🔬 mesoscale physics

Interplay of interlayer distance and in-plane lattice relaxations in encapsulated twisted bilayers

This paper presents a theoretical model demonstrating that the rigidity of encapsulation interfaces significantly influences lattice relaxation in twisted bilayers, specifically raising the critical twist angle for the transition between weak and strong relaxation regimes and enabling better alignment with experimental data.

Original authors: V. V. Enaldiev

Published 2026-02-09

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CC BY 4.0

Original authors: V. V. Enaldiev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have two sheets of a very sticky, patterned wallpaper. If you place one sheet directly on top of the other but twist them slightly, the patterns don't line up perfectly. Instead, they create a giant, repeating "shadow" pattern called a moiré pattern.

In the world of quantum materials, scientists twist these atomic layers to create new electronic properties. However, atoms are lazy; they want to settle into the most comfortable, energy-saving position. So, when you twist these layers, the atoms don't just stay put—they shuffle around, stretching and squeezing to find the best fit. This shuffling is called lattice relaxation.

The Problem: The "Floating" vs. The "Sandwich"

For a long time, scientists studied these twisted layers as if they were floating in a vacuum (suspended). They knew that at certain small angles, the atoms would shuffle a lot (strong relaxation), creating distinct islands of perfect alignment separated by walls of stress. At larger angles, they wouldn't shuffle much (weak relaxation).

But in real experiments, these layers aren't floating. They are usually sandwiched between other protective layers (like hexagonal boron nitride) to keep them stable. This is called encapsulation.

The paper asks: Does this sandwich change how the atoms shuffle?

The Discovery: The "Stiff Sandwich" Effect

The author, V. V. Enaldiev, built a mathematical model to answer this. He realized that the protective "bread" of the sandwich (the encapsulation) acts like a stiff constraint.

Here is the analogy:

The Twisted Layers: Imagine two soft, squishy rubber mats with a honeycomb pattern. When you twist them, the honeycombs try to snap into a perfect alignment.
The Encapsulation: Now, imagine you press these mats between two very hard, rigid boards.
The Result: In the middle (where the mats touch), the rubber wants to squish up and down to find the perfect fit. But the hard boards on top and bottom say, "No, stay flat!" The boards resist the mats from moving up and down.

The paper finds that because the "boards" (encapsulation) are stiff, they suppress the vertical movement of the atoms. The atoms can't squeeze as much as they want to.

The Main Finding: Changing the "Tipping Point"

Because the atoms can't squeeze as easily, it takes a smaller twist angle to force them to start shuffling horizontally to find their comfort zone.

Think of it like a seesaw:

Suspended (Floating): The atoms are free to move up and down. They only start shuffling horizontally when the twist is very small (around 1° to 2.5°).
Encapsulated (Sandwiched): The atoms are pinned down vertically. Because they can't use the "up and down" trick to save energy, they are forced to shuffle horizontally earlier (at a larger twist angle).

The paper calculates that for a perfectly rigid sandwich, this "tipping point" (where the atoms start shuffling significantly) shifts from about 3.8° to 4.5°.

Why This Matters

The author shows that by adjusting just one number in his model (representing how stiff the sandwich is), his predictions match real-world experiments perfectly.

Real-world proof: Experiments showed that twisted layers in a sandwich behave differently than floating ones.
The model's success: The model explains why: The sandwich makes the layers "stiffer" vertically, which changes the angle at which the atoms decide to rearrange themselves.

In a Nutshell

This paper explains that when you wrap twisted atomic layers in a protective shell, the shell acts like a stiff clamp. This clamp stops the atoms from moving up and down, forcing them to rearrange their side-to-side positions at different angles than they would if they were floating freely. This simple change in "stiffness" explains why real experiments look different from old theories that ignored the protective shell.

Technical Summary: Interplay of Interlayer Distance and In-Plane Lattice Relaxations in Encapsulated Twisted Bilayers

Problem Statement
Twisted van der Waals heterostructures, particularly twisted bilayers of transition metal dichalcogenides (TMDs), exhibit mesoscale moiré superlattices that significantly alter electronic and optoelectronic properties. The formation of these superlattices drives atomic relaxation to minimize total energy, transforming rigid moiré patterns into arrays of commensurate domains separated by strain-absorbing domain walls. While the crossover from rigid moiré patterns to relaxed domain structures has been characterized for suspended bilayers (with crossover angles $\theta^* \approx 2.5^\circ$ for parallel and $\approx 1^\circ$ for anti-aligned TMDs), recent experiments indicate that encapsulating TMD bilayers in hexagonal boron nitride (hBN) enhances lattice relaxation and shifts these crossover angles to higher values. Existing theoretical models for lattice relaxation do not account for the mechanical constraints imposed by encapsulation layers, specifically the modulation of interlayer distances at the interface.

