Long-range spatial extension of exciton states in van der Waals heterostructure

This study demonstrates that narrow photoluminescence lines in a MoSe$_2$/WSe$_2$ van der Waals heterostructure correspond to spatially indirect excitons localized in a moiré potential with weak disorder, exhibiting a macroscopic spatial extension of several micrometers that deviates from a random potential landscape.

The Gist

Imagine you are looking at a vast, flat landscape made of two ultra-thin sheets of material (like graphene, but with different properties) stacked on top of each other. In this world, tiny particles called excitons (which are pairs of an electron and a hole, acting like a single unit of light) are running around.

Usually, when these excitons get stuck in a "pothole" or a local dip in the landscape, they stop moving and glow with a very specific, sharp color. In most materials, these potholes are tiny, random, and chaotic—like a field full of randomly scattered pebbles. If an exciton gets stuck in one, it can only wiggle around in a space smaller than a human hair (nanometers).

The Big Discovery:
This paper reports something surprising. The researchers found that in their specific "sandwich" of materials (MoSe2 and WSe2), the excitons aren't getting stuck in tiny, random potholes. Instead, they are getting stuck in giant, organized valleys that stretch for miles (well, several micrometers, which is huge for the atomic world).

Here is the breakdown using simple analogies:

1. The "Moiré" Pattern: The Perfect Grid

When you stack two sheets of paper with a slight twist, you see a new, larger pattern emerge where the lines overlap. This is called a Moiré pattern.

• The Analogy: Imagine two window screens placed over each other at a slight angle. You see a giant, repeating grid of dark and light spots.
• In the Experiment: The researchers twisted their two material sheets just right to create this giant, repeating grid. This grid acts like a perfect "parking lot" for the excitons.

2. The "Narrow Lines": The Singing Voice

When the researchers shined a light on the material and looked at the color of the light coming back (the spectrum), they saw "narrow lines."

• The Analogy: Think of a choir. If everyone sings slightly different notes, you hear a loud, blurry roar (a broad line). But if a few people sing the exact same perfect note, you hear a sharp, clear, piercing tone (a narrow line).
• The Finding: These sharp tones mean the excitons are all sitting in very specific, identical "parking spots" within that giant grid.

3. The "Vanishing Act": The Traffic Jam

The researchers did something clever: they turned up the volume (added more excitons).

• The Analogy: Imagine a quiet library where people are sitting in specific, quiet corners (the narrow lines). As soon as you fill the library with thousands of people, the specific quiet corners disappear, and the whole room becomes a noisy crowd (a broad line).
• The Twist: They noticed that the moment these "quiet corners" (narrow lines) disappeared, the excitons started running across the sample.
• The Conclusion: The narrow lines only exist when the excitons are stuck (localized). Once they get enough energy to move, they stop being stuck in those specific spots, and the sharp lines vanish.

4. The "Giant Stuck" Surprise

Here is the real magic. Usually, when something is "stuck" in a material, it's stuck in a tiny, random spot (like a marble stuck in a crack in a sidewalk).

• The Old Way: In other materials, these "stuck" excitons are only a few nanometers wide. They are like a person stuck in a single room.
• The New Way: In this experiment, the "stuck" excitons were macroscopic. They stretched for several micrometers—covering about 10% of the entire sample!
• The Analogy: Imagine a person getting "stuck" in a valley. In a normal material, the valley is the size of a shoebox. In this material, the valley is the size of a football field, and the person is stuck in the middle of it, unable to leave that specific field, even though the field is huge.

Why Does This Matter?

This tells us that the "landscape" the excitons live on isn't a messy, random junkyard. It's a highly organized, giant grid (the Moiré pattern) with only very slight imperfections (weak disorder).

Because the "parking lots" are so huge and organized, the excitons can travel incredibly far without getting lost. It's like having a highway system where the cars (excitons) can zip along for miles without hitting a single pothole.

In Summary:
The researchers found that by twisting two atomic sheets just right, they created a giant, orderly "parking grid" for light-particles. These particles can get "stuck" in these parking spots, but the spots are so large (football-field sized) that they prove the material is incredibly clean and organized. This discovery opens the door to building super-fast, efficient electronic devices that use light-particles to carry information over long distances without losing energy.

Technical Summary

1. Problem Statement

In low-dimensional semiconductors, photoluminescence (PL) spectra often exhibit narrow emission lines ($\lesssim 1$ meV linewidth). These lines are typically attributed to excitons localized in local minima of a random potential energy landscape caused by fluctuations in the local environment (e.g., interface roughness, strain, or material defects).

• The Limitation: In conventional systems (like GaAs quantum wells) and many van der Waals (vdW) heterostructures, the spatial extension of these localized states is typically limited to the nanometer scale, often below the diffraction limit of optical microscopy ($\sim 1$ $\mu$m).
• The Uncertainty: In transition-metal dichalcogenide (TMD) heterostructures, narrow lines have been observed in various contexts: electrostatic traps, random potentials, strain-induced traps, and moiré superlattices. It has been unclear whether the narrow lines in moiré systems arise from strong disorder (random potential) or from the periodic moiré potential itself. A key distinction is that random potentials typically yield nanometer-scale localization, whereas periodic potentials can support macroscopic extension if disorder is weak.
• The Goal: To determine the spatial extent of exciton states associated with narrow PL lines in a MoSe$_2$/WSe$_2$ heterostructure and to distinguish whether their origin is a random disorder potential or a weakly disordered moiré potential.

