the structure. First-principles data (dotted lines) are well reproduced by the $p_z\text{−}d$ model (solid lines). (c) The corresponding splittings of the valence (val) and conduction (cond) Dirac states of graphene. The main fit parameters are $E_0=-430.89$ meV, $\tilde{\mathrm{\Delta }}=21.45$ meV, ${\tilde{\lambda }}_{\mathrm{ex}}^{\mathrm{A}}=-7.63$ meV, ${\tilde{\lambda }}_{\mathrm{ex}}^{\mathrm{B}}=8.95$ meV, $E_{\mathrm{u}}=-279.41$ meV, $E_{\mathrm{v}}=-19.37$ meV, $E_{\mathrm{w}}=282.31$ meV, ${\mathrm{w}}_{\downarrow }^{\mathrm{A}}=48.44$ meV. The most relevant parameters are obtained by a least-squares fit, minimizing the difference between the model and the DFT data for a fitting range from $\mathrm{K}$ towards the $\mathrm{\Gamma }$ point for $k$ points up to $20\times {10}^{-3}/\AA$. By performing the fit towards the $\mathrm{M}$ point, one would obtain slightly different (by at most 5%) parameters. From the band structure and from the fact that we limit our fitting range, we find that no additional hybridization parameters $j_{\uparrow ,\downarrow }^{\mathrm{A},\mathrm{B}}$ are necessary to fit our band structure, except for the mentioned ones. The Fermi velocity to match the slope away from the $\mathrm{K}$ point is $v_{\mathrm{F}}=0.812\times {10}^6$ m/s, which corresponds to a nearest-neighbor hopping parameter of $t=2.48$ eV, slightly smaller than the commonly used value of 2.6 eV [9, 11, 16] due to the larger lattice constant used here.

Figure 4

Spin-polarized band structure of the graphene/hBN/Co heterostructure for two layers of hBN $({\mathrm{AA}}^{\prime }$ stacking). (a) Band structure in the vicinity of the Dirac point with labels for the main orbital contributions. The inset shows a close-up of the conduction Dirac states to visualize the reversal of the spin states. (b) The fit to the $p_z\text{−}d$ model with a side view of the structure for two layers of hBN. DFT data (dotted lines) are well reproduced by the $p_z\text{−}d$ model (solid lines). (c) The corresponding splittings of the valence and conduction Dirac states. The fit parameters are $E_0=-352.65$ meV, $\tilde{\mathrm{\Delta }}=41.02$ meV, ${\tilde{\lambda }}_{\mathrm{ex}}^{\mathrm{A}}=0.096$ meV, ${\tilde{\lambda }}_{\mathrm{ex}}^{\mathrm{B}}=-0.512$ meV, $E_{\mathrm{u}}=-357.12$ meV, $E_{\mathrm{v}}=-114.75$ meV, $E_{\mathrm{w}}=207.34$ meV, ${\mathrm{v}}_{\uparrow }^{\mathrm{B}}=41.67$ meV. The Fermi velocity to match the slope away from the $\mathrm{K}$ point is $v_{\mathrm{F}}=0.820\times {10}^6$ m/s. All other parameters are zero for the same fitting range as for the one-layer case.

Figure 5

Influence of the number of hBN layers on the proximity-induced parameters for the graphene/hBN/Co structure, using the $p_z$ model at the $\mathrm{K}$ point. Dependence of (a) the proximity gap $\mathrm{\Delta }$, (b) the exchange parameters ${\lambda }_{\mathrm{ex}}^{\mathrm{A}}$, and (c) ${\lambda }_{\mathrm{ex}}^{\mathrm{B}}$ on the number of hBN layers for different lattice constants or an additional Hubbard parameter of $U=1.0$ eV. Parameter values for two (three) layers of hBN were increased by a factor of 10 (100) for better visualization, as indicated.

