🔬 mesoscale physics

Universal Thickness-Dependent Absorption in Solids at the Nanoscale: Anomalous Enhancement in the Ultrathin Limit

Through systematic studies of 2D semiconductors, this paper reveals a universal, non-monotonic, and anomalous thickness-dependent absorption behavior in solids at the nanoscale that deviates significantly from the Beer-Lambert law due to electromagnetic interference effects, with implications for a wide range of materials and optoelectronic applications.

Original authors: Bhumika Chauhan, Nikhil Singh, Subhrajit Dalai, Abhisek Saidarsan, Sayantan Patra, Sourabh Jain, Aparna Deshpande, Ashish Arora

Published 2026-07-14

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CC BY 4.0

Original authors: Bhumika Chauhan, Nikhil Singh, Subhrajit Dalai, Abhisek Saidarsan, Sayantan Patra, Sourabh Jain, Aparna Deshpande, Ashish Arora

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 a stack of ultra-thin, magical sheets of material—so thin they are measured in billionths of a meter. If you were a classic physicist following the old rulebook (called the Beer-Lambert law), you'd expect these sheets to act like a simple sponge: the thicker the sponge, the more water (or in this case, light) it soaks up. You'd think if you double the thickness, you double the absorption. It's a straight line, a monotonic climb.

But when the researchers at IISER Pune looked closely at these nanoscale sheets, they found the universe was playing a trick on them. Instead of a steady climb, the absorption of light danced a chaotic, wiggly dance.

The Great Light Dance
The team studied two specific types of "magic sheets" called MoSe2 and WS2. They peeled them off crystals to make layers ranging from a single atom thick (about 0.65 nm) all the way up to 200 nm. As they measured how much light these layers swallowed, they didn't see a smooth ramp. Instead, they saw a rollercoaster.

As the layers got thicker, the total light absorption didn't just go up; it jumped up, then flattened out, then dipped down, then jumped again. It was a non-monotonic, oscillatory behavior. In the thinnest layers—specifically those under 10 nm thick—the deviation from the old "sponge rule" was massive. For WS2, the absorption was off by more than 50% compared to what the old law predicted. For MoSe2, the difference was a staggering 150%.

Why the Old Rulebook Failed
Why did the sponge analogy fail? The paper argues that these thin crystals aren't just soaking up light; they are conducting a complex light show involving interference.

Think of the light waves hitting the crystal like a surfer hitting a wave. When the light hits the top surface of the crystal, some bounces off. But some goes through, hits the bottom surface, bounces back up, and then hits the top surface again. These waves are like two surfers trying to ride the same wave. Sometimes they crash into each other and cancel out (destructive interference), and sometimes they boost each other up (constructive interference).

Because the crystal is so thin, these "bounces" happen almost instantly, creating a standing wave effect between the top and bottom surfaces. This electromagnetic interference causes the absorption to spike and dip depending on the exact thickness of the sheet. It's not about the material being "greedier" for light; it's about the light waves arguing with themselves inside the tiny slab.

What It's NOT
The researchers were very careful to rule out some popular theories. They explicitly stated that this wild behavior is not caused by "super-radiant coupling" of excitons (a fancy way of saying excited electrons teaming up to glow brighter). They also showed that the effect doesn't depend on specific resonances like plasmons or excitons being present. Even if you mathematically remove the exciton features from their calculations, the wiggly, oscillating absorption pattern remains. The cause is purely the interference of light waves bouncing between the surfaces.

Is This Just a Fluke?
The team didn't stop at just two materials. They used a powerful computer model called the Generalized Transfer-Matrix (GTM) method to simulate what would happen with other solids. They ran the numbers for conventional semiconductors like Silicon (Si) and Gallium Arsenide (GaAs), 2D magnetic materials like CrSBr, and even metals like Gold (Au) and Silver (Ag).

The result? The wiggly, oscillating behavior showed up for all of them. Whether it's a semiconductor, a metal, or a magnetic material, if you make it thin enough (in the sub-wavelength regime), the light absorption will dance to the tune of interference, not the old sponge rule.

The Bottom Line
The paper demonstrates that for solids at the nanoscale, the relationship between thickness and light absorption is universal but wildly non-intuitive. It is a measured reality in MoSe2 and WS2, and a robust simulation for a wide variety of other materials. This discovery is vital for anyone trying to build the next generation of flexible, ultra-thin solar cells or light detectors. If you design these devices assuming the old "thicker is always better" rule, you might miss the sweet spot where the light absorption actually peaks. The light isn't just being absorbed; it's being choreographed.

