Proximity effects in graphene on monolayers of transition-metal phosphorus trichalcogenides $M\mathrm{P}{X}_3$ $(M:\mathrm{Mn}, \mathrm{Fe}, \mathrm{Ni}, \mathrm{Co}, \mathrm{and} X: \mathrm{S}, \mathrm{Se})$

We investigate the electronic band structure of graphene on a series of two-dimensional magnetic transition-metal phosphorus trichalcogenide monolayers, ${M\text{P}X}_3$ with $M=\{\mathrm{Mn},\mathrm{Fe},\mathrm{Ni},\mathrm{Co}\}$ and $X=\{\mathrm{S},\mathrm{Se}\}$, with first-principles calculations. A symmetry-based model Hamiltonian is employed to extract orbital parameters and sublattice resolved proximity-induced exchange couplings (${\lambda }_{\text{ex}}^{\text{A}}$ and ${\lambda }_{\text{ex}}^{\text{B}}$) from the low-energy Dirac bands of the proximitized graphene. Depending on the magnetic phase of the ${M\text{P}X}_3$ layer (ferromagnetic and three antiferromagnetic ones), completely different Dirac dispersions can be realized with exchange splittings ranging from 0 to 10 meV. Remarkably, not only the magnitude of the exchange couplings depends on the magnetic phase, but also the global sign and the type. Important, one can realize uniform $({\lambda }_{\text{ex}}^{\text{A}}\approx {\lambda }_{\text{ex}}^{\text{B}})$ and staggered $({\lambda }_{\text{ex}}^{\text{A}}\approx -{\lambda }_{\text{ex}}^{\text{B}})$ exchange couplings in graphene. From selected cases we find that the interlayer distance, as well as a transverse electric field, are efficient tuning knobs for the exchange splittings of the Dirac bands. More specifically, decreasing the interlayer distance by only about 10%, a giant fivefold enhancement of proximity exchange is found, while applying few V/nm of electric field, provides tunability of proximity exchange by tens of percent. We have also studied the dependence on the Hubbard $U$ parameter and find it to be weak. Moreover, we find that the effect of SOC on the proximitized Dirac dispersion is negligible compared to the exchange coupling.

Figure 1

(a) and (b) Top and side view of the graphene/${M\text{P}X}_3$ heterostructure. Dashed line in (a) defines the heterostructure unit cell and in (b) we define the interlayer distance $d_{\mathrm{int}}$. Different colors correspond to different atom types. Depending on the ${M\text{P}X}_3$ crystal, the hexagonal $M$ lattice can show the (c) ferromagnetic, (d) antiferromagnetic Néel, (e) antiferromagnetic zigzag, and (f) antiferromagnetic stripy phase. Red (blue) spheres indicate magnetization of the $M$ atom along $z$ ($-z$) direction.

Figure 2

(a) DFT-calculated band structure of the graphene/${\mathrm{MnPS}}_3$ heterostructure along the high-symmetry path $M\text{−}K\text{−}\mathrm{\Gamma }$ without SOC, when the ${\mathrm{MnPS}}_3$ is in the ferromagnetic phase. Red solid (blue dashed) lines correspond to spin up (spin down). (b) Zoom to the DFT-calculated (symbols) low energy Dirac bands near the $K$ point with a fit to the model Hamiltonian (solid line). (c) The splitting of valence and conduction band.

Figure 3

Same as Fig. 2, but for the ${\mathrm{MnPS}}_3$ in the antiferromagnetic Néel phase.

Figure 4

Same as Fig. 2, but for the ${\mathrm{MnPS}}_3$ in the antiferromagnetic zigzag phase.

Figure 5

Same as Fig. 2, when the ${\mathrm{MnPS}}_3$ is in the antiferromagnetic stripy phase.

Figure 6

The average proximity exchange coupling ${\lambda }_{\text{ex}}$ as a function of the substrate material and the magnetic phase (FM = ferromagnetic, AFM = antiferromagnetic).

Figure 7

Interlayer distance dependence of (a) the staggered potential $\mathrm{\Delta }$, (b) the total energy $E_{\mathrm{tot}}$, (c) the spin dependent hopping parameters $t_{\uparrow }$ and $t_{\downarrow }$, and (d) the sublattice resolved proximity exchange parameters ${\lambda }_{\text{ex}}^{\text{A}}$ and ${\lambda }_{\text{ex}}^{\text{B}}$ for the graphene/${\mathrm{MnPS}}_3$ heterostructure when the ${\mathrm{MnPS}}_3$ is in the antiferromagnetic Néel phase.

Figure 8

Hubbard $U$ dependence of (a) the staggered potential $\mathrm{\Delta }$, (b) the band offset of the Dirac point $E_{\text{D}}$, with respect to the ${\mathrm{MnPS}}_3$ conduction band edge, $E_{\text{CBE}}$, at the $K$ point, (c) the spin dependent hopping parameters $t_{\uparrow }$ and $t_{\downarrow }$, and (d) the sublattice resolved proximity exchange parameters ${\lambda }_{\text{ex}}^{\text{A}}$ and ${\lambda }_{\text{ex}}^{\text{B}}$ for the graphene/${\mathrm{MnPS}}_3$ heterostructure when the ${\mathrm{MnPS}}_3$ is in the antiferromagnetic Néel phase.

Figure 9

Transverse electric field dependence of (a) the staggered potential $\mathrm{\Delta }$, (b) the band offset of the Dirac point $E_{\text{D}}$, with respect to the ${\mathrm{MnPS}}_3$ conduction band edge $E_{\text{CBE}}$ at the $K$ point, (c) the spin dependent hopping parameters $t_{\uparrow }$ and $t_{\downarrow }$, and (d) the sublattice resolved proximity exchange parameters ${\lambda }_{\text{ex}}^{\text{A}}$ and ${\lambda }_{\text{ex}}^{\text{B}}$ for the graphene/${\mathrm{MnPS}}_3$ heterostructure when the ${\mathrm{MnPS}}_3$ is in the antiferromagnetic Néel phase.

Figure 10

(a) DFT-calculated low energy Dirac bands near the $K$ point for the graphene/${\mathrm{MnPS}}_3$ heterostructure, when ${\mathrm{MnPS}}_3$ is in the ferromagnetic phase and SOC is included. The symbols are color coded by their $s_z$ spin expectation value. (b) Same as (a) but near the $K^{\prime }$ point.