Bilayer Stacking Engineering

• Bilayer stacking engineering is the systematic manipulation of two atomically thin layers to modulate symmetry, interlayer coupling, and energy landscapes.
• It enables tuning of electronic, optical, and magnetic properties by adjusting twist angles, translations, strain, and pressure in van der Waals systems.
• This approach underpins advances in designing ferroelectric, altermagnetic, excitonic, and ion-diffusive states for next-generation electronic and energy devices.

Bilayer stacking engineering is the controlled manipulation of the relative registry of two atomically thin layers through lateral translation, rotation, twist angle, layer inversion, strain, charging, or pressure in order to tune interlayer coupling and the emergent properties of the bilayer. In van der Waals systems, the weak interlayer bond makes stacking order a primary design variable rather than a fixed crystallographic detail. As a result, the same chemical composition can realize distinct potential-energy landscapes, minibands, polar states, magnetic ground states, excitonic spectra, and ion-transport pathways depending on whether the bilayer is assembled in AA-, AB-, BA-, AA′-, 2H-, 3R-, or moiré-modulated registries (Pakdel et al., 2023, Yao et al., 2020, Ji et al., 2022, Fukuzawa et al., 29 Mar 2026).

1. Geometric and symmetry foundations

The basic object of bilayer stacking engineering is the relative displacement field between two layers. For aligned bilayers, stacking is specified by an in-plane translation that selects a registry such as AA or AB; for rotated bilayers, a twist angle $\theta$ generates a moiré pattern whose local registries vary continuously across the supercell. In homobilayers of lattice constant $a$, the moiré length is

$\lambda = \frac{a}{2\sin(\theta/2)},$

so small $\theta$ produces large supercells in which AA-like, AB-like, and saddle-point environments recur periodically (Yao et al., 2020). In twisted bilayer graphene, this geometric modulation creates a long-period moiré superlattice; in twisted TMD bilayers, the same principle applies with material-specific high-symmetry stackings such as 2H and 3R (Fukuzawa et al., 29 Mar 2026, Li et al., 2021).

Symmetry is the organizing principle that determines which physical responses stacking can create or remove. The general theory of bilayer stacking ferroelectricity models stacking by an operator $O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$ acting on the upper layer and shows that stacking can annihilate inversion or mirror symmetries and thereby generate in-plane, out-of-plane, or combined polarization states (Ji et al., 2022). A crucial rule is that a pure translation $O=(E\mid\mathbf t)$ leaves inversion centers of a centrosymmetric monolayer intact; accordingly, sliding alone cannot turn on out-of-plane polarization for a centrosymmetric monolayer (Ji et al., 2022). An analogous symmetry filtering appears in bilayer altermagnetism: the general stacking theory for altermagnetism identifies exactly seven bilayer point groups that permit altermagnetic spin splitting, namely $C_2$, $D_2$, $D_3$, $D_4$, $a$0, $a$1, and $a$2 (Pan et al., 2024).

These geometric and symmetry notions already imply two common corrections to oversimplified usage. First, bilayer stacking engineering is not synonymous with twistronics: sliding, flipping, heterostrain, pressure, and electrostatic perturbations are equally legitimate stacking knobs in the literature surveyed here (Georgoulea et al., 2022, Bacaksiz et al., 2016). Second, the physically relevant variable is often not a single global stacking label but a spatially varying registry field, especially in moiré systems and reconstructed small-angle bilayers (Gong et al., 2013, Enaldiev et al., 2019).

2. Energetic landscapes, stability criteria, and reconstruction

At the materials-discovery scale, stacking engineering is governed by the interlayer binding landscape. A high-throughput DFT workflow constructed 8451 unique homobilayer stacking configurations from 1052 dynamically and thermodynamically stable monolayers drawn from C2DB by enumerating unit-cell-preserving point-group operations and in-plane translations of an AA reference bilayer (Pakdel et al., 2023). The binding energy per area was defined as

$a$3

and candidate realizable stackings were filtered by the criterion $a$4 together with a 2D Hessian slide-stability test (Pakdel et al., 2023). This produced 2586 thermodynamically and mechanically stable bilayers. Validation against 247 naturally occurring van der Waals bulk crystals showed that 73% of experimentally observed stackings are the global minimum and 74% lie within $a$5 (Pakdel et al., 2023).

