Origin of trapped intralayer Wannier and charge-transfer excitons in moiré materials

This paper resolves discrepancies between continuum and ab initio models of moiré excitons by employing an atomistic Bethe-Salpeter equation framework to demonstrate that hBN encapsulation critically influences the competition between Wannier and charge-transfer characters, thereby determining the nature of the lowest-energy bright excitons in WS$_2$/WSe$_2$ heterobilayers and twisted WSe$_2$ homobilayers.

The Gist

Imagine a world made of ultra-thin, sticky sheets of material (like graphene or special metals) stacked on top of each other. When you stack these sheets and twist them slightly, they don't line up perfectly. Instead, they create a giant, repeating pattern of ripples and bumps, much like the interference pattern you see when you overlap two window screens. Scientists call this a Moiré pattern.

In this paper, the researchers are studying what happens to tiny particles of light and electricity called excitons when they get trapped inside these ripples. Think of an exciton as a "dance couple": an electron (negative charge) and a hole (positive charge) that are holding hands and dancing together.

Here is the simple breakdown of what the paper discovered:

1. The Great Debate: Where are the dancers?

For a long time, scientists argued about how these exciton couples behave in these twisted materials.

• Theory A said they act like standard dancers (called "Wannier excitons") who stay close together and move around freely within a specific spot.
• Theory B (based on massive computer simulations) said some of them act like "long-distance couples" (called "charge-transfer excitons"), where the electron and hole are far apart, almost like they are dancing in different rooms but still holding hands.

The problem was that the computer models didn't match what scientists saw in real experiments. The theories predicted the dancers should be in one place, but experiments showed them in another.

2. The Missing Ingredient: The "Blanket"

The authors realized the computer models were missing a crucial piece of the puzzle: environmental screening.

Imagine the Moiré material is a dancer on a stage. In many computer models, the stage is empty. But in real experiments, the material is usually wrapped in a protective "blanket" made of a material called hBN (hexagonal boron nitride).

• Without the blanket: The electric forces between the electron and hole are too strong and messy. The computer models get confused, and the dancers behave strangely.
• With the blanket: The hBN acts like a dampener or a filter. It softens the electric forces between the dancers. When the researchers added this "blanket" to their calculations, the computer models suddenly matched the real-world experiments perfectly.

3. The New Discovery: It Depends on the Twist

Once they fixed the "blanket" issue, they found something surprising: The type of dance depends entirely on how the sheets are stacked.

• Scenario A (WS2/WSe2 Heterobilayer): When they stack two different materials (like a WSe2 sheet on a WS2 sheet), the lowest-energy excitons are the standard "close-couple" dancers (Wannier type). They stay trapped in the deepest part of the Moiré ripple.
• Scenario B (Twisted WSe2 Homobilayer): When they stack the same material on itself but twist it at a specific angle (about 57.7 degrees), the story changes. Even though the ripple pattern looks almost identical to Scenario A, the lowest-energy exciton becomes a "long-distance couple" (charge-transfer type).

The Analogy: Imagine two identical-looking rooms. In one room, the furniture is arranged so a couple must sit right next to each other. In the other room, the furniture is arranged so the couple is forced to sit on opposite sides of the room, yet they still feel connected. The paper shows that the specific atomic arrangement (the "furniture") dictates whether the couple stays close or drifts apart.

4. The "Adiabatic Switching" Experiment

To understand why this happens, the researchers played a game of "slow motion." They took a perfectly relaxed, twisted structure and slowly "un-relaxed" it (like taking the tension out of a spring).

• They found that as the atomic structure changes slightly, the energy levels of the excitons shift.
• They discovered a subtle competition: The exciton wants to sit where the energy gap is smallest, but it also wants to stay close to its partner. The environment (the hBN blanket) tips the balance of this competition.

Summary

The paper solves a mystery by showing that you cannot accurately predict how these light-particles behave without accounting for the protective "blanket" (hBN) used in real experiments.

Once that is accounted for, they reveal that the nature of these excitons isn't fixed; it's a delicate balance between the specific atomic stacking of the materials and the environment they are in. Sometimes they are tight-knit couples, and sometimes they are long-distance partners, depending entirely on how the layers are twisted and what is wrapped around them.

This gives scientists a powerful new tool: by changing the twist angle or the surrounding materials, they can now deliberately design whether these excitons stay close together or separate, which is a key step for building future optical devices.

