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Dark exciton signatures in the infrared transient absorption of MoS $_2$ monolayer

This paper presents a fully ab initio scheme based on $GW$ +BSE to compute exciton-exciton transient absorption spectra, demonstrating that this technique can detect various dark exciton populations in monolayer MoS $_2$ by revealing complex transition signatures arising from multiple valleys and spin states that differ significantly from standard $\mathbf{q}=\Gamma$ interpretations.

Original authors: Tian-Xiang Qian, Marco D'Alessandro, Claudio Attaccalite, Tian-Yi Cai, Sheng Ju, Davide Sangalli

Published 2026-09-01

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CC BY 4.0

Original authors: Tian-Xiang Qian, Marco D'Alessandro, Claudio Attaccalite, Tian-Yi Cai, Sheng Ju, Davide Sangalli

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

In the microscopic world of two-dimensional materials, light does not just bounce off surfaces; it creates fleeting, bound pairs of electrons and holes that behave like tiny, self-contained atoms. These pairs, known as excitons, are the primary actors in how these ultra-thin semiconductors interact with light. While some of these excitons are "bright," meaning they can easily absorb or emit light and be seen by standard instruments, a vast population of "dark" excitons remains hidden. These dark variants are forbidden from interacting with light directly due to specific quantum rules regarding their spin or momentum, making them invisible to traditional optical measurements. Yet, these invisible particles play a crucial role in the energy flow and speed of future electronic devices. Understanding them is essential for designing materials that can operate at the incredibly fast speeds required for next-generation computing, but their elusive nature has long made them a blind spot in the field.

A team of researchers has now developed a new way to see these hidden particles by listening to the whispers they leave behind when they are already excited. Instead of trying to create an exciton with a laser pulse and watch it appear, the scientists focused on what happens after a pulse has already created a crowd of them. In this scenario, a second, weaker laser pulse is used to probe the existing excitons, nudging them from one energy state to another. This technique, known as transient absorption, allows the researchers to detect transitions between excitonic states, including those involving the dark, invisible varieties. By applying this method to a single layer of molybdenum disulfide, a material widely studied for its potential in optoelectronics, the team constructed a detailed theoretical map of how these particles behave. Their work reveals that the signals observed in experiments are not simple snapshots of a single type of transition, but rather a complex chorus of interactions occurring across different regions of the material's internal structure.

The researchers built a sophisticated computer model based on the fundamental laws of quantum mechanics to simulate these interactions from the ground up. They calculated how excitons move and interact within the material, paying close attention to the fact that these particles can exist with different amounts of momentum, a property that determines how they travel through the crystal lattice. In their simulation, they accounted for the fact that after an initial laser pulse, the excitons settle into a thermal distribution, meaning they spread out across various energy levels and locations within the material's structure. The model showed that the resulting signal is a sum of contributions from four distinct valleys in the material's energy landscape, named after the Greek letters Gamma, K, M, and Q. While previous interpretations of similar experiments assumed that the observed signals came solely from excitons at a single, specific point, this new analysis demonstrates that the signal is actually a blend of activity from all four of these valleys.

When the researchers compared their simulated results with existing experimental data, they found a striking agreement in the overall shape and position of the spectral peaks, provided they adjusted for a known difference in energy caused by the material's environment. The simulations revealed that the main features in the signal arise from a superposition of transitions. At lower energies, the prominent peak is formed by a combination of transitions from both bright and dark excitons located at the Gamma and K valleys, with smaller contributions from the M and Q valleys. The researchers found that the intensity of the signal from each valley is not determined by how strongly that valley interacts with light, but rather by how many excitons are actually present there. Because the dark excitons at the Gamma and K valleys are more numerous in the thermal distribution, they dominate the signal, even though their individual transition strengths are similar to those in other valleys.