Problem Statement
The optical absorption of thin solid films is traditionally expected to follow the Beer-Lambert law, implying a monotonic increase in absorption as layer thickness increases. While this intuition holds for many bulk materials and specific previous studies on solids like silicon with silica nanoparticles, $V_2O_5$ , $SiO_2$ , $MgF_2$ , and $TiN$, a comprehensive understanding of thickness-dependent absorption in the sub-wavelength regime (nanoscale) has been lacking. Specifically, it was unclear whether the strong interference effects inherent to ultrathin films lead to deviations from this law, particularly in the context of emerging low-dimensional materials like van der Waals (vdW) semiconductors. Previous observations of oscillatory absorption in specific excitonic lines (e.g., in $WSe_2$ and $MoSe_2$ ) were attributed to super-radiant coupling of excitonic emitters, but a general, universal behavior across the spectral range and across different material classes remained unexplored.

Methodology
The authors employed a combined experimental and theoretical approach to investigate layer-thickness-dependent absorption in semiconducting $MoSe_2$ and $WS_2$ across the visible to near-infrared spectral range.

Experimental: Thin crystals of $MoSe_2$ and $WS_2$ were prepared via standard exfoliation and viscoelastic dry-transfer onto 0.5 mm thick c-cut sapphire substrates. Spatially resolved reflectance ( $R$ ) and transmittance ( $T$ ) spectra were measured at normal incidence and room temperature using a home-built setup. Absorption ( $A$ ) was deduced using the relation $A = 1 - R - T$ . Crystal thicknesses were estimated via optical microscopy color comparison and accurately measured using Atomic Force Microscopy (AFM), covering a range from monolayers (approx. 0.65 nm) up to ~200 nm.
Theoretical: The experimental data were analyzed using a Generalized Transfer-Matrix (GTM) method. The complex refractive indices ( $\tilde{n} = n + ik$ ) of the materials were extracted by fitting the GTM model to the experimental $R$ and $T$ spectra of a 12 nm thick crystal. These refractive indices were then used to simulate absorption spectra for varying thicknesses. To isolate the role of excitons, the authors also performed calculations where excitonic features were manually removed from the refractive indices.
Extension: The theoretical framework was extended to calculate integrated absorption for a wide variety of solids, including conventional semiconductors ($Si, GaAs, GaN, InP$), 2D magnetic materials ( $CrSBr, NiPS_3$ ), and (semi)metals ($Al, Ag, Au, c-HOPG$), using refractive indices from existing literature.

Key Results

Non-Monotonic Absorption: Contrary to the Beer-Lambert law, the integrated absorption (area under the absorption curve normalized by spectral bandwidth) does not increase monotonically with thickness. Instead, it exhibits pronounced oscillatory features.
Anomalous Enhancement: In the ultrathin limit (below 10 nm), a sharp anomalous increase in absorption is observed. The deviation from Beer's law is significant, exceeding 50% in $WS_2$ and 150% in $MoSe_2$ at thicknesses below 10 nm.
Universality: The oscillatory behavior is independent of specific optical resonances. Calculations where excitonic resonances were removed from the refractive indices yielded the same qualitative oscillatory behavior, indicating the effect is not driven by super-radiant coupling of excitons or plasmons.
Broad Applicability: The theoretical extension demonstrates that this universal absorption behavior applies to diverse material classes, including conventional semiconductors, 2D magnetic materials, and metals, regardless of whether they are freestanding or on a substrate.
Mechanism: The observed deviations and oscillations are attributed to electromagnetic interference effects occurring between the multiple interfaces of the thin crystals (air-crystal, crystal-substrate, and substrate-air), rather than intrinsic quantum many-body effects.

Significance and Claims
The paper claims to provide fundamental insights into light-matter interactions in solids at the nanoscale, challenging the intuitive expectation of monotonic absorption growth. The authors assert that this universal behavior is critical for the optimal design of next-generation absorption-based flexible optoelectronic devices, such as photodiodes, photovoltaics, and phototransistors, where precise control over film thickness is essential for efficiency. By demonstrating that absorption in the ultrathin limit can deviate drastically from standard models, the work suggests that device engineering must account for these interference-induced oscillations to achieve peak performance. The findings are presented as a correction to previous assumptions and a necessary foundation for future device optimization in the era of 2D materials and flexible electronics.

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