2D Bilayer ScI₂: Tunable Multiferroicity

• 2D bilayer ScI₂ is a layered van der Waals material featuring tunable multiferroic states, including stacking-dependent magnetism and valley polarization.
• Interlayer sliding and rotation enable precise control of magnetic coupling, ferroelectric polarization, and topological phase transitions.
• Stacking-induced symmetry breaking drives orbital hybridization and ferroelectric effects, paving the way for next-generation spintronic and valleytronic devices.

Two-dimensional (2D) bilayer scandium diiodide (ScI₂) is a layered van der Waals material that exhibits tunable multiferroic states through interlayer sliding and rotation. The system is characterized by stacking-dependent interlayer magnetic coupling, emergent ferroelectricity, and spontaneous valley polarization, arising from the interplay of orbital hybridization, superexchange interaction, symmetry breaking, and spin–orbit coupling. Bilayer ScI₂ serves as a prototypical platform for engineering reconfigurable spintronic, ferroelectric, and valleytronic devices at the nanoscale due to its highly sensitive structural-property relationships.

1. Interlayer Magnetic Coupling and Stacking Dependence

Each monolayer of ScI₂ is ferromagnetic (FM). The interlayer magnetic coupling is determined by the stacking geometry:

• AA stacking (aligned Sc atoms, vertical displacement): The dominant vertical superexchange occurs via Sc 3d₍z²₎ and I 5p₍z₎ orbitals, favoring antiferromagnetic (AFM) coupling with a nearest-neighbor interlayer exchange $J_{\text{inter-1}} \simeq +2.564$ meV.
• AB/BA stackings (lateral displacement): Hybridization between Sc 3d₍z²₎ and orbitals with in-plane character (d₍xy₎, d₍x²-y²₎) via I 5p₍x₎ and 5p₍y₎ leads to FM coupling ($J_{\text{inter-1}} \simeq -0.249$ meV, $J_{\text{inter-2}} \simeq -0.186$ meV).
• Antialigned stackings (180° rotation): FM coupling is found for AA*, while AB* and BA* show AFM coupling.

This interplay is captured by the effective Heisenberg spin Hamiltonian:

$$H = -J_1 \sum_{\langle ij \rangle} \vec{S}_i \cdot \vec{S}_j - J_{\text{inter-1}} \sum_{\langle ij \rangle} \vec{S}_i \cdot \vec{S}_j - J_{\text{inter-2}} \sum_{\langle\langle ij \rangle\rangle} \vec{S}_i \cdot \vec{S}_j$$

where $J_1 < -67$ meV is the strong intralayer FM exchange. The orbital-resolved exchange mechanisms and stacking-induced sign reversals of $J_{\text{inter}}$ enable reversible switching of interlayer magnetic order through mechanical manipulation of the stacking (Pan et al., 18 Oct 2025).

2. Stacking-Induced Ferroelectricity

Ferroelectricity in bilayer ScI₂ is stacking-engineered and not intrinsic to individual monolayers. In AB and BA stackings of aligned layers, both mirror ($M_z$) and inversion ($P$) symmetries are broken. Interlayer orbital hybridization results in asymmetric charge redistribution, with planar-averaged electrostatic potential differences $\Delta\varphi \approx \pm 29.78$ meV between the layers (as measured for AB, BA stackings). The Berry-phase calculation yields a spontaneous out-of-plane polarization $P_z \approx 0.18 \times 10^{-12}$ C/m for these stackings, which maximizes the ferroelectric response.

Stacking operations, modeled as combinations of rotation and in-plane translation, dictate whether the bilayer is polar. Group-theoretical analysis leads to polarization selection rules:

• For a bilayer symmetry group $G_B$, polarization survives only those symmetry operations $R$ for which $R \vec{P} \neq \vec{P}$.
• Mirror and inversion loss produce out-of-plane and/or in-plane polarization; interlocking effects may permit deterministic switching of $P_z$ via applied in-plane electric fields (Ji et al., 2022).

3. Valley Polarization and Inversion Symmetry Breaking

The valley degree of freedom in ScI₂ is manipulated by inversion symmetry breaking and spin–orbit coupling (SOC):

• In AA stacking with AFM interlayer coupling, C₂y and time-reversal symmetry ($T$) protect valley degeneracy. Rotation of spin orientation to the out-of-plane direction (i.e., breaking $M_z$) in presence of SOC splits the valleys at $K$/$K'$ by nearly 100 meV.
• In AB/BA stackings, coexisting ferroelectric distortion further breaks inversion symmetry, leading to clear valley splitting in spin-resolved bands. The sign of valley polarization reverses under 180° stacking rotation between AB and BA.
• The valley polarization $\Delta E(K) \simeq 100$ meV at $K$/$K'$ arises both from stacking-induced inversion symmetry breaking and AFM interlayer coupling in tandem with SOC (Pan et al., 18 Oct 2025).