2. Methodology

The researchers utilized a MoSe$_2$/WSe$_2$ van der Waals heterostructure with a specific twist angle ($\delta\theta \approx 1.1^\circ$), creating a moiré superlattice with a period of $b \approx 17$ nm.

• Sample Structure: A monolayer of MoSe$_2$ was stacked on a monolayer of WSe$_2$, encapsulated by hexagonal boron nitride (hBN) layers. The twist angle creates spatially indirect excitons (IXs), where electrons and holes are confined in separate layers.
• Optical Spectroscopy:
  • PL Spectra: Measured using a continuous-wave Ti:sapphire laser ($E_{ex} = 1.689$ eV) and a high-resolution spectrometer (0.2 meV resolution).
  • Density Dependence: Excitation power ($P_{ex}$) was varied from 1 $\mu$W to 2000 $\mu$W to study the evolution of narrow lines versus the broad background.
  • Magnetic Field: Circularly polarized excitation and PL were used in magnetic fields up to 8 T to measure the excitonic $g$-factor, identifying the specific atomic registry (stacking site) of the excitons.
• Spatial Mapping:
  • $x$-Energy Maps: The laser spot was defocused to $\sim 25$ $\mu$m to weakly excite the entire sample. The PL signal was collected with a slit integrated over 1.3 $\mu$m in the $y$-direction while scanning in $x$.
  • $x$-$y$ Maps: By scanning the slit position in $y$ and collecting data in $x$, the team constructed 2D spatial maps of specific narrow-line energies.
  • Data Processing: The broad Gaussian background (representing delocalized or ensemble-averaged excitons) was subtracted to isolate the signal from the narrow lines.

3. Key Contributions

1. Discovery of Macroscopic Extension: The paper provides the first experimental evidence that narrow PL lines in TMD heterostructures can correspond to exciton states extending over several micrometers (macroscopic scale), covering up to 10% of the sample area.
2. Correlation with Transport: The study establishes a direct correlation between the disappearance of narrow lines and the onset of long-range exciton transport. As exciton density increases, narrow lines vanish, and transport begins, indicating that the narrow lines represent localized states that block transport until a critical density is reached.
3. Differentiation of Potential Landscapes: By demonstrating that these localized states are macroscopically extended, the authors rule out a strong random disorder potential as the primary cause. Instead, they conclude the excitons are confined in a moiré potential with weak disorder.
4. Site-Specific Identification: The measurement of a consistent $g$-factor ($g \approx -15.5$) across all narrow lines confirms that these states originate from a specific atomic registry ($H_h^h$ site) within the moiré unit cell, further supporting the moiré origin over random strain or defects.

4. Key Results

• Spectral Behavior:
  • At low densities, the PL spectrum shows a broad background with distinct narrow lines ($\lesssim 1$ meV).
  • The energies of these narrow lines remain fixed as density increases, unlike the broad background which shifts monotonically due to mean-field interactions (capacitor effect).
  • At high densities, the narrow lines vanish, and the spectrum is dominated by the broad line.
• Transport Correlation:
  • The relative intensity of narrow lines decreases as the exciton transport decay length ($d_{1/e}$) increases.
  • The disappearance of narrow lines coincides with the onset of long-range transport, confirming that the narrow lines correspond to localized states.
• Spatial Extension:
  • $x$-Energy Maps: Show vertical modulated lines, indicating that specific narrow-line energies persist over long distances along the $x$-axis.
  • $x$-$y$ Maps: Reveal that individual exciton states associated with narrow lines extend over distances of several micrometers and cover areas up to $\sim 10\%$ of the measured sample.
  • Different narrow lines correspond to different spatial regions, indicating that the local environment fluctuates slightly across the sample but remains ordered enough to allow long-range extension.
• $g$-Factor Analysis:
  • All narrow lines exhibit a consistent $g \approx -15.5 \pm 0.7$.
  • This value matches theoretical predictions for the $H_h^h$ stacking site in MoSe$_2$/WSe$_2$ moiré potentials. This implies that the localization is not due to random defects (which would vary the registry) but is tied to the periodic moiré lattice.

5. Significance

• Fundamental Physics: The results challenge the conventional view that narrow PL lines in semiconductors always imply nanometer-scale localization due to strong disorder. They demonstrate that weakly disordered periodic potentials (moiré lattices) can host localized states with macroscopic spatial extension.
• Implications for Superfluidity: The existence of long-range localized states in a weakly disordered moiré potential is a prerequisite for exciton superfluidity. Strong disorder destroys superfluidity, but the weak disorder revealed here suggests a pathway to observing long-range ballistic transport and superfluid phases in TMD heterostructures.
• Device Applications: Understanding the spatial extent and localization mechanisms of excitons in moiré systems is crucial for designing quantum emitters, excitonic circuits, and devices based on correlated exciton phases (e.g., exciton crystals or Bose-Einstein condensates).
• Methodological Advance: The study demonstrates the necessity of high-resolution spatial mapping to distinguish between random disorder and ordered moiré potentials, moving beyond standard optical resolution limits.

In summary, this work identifies a regime where excitons are localized by a moiré potential rather than random disorder, resulting in a unique state of matter where localized excitons possess macroscopic spatial coherence, facilitating efficient transport and opening new avenues for studying quantum many-body phenomena in 2D materials.