Methodology
The authors develop a theoretical model describing lattice relaxation in twisted TMD bilayers that explicitly incorporates encapsulation effects. The approach involves:

Adhesion Energy Formulation: Utilizing a stacking-dependent adhesion energy density, $W_{ad}(r_0, d)$ , which depends on the in-plane offset $r_0$ and interlayer distance $d$ .
Encapsulation Modeling: Treating the twisted bilayer as embedded between multilayer slabs (encapsulation). The interaction between the TMD bilayer and the encapsulation slabs is modeled via a harmonic potential characterized by an effective parameter $k$ , representing the rigidity of the bilayer/slab interface.
Interlayer Distance Modulation: Deriving a new expression for the interlayer distance modulation, $d_{enc}(r_0)$ , by balancing the stacking-dependent pressure from the twisted layers ( $P_{in}$ ) against the restoring pressure from the encapsulation slabs ( $P_{out}$ ). This results in a suppression of interlayer distance variation compared to suspended bilayers.
In-Plane Relaxation Calculation: Minimizing a total energy functional comprising in-plane elastic energy and local adhesion energy (using the modified $d_{enc}$ ) to determine in-plane displacement fields. The calculations employ Fourier series expansions over the moiré superlattice reciprocal vectors.
Quantification: Analyzing the strength of lattice relaxation using the ratio $R_\nu(\theta)$ , defined as the area of XX-stacked regions relative to the moiré supercell area, across various twist angles and interface rigidity parameters.

Key Contributions and Results

Model Development: The paper introduces a single phenomenological parameter, $k$ , which characterizes the rigidity of the interface between the twisted bilayer and the encapsulation material. This parameter governs the suppression of interlayer distance modulation.
Effect of Encapsulation: The model demonstrates that encapsulation suppresses the range of accessible interlayer distances. This suppression increases the energy difference between energetically favorable (e.g., rhombohedral 3R) and unfavorable (XX) stackings.
Shift in Crossover Angle: A primary result is that increasing the interface rigidity ( $k$ ) raises the crossover twist angle ( $\theta^*$ ) between weak and strong relaxation regimes. For suspended bilayers ( $k=0$ ), $\theta^*$ is calculated to be $\approx 3.8^\circ$ . For perfectly rigid interfaces ( $k \gg 4\varepsilon$ ), $\theta^*$ increases to $\approx 4.5^\circ$ .
Quantitative Agreement: By adjusting the rigidity parameter to model a perfectly rigid interface (likely due to lattice incommensurability between TMD and hBN), the model predicts a shift in the crossover angle of $\Delta\theta^* \approx 0.75^\circ$ . This aligns well with experimental observations of $\Delta\theta^* \approx 1^\circ$ for hBN-encapsulated parallel-aligned WSe $_2$ bilayers.
Regime Behavior: The study confirms that in the strong relaxation regime (small twist angles), the size of XX-stacked regions becomes independent of the twist angle, following a $\theta^2$ scaling for the relaxation strength parameter, a behavior that remains universal across different encapsulation strengths.

Significance
The paper claims that its significance lies in providing a theoretical framework that bridges the gap between models of suspended bilayers and experimental results on encapsulated systems. By identifying that the rigidity of the encapsulation interface suppresses interlayer distance modulation, the authors explain why encapsulated systems exhibit enhanced lattice relaxation and higher crossover twist angles. The model successfully achieves quantitative agreement with existing experimental data by treating the interface as effectively rigid, suggesting that lattice incommensurability at the TMD/hBN interface is a critical factor in determining the mechanical response of these heterostructures. The authors note that this framework can be extended to other twistronic heterostructures, such as WX $_2$ /MoX $_2$ systems, to predict changes in strain magnitude and domain wall network behavior.

The Problem: The "Floating" vs. The "Sandwich"

The Discovery: The "Stiff Sandwich" Effect

The Main Finding: Changing the "Tipping Point"

Why This Matters

In a Nutshell

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