🔬 materials science

Disorder-induced symmetry breaking in moiré bands of marginally twisted bilayer MoS $_2$

Using scanning tunneling spectroscopy and continuum model calculations, this study reveals that electrostatic disorder and structural relaxation are critical in breaking symmetry and shaping the electronic structure of marginally twisted bilayer MoS $_2$ , explaining unexpected energy differences between stacking regions.

Original authors: Pablo Reséndiz-Vázquez, Christophe de Beule, Thi-Hai-Yen Vu, Kaijian Xing, Daniel McEwen, Daniel Bennett, Liangtao Peng, Héctor González-Herrero, Shaffique Adam, Mark T. Edmonds, Michael S. Fuhrer

Published 2026-02-09

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Pablo Reséndiz-Vázquez, Christophe de Beule, Thi-Hai-Yen Vu, Kaijian Xing, Daniel McEwen, Daniel Bennett, Liangtao Peng, Héctor González-Herrero, Shaffique Adam, Mark T. Edmonds, Michael S. Fuhrer

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 special, ultra-thin fabric called molybdenum disulfide (MoS₂). In a perfect world, if you stack these two sheets perfectly on top of each other, they create a smooth, uniform surface for electrons (the tiny particles that carry electricity) to travel through.

But in the real world, things are rarely perfect. In this study, scientists took two sheets of this fabric and twisted them slightly—just a tiny fraction of a degree (about 0.95 degrees). When you twist two honeycomb-patterned sheets like this, they create a giant, new pattern called a Moiré pattern. Think of it like holding two window screens slightly out of alignment; you see a new, larger pattern of light and dark spots appear where the wires overlap.

The "Flat" Highway and the "Bumpy" Road

Usually, when scientists create these twisted patterns, they are looking for "flat bands." Imagine an electron highway that is perfectly flat. On a flat road, cars (electrons) move slowly and can interact with each other in very interesting, complex ways, potentially leading to new states of matter like superconductivity (electricity flowing with zero resistance).

However, this paper asks a crucial question: What happens when the road isn't perfectly smooth?

The researchers discovered that even though the twist angle was set up to be symmetrical (meaning the left side should look exactly like the right side), the electrons were behaving differently on one side compared to the other.

The Mystery of the "Ghost" Charge

Using a super-powerful microscope called a Scanning Tunneling Microscope (STM), which acts like a blind person's cane feeling the surface of the material atom by atom, the team found something strange.

They measured the energy required for electrons to jump onto the "road" (the conduction band) or fall off it (the valence band). They expected the energy to be the same in two specific types of overlapping areas (called MX and XM regions) because the setup was symmetrical.

But it wasn't. There was a difference of about 15 "electron-volts" (a tiny unit of energy). It was as if the road was 15 steps higher on one side of the street than the other, even though the street was supposed to be flat.

The Culprit: Invisible "Potholes"

The paper concludes that this unevenness wasn't caused by the twist itself, but by disorder—specifically, invisible "potholes" in the fabric.

In these materials, atoms are sometimes missing. The most common missing piece is a sulfur atom, leaving behind a tiny hole. These holes act like tiny magnets that trap electric charge. The researchers realized that these missing atoms were scattered randomly throughout the material, creating a messy, invisible electric field.

They built a mathematical model to prove this. They treated the missing atoms like random static charges scattered on a table. When they calculated how these charges would affect the electrons, the math matched their microscope pictures perfectly. It turns out that even a relatively small number of these missing atoms (about 100 billion per square centimeter) is enough to break the symmetry and tilt the energy landscape.

The "Magic" of the Electric Field

To confirm this, the scientists simulated what would happen if they applied a tiny, controlled electric field to the material—mimicking the effect of those random missing atoms.

Before the field: The electron paths were balanced and symmetrical, like a calm lake.
With the field: The lake tilted. The "flat" bands split apart. One side became lower in energy, and the other became higher.

This simulation matched their real-world measurements exactly. The "tilt" caused by the disorder was strong enough to completely change the character of the electron paths, turning a balanced, symmetrical system into a lopsided one.

The Takeaway

The main lesson from this paper is simple but profound: In the microscopic world of twisted materials, "messiness" matters.

You cannot assume that just because you twist two sheets of material perfectly, the result will be perfect. Tiny, random defects (like missing atoms) act like invisible hands that push and pull on the electrons, breaking the symmetry and changing how electricity flows. If scientists want to build future quantum devices using these materials, they have to account for these tiny imperfections, because they are powerful enough to rewrite the rules of the game.