For small twist angles, the naive rigid-moiré picture fails because the lattice reconstructs. In marginally twisted TMD homobilayers, a multiscale model combining DFT-parametrized adhesion and elasticity predicts a crossover to domain reconstruction below $a$6 for parallel bilayers and below $a$7 for antiparallel bilayers (Enaldiev et al., 2019). Below these thresholds, large 3R or 2H domains form and are separated by partial or full screw dislocations, with domain sizes scaling as $a$8 and $a$9 (Enaldiev et al., 2019). This reconstruction is central to the interpretation of low-angle moiré experiments because the local registries occupy unequal areas after relaxation.

Stacking can also be reconfigured by elastic mismatch. In bilayer graphene subjected to uniaxial heterostrain, DFT and a simple energy model indicate that above a critical strain of about $\lambda = \frac{a}{2\sin(\theta/2)},$0 the ground state shifts from uniformly strained AB stacking to a non-uniform mixed-stacking state in which the free layer remains effectively unstrained (Georgoulea et al., 2022). The simple estimate equates $\lambda = \frac{a}{2\sin(\theta/2)},$1 and the average stacking cost $\lambda = \frac{a}{2\sin(\theta/2)},$2, yielding $\lambda = \frac{a}{2\sin(\theta/2)},$3, while the AGNR/MLG DFT model gives a sign change in $\lambda = \frac{a}{2\sin(\theta/2)},$4 near $\lambda = \frac{a}{2\sin(\theta/2)},$5 (Georgoulea et al., 2022).

Other systems show that stacking energetics can be unusually soft. In bilayer SnS$\lambda = \frac{a}{2\sin(\theta/2)},$6, the interlayer coupling is weaker than in typical TMDs, AA, A′B, and AB stackings differ by only 0, 1, and 6 meV per SnS$\lambda = \frac{a}{2\sin(\theta/2)},$7 formula unit, and the most favorable stacking can be switched by charging or by a loading pressure exceeding 3 GPa (Bacaksiz et al., 2016). In IDB-engineered bilayer MoSe$\lambda = \frac{a}{2\sin(\theta/2)},$8, experimentally realized low-symmetry registries exhibit a nearly linear relation between stacking energy and interlayer distance over the relaxed set of stackings (Hong et al., 2017). These results emphasize that the stacking phase space is often shallow enough for external fields, defects, or reconstruction to compete with nominal ground-state registries.

3. Electronic, transport, and optical structure control

The best-known consequence of stacking engineering is band-structure reconstruction. In graphene/hBN moiré heterobilayers, the lattice mismatch produces a weak periodic potential with moiré period $\lambda = \frac{a}{2\sin(\theta/2)},$9–14 nm, second-generation Dirac cones, mini-gaps of order 10–30 meV, and Hofstadter-butterfly transport signatures (Yao et al., 2020). In twisted bilayer graphene, the evolution from intermediate-angle van Hove singularities to magic-angle flat bands is controlled by the reduction of $\theta$0 and the corresponding increase of the dimensionless coupling $\theta$1; the first magic angle in the simplest continuum description occurs at $\theta$2, where nearly flat minibands emerge near charge neutrality (Yao et al., 2020).

High-throughput computational stacking extends this logic far beyond graphene. For 2586 stable homobilayers, stacking-dependent calculations revealed 349 bilayers that change electronic type and 126 emergent direct-gap semiconductors derived from indirect-gap monolayers (Pakdel et al., 2023). In polar bilayers, the band-gap renormalization follows the empirical relation $\theta$3, directly connecting stacking-controlled polarization potential steps to the bilayer electronic spectrum (Pakdel et al., 2023). The same database study showed that vibrational signatures are equally stacking-sensitive: low-frequency shear and breathing Raman modes shift by up to several cm$\theta$4 and vary in symmetry-dependent intensity with stacking (Pakdel et al., 2023).