4. Hybrid-Order Topological Phase Transitions via Sliding Ferroelectricity

Ferroelectric layer sliding in magnetic bilayer ScI₂ offers control over the topological quantum state:

• In an AA-stacked bilayer AFM system, the ground state is a second-order topological insulator (SOTI), characterized by corner charges protected by threefold rotational symmetry.
• Sliding one layer induces a ferroelectric polarization potential $P_{(\alpha,s)}$ that decouples spin channels and breaks the $\{C_2 \mid m_z\}$ symmetry. This causes asynchronous band evolution: one spin channel may undergo band inversion and transition to a first-order (Chern or QAHI) phase, while the other remains in the SOTI regime—a spin-hybrid-order topological insulator (SHTI) emerges.
• The multiphase sequence (SOTI $\to$ SHTI $\to$ QSHI $\to$ QAHI $\to$ normal insulator) can be driven by strain ($\varepsilon$) and sliding-induced potential ($p$), with topological indices $$I = (Q^\uparrow \oplus C^\uparrow, Q^\downarrow \oplus C^\downarrow)$$ labeling each phase.
• The anomalous Nernst effect (ANE) is a robust experimental probe of these transitions, as the ANC exhibits spin-dependent peak structures reflecting the underlying Berry curvature evolution (Yang et al., 2 Jun 2025).

5. Multiferroic Coupling and Electronic Structure Engineering

Interlayer sliding and rotation facilitate reconfigurable coupling between magnetic, ferroelectric, and valley orders:

• Stacking configuration governs the sign and magnitude of interlayer exchange, ferroelectric polarization, and valley splitting, by modifying orbital overlap and symmetry environment.
• Ferroelectricity can be switched by mechanical or electric means, allowing nonvolatile control of both magnetic and valleytronic responses. The interlocking of in-plane and out-of-plane polarization enables deterministic manipulation via applied fields, potentially yielding multi-state memory cells and spin–charge cross-coupling (Ji et al., 2022, Pan et al., 18 Oct 2025).
• Band structure engineering is realized through stacking-dependent interlayer hybridization and polarization step ($\Phi_p$), facilitating transitions between direct and indirect gap semiconductors, or metal-to-insulator transitions. The emergent electronic states are highly sensitive to stacking operations (Pakdel et al., 2023).

6. Raman and Vibrational Fingerprints of Stacking Configurations

Variations in stacking induce distinctive fingerprints in vibrational spectra:

• Out-of-plane breathing and in-plane shear modes are sensitive to stacking order and can be resolved via Raman spectroscopy. Breathing mode frequency differences on the order of $\sim2$ cm⁻¹ are typical between AA and AB configurations in layered materials.
• Intralayer phonon modes exhibit Davydov splitting, scaling as $$\Delta\omega_{\text{split}} \sim 2 \Delta\omega_{\text{shift}}$$. These shifts, measurable in Raman spectra, provide a nondestructive route to verify stacking in devices and paper slidetronic switching (Pakdel et al., 2023).

7. Implications for Multifunctional Device Design

The stacking-tunable ferroic orders in bilayer ScI₂ underpin design of next-generation nanodevices:

• Magnetic, ferroelectric, and valley properties are mutually reconfigurable through sliding and rotation, enabling logic, memory, and valleytronic functions in a single material system without the need for compositional heterostructures (Pan et al., 18 Oct 2025).
• Sliding ferroelectricity and the associated layer-resolved topological states offer routes to energy-efficient, non-volatile memory and robust quantum devices, with the ANE as a viable readout mechanism (Yang et al., 2 Jun 2025).
• The slidetronics paradigm—mechanically controlling device states by layer translation—finds strong support in both computational and experimental stacking studies, suggesting that bilayer ScI₂ and its chemical analogs are promising candidates for multifunctional and reconfigurable nanoelectronics (Pakdel et al., 2023).

Summary Table: Stacking-Dependent Properties in Bilayer ScI₂

Stacking	Interlayer Magnetism	Ferroelectricity	Valley Polarization
AA (aligned)	AFM	None	Absent
AB / BA (aligned)	FM	Maximal	Present, sign flips
AA* (rotated)	FM	None	Absent
AB* / BA* (rotated)	AFM	None	Present

The intricate coupling among stacking geometry, orbital interactions, symmetry breaking, and spin–orbit coupling in bilayer ScI₂ enables precise modulation of multiferroic states, positioning this material system at the forefront for advanced studies and applications in nanoscale multifunctional devices (Pan et al., 18 Oct 2025, Yang et al., 2 Jun 2025, Ji et al., 2022, Pakdel et al., 2023).