Tunable Interlayer Charge-transfer States in MoSe$_2$/WS$_2$ Moiré Superlattices

This study combines first-principles calculations and optical spectroscopy to demonstrate how tuning the vertical electric field in electron-doped MoSe$_2$/WS$_2$ moiré superlattices switches the band alignment from Type-I to Type-II, enabling precise control over interlayer charge-transfer states and the realization of a tunable Fermi-Hubbard model with predicted correlated charge-ordered states at integer and fractional fillings.

The Gist

Imagine you have two very thin, magical sheets of paper made of special materials (MoSe₂ and WS₂). When you stack them on top of each other and twist them slightly, they don't just sit flat; they create a giant, repeating pattern of hills and valleys, much like the ripples you see when you overlap two fishing nets. Scientists call this a "Moiré superlattice."

This paper is about what happens when you put extra electrons (tiny negative charges) into this pattern and then push them around with an electric field. Here is the story of what the researchers found, explained simply:

1. The Playground: A Lattice of Hills and Valleys

Think of the Moiré pattern as a giant honeycomb playground. In this playground, there are two main types of "seats" where electrons can sit:

• The "M" Seats: Located in the top layer (MoSe₂).
• The "W" Seats: Located in the bottom layer (WS₂).

Normally, without any outside help, all the electrons prefer to sit in the "M" seats because they are more comfortable there.

2. The Magic Switch: The Electric Field

The researchers built a device that acts like a dimmer switch for an electric field. By turning this switch up or down, they could change the "comfort level" of the seats.

• Low Switch: The "M" seats are still the most comfortable.
• High Switch: The "W" seats become just as comfortable, or even more comfortable, than the "M" seats.

3. The Dance of Electrons (Charge Transfer)

The researchers added electrons one by one into this playground and watched how they moved. They used a special "flashlight" (optical spectroscopy) that glows differently depending on where the electrons are sitting.

• The First Electron: It happily sits in an "M" seat.
• The Second Electron: This is where it gets interesting.
  • If the electric switch is low, the second electron is forced to sit in the same "M" seat as the first one. They pair up tightly (like two people huddling in a small chair), which stops the "flashlight" from glowing in a specific way.
  • If the electric switch is high, the second electron decides, "That seat is full; I'll go sit in a 'W' seat in the bottom layer instead!" This is called interlayer charge transfer. The electron literally jumps from the top layer to the bottom layer.

4. The "Trion" and the "Exciton" (The Glowing Clues)

To see where the electrons were, the scientists looked for two types of glowing signals:

• The "Trion" (LET): This is like a glowing trio: an electron, a "hole" (a missing electron), and an extra electron. The researchers found that this glow only appears when an electron is sitting in an "M" seat. If the electron jumps to a "W" seat, this glow disappears.
• The "Exciton" (EX): This is a different kind of glow that appears when the "M" seats are completely full (two electrons in every "M" seat).

By watching these glows turn on and off, the scientists could map exactly where every electron was sitting. They discovered they could precisely control the electrons, making them jump between the top and bottom layers just by turning a knob.

5. The Crowd Dynamics (Correlated States)

When they added even more electrons (filling the playground to 1.5 or 2 times its capacity), the electrons started behaving like a crowd at a concert. They didn't just sit randomly; they organized themselves into specific patterns to avoid bumping into each other (due to their natural repulsion).

• At certain filling levels, the electrons formed a "stripe" pattern.
• At other levels, they formed a perfect checkerboard.

The researchers used computer simulations to show that these patterns are caused by the electrons pushing against each other, creating a "correlated" state where the whole group moves in sync.

Summary

In short, this paper shows that by stacking two layers of 2D material and twisting them, scientists created a controllable playground. They proved they could use an electric field to force electrons to jump between layers, effectively building a switchable "honeycomb" or "triangular" lattice. This allows them to create and study complex quantum states where electrons organize themselves in fascinating, predictable patterns, all observed through the unique way the material glows under a light.

Technical Summary

Technical Summary: Tunable Interlayer Charge-transfer States in MoSe$_2$/WS$_2$ Moiré Superlattices

Problem Statement
Moiré superlattices formed by transition metal dichalcogenide (TMD) heterobilayers, specifically MoSe$_2$/WS$_2$, offer a versatile platform for investigating strongly correlated electronic, excitonic, and topological phenomena. While previous studies have established the existence of correlated spin and charge physics in these systems—such as incompressible Wigner-Mott states and kinetic magnetism near Mott insulator states—the nature of multiple emergent optical transitions in the electron-doped regime remained unclear. Specifically, the strong dependence of these optical transitions on vertical electric fields and the underlying mechanisms of electron localization and interlayer charge transfer required a comprehensive investigation.