Excitonic structure in semiconductor bilayers is similarly registry-specific. In WSe$\theta$5 homobilayers, 2H stacking retains inversion symmetry and yields a single A-exciton resonance at $\theta$6 eV in differential reflectivity, whereas 3R stacking breaks inversion and splits the direct intralayer A exciton into two peaks, $\theta$7 and $\theta$8, separated by $\theta$9 meV (Li et al., 2021). Photoluminescence sidebands also differ qualitatively: 2H bilayers exhibit phonon sidebands $O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$0–$O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$1, while 3R bilayers show both $O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$2–$O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$3 and additional $O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$4–$O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$5 features associated with distinct indirect excitons and distinct magnetic $O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$6-factors (Li et al., 2021). In bilayer MoSe$O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$7, low-symmetry stackings generated by inversion-domain boundaries drive a crossover between K-dominated direct gaps and $O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$8-dominated indirect gaps, accompanied by the appearance of stacking-dependent band-tail states in scanning tunneling spectroscopy (Hong et al., 2017).

The design space is not limited to hexagonal Dirac or valley materials. In Pd$O(\theta,\mathbf t)=\{R(\theta)\mid \mathbf t\}$9O$O=(E\mid\mathbf t)$0Cl$O=(E\mid\mathbf t)$1 kagome bilayers, first-principles calculations found strong stacking-dependent gap modulation from 0.08 to 0.76 eV across AA, AA′, AB, and AB′ registries, with the AB′ configuration being the most stable (Yang et al., 18 Dec 2025). Carrier effective masses span 2.39–6.35 $O=(E\mid\mathbf t)$2 for electrons and 0.67–1.55 $O=(E\mid\mathbf t)$3 for holes, and uniaxial or biaxial strain applied to AB′ produces non-monotonic gap tuning and strongly asymmetric hole-mass modulation (Yang et al., 18 Dec 2025). This suggests that stacking engineering can be combined with mechanical deformation to set both the baseline transport regime and its in situ tunability.

4. Ferroelectric, magnetic, valley, and altermagnetic states

One of the central developments in the field is the recognition that stacking can create ferroic order from non-ferroic or weakly ferroic monolayers. In parallel-stacked bilayer hBN, AB and BA registries are energetically degenerate, non-centrosymmetric, and carry equal but opposite out-of-plane dipoles (Yasuda et al., 2020). Dual-gated graphene detectors measured a polarization

$O=(E\mid\mathbf t)$4

in agreement with Berry-phase theory, and coercive fields of about 0.10 V/nm and 0.12 V/nm for backward and forward switching (Yasuda et al., 2020). A small twist of about $O=(E\mid\mathbf t)$5 converts the system into a moiré ferroelectric with triangular AB and BA domains, reducing the coercive field by a factor of $O=(E\mid\mathbf t)$6 to $O=(E\mid\mathbf t)$7 V/nm and broadening the switching over $O=(E\mid\mathbf t)$8 V/nm (Yasuda et al., 2020). The ferroelectric response persists up to 300 K, with no measurable decay after more than 30 days at zero gate bias (Yasuda et al., 2020).

The general theory of bilayer stacking ferroelectricity places such examples in a wider symmetry classification and predicts that properly stacked bilayers of identical monolayers can become purely in-plane polar, purely out-of-plane polar, or combined-polar even when the monolayer itself is non-polar (Ji et al., 2022). In bilayer CrI$O=(E\mid\mathbf t)$9, a centrosymmetric monolayer parent, first-principles calculations found six equivalent shifted minima with combined polarization; at $C_2$0 the in-plane component is $C_2$1 and the out-of-plane component is $C_2$2 (Ji et al., 2022). The lowest switching path rotates $C_2$3 by $C_2$4 while flipping $C_2$5, with a barrier of about 6.4 meV/f.u., so the out-of-plane polarization is interlocked with the in-plane polarization and can be driven deterministically by an in-plane electric field (Ji et al., 2022).