Figure 6

Conduction $\mathrm{\Delta }{E}_{\mathrm{cond}}$ and valence $\mathrm{\Delta }{E}_{\mathrm{val}}$ band splittings of the three individual hBN layers at the $\mathrm{K}$ point. Values are obtained by identifying the spin-split hBN conduction and valence bands of the three individual layers in the band structure of the graphene/hBN/Co heterostructure for three layers of hBN.

Figure 7

Influence of the electric field on the proximity-induced parameters for the graphene/hBN/Co structure for one hBN layer, using the minimal $p_z$ model at the $\mathrm{K}$ point. Dependence of the (a) Dirac energy $E_{\mathrm{D}}$ and the proximity gap $\mathrm{\Delta }$ and (b) the exchange parameters ${\lambda }_{\mathrm{ex}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{ex}}^{\mathrm{B}}$ on the applied transverse electric field.

Figure 8

Influence of the electric field on the proximity-induced parameters for the graphene/hBN/Co structure for two hBN layers, using the minimal $p_z$ model at the $\mathrm{K}$ point. Dependence of the (a) Dirac energy $E_{\mathrm{D}}$ and the proximity gap $\mathrm{\Delta }$ and (b) the exchange parameters ${\lambda }_{\mathrm{ex}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{ex}}^{\mathrm{B}}$ on the applied transverse electric field. (c) and (d) The calculated spin-resolved band structure projected on the graphene states in the vicinity of the Dirac point for two different field strengths, illustrating the reversal of the valence spin states at the $\mathrm{K}$ point.

Figure 9

Influence of the number of Co layers on the band structure for the graphene/hBN/Co system for one hBN layer, using the minimal $p_z$ model at the $\mathrm{K}$ point. Dependence of (a) the proximity gap $\mathrm{\Delta }$ and the exchange parameters ${\lambda }_{\mathrm{ex}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{ex}}^{\mathrm{B}}$, as well as (b) the Dirac energy $E_{\mathrm{D}}$, on the number of Co layers.

Figure 10

Structure of graphene/hBN/Ni, with labels for the different atoms. (a) Top view of the structure, with one unit cell emphasized by the dashed line. (b) Side view with stacking configuration: ${\mathrm{C}}_{\mathrm{B}}$ is over boron, and ${\mathrm{C}}_{\mathrm{A}}$ is over the hBN ring. Nitrogen is at the top site above Ni, and boron is above the fcc site of Ni. The indicated distances are measured between graphene/Ni and the nitrogen atom of hBN since the hBN layer is slightly corrugated by $\mathrm{\Delta }z=0.101\AA$, with the boron atom closer to the Ni surface. Numbers in parentheses indicate the Ni layer.

Figure 11

Spin-polarized band structure of the graphene/hBN/Ni heterostructures for one layer of hBN. (a) Band structure in the vicinity of the Dirac point with labels for the main orbital contributions. Labels $E_j,j=\mathrm{u},\mathrm{v},\mathrm{w}$, are the energy bands, which correspond to the Ni $d$ states used to fit the $p_z\text{−}d$ model Hamiltonian in Eq. (5). (b) The fit to the $p_z\text{−}d$ model with a side view of the structure. First-principles data (dotted lines) are well reproduced by the model (solid lines). (c) The corresponding splittings of the valence (val) and conduction (cond) Dirac states of graphene. The fit parameters are $E_0=-527.98$ meV, $\tilde{\mathrm{\Delta }}=22.98$ meV, ${\tilde{\lambda }}_{\mathrm{ex}}^{\mathrm{A}}=-1.25$ meV, ${\tilde{\lambda }}_{\mathrm{ex}}^{\mathrm{B}}=8.17$ meV, $E_{\mathrm{u}}=-272.58$ meV, $E_{\mathrm{v}}=-201.27$ meV, $E_{\mathrm{w}}=-158.17$ meV, ${\mathrm{v}}_{\uparrow }^{\mathrm{B}}=10.15$ meV, ${\mathrm{w}}_{\downarrow }^{\mathrm{B}}=4.19$ meV. The Fermi velocity to match the slope away from the $\mathrm{K}$ point is $v_{\mathrm{F}}=0.81\times {10}^6$ m/s. The fit parameters are again obtained in the same way as for the Co case.

