🔬 materials science

Two-dimensional structures formed by pair potentials: An analytical and numerical study

This study analytically and numerically demonstrates that the $1/N$ scaling law for thin film excess energy breaks down in the monolayer limit due to long-range interatomic interactions, thereby proving that non-van der Waals two-dimensional materials can be formed using pair potentials alone.

Original authors: Shota Ono

Published 2026-10-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shota Ono

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 world of materials science, scientists are constantly searching for ways to create incredibly thin, two-dimensional sheets of matter. Some of these materials, like graphene, are made from layers that naturally stack on top of each other, held together by weak, gentle forces. These are known as van der Waals materials, and they are relatively easy to peel apart into single sheets. However, many other useful materials, such as metals or certain semiconductors, do not have these natural layers. Their atoms are locked together in a solid, three-dimensional block by strong, directional bonds. For a long time, it was unclear if it was even possible to isolate a single, stable layer from these non-layered structures without them falling apart. The key to predicting whether such a material could exist as a standalone sheet lies in understanding how its energy changes as it gets thinner. If the energy behaves in a predictable way as layers are removed, the material likely cannot exist as a single sheet. But if the energy shifts unexpectedly at the very thinnest limit, it suggests that a stable, two-dimensional version might be possible.

A researcher at the Muroran Institute of Technology in Japan has taken a fresh look at this problem by stripping away the complex details of real-world chemistry to focus on the fundamental rules of how atoms attract and repel one another. Instead of using the heavy, complex calculations usually required to model specific metals or compounds, the study used a simplified approach based on "pair potentials." This is a method that assumes the energy between atoms depends only on the distance between any two of them, ignoring the more complicated interactions that happen when three or more atoms are involved. By building a mathematical model of a thin film made of these simple interacting atoms, the researcher could analytically trace exactly how the energy of the film changes as the number of layers decreases from a thick block down to a single sheet.

The study began by establishing a baseline rule that holds true for most thick films. In these thicker stacks, the extra energy caused by having a surface drops off in a very predictable pattern: as the number of layers doubles, the extra energy per layer is cut in half. This is a standard scaling law that works well when the forces between atoms are short-ranged, meaning an atom only really feels the pull or push of its immediate neighbors. In such cases, the math predicts that even a single layer would follow this same smooth trend. However, the researcher found that this rule breaks down when the forces between atoms are long-ranged. In these scenarios, an atom feels the influence of neighbors that are several layers away, not just the ones touching it. When this happens, the energy of a single layer does not fit the smooth curve predicted by the thicker films; instead, it drops lower than expected.

To confirm that this theoretical breakdown was real and not just a quirk of the math, the researcher ran computer simulations using two well-known models for how atoms interact: the Lennard-Jones potential and the Morse potential. These models are like different sets of rules for how atoms behave, with parameters that can be adjusted to make the forces between them either short-ranged or long-ranged. When the simulations used short-range forces, the results matched the standard rule perfectly, even for a single layer. But when the parameters were adjusted to create long-range forces, the simulations showed a clear deviation. The energy of the single layer was significantly lower than what the standard rule would predict. This deviation was quantified by a specific value that measures how steeply the energy curve bends at the single-layer limit. For short-range forces, this value was close to -1, indicating perfect adherence to the standard rule. For long-range forces, the value shifted to numbers like -0.8 or -0.47, signaling a distinct break from the expected pattern.

The findings suggest that the ability to form stable, two-dimensional materials from non-layered bulk structures does not necessarily require complex, multi-atom chemical bonds. Instead, it may be possible to create these structures using only simple, pairwise interactions, provided those interactions are long-ranged enough. This offers a new, simpler way to think about why certain materials like plumbene (a single layer of lead) or goldene (a single layer of gold) can exist, while others cannot. The study implies that the "stickiness" of the forces holding the atoms together, specifically how far that stickiness reaches, is the critical factor. If the forces reach far enough to connect atoms across multiple layers, the material can stabilize itself as a single sheet. While the study did not account for the effects of the environment, such as the strain from a substrate or the presence of other materials, it provides a foundational insight. It suggests that the path to discovering new two-dimensional materials might be found by looking for systems where the atomic interactions extend well beyond their immediate neighbors, allowing the delicate balance of a single layer to hold together against the pull of the bulk.