Stacking also controls interlayer magnetism. In bilayer ScI$C_2$6 with AA, AB, and BA registries, DFT+$C_2$7 mapping to an anisotropic Heisenberg Hamiltonian showed strong intralayer ferromagnetism, $C_2$8 meV, and weak but sign-tunable interlayer exchange: AA and BA are ferromagnetically coupled, while AB is antiferromagnetically coupled (Sarkar et al., 12 Feb 2026). All three have an out-of-plane easy axis and retain ordering temperatures in the range 369–372 K from finite-size scaling of Monte Carlo simulations (Sarkar et al., 12 Feb 2026). A related study that explicitly considered aligned and antialigned stackings found that sliding and rotation in bilayer ScI$C_2$9 can also induce ferroelectric polarization, reaching $D_2$0 C/m at AB and BA, and produce spontaneous valley polarization of about 100 meV in the presence of out-of-plane magnetization and SOC (Pan et al., 18 Oct 2025).

Valley control without time-reversal breaking has been formulated as bilayer stacking ferrovalley. Group-theory analysis showed that breaking threefold or fourfold rotation by interlayer translation can split non-time-reversal-invariant valleys while preserving time-reversal symmetry, and first-principles calculations demonstrated this in RhCl$D_2$1 and InI bilayers with valley polarizations of 39 meV and 326 meV, respectively (Yu et al., 2023). Sliding switches the favored valley and thus provides a non-volatile valley-control mechanism (Yu et al., 2023).

A further symmetry-controlled phase is altermagnetism. The general stacking theory predicts altermagnetic spin splitting when the bilayer point group belongs to the seven allowed classes and the stacking operator preserves antiferromagnetic interlayer coupling (Pan et al., 2024). Bilayer MnPS$D_2$2 provides a concrete realization: depending on lateral shift $D_2$3 and whether the two Néel patterns are parallel or antiparallel, the bilayer switches between Type II collinear antiferromagnetism and Type III altermagnetism with momentum-dependent spin splitting and zero net magnetization (González et al., 22 May 2025). Here the decisive variable is whether the remaining symmetry exchanging opposite spins is a degeneracy-enforcing operation such as $D_2$4 or a nonsymmorphic twofold rotation such as $D_2$5 (González et al., 22 May 2025).

5. Moiré-ionics and electrochemical functionality

Bilayer stacking engineering has recently expanded beyond electronic structure into ion transport. In twisted bilayer graphene, the local potential experienced by an intercalated Li atom depends strongly on the local atomic environment around the interlayer site, so twisting creates a spatially modulated potential-energy surface unavailable in uniform AA or AB stacking (Fukuzawa et al., 29 Mar 2026). The Li intercalation energy is defined as

$D_2$6

and the diffusion barrier is extracted from the PES as

$D_2$7

Dense-grid first-principles PES mapping showed that AA-like regions provide deep minima but high barriers, $D_2$8 eV, whereas AB-like regions provide shallow minima but very low barriers, $D_2$9 eV (Fukuzawa et al., 29 Mar 2026).

The key result is that an intermediate twist can resolve the conventional thermodynamic-kinetic trade-off. Among the studied coincidence-site-lattice structures, $D_3$0 at $D_3$1 simultaneously gives the most favorable intercalation energy, $D_3$2 eV, and the lowest diffusion barrier in the twisted series, $D_3$3 eV (Fukuzawa et al., 29 Mar 2026). Its moiré cell contains extended AB-like diffusion corridors together with sufficiently deep AA-like trapping pockets, so the same superlattice supports both strong Li binding and facile migration (Fukuzawa et al., 29 Mar 2026).

