Stacking-Controlled Magnetic Exchange and Magnetoelectric Coupling in Bilayer CrI$_2$

TL;DR

This study demonstrates stacking-controlled magnetism in CrI$_2$ bilayers using first-principles calculations. By mapping stacking registries (direct and indirect) and extracting full magnetic exchange tensors, it shows that intralayer exchange dominates, while stacking symmetry selects anisotropic terms and enables Dzyaloshinskii–Moriya interactions and magnetoelectric coupling in non-centrosymmetric registries. The BA$^ extPrime$ registry is the bilayer ground state with antiparallel interlayer alignment, and in-plane exchange channels strengthen by about $6$–$10 ext{%}$ upon bilayer formation; controlled sliding between registries is energetically feasible ($25$–$50$ meV/fu) and can reversibly switch polarization, enabling reconfigurable spintronic functionality. The work links stacking-induced symmetry breaking to measurable magnetoelectric effects, with polarization up to roughly $10~ extmu ext{C}/ ext{cm}^2$ and exchange-driven spin splittings of a few meV, suggesting practical routes for electrically tunable antiferromagnetic states and chiral spin textures in van der Waals devices.

Abstract

We use a first-principles calculations approach to reveal the electronic and magnetic properties of chromium diiodide (CrI$_2$) bilayers and establish a hierarchy of magnetic interactions across stable registries. The monolayer presents a x-stripe antiferromagnetic ground state, while in bilayers the BA$^\prime$ stacking is the global minimum with antiparallel interlayer magnetic alignment. Bilayer configurations strengthen the exchange in the plane by 6 % to 10 %, while the exchange between layers is registry-dependent. The symmetry of each stacking configuration allows for anisotropic interactions. Dzyaloshinskii-Moriya terms appear in structures without inversion symmetry, which in this case also generates in-plane polarizations of up to $\sim$ 10 $μ$C/cm$^2$, resulting in direct magnetoelectric coupling that is absent in centrosymmetric monolayers. Thus, stacking acts both as a selector of exchange anisotropy and as a driver of magnetoelectricity. Our results show that bilayer CrI$_2$ can be mechanically reconfigured through interlayer sliding, with energy differences between stacking orders (25-50 meV/f.u.) that are compatible with experimental actuation. Tunable magnetism and register-dependent polarization offer promising opportunities for novel spintronic devices, where structural transitions can affect both magnetic states and electric dipoles.

Figures (11)

• Figure 1: Atomic structures of CrI$_2$ in monolayer and bilayer forms. (a) Top view and (b) side view of the relaxed monolayer. (c,d) Top and side views of the AA-stacked bilayer. The inset in (c) shows the small in-plane displacement ($\delta_{xy}$) along the $x$ direction that emerges after relaxation, breaking the ideal registry.(e,f) Top and side views of the AA$^\prime$ bilayer, constructed by inverting one layer through the Cr plane followed by a vertical shift. In panels (c--f), different colors distinguish the top (green) and bottom (orange) Cr layers, while I atoms are shown in dark and light gray. The side views identify the interlayer distance $\delta_z$, defined between inner iodine planes. The Cartesian coordinate system is defined as $\hat{a} \parallel \hat{x}$, $\hat{b} \parallel \hat{y}$, $\hat{c} \parallel \hat{z}$.
• Figure 2: Electronic band structures of monolayer CrI$_2$ in the stripe-type antiferromagnetic AF$_x$(left) and ferromagnetic (right) configurations. Calculations are shown along the high-symmetry path $\Gamma$-X-M-Y-$\Gamma$, where $\Gamma=(0,0)$, X=$(1/2,0)$, M=$(1/2,1/2)$, and Y=$(0,1/2)$ in reciprocal lattice units. The Fermi level is set to zero (represented by a horizontal line).
• Figure 3: Electronic band structures of bilayer CrI$_2$ in the most stable magnetic configurations: direct AA ($\uparrow\downarrow/\uparrow\downarrow$) (left) and BA$^\prime$ ($\uparrow\downarrow/\downarrow\uparrow$) (right). The calculations are performed along the high-symmetry path $\Gamma$--X--M--Y--$\Gamma$, with $\Gamma=(0,0)$, X=$(1/2,0)$, M=$(1/2,1/2)$, and Y=$(0,1/2)$ in reciprocal lattice units. The Fermi level is set to zero (represented by the horizontal dashed line).
• Figure 4: Controlled in-plane displacement and magnetic configuration considering AFM monolayers. We use a subscript to label the magnetic state: “$+$” for the parallel $\uparrow\downarrow/\uparrow\downarrow$ and “$-$” for the antiparallel $\uparrow\downarrow/\downarrow\uparrow$ configurations. Starting from the AA$_{+}$ and AA$_{+}$($^{\prime}$) reference, the top schematics illustrate the highly symmetric registries AA($^{\prime}$), AB($^{\prime}$), and BA($^{\prime}$). Panels (a)-(b) show the stacking relative energy $\Delta E=E-E_{\mathrm{GS}}$ as a function of the lateral shift $(\Delta x/a,\, \Delta y/b)$ for AA- and AA$^{\prime}$-based configurations, respectively. Panels (c)-(d) display the corresponding variation of the interlayer separation $\delta z$, defined as the distance between the inner iodine planes of opposite layers.
• Figure S1: Collinear antiferromagnetic configurations of the CrI$_2$ monolayer in a rectangular supercell. The stripe antiferromagnetic ground state AF$_x$(left), the zigzag antiferromagnetic configuration AF$_z$(center), and the stripe antiferromagnetic configuration AF$_y$(right). Numbers in parentheses indicate the energy difference relative to the AF$_x$ ground state, expressed in meV per Cr atom. Blue (red) spheres denote Cr atoms with spin up (down), while black spheres represent iodine atoms.