Methodology
The study employs a combined approach of large-scale first-principles calculations and optical reflection spectroscopy on angle-aligned (both 60° H-type and 0° R-type) MoSe$_2$/WS$_2$ heterobilayers.

• Device Fabrication: The researchers fabricated dual-gate devices where monolayer MoSe$_2$ and WS$_2$ are encapsulated by hexagonal boron nitride (hBN). Few-layer graphite serves as the top gate, while gold or platinum acts as the bottom gate, enabling independent control of electron doping ($n$) and vertical electric field ($E$).
• Optical Spectroscopy: Reflection contrast (RC) spectroscopy and magnetic circular dichroism (MCD) measurements were performed at cryogenic temperatures (down to 2 K) across a wide range of electron doping ($n/n_0 = 0$–$4$) and vertical electric fields.
• Theoretical Modeling:
  • First-Principles Calculations: Structural relaxation, electronic band structures, and optical absorption spectra were computed using Density Functional Theory (DFT) and many-body perturbation theory (GW-Bethe-Salpeter equation). To handle the large moiré superlattice size, the Pristine Unit-Cell Matrix Projection (PUMP) approach was utilized to construct the BSE kernel.
  • Monte Carlo Simulations: A classical Monte Carlo simulation of a two-site model (M and W sites) on an effective honeycomb lattice was performed. This model included long-range screened Coulomb interactions to analyze the doping dependence of electric-field susceptibility and predict charge-ordered states.

Key Contributions and Results

1. Identification of Emergent Optical Probes:
   The study identifies and characterizes specific moiré excitonic states that serve as sensitive optical probes for electron localization:

   • Low-Energy Moiré Trion (LET): A bound state of a doped electron and an intralayer exciton, localized at the MoSe$_2$ M site (BSe/S stacking in H-type, AA stacking in R-type).
   • Emergent Moiré Exciton (EX): A higher-energy excitonic state that emerges when the M site is fully occupied (two electrons), corresponding to a fully filled $c_1$ band.
   • High-Energy Moiré Trion (HET): A higher-energy trion-like state associated with the repulsive polaron, which tracks the LET state's behavior.

2. Electric-Field Tunable Interlayer Charge Transfer:
   The researchers demonstrate precise control over interlayer electron localization by tuning the vertical electric field, effectively switching the system between Type-I and Type-II band alignments.

   • H-type (60°) Superlattices: At low doping ($n/n_0 < 1$), electrons localize at the M site. As doping increases to $1 < n/n_0 < 3$, the second electron can be switched between the M site (MoSe$_2$) and the W site (WS$_2$, AB stacking) by varying the electric field. This creates a tunable honeycomb lattice for the second electron.
   • R-type (0°) Superlattices: Similar charge-transfer behaviors are observed but with different critical electric fields. The study maps transitions for $n/n_0 = 2, 3,$ and $4$, showing that electrons sequentially transfer to the WS$_2$ layer (W site) as the electric field increases, leaving the MoSe$_2$ M site vacant for subsequent electrons.

3. Correlated Charge-Ordered States:
   By analyzing the doping dependence of the electric-field susceptibility ($\alpha$) derived from the intensity evolution of LET and EX peaks, the study reveals non-monotonic behavior inconsistent with non-interacting models.

   • Monte Carlo Simulations: These simulations predict the emergence of correlated charge-ordered states at integer and fractional fillings.
   • Specific Orders: A polarized charge-ordered state is identified at $n/n_0 = 2$ (half-filling of the effective honeycomb lattice). Evidence for a stripe-like charge pattern breaking $C_3$ rotational symmetry is found near $n/n_0 = 3/2$, and a charge-ordered state at $n/n_0 = 4/3$ is also predicted.

4. Realization of a Tunable Fermi-Hubbard Model:
   The ability to switch electron localization between M and W sites allows the system to realize a Fermi-Hubbard model on an effective honeycomb lattice with a tunable charge-transfer band. The competition between the conduction band offset ($\Delta$) and onsite Coulomb repulsion ($U$) dictates the critical electric fields for these transitions.

Significance
The paper provides a holistic understanding of emergent optical excitations and correlated charge-transfer states in electron-doped MoSe$_2$/WS$_2$ moiré superlattices. By establishing moiré trions and emergent excitons as reliable optical probes, the work enables the precise mapping of electron localization profiles. The findings demonstrate that these systems can host tunable lattice geometries (triangular to honeycomb) and correlated insulating phases, including potential ferroelectric and Wigner-molecule states. This work lays the groundwork for future experiments utilizing moiré excitations to explore electron localization and correlated physics in TMD moiré superlattices.