Figure 12

Spin-polarized band structure of graphene/hBN/Ni heterostructures for two layers of hBN $({\mathrm{AA}}^{\prime }$ stacking). (a) Band structure in the vicinity of the Dirac point with labels for the main orbital contributions. The inset shows a close-up of the conduction Dirac states to visualize the reversal of the spin states. (b) The fit to the $p_z\text{−}d$ model with a side view of the structure for two layers of hBN. First-principles data (dotted lines) are well reproduced by the $p_z\text{−}d$ model (solid lines). (c) The corresponding splittings of the valence and conduction Dirac states. The fit parameters are $E_0=-435.76$ meV, $\tilde{\mathrm{\Delta }}=42.88$ meV, ${\tilde{\lambda }}_{\mathrm{ex}}^{\mathrm{A}}=0.080$ meV, ${\tilde{\lambda }}_{\mathrm{ex}}^{\mathrm{B}}=-1.44$ meV, $E_{\mathrm{u}}=-363.36$ meV, $E_{\mathrm{v}}=-309.27$ meV, $E_{\mathrm{w}}=-251.05$ meV, ${\mathrm{v}}_{\uparrow }^{\mathrm{B}}=32.67$ meV. The Fermi velocity to match the slope away from the $\mathrm{K}$ point is $v_{\mathrm{F}}=0.824\times {10}^6$ m/s. All other parameters are zero for the same fitting range as for the one-layer case.

Figure 13

Influence of the number of hBN layers on the proximity-induced parameters for the graphene/hBN/Ni structure, using the $p_z$ model at the $\mathrm{K}$ point. Dependence of (a) the proximity gap $\mathrm{\Delta }$, (b) the exchange parameters ${\lambda }_{\mathrm{ex}}^{\mathrm{A}}$, and (c) ${\lambda }_{\mathrm{ex}}^{\mathrm{B}}$ on the number of hBN layers for different lattice constants or an additional Hubbard parameter of $U=1.0$ eV. Parameter values for two (three) layers of hBN were increased by a factor of 10 (100) for better visualization.

Figure 14

Conduction $\mathrm{\Delta }{E}_{\mathrm{cond}}$ and valence $\mathrm{\Delta }{E}_{\mathrm{val}}$ band splittings of the three individual hBN layers at the $\mathrm{K}$ point. Values are obtained by identifying the spin-split hBN conduction and valence bands of the three individual layers in the band structure of graphene/hBN/Ni heterostructure for three layers of hBN.

Figure 15

Influence of the electric field on the proximity-induced parameters for the graphene/hBN/Ni structure for one hBN layer, using the $p_z$ model at the $\mathrm{K}$ point. Dependence of the (a) Dirac energy $E_{\mathrm{D}}$ and the proximity gap $\mathrm{\Delta }$ and (b) the exchange parameters ${\lambda }_{\mathrm{ex}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{ex}}^{\mathrm{B}}$ on the applied transverse electric field.

Figure 16

Influence of the electric field on the proximity-induced parameters for the graphene/hBN/Ni structure for two hBN layers, using the $p_z$ model at the $\mathrm{K}$ point. Dependence of the (a) Dirac energy $E_{\mathrm{D}}$ and the proximity gap $\mathrm{\Delta }$ and (b) the exchange parameters ${\lambda }_{\mathrm{ex}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{ex}}^{\mathrm{B}}$ on the applied transverse electric field.

Figure 17

Influence of the number of Ni layers on the band structure for the graphene/hBN/Ni system for one hBN layer, using the $p_z$ model at the $\mathrm{K}$ point. Dependence of (a) the proximity gap $\mathrm{\Delta }$ and the exchange parameters ${\lambda }_{\mathrm{ex}}^{\mathrm{A}}$ and ${\lambda }_{\mathrm{ex}}^{\mathrm{B}}$ and (b) the Dirac energy $E_{\mathrm{D}}$ on the number of Ni layers.