The study also clarified the specific nature of the transitions responsible for the observed peaks. The researchers identified that the main signals correspond to excitons jumping from their lowest energy state to higher, excited states, much like an electron moving to a higher rung on a ladder. Specifically, the primary peaks were traced to transitions where an exciton moves from a 1s state to a 2p state, and from a 1s state to a 3p state. Crucially, the analysis showed that these transitions involve both spin-conserving excitons and spin-flipping dark excitons. The researchers found that the signal is not just a simple jump from a bright state to a dark one, but a complex mixing of transitions where the spin of the electron changes or stays the same, depending on the specific path taken. This level of detail was previously inaccessible because standard models treated the material as if all excitons were located at a single point with zero momentum.

One of the most significant findings of this work is the correction of a long-held assumption about how these signals should be interpreted. Earlier studies had attributed the features in the transient absorption spectra to transitions occurring only at the Gamma point, where the exciton momentum is zero. However, the new simulations demonstrate that this view is incomplete. The signal is actually a weighted average of transitions occurring across a wide range of momenta within the different valleys. For instance, the peaks observed at higher energies, which were previously thought to be simple transitions at the Gamma point, are shown to be dominated by transitions from dark excitons at the Gamma and K valleys, with the specific energy of the peak shifting slightly depending on the exact momentum of the exciton involved. This distinction is vital because it changes the understanding of the material's internal dynamics, showing that the "dark" particles are not just passive bystanders but active participants in the optical response.

The researchers also noted that the relative strength of the signals from different valleys is dictated by the temperature of the exciton population. At lower temperatures, the excitons are concentrated in the lowest energy states, leading to a sharp, intense signal. As the temperature rises, the excitons spread out to higher energy states and different valleys, causing the signal to broaden and lose intensity. This temperature dependence was successfully reproduced in the simulations, further validating the model's accuracy. The work confirms that the transient absorption technique is a powerful tool for probing both bright and dark excitons, but it also highlights that interpreting the data requires a comprehensive model that accounts for the full complexity of the material's electronic structure. Without such a model, the rich information encoded in the signal remains obscured, and the true nature of the dark excitons remains hidden.

By providing a fully theoretical framework that matches experimental observations, this study offers a new lens through which to view the behavior of two-dimensional semiconductors. It demonstrates that the optical properties of these materials are not just a reflection of their most visible components, but are deeply influenced by the vast, hidden population of dark excitons. The ability to disentangle these contributions opens the door to more precise control over the flow of energy in future devices. The researchers conclude that while the experimental signals are rich and complex, they can be decoded with the right theoretical tools, revealing a landscape of interactions that was previously invisible. This approach not only explains the current data for molybdenum disulfide but also sets a precedent for understanding similar materials, ensuring that the design of future ultra-fast optical devices is based on a complete picture of the quantum world at play.

Technical Summary: Dark Exciton Signatures in the Infrared Transient Absorption of MoS2 Monolayer

Problem Statement
Dark excitons play a central role in the non-equilibrium dynamics of two-dimensional (2D) semiconductors, particularly transition-metal dichalcogenides (TMDCs) like MoS2. However, characterizing these states remains challenging. While time-resolved angle-resolved photoemission spectroscopy (TR-ARPES) offers a direct measurement, it is limited by complex apparatus requirements, low signal-to-noise ratios in 2D materials, and resolution constraints. Optical measurements are more accessible but traditionally struggle to detect dark excitons due to optical selection rules (symmetry constraints, spin conservation, and zero momentum transfer). Previous interpretations of inter-exciton transient absorption (exc-tr-abs) experiments in TMDCs have largely relied on simplified models assuming transitions occur solely at the $\Gamma$ point ( $q=0$ ) between bright $1s$ and $p$ -like excitons. This approach overlooks the rich landscape of dark excitons, including symmetry-forbidden, spin-forbidden, and finite-momentum ( $q \neq 0$ ) states, which become populated after the initial laser excitation and subsequent relaxation.

