Insight into interlayer magnetic coupling in $1T$-type transition metal dichalcogenides based on the stacking of nonmagnetic atoms

The interlayer coupling in two-dimensional (2D) magnetic materials is significantly important in determining the properties of 2D materials and applications of related devices. However, the mechanism that determines the interlayer magnetic coupling has only been comprehensively studied in $\mathrm{Cr}{\mathrm{I}}_3$ and that of transition metal dichalcogenides is still blurred. In this work, through first-principle calculations, we find in 2D magnetic bilayer that the interlayer magnetic coupling is determined by the stacking order of interlayer nonmagnetic atoms, accompanied by the transition between half metal and semiconductor in $\mathrm{Mn}{\mathrm{S}}_2$. The nonmagnetic atoms bridge the interlayer coupling and the stacking order of nonmagnetic atoms determines the interlayer coupling by altering the interlayer $p_z$-orbital bonding. Adjusting the structure of interlayer nomagnetic atoms by biaxial strain can also tune the interlayer coupling. The perspective proposed in our work, from the stacking order of nonmagnetic atoms, to understand the interlayer magnetic coupling is also applicable in other 2D materials.

Figure 1

Structure and band structure of $\mathrm{Mn}{\mathrm{S}}_2$. (a) Top view and side view of monolayer $\mathrm{Mn}{\mathrm{S}}_2$. The black dotted rhombuses highlight the primitive cell. (b) Top view of $\mathrm{Mn}{\mathrm{S}}_2$ bilayer in AB stacking and AC stacking. Blue (gray) and orange (pink) balls represent Mn atoms and S atoms from top (bottom) layer. The green arrows represent the direction of translation. (c) Band structure of monolayer $\mathrm{Mn}{\mathrm{S}}_2$, both spin-up (red) and spin-down (blue) bands are exhibited.

Figure 2

Stacking-dependent interlayer coupling in $\mathrm{Mn}{\mathrm{S}}_2$. (a)–(f) Structure of interlayer S atoms and side view of bilayer in different stacking orders (a)–(c) and after rotation (d)–(f). The green and magenta triangles represent the lower S atoms from top layer and the upper S atoms from bottom layer, respectively. Red and green arrows represent the magnetic moments of Mn atoms. The interlayer distance of the ground magnetic state is also shown. (g) Interlayer coupling energy defined as the difference between interlayer ferromagnetic coupling and antiferromagnetic coupling $E_{\mathrm{FM}}\text{–}{E}_{\mathrm{AFM}}$ and stacking energy defined as the energy difference between AA stacking and particular stacking order $E_{\mathrm{AA}}\text{–}{E}_{\mathrm{stack}}$ under different stacking orders. (h) Interlayer coupling energy and stacking energy under different stacking orders with rotational transform.

Figure 3

DCD, interlayer exchange mechanism, and band structure of $\mathrm{Mn}{\mathrm{S}}_2$ in different stacking. (a)–(d) Spin-dependent DCDs of $\mathrm{Mn}{\mathrm{S}}_2$ in (a) AA, (b) AC stacking, (c) $\mathrm{A}{\mathrm{A}}^{\mathrm{R}}$ stacking, and (d) AB (isosurface value of $0.0001e/\mathrm{boh}{\mathrm{r}}^3$). Left and right DCDs in each figure represent spin-down and spin-up component, respectively. Red and green isosurface contours correspond to charge accumulation and depletion. Blue (gray) and orange (pink) balls represent Mn atoms and S atoms from top (bottom) layer. (e), (f) Interlayer coupling mechanism of (e) AA stacking and (f) AC stacking. (g), (h) Band structure of (g) AA and (h) $\mathrm{A}{\mathrm{A}}^{\mathrm{R}}$ stacking. Both spin-up (red) and spin-down (blue) bands are exhibited; degenerated bands are shown in purple.

Figure 4

Strain-tunable interlayer coupling in $\mathrm{Mn}{\mathrm{S}}_2$. (a) Interlayer structure of $\mathrm{Mn}{\mathrm{S}}_2$ in AA stacking after rotation with defined interlayer distance $d$ (S–S) and interlayer bond angle θ. (b) DCD of bilayer $\mathrm{Mn}{\mathrm{S}}_2$ (isosurface value of $0.0001e/\mathrm{boh}{\mathrm{r}}^3$); red and green isosurface contours represent charge accumulation and reduction, respectively. (c) Interlayer coupling energy and interlayer bond angle θ under different biaxial strain. (b) Interlayer coupling energy as a function of interlayer bond angle under two different interlayer distances. (e) Interlayer coupling energy as a function of interlayer distance under two different interlayer bond angles.

Figure 5

Stacking-dependent interlayer coupling in ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$. (a) Top view of ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ monolayer. Two shift directions $\vec{a}$ and $\vec{b}$ are labeled as red and blue vectors, respectively. The black dotted rhombuses highlight the primitive cell. (b) Stacking configurations of Cr atoms under various stacking orders. Gray balls and blue balls represent the Cr atoms from bottom and top layer, respectively. (c)–(f) Interlayer structure of Te atoms in (c) AB stacking, (d) $\mathrm{A}{\mathrm{C}}_1$ stacking, (e) $\mathrm{A}{\mathrm{B}}^{\mathrm{R}}$ stacking, and (f) $\mathrm{A}{{\mathrm{C}}_1}^{\mathrm{R}}$ stacking. Green and magenta balls correspond to lower Te atoms from top layer and upper Te atoms from bottom layer, respectively. (g) Stacking energy and interlayer coupling energy under different stacking orders. (h) Interlayer coupling energy under different stacking orders with rotational transform.