🔬 mesoscale physics

Layer-tunable Hubbard bands probed via moiré excitons in MoSe $_2$ /WS $_2$ heterostructures

This study demonstrates that vertical electric fields can deterministically tune layer-specific Hubbard parameters in dual-gated MoSe $_2$ /WS $_2$ heterobilayers, enabling the quantitative extraction of layer-dependent on-site Coulomb repulsions and the stabilization of generalized Wigner crystal and stripe phases by shifting the ground state to the WS $_2$ layer.

Original authors: Hongyu Yao, Qiao Li, Chih-En Hsu, Takashi Taniguchi, Kenji Watanabe, Hung-Chung Hsueh, Zhenglu Li, Andrew Y. Joe

Published 2026-06-25

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CC BY 4.0

Original authors: Hongyu Yao, Qiao Li, Chih-En Hsu, Takashi Taniguchi, Kenji Watanabe, Hung-Chung Hsueh, Zhenglu Li, Andrew Y. Joe

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 a microscopic dance floor made of two different types of sticky tiles stacked on top of each other. One tile is made of a material called MoSe2 and the other is WS2. Because the tiles have slightly different patterns, when you stack them at a specific angle (about 60 degrees), they create a giant, repeating pattern of "dance circles" called a Moiré superlattice.

Think of these dance circles as tiny parking spots for electrons (the dancers). In this paper, the researchers figured out how to control exactly which parking spots the electrons choose to sit in, and how much they "push" against each other when they get crowded.

Here is the breakdown of their discovery using simple analogies:

1. The "Spot Check" Cameras (Moiré Excitons)

To see what the electrons are doing without touching them, the researchers used special light signals called moiré excitons.

Analogy: Imagine two different types of security cameras watching the dance floor.
- Camera A (XM1): This camera only watches the "MoSe2" side of the parking spots. It is very sensitive to what happens specifically on that side.
- Camera B (XM2): This camera watches the "WS2" side. It acts like a giant sensor that feels the overall "crowdedness" of the room.

By watching how these cameras react when they shine light on the dance floor, the scientists could tell exactly where the electrons were sitting.

2. The "Pushiness" of the Dancers (Hubbard Bands)

In physics, when electrons get close, they don't like to be in the same spot because they repel each other (like magnets with the same pole facing each other). This "pushiness" is called Coulomb repulsion (or Hubbard U).

The Discovery: The researchers found that the "pushiness" is different depending on which layer the electron is in.
- In the MoSe2 layer, the electrons are very "pushy." They hate being in the same spot, so it takes a lot of energy to force two of them together. (Think of this as a very strict bouncer).
- In the WS2 layer, the electrons are more relaxed. They are less "pushy" and can get closer together more easily. (Think of this as a more lenient bouncer).

They measured this exactly: The "pushiness" in MoSe2 is about 60 meV, while in WS2 it is only 30 meV.

3. The "Remote Control" (Electric Field)

The most exciting part is that the researchers built a remote control for this system. By applying a vertical electric field (like a gentle wind blowing up or down through the layers), they could change the rules of the dance floor.

The Trick: They could use this "wind" to lower the cost of parking in the WS2 layer below the cost of parking in the MoSe2 layer.
The Result: Suddenly, the electrons stopped sitting in the "strict" MoSe2 spots and moved to the "relaxed" WS2 spots. This allowed them to completely reorder the energy levels of the system just by turning a knob.

4. New Dance Patterns (Wigner Crystals and Stripes)

Once the electrons moved to the more relaxed WS2 layer, something magical happened. Because they weren't fighting each other as hard (lower "pushiness"), they started organizing themselves into very specific, orderly patterns, even when there wasn't a full parking spot for everyone.

The Patterns: They saw the electrons form Generalized Wigner Crystals (a crystal-like grid of electrons) and Stripe Phases (electrons lining up in rows).
The Analogy: Imagine if the dancers, instead of just filling up the parking spots one by one, decided to hold hands and form a perfect triangle or a straight line across the floor. This only happened because the "pushiness" was low enough to let them coordinate.

Summary

In short, this paper shows that by stacking two specific materials and using an electric field as a "remote control," scientists can:

Measure exactly how much electrons repel each other in different layers.
Force electrons to switch layers.
Create new, complex states of matter (like crystals and stripes) that only exist when the electrons are in the "relaxed" layer.

This gives scientists a powerful new tool to design materials where they can precisely control how electrons interact, simply by adjusting an electric field.