Technical Summary

Technical Summary: Origin of Trapped Intralayer Wannier and Charge-Transfer Excitons in Moiré Materials

Problem Statement
Moiré materials, particularly transition metal dichalcogenide (TMD) heterobilayers like WS₂/WSe₂, offer a platform for engineering excitons with potential applications in optoelectronics and quantum information. While continuum models have been widely used to study these systems due to their efficiency, they often disagree with ab initio many-body approaches regarding the microscopic nature of intralayer excitons. Specifically, in nearly aligned WS₂/WSe₂ heterobilayers, experiments reveal three absorption peaks around 1.7 eV separated by approximately 90 meV. However, theoretical interpretations of the exciton wavefunctions remain debated: some continuum models predict all peaks are Wannier-type (localized), while large-scale ab initio GW-BSE calculations suggest a mix of Wannier and charge-transfer characters. Furthermore, existing theories fail to quantitatively reproduce the experimentally observed energy separations (~90 meV vs. predicted ~180–200 meV) and the redshift of the lowest bright exciton. A unified, quantitatively accurate explanation for these features and their microscopic origins has been lacking.

Methodology
The authors employ an atomistic, quantum-mechanical framework based on the Bethe-Salpeter Equation (BSE) using Wannier functions as the electronic structure basis. This approach allows for the simulation of experimentally relevant moiré system sizes (up to ~8.2 nm lattice constants), which were previously inaccessible to standard ab initio methods.

• Structural Modeling: Moiré structures were generated using the TWISTER package, with atomic relaxations calculated via classical interatomic potentials (Stillinger-Weber and Kolmogorov-Crespi) in LAMMPS.
• Electronic Structure: Calculations were performed using SIESTA with localized atomic orbitals. Wannier functions were generated using WANNIER90 via a one-shot projection method.
• Exciton Calculation: The BSE was solved using an improved version of the PyMEX package. The authors introduced an accurate interpolation scheme based on minimal-distance replica selection to handle larger systems efficiently.
• Screening: Crucially, the study incorporates dielectric screening from hexagonal boron nitride (hBN) encapsulation using the Rytova-Keldysh potential. Parameters for the screening length ($r_0$) and effective dielectric constant ($\epsilon_{env}$) were derived directly from ab initio calculations.
• Adiabatic Switching: To understand the origin of exciton character, the authors systematically reduced atomic relaxation amplitudes (adiabatic switching) to isolate the effects of structural relaxation and environmental screening on the excitonic spectrum.

Key Contributions and Results

1. Resolution of Theory-Experiment Discrepancies: The study demonstrates that including dielectric screening from hBN encapsulation is essential to reproduce experimental observations. Without hBN screening, the calculated energy separation between peaks is ~137 meV (vs. experimental ~90 meV), and the lowest bright exciton exhibits a blueshift rather than the observed redshift. With hBN screening, the calculated separations (47 meV and 39 meV) and the redshift (11 meV) align closely with experimental data.
2. Nature of Exciton Characters: The analysis reveals a subtle competition between local stacking-dependent intralayer bandgaps and electron-hole Coulomb interactions.
   • Peak I (Lowest Energy): Identified as a trapped Wannier exciton localized in the AA stacking region (W above W, Se above S). This localization agrees with GW-BSE results but contradicts improved continuum models that lack hBN screening.
   • Peak II: A trapped exciton with a small charge-transfer character.
   • Peak III (Highest Energy): Exhibits strong charge-transfer character and spatial delocalization when hBN screening is included. Without hBN screening, this charge-transfer character vanishes, and the exciton becomes significantly more localized.
3. Tunability via Stacking and Screening: The authors show that the ordering of Wannier and charge-transfer characters can be tuned by stacking and environmental screening.
   • In parallel-aligned WS₂/WSe₂, the lowest bright exciton remains Wannier-like regardless of hBN screening.
   • In a 57.7° twisted WSe₂ homobilayer (with a moiré size nearly identical to the aligned heterobilayer), the lowest bright exciton is charge-transfer-like when hBN screening is included, but becomes Wannier-like without it.
4. Mechanistic Insight: Through adiabatic switching, the study establishes that atomic relaxation drives direct-to-indirect bandgap transitions. The interplay between these structural changes and tunable electron-hole interactions (via screening) dictates whether excitons are trapped Wannier states or delocalized charge-transfer states.

Significance
The paper establishes atomistic modeling as a powerful tool for understanding and controlling excitonic phenomena in moiré materials. By resolving the discrepancies between continuum models and ab initio approaches, the authors provide a unified explanation for the experimentally observed moiré exciton features. The work highlights that dielectric screening is not merely a quantitative correction but a qualitative factor that determines the fundamental character (Wannier vs. charge-transfer) of the excitons. These findings pave the way for the atomistic design of excitonic states in van der Waals heterostructures and suggest that the interplay between Coulomb engineering and moiré relaxation remains an underexplored avenue for shaping exciton characteristics. The authors note that while their approach is computationally feasible for large systems, it is primarily motivated by computational simplicity in isolating the WSe₂ electronic states, rather than inherent limitations of the method.