Intralayer Exchange Coupling in 2D Materials

• Intralayer exchange coupling is the quantum-mechanical interaction within a single atomic layer that drives exciton hybridization and magnetic ordering.
• It is modeled using techniques like GW+BSE and DFT, with experimental methods such as magnetometry and NV-center scanning offering precise measurements.
• This mechanism is pivotal for tuning optical responses, 2D ferromagnetism, and exchange bias in advanced spintronic and quantum devices.

Intralayer exchange coupling refers to the spin, orbital, or particle-exchange processes that occur within a single atomic or molecular layer in low-dimensional materials, leading to hybridization, mode-splitting, or magnetic ordering. This coupling is a fundamental mechanism in quantum magnets, 2D semiconductors, and van der Waals heterostructures, controlling properties such as ferromagnetism, magnon–magnon hybridization, and exciton mixing.

1. Microscopic Origins and Definitions

Intralayer exchange coupling is governed by quantum-mechanical exchange interactions within an individual layer or sublattice, typically mediated either via direct overlap of atomic orbitals or through ligand-assisted superexchange. In the case of monolayer and bilayer transition-metal dichalcogenides (TMDCs) such as MoS₂, the lowest-energy optical excitations ("A" and "B" excitons) arise from electron–hole pairs bound within the same valley and monolayer. Here, intralayer (intravalley) exchange coupling corresponds to a Coulomb-mediated virtual exchange of electron–hole constituents between two excitons, resulting in an off-diagonal matrix element in the excitonic Hamiltonian that mixes A- and B-type excitonic character (Sponfeldner et al., 2021).

In 2D magnets, such as MnBi₂Te₄ and CrPS₄, the intralayer exchange constant $J_\text{intra}$ quantifies the effective nearest-neighbor (and longer-range) coupling between magnetic moments within the same atomic layer, set by transition-metal–ligand–transition-metal exchange pathways, anisotropy, and the local electronic structure (Shao et al., 2020, Wang et al., 3 Dec 2025). In compensated ferrimagnetic insulators, such as GdIG, the same mechanism is responsible for hybridization between sublattice magnon modes, yielding exchange-enhanced couplings (Liensberger et al., 2019).

2. Theoretical Formulation and Model Hamiltonians

The underlying physics of intralayer exchange coupling is described by model Hamiltonians appropriate to the system under study.

• Excitonic Systems: The Bethe–Salpeter equation (BSE) with GW-corrected band structures is used to obtain excitonic eigenstates in TMDCs. The Hamiltonian includes a direct electron–hole attraction $K^d$ and an unscreened exchange kernel $K^x$ that mediates intralayer exchange:

$$\left( E_{c k}-E_{v k} \right) A^{v c k}_n + \sum_{v' c' k'} \left[ K^d_{v c k, v' c' k'} - K^x_{v c k, v' c' k'} \right] A^{v' c' k'}_n = \hbar \Omega_n A^{v c k}_n$$

The off-diagonal term $⟨X_A|K^x|X_B⟩$ quantifies the intralayer A–B exciton coupling (Sponfeldner et al., 2021).

• Magnetic Systems: For layered magnets, the lattice spin Hamiltonian is written:

$$H = -\sum_{\langle i,j \rangle} J_{ij} \mathbf{S}_i \cdot \mathbf{S}_j$$

with $J_{ij} = J_\text{intra}$ for spins within a layer, and $J_{ij} = J_\text{inter}$ for interlayer pairs. In CrPS₄, an extended Hamiltonian includes nearest- and next-nearest neighbor intralayer exchange ($J_1, J_2$), interlayer coupling $J_c$, uniaxial anisotropy $K$, and Zeeman terms (Wang et al., 3 Dec 2025).

• Ferrimagnets: For compensated systems, a two-sublattice spin Hamiltonian models the coupled Gd and Fe nets:

$$\mathcal H = -J\,\mathbf M_A \cdot \mathbf M_B - K(M_{Az}^2 + M_{Bz}^2) + K_a(M_{Ax}^2 + M_{Bx}^2) - \mu_0 H_0 (M_{Az} + M_{Bz})$$

where $J \gg K \gg |K_a|$ controls the exchange-enhanced magnon–magnon coupling (Liensberger et al., 2019).

3. Mechanisms and Quantitative Parameters

The intralayer exchange mechanism is determined by the specifics of the electronic structure and atomic geometry.

• TMDC Exciton Mixing: In bilayer MoS₂, the exchange element $⟨A|K^x|B⟩$ is responsible for A–B exciton mixing, with GW+BSE calculations yielding values of $-15$ to $-18$ meV (negative sign). Experimentally, an effective A–B exchange of $-16.6 \pm 2.5$ meV is inferred from avoided crossing in optical susceptibility measurements (Sponfeldner et al., 2021).
• Magnetic Topological Insulators: In MnBi₂Te₄, DFT calculations for monolayer intralayer exchange yield $J_1 = -0.94$ meV (FM), $J_2 = +0.14$ meV (AFM), $J_3 = -0.038$ meV (FM), with ferromagnetism arising from hybridization between Bi $6p$ and Te $5p$ orbitals. The presence of empty Bi $p$ states stabilizes FM exchange contrary to the Goodenough-Kanamori rule, as confirmed by tight-binding and DFT analysis (Li et al., 2020). In superlattice MnBi₂Te₄/(Bi₂Te₃)ₙ, pressure and Sb-doping can tune $J_\text{intra}$ up to $\sim$10% reduction at 3 GPa, leading to transitions between AFM and 2D FM phases (Shao et al., 2020).
• 2D Antiferromagnets and Exchange Bias: In CrPS₄, $J_1 \approx 1.8$ meV and $J_2 \approx 0.2$ meV drive strong intralayer ferromagnetic correlations. Interlayer antiferromagnetic exchange $J_c \approx -0.07$ meV leads to a lateral exchange bias field controlled by microstructure (perimeter-area ratio), further modulated by domain-wall energetics and anisotropy (Wang et al., 3 Dec 2025).
• Magnon-Magnon Hybridization: In GdIG, the coupling constant $g$ for intralayer (sublattice) magnon–magnon coupling is $g/2\pi \approx 0.92$ GHz (easy-axis) and up to $6.4$ GHz (hard-axis), corresponding to $g/\omega \sim 30\%-40\%$, enabled by an exchange-enhancement factor $\sqrt{J/K} \gg 1$ (Liensberger et al., 2019).