Technical Summary: Layer-tunable Hubbard bands probed via moiré excitons in MoSe2/WS2 heterostructures

Problem Statement
Moiré superlattices in transition metal dichalcogenide (TMD) heterostructures offer a promising platform for engineering strongly correlated quantum states, such as Mott insulators and Wigner crystals. However, a significant experimental challenge remains: the quantitative determination and in-situ tuning of the underlying Hubbard parameters (specifically the on-site Coulomb repulsion, $U$ ) within these systems. While excitonic responses have been shown to track correlated phases, the specific role of layer-dependent Hubbard interactions in governing state formation, and how electric-field-driven reordering of layer-specific Hubbard bands manifests optically, has remained largely unexplored.

Methodology
The authors investigated a dual-gated, near 60°-aligned (H-type) MoSe2/WS2 heterobilayer encapsulated in hexagonal boron nitride (hBN). The device features independent top and bottom graphite gates, allowing for full control over both the carrier density ( $n$ ) and the vertical electric field ( $E$ ).

The study employed differential reflectance spectroscopy at cryogenic temperatures (1.65 K) to probe two spatially distinct moiré excitons:

XM1: A tightly localized intralayer exciton in MoSe2, with both electron and hole localized at the BS/Se moiré site.
XM2: An intralayer charge-transfer exciton in MoSe2, where the electron is at the BS/Se site and the hole is at the BW/Mo site.

These excitons served as local optical probes sensitive to the local charge environment and dielectric screening. The researchers systematically varied the carrier filling factor ( $\nu$ ) and the vertical electric field to map the evolution of these excitonic resonances. The experimental findings were supported by first-principles calculations using Density Functional Theory (DFT) and GW-Bethe-Salpeter Equation (GW-BSE) methods to model the moiré potential and excitonic spectrum.

Key Results

Layer-Specific Hubbard Band Ordering: By tracking the spectral shifts of XM1 and XM2 as a function of doping at zero electric field, the authors established a sequential filling order of the Hubbard bands. The first electron fills the lower Hubbard band of MoSe2 ( $C^{\downarrow}_{MoSe2}$ ), the second fills the lower Hubbard band of WS2 ( $C^{\downarrow}_{WS2}$ ), and the third fills the upper Hubbard band of MoSe2 ( $C^{\uparrow}_{MoSe2}$ ). This ordering was deduced from the abrupt redshift of XM1 (indicating trion formation in MoSe2) followed by its recovery, while XM2 acted as a dielectric sensor reflecting the compressibility of the Fermi sea versus Mott insulating states.
Quantitative Extraction of Hubbard $U$ : By applying a vertical electric field to tune the relative energy alignment of the layers, the authors identified critical electric fields where interlayer charge transfers occurred (e.g., shifting the WS2 lower Hubbard band below the MoSe2 lower Hubbard band). Converting these critical field intervals into energy shifts allowed for the quantitative extraction of layer-dependent on-site Coulomb repulsions:
- $U_{MoSe2} \approx 60$ meV
- $U_{WS2} \approx 30$ meV
  The larger $U$ in MoSe2 indicates stronger spatial confinement at the BS/Se site compared to the AB sites in WS2.
Electric-Field Induced Phase Transitions: Tuning the system to a type-II band alignment (where the WS2 lower Hubbard band becomes the ground state) revealed new correlated phases. In this regime, the reduced on-site repulsion in WS2 allowed inter-site Coulomb interactions to dominate.
Observation of Fractional States: In the WS2-dominated regime, the authors observed pronounced intensity enhancements in the XM1 signal at fractional fillings of 1/3, 1/2, and 2/3. Temperature-dependent measurements confirmed these features correspond to correlated charge-ordered phases:
- 1/3 and 2/3 fillings: Generalized Wigner Crystal (GWC) states.
- 1/2 filling: A stripe phase.
  These fractional states exhibited lower thermal stability (melting temperatures $\sim$ 25–35 K) compared to integer Mott states, consistent with the competition between on-site $U$ and nearest-neighbor interactions ( $V$ ).

Significance and Claims
The paper establishes vertical electric fields as a deterministic tuning knob for layer-selective Hubbard physics. By demonstrating the ability to quantitatively extract layer-specific Hubbard parameters ( $U$ ) and actively reorder the Hubbard band hierarchy, the work provides a direct route to controlling complex many-body phases in moiré materials. The authors claim that their results enable device-level control over the transition between different correlated states (e.g., Mott insulators, Wigner crystals, and stripe phases) by simply adjusting the electric field to favor electron occupation in either the MoSe2 or WS2 layer. This approach offers a new paradigm for engineering strongly interacting states where the relative strength of on-site versus inter-site interactions can be tuned in-situ.

1. The "Spot Check" Cameras (Moiré Excitons)

2. The "Pushiness" of the Dancers (Hubbard Bands)

3. The "Remote Control" (Electric Field)

4. New Dance Patterns (Wigner Crystals and Stripes)

Summary

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