2026

Figure 1 | From left to right: a regular array of dots, a slightly rotated array, and the pattern formed by superimposing the two. The pale regions on the right help reveal the periodic changes in local alignment.

In the laboratory, researchers can stack two atomically thin sheets of material. Within each sheet, the atoms form an orderly lattice. If the atomic spacing or orientation differs even slightly between the sheets, the relative positions of atoms in the two layers vary in a pattern that repeats over a much greater distance. This larger repeating structure is called a moiré superlattice.

The way the atoms line up affects an electron’s energy. An electron in a region where the atoms in the two layers align may have a different energy from one in a region where they are offset. Plot these energies as a map, and hills and valleys appear; electrons generally favour the lower-energy regions. This periodically varying energy landscape is the moiré potential. By choosing the materials and how they are stacked, researchers can reshape this landscape and explore a rich variety of physical behaviour. Moiré superlattices have therefore become an important platform for studying the collective behaviour of electrons.

Our paper concerns two atomically thin semiconductors, WSe₂ and WS₂, which form a moiré superlattice when stacked together. Light shining on such a semiconductor can create an important type of excited state: an exciton. When the semiconductor absorbs light of a suitable energy, an electron can move into a higher-energy state, leaving its original state empty. In the crystal, this vacancy behaves like a positively charged particle, known as a hole. The negatively charged electron and positively charged hole attract one another; when they bind together, they form an exciton. What happens to electrons and holes matters for everyday devices. Organic light-emitting diode displays, widely used in phones, rely on electrons and holes meeting to form excitons, which then emit light of different colours. Light also creates excitons in organic solar cells. There, the electron and hole must subsequently separate so that their charges can be collected as an electric current. How tightly an exciton is bound, and how long it survives, both affect the performance of these devices.

For physicists, excitons in atomically thin semiconductors are particularly useful because they leave clear signatures in optical spectra. Researchers vary the energy of the incident light and record the material’s response. Resonance peaks in the resulting curve correspond to quantum states that light can excite. The positions and strengths of these peaks, and how they change with external conditions, all carry information about excitons.

Figure 2 | A conceptual illustration of an exciton. The blue and orange glows represent the electron and hole, respectively, while the curved lines represent the Coulomb attraction between them.

An exciton in a conventional material is somewhat like a hydrogen atom. In hydrogen, an electron is attracted to the atomic nucleus; in an exciton, it is attracted to a hole. Different exciton states can have different energies and spatial shapes. Some are compact, while others spread out over a larger region.

In a moiré superlattice, these spatial distributions become more complex. The electron and hole may experience different energy landscapes: a valley that attracts the electron need not be a favoured location for the hole. At the same time, Coulomb attraction draws them towards one another. The schematic from our paper brings these effects together. The blue and red landscapes belong to the electron and hole, respectively, while the purple connection represents their mutual attraction. Together, these effects determine the shape of the exciton.

Figure 3 | From Fig. 1(b) of the paper. The blue and red landscapes represent the moiré potentials experienced by the electron and hole, respectively; the purple curve represents Coulomb attraction. The three circles below show different local stacking arrangements within a moiré period.

Consider a relatively compact exciton first. The electron and hole are mainly distributed in nearby regions, where their Coulomb attraction binds them strongly. If another valley in the electron’s moiré potential lies a little farther away, an additional possibility comes into play. Staying close to the hole gives the electron a stronger attraction, while moving towards the more distant valley lowers its energy in the moiré potential. Both effects help determine the quantum state, and the exciton’s spatial distribution adjusts accordingly. If the potential well becomes deep enough, an exciton can form in which the electron and hole favour different regions while remaining bound by Coulomb attraction. A state with this separation of charge is called a charge-transfer exciton. This example shows how strongly the moiré potential can influence an exciton’s spatial distribution.

The relative twist angle and lattice mismatch between the layers are usually set when the sample is made, fixing the moiré potential as well. Changing that potential within the same device to control its exciton states is therefore a considerable challenge. Pressure offers another way to tune the system. By gradually increasing the pressure on a single sample while recording its optical spectrum, we can continuously follow the evolution of its excitons. Comparing excitons under different moiré potentials in this way also reduces the complications caused by differences between samples.