4. Experimental Approaches and Control

Quantitative extraction and tuning of intralayer exchange coupling utilize complementary experimental and computational techniques:

• DFT and Many-body Methods: GW+BSE for TMDCs; GGA+$U$ DFT for magnetic insulators (plane-wave cutoff, k-point meshes, projected density of states).
• Magnetometry and Transport: Zero-field/field-cooled susceptibility, anomalous Hall effect, and broadband ferromagnetic resonance are used to map $T_N$, saturation field, and magnon spectra (Shao et al., 2020, Liensberger et al., 2019).
• NV-center Scanning Magnetometry: High-sensitivity NV centers resolve nanoscale magnetization patterns, coercive fields, and domain wall energetics in ultrathin crystals, enabling the measurement of sheet magnetization, domain switching fields, and bias fields $H_\text{EB}$ (Wang et al., 3 Dec 2025).
• Tuning by Pressure and Doping: Hydrostatic pressure compresses bond lengths and modifies exchange overlap, while chemical substitution (e.g., Sb for Bi) contracts the lattice, both systematically adjusting $J_\text{intra}$ and associated phase transitions (Shao et al., 2020).

5. Role in Optical, Magnetic, and Spintronic Phenomena

Intralayer exchange coupling underpins diverse quantum phenomena:

• Exciton Hybridization and Optical Susceptibility: In bilayer TMDCs, exchange-induced mixing of A and B excitons produces avoided crossings and characteristic redistribution of oscillator strengths in optical spectra. The sign of the coupling (negative for A–B) results in inductive mixing and branch suppression, while positive coupling (IE–B) leads to capacitive enhancement (Sponfeldner et al., 2021).
• Exchange Bias and Domain Engineering: In CrPS₄, lateral intralayer exchange at interfaces between odd/even layer patches generates tunable exchange bias fields ($H_\text{EB}$) scaling as perimeter/area. Multilevel bias structures and precise domain switching are enabled by controlling the balance of intralayer and interlayer exchanges (Wang et al., 3 Dec 2025).
• Ultrastrong Coupling of Collective Modes: In compensated ferrimagnets, exchange-enhanced intralayer coupling produces nontrivial hybridization between magnon modes, leading to avoided crossings and controllable strong to ultrastrong coupling regimes as a function of field orientation and temperature (Liensberger et al., 2019).
• Tunable Magnetic Phases: Intralayer exchange is a tunability axis for accessing new quantum phases such as quasi-2D ferromagnetism, quantum anomalous Hall insulator (QAHE), and axion-insulator phases in MnBi₂Te₄-based heterostructures (Shao et al., 2020, Li et al., 2020).

6. Representative Numerical Values and Materials Comparison

Key microscopic and emergent parameters for selected material systems are summarized below:

System	$J_\text{intra}$/meV	Physical Manifestation	Reference
MoS₂ bilayer (A–B ex)	$-15$ to $-18$	Exciton mixing, optical spectra	(Sponfeldner et al., 2021)
MnBi₂Te₄ monolayer	$-0.94$ (NN, FM)	FM septuple-layer ground state	(Li et al., 2020)
CrPS₄ (2D AFM)	$1.8$ (NN), $0.2$ (NNN)	Lateral exchange bias, domain control	(Wang et al., 3 Dec 2025)
MnBi₄Te₇, $n=1$	$10.5$ ($P=0$), $\alpha=-0.37$/GPa	Pressure-tuned AFM/QAHE	(Shao et al., 2020)
GdIG (ferrimagnet)	—	$g/\omega\sim 0.3$–$0.4$	(Liensberger et al., 2019)

Reduction of $J_\text{intra}$ by $\sim10\%$ at 3 GPa is observed for both MnBi₄Te₇ and MnBi₆Te₁₀, directly correlating with macroscopic phase transitions and transport signatures (Shao et al., 2020). In CrPS₄, lateral bias fields up to $|H_\text{EB}|\sim 52$ mT are realized intrinsically in planar geometries (Wang et al., 3 Dec 2025).

7. Implications and Future Directions

The precise control and understanding of intralayer exchange coupling offers a platform for the design of emergent phases and device functionalities in low-dimensional quantum materials. Pressure, chemical modification, and nanostructure engineering enable real-time tuning of $J_\text{intra}$, thereby permitting manipulation of magnetism, exciton resonances, and topological bandgaps. There is a growing paradigm in leveraging intralayer exchange in the development of spintronic and quantum anomalous Hall devices, as well as in tailoring multilevel exchange bias and hybridization phenomena for advanced information storage and computation (Shao et al., 2020, Wang et al., 3 Dec 2025). A plausible implication is that continued advances in experimental probes (scanning NV magnetometry, high-resolution optical spectroscopy) and first-principles modeling will enable further refinement of microscopic spin and exciton Hamiltonians, providing quantitative predictions and control at the atomic scale.