Problem Statement
Twisted transition-metal dichalcogenides (tTMDs) host highly tunable moiré potentials and flat bands that support strongly correlated electronic phases. While theoretical models predict rich emergent quantum phenomena in these systems, the specific role of intrinsic disorder in shaping these properties remains largely unresolved. Unlike twisted bilayer graphene, which requires a specific "magic angle" for flat bands, small-angle tTMDs exhibit continuously decreasing bandwidths near parallel (0°) and antiparallel (60°) stacking, offering a broad platform for investigating quantum phases. However, intrinsic disorder—such as point defects and charge puddles—is an unavoidable feature of two-dimensional materials. The microscopic nature of disorder in tTMDs and its impact on the electronic structure of moiré superlattices, particularly regarding symmetry breaking, has received limited attention.

Methodology
The authors investigated marginally twisted bilayer MoS $_2$ (tb-MoS $_2$ ) with a twist angle of $\theta \approx 0.95^\circ$ using scanning tunneling microscopy (STM) and spectroscopy (STS).

Experimental: Samples were fabricated using a tear-and-stack technique to create bilayers on a graphite/SiO $_2$ substrate. STM topography was used to characterize the moiré superlattice, while STS was employed to map the local density of states (LDOS). The researchers measured differential conductance ($dI/dV$) spectra at high-symmetry stacking regions (MM, MX, XM, and domain walls) and generated spatial "onset maps" to visualize the energy at which the LDOS rises above a noise floor for both valence and conduction bands.
Theoretical: The study utilized a continuum model for twisted MoS $_2$ that incorporates atomic relaxation effects derived from first-principles density functional theory (DFT) calculations. To model the experimental observations, the authors developed an electrostatic disorder model treating the system as alternating dielectric layers over a metallic graphite substrate. This model accounted for randomly distributed charged defects (specifically sulfur vacancies) to calculate the resulting electrostatic potential and its effect on the moiré bands.

Key Results

Symmetry-Breaking Onset Shifts: STS measurements revealed an unexpected splitting in the onset energies of the valence and conduction bands between mirror-symmetric MX and XM stacking regions. Specifically, a difference of $15 \pm 4$ meV was observed for the valence band and $15 \pm 6$ meV for the conduction band. In a pristine, symmetric system, these regions should exhibit identical onset energies.
Spatially Correlated Disorder: The onset maps displayed spatially correlated variations in energy on length scales exceeding the moiré period. Autocorrelation analysis of these maps showed a radial dependence consistent with a $1/r^3$ potential, characteristic of dipoles formed by charged defects and their image charges in the substrate.
Defect Density Estimation: By fitting the autocorrelation data to the electrostatic disorder model, the authors inferred a background random charge density of approximately $1 \times 10^{12}$ cm $^{-2}$ to $3 \times 10^{11}$ cm $^{-2}$ . This range aligns with reported densities of sulfur vacancies in mechanically exfoliated MoS $_2$ monolayers.
Theoretical Agreement: Continuum model calculations incorporating an interlayer electric displacement field (simulating the potential from the inferred disorder) reproduced the experimental results. The calculations showed that the electric field induces layer polarization in the moiré bands, causing a splitting of the onset energies between MX and XM sites. The calculated local density of states, when broadened to account for disorder and thermal effects, matched the experimental onset maps and spectral shifts quantitatively.

Significance and Claims
The paper claims that even modest levels of electrostatic disorder, inherent to exfoliated TMDs, play a critical role in determining the electronic structure of moiré materials at the nanoscale. The primary significance of these findings is the demonstration that disorder can induce symmetry breaking in the moiré bands, transforming the character of the electronic states. Specifically, the topmost valence bands, which are theoretically predicted to be unpolarized and graphene-like in a pristine system, are shown to become completely layer-polarized and widely separated in energy due to the presence of charged defects.

The authors conclude that these disorder-induced effects occur at an energy scale of tens of meV, which is comparable to the bandwidths of the flat bands themselves. Consequently, the paper asserts that care must be taken when interpreting experimental results in tTMDs, as native disorder may fundamentally alter the realization of specific flat band topologies and correlated phases. The work highlights that structural relaxation and electrostatic disorder must be simultaneously accounted for to achieve quantitative agreement between theory and experiment in these systems.

The "Flat" Highway and the "Bumpy" Road

The Mystery of the "Ghost" Charge

The Culprit: Invisible "Potholes"

The "Magic" of the Electric Field

The Takeaway

More like this