This ionic version of stacking engineering is also computationally tractable through local-environment descriptors. Using the Smooth Overlap of Atomic Positions descriptor with cutoff $D_3$4, the local atomic density around a query point is expanded into an 84-dimensional rotationally invariant vector, and a ridge-regression model maps this descriptor to PES energies (Fukuzawa et al., 29 Mar 2026). Within one twist angle, the model reaches $D_3$5 and MAE $D_3$6 eV; in cross-structure prediction, training on one twist and predicting another still yields $D_3$7 and MAE $D_3$8 eV (Fukuzawa et al., 29 Mar 2026). The resulting “Moiré-Ionics” workflow combines monolayer adsorption screening, DFT PES mapping on a small number of twists, SOAP-based PES prediction on new twists, and ranking by a figure of merit such as

$D_3$9

followed by DFT validation of top candidates (Fukuzawa et al., 29 Mar 2026). A plausible implication is that bilayer stacking engineering provides a general route to electrochemical optimization in layered hosts, not only a band-structure design method.

6. Fabrication, characterization, and dynamic reconfiguration

The experimental viability of bilayer stacking engineering depends on controlling orientation and registry during fabrication. A gold-epitaxy-assisted exfoliation method for TMD monolayers exploits the strong preference of Au(111) overlayers for Mo-terminated zigzag steps, with step-contact energies $D_4$0 eV/Å compared with $D_4$1 eV/Å and $D_4$2 eV/Å (Li et al., 2021). Because the Au “gripper” locks to a known crystallographic edge, the resulting monolayers have armchair-edge orientations known to better than $D_4$3, enabling direct assembly of 3R ($D_4$4) and 2H ($D_4$5) bilayers (Li et al., 2021). Reported performance includes monolayer yield above 95% of Au-covered area, flake sizes of 200–500 $D_4$6m, demonstrated twist-angle error below $D_4$7, and wafer-scale exfoliation up to 2 inches (Li et al., 2021).

Several studies show that stacking is dynamically reconfigurable after fabrication. In twisted bilayer graphene, thermal annealing at 200 $D_4$8C for 12 hours drives an incommensurate-to-AB transition via macroscopic self-rotation once interlayer contaminants are mobilized and expelled by self-cleaning (Zhu et al., 2016). Atomistic estimates quoted in that study give an interlayer van der Waals energy gain of about 0.13 meV per carbon atom for the incommensurate-to-AB transition and an energy barrier $D_4$9 meV/atom, sufficient for self-rotation on laboratory timescales at $a$00 K (Zhu et al., 2016). The structural transition is accompanied by a reciprocal-space topological transition from decoupled Dirac cones toward the parabolic low-energy spectrum of AB bilayer graphene (Zhu et al., 2016).

In bilayer MoS$a$01, atomically sharp AA′–AB stacking boundaries can be nucleated and expanded in situ under 80 keV STEM at 350–400 $a$02C, permitting direct observation of boundary motion and defect-mediated stacking conversion (Yan et al., 2017). DFT nudged-elastic-band calculations give S-atom migration barriers from about 0.15 to 1.05 eV per atom, and the relaxed boundary supercells host metallic in-gap states localized at the wall (Yan et al., 2017). In bilayer MoSe$a$03, inversion-domain boundaries introduced during Se-rich MBE growth generate fractional lattice translations and thereby stabilize large-area uniform low-symmetry stacking domains that are difficult to access in pristine bilayers (Hong et al., 2017). These experiments show that stacking engineering can proceed not only by deterministic transfer but also by growth templating, defect patterning, thermal activation, and in situ beam-assisted reconstruction.

Taken together, the literature establishes bilayer stacking engineering as a general framework for programming the interlayer degree of freedom in two-dimensional matter. The same operations that set local registry also set moiré reconstruction, interlayer hybridization, polarization, exchange pathways, exciton multiplicity, valley splitting, and ion-diffusion topology. A recurrent lesson across graphene, hBN, TMDs, halides, kagome bilayers, and electrochemical hosts is that stacking should be treated as an active thermodynamic and symmetry variable rather than a passive structural descriptor (Pakdel et al., 2023, Ji et al., 2022, Fukuzawa et al., 29 Mar 2026).