Methodology
The authors develop a fully ab initio scheme based on the GW approximation plus the Bethe-Salpeter Equation (GW+BSE) to model exc-tr-abs spectra from arbitrary, populated exciton distributions. Key methodological advancements include:

Inclusion of Finite- $q$ Excitons: The framework explicitly accounts for excitonic states with finite center-of-mass momentum, moving beyond the $q=0$ approximation.
Locally Smooth Gauge Formulation: To accurately compute exciton-exciton transition dipoles ( $\mu_{\lambda \lambda'}$ ) in the length gauge, the authors employ a locally smooth gauge construction based on singular value decomposition (SVD). This approach ensures consistent phases for wave functions across neighboring $k$ -points, allowing for the reliable evaluation of derivatives required for intra-band contributions and Berry connections.
Thermal Population Weighting: The total response function is calculated by summing over all populated initial excitonic states, weighted by a Boltzmann distribution ( $N_{\lambda i}(q)$ ) that reflects the thermal relaxation of the system after the pump pulse.
Valley Decomposition: The signal is decomposed into contributions from four distinct valleys in the exciton dispersion: $\Gamma$ , $K$ , $M$ , and $Q$ .

Key Contributions

Theoretical Framework: The work provides a rigorous numerical method to compute exc-tr-abs spectra that captures the interplay between bright and dark excitons, including spin-flip and finite-momentum states, within a single consistent gauge.
Reinterpretation of Experimental Data: The study challenges the prevailing interpretation of exc-tr-abs signals in MoS2, which previously attributed spectral features solely to transitions at the $\Gamma$ point.
Valley-Resolved Dynamics: The authors demonstrate that the observed transient absorption signal is a superposition of contributions from multiple valleys ( $\Gamma, K, M, Q$ ), with the relative weight of each valley dictated primarily by the thermal population of excitonic states rather than intrinsic dipole intensity differences.

Results
Applied to monolayer MoS2, the calculations yield the following findings:

Temperature Dependence: The exc-tr-abs spectrum exhibits strong temperature dependence. At low temperatures (77 K), the signal is dominated by a sharp peak around 0.22 eV, corresponding to transitions from the lowest-energy excitonic states. As temperature increases (300 K, 1000 K), the peak broadens and shifts, reflecting the thermal occupation of higher-energy and finite- $q$ excitonic states across different valleys.
Valley Contributions: The signal arises from a sum of contributions from the $\Gamma$ , $K$ , $M$ , and $Q$ valleys. While the intrinsic exciton-exciton dipole intensities are comparable across these valleys, the $\Gamma$ and $K$ valleys dominate the experimental signal at 300 K due to their higher thermal population.
Transition Identification:
- The main low-energy peak (~0.22 eV) is identified as a superposition of $1s \to 2p$ transitions. Specifically, it results from a complex interplay of spin-conserving ( $A_{1s} \to A_{2p}$ ) and spin-flip ( $A^*_{1s} \to A^*_{2p}$ ) transitions occurring at both $\Gamma$ and $K$ valleys, alongside contributions from $M$ and $Q$ valleys.
- Higher-energy features (~0.31–0.40 eV), previously attributed to $1s \to 3p$ transitions at $q=\Gamma$ , are reinterpreted in this work. The calculations show these peaks are predominantly driven by $A^*_{1s} \to A^*_{3p}$ transitions (spin-flip) at finite $q$ points near the $\Gamma$ and $K$ valleys.
Agreement with Experiment: After a minor energy shift correction (46 meV) to account for substrate screening effects not present in the free-standing calculation, the theoretical spectra show excellent agreement with experimental pump-probe data regarding the positions and relative intensities of the main spectral features.

Significance
The paper claims that exc-tr-abs spectroscopy serves as a sensitive probe for the dynamics of both dark and bright excitons. However, the authors emphasize that the interpretation of these rich experimental signals is non-trivial. The study demonstrates that a comprehensive understanding requires accurate theoretical modeling that goes beyond single-valley, zero-momentum approximations. By disentangling the contributions from different valleys and excitonic symmetries, the proposed ab initio scheme provides a necessary tool to guide the interpretation of future ultrafast optical experiments in 2D semiconductors.

Technical Summary: Dark Exciton Signatures in the Infrared Transient Absorption of MoS2 Monolayer

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