In the experiment, the tiny moiré device is placed in a diamond anvil cell, where two opposing diamonds apply pressure to the sample. Because diamond is transparent, light can pass through it, illuminate the sample and be reflected back, carrying information about the sample’s optical response. This allows spectra to be measured while the sample remains under high pressure. Gates on the device also let us adjust the number of electrons or holes and observe how different excitons respond to changes in the surrounding charge.

At low pressure, several clear exciton peaks are already present in the experimental spectrum. One type of exciton has an electron and hole distributed relatively close together, while another has a pronounced charge-transfer character. As the pressure rises, the positions and strengths of these peaks gradually change. A more striking change appears above about 3 GPa: an additional resonance emerges at higher energy, which we label X₄. This new signal reveals a new exciton state in the compressed superlattice that can be distinguished in the optical spectrum.

To understand where it comes from, we first need to consider what pressure does to the atomic layers. The two semiconductor sheets move closer together, strengthening their influence on one another. The atoms also shift slightly, adjusting their local arrangement. A moiré period already contains several different stacking regions, each with a different energy. By locally stretching and contracting, the layers allow energetically favourable arrangements to occupy more space. This adjustment is called lattice reconstruction. Pressure strengthens the reconstruction and changes the local strain in different regions, which in turn affects the energies of electrons and holes.

The calculations show that the reduced layer separation and lattice reconstruction together deepen the moiré potential. Imagine a landscape in which the height difference between hills and valleys steadily increases. The regions favoured by the electron and hole become more attractive, strengthening the competition between the moiré potential and Coulomb binding. Figure 4 shows how the moiré potential changes with pressure.

Figure 4 | From Fig. 3(a, b) of the paper. Left: the calculated moiré potentials for holes and electrons at 0 and 4 GPa. Right: the potential-well depths as a function of pressure. Solid lines are obtained from large-scale first-principles calculations; dashed lines are obtained using a local-stacking approximation.

Getting from the atomic arrangement under pressure to the exciton spectrum presents a computational challenge. A moiré superlattice contains thousands of atoms, and an exciton consists of an electron and a hole whose behaviour is correlated. Describing it requires us to consider where both particles might be found, as well as how changes in their positions affect their attraction. Carrying out a direct first-principles calculation of these excited states across the entire superlattice would demand enormous computational resources. Comparing states at several pressures makes the task still more demanding.

We connect these different scales in stages. First, atomistic simulations determine how the layers reconstruct under pressure. We then perform first-principles calculations on the reconstructed superlattice to obtain the moiré potentials experienced separately by the electron and hole. Starting from the material’s atomic structure and quantum mechanics, this step provides the basis for the exciton calculations that follow. We then incorporate these potentials into a continuum model, describing electrons and holes on a smoother spatial scale. Solving the Bethe–Salpeter equation (BSE) gives the exciton states that they form together.

With this model, the new spectral peak acquires a more concrete explanation. Return to the comparison with hydrogen: an electron and a hole bound by Coulomb attraction can form a series of internal quantum states known as Wannier–Rydberg excitons. As in hydrogen, the most compact, low-energy state of a Wannier exciton is usually labelled 1s. Higher excited states can extend farther in space and have different shapes. The moiré potential wells also support a series of charge-transfer exciton states. As pressure increases, states from these two different series can mix, giving rise to the new exciton peak.

The Wannier–Rydberg state involved in this mixing is the 3d state. In the two-dimensional system considered in our paper, fixing the hole’s position reveals a ring-shaped probability distribution for the electron. Its spatial extent is also close to the typical electron–hole separation in a charge-transfer state. The ring therefore reaches the regions favoured by the electron in the charge-transfer state, providing substantial spatial overlap between the two states.

Pressure further changes their relative energies. As the moiré potential deepens, the charge-transfer state experiences a greater drop in energy, bringing it closer to the 3d state. This convergence in energy, together with the spatial overlap of their wavefunctions, allows the two states to mix strongly.

Figure 5 | From Fig. 3(c–f) of the paper. The first three panels illustrate how exciton states change as the moiré potential strengthens: the charge-transfer (CT) state and the 3d state approach one another in energy and mix. The rightmost panel shows the calculated electron distribution of the 3d state without the moiré potential, with the locations of CT states marked to illustrate spatial overlap between the two states.