Technical Summary: Two-Dimensional Structures Formed by Pair Potentials

Problem Statement
The formation of two-dimensional (2D) materials derived from non-layered bulk structures, known as non-van der Waals (non-vdW) 2D materials (e.g., silicene, plumbene, goldene), remains a challenge in bridging theoretical predictions with experimental synthesis. A critical indicator for predicting the stability of such materials is the thickness dependence of the excess energy ( $\Delta E$ ) of thin films relative to the bulk. Previous work established that a breakdown of the $1/N$ scaling law (where $N$ is the number of layers) in the monolayer limit serves as a useful descriptor for identifying non-vdW 2D materials. However, while first-principles calculations can accurately predict these energies, the underlying physical mechanisms governing this scaling law breakdown were not fully understood analytically. Specifically, it was unclear whether pair potentials alone could account for the deviation from the $1/N$ scaling law observed in real materials, or if many-body interactions were strictly necessary.

Methodology
The study employs a combined analytical and numerical approach to investigate the excess energy of $N$ -layer thin films formed by pair potentials.

Analytical Model: The author assumes a monatomic crystal with a close-packed structure (triangular lattices stacked along the surface normal) where interatomic interactions are governed by a pair potential $v(R)$ . The total energy of an $N$ -layer film is derived as a sum of intralayer and interlayer interaction energies. The excess energy $\Delta E(N)$ is defined as the difference between the energy per layer of the film and the bulk energy. The analysis distinguishes between short-range and long-range interatomic interactions to determine their effect on the scaling behavior at $N=1$ .
Numerical Simulations: To validate the analytical theory, numerical simulations were conducted using the Atomic Simulation Environment (ASE). Two specific pair potentials were tested: the Lennard-Jones (LJ) potential and the Morse potential. The range of these interactions was systematically tuned by varying the cutoff radius ( $R_c$ ) for the LJ potential and the exponential prefactor ( $\rho_0$ ) for the Morse potential. The study examined $N$ -layer films derived from the (111) surface of a face-centered cubic structure (ABC stacking) without geometry relaxation to isolate the effects of interaction range.

Key Contributions and Results

Analytical Derivation of Scaling Breakdown: The paper analytically demonstrates that the validity of the $1/N$ scaling law for excess energy depends critically on the range of interatomic interactions.
- For short-range interactions, where only adjacent layers interact significantly, the excess energy follows the $1/N$ scaling law even in the monolayer limit ( $N=1$ ). In this regime, the extrapolated value $\Delta E_{ext}^1$ matches the actual monolayer energy $\Delta E(1)$ .
- For long-range interactions, where interactions extend beyond nearest neighbors, the $1/N$ scaling law breaks down. The analytical derivation shows that the actual excess energy at $N=1$ is lower (more negative) than the value extrapolated from the $1/N$ trend. This results in a "downward deviation," quantified by the parameter $p_1 > -1$ (where $p_1$ is the slope of $\ln \Delta E$ vs. $\ln N$ at $N=1$ ).
Numerical Confirmation: Simulations using LJ and Morse potentials confirmed the analytical predictions.
- When interaction ranges were short (e.g., $R_c = 2\sigma$ for LJ, $\rho_0 = 8$ for Morse), the system adhered to the $1/N$ scaling law ( $p_1 \approx -1$ ).
- As the interaction range increased (e.g., $R_c = 10\sigma$ for LJ, $\rho_0 = 3$ for Morse), a clear downward deviation was observed ( $p_1 = -0.8$ and $-0.47$, respectively).
- The study further noted that this deviation occurs even without structural relaxation, suggesting that geometry relaxation has a minor impact on the scaling law itself.

Significance and Claims
The primary significance of this work is the establishment that non-vdW 2D structures can theoretically be formed by pair potentials alone, provided that the interatomic interactions are sufficiently long-ranged. The breakdown of the $1/N$ scaling law in the monolayer limit is not exclusively a result of complex many-body effects (such as covalent bonding or chemical bonds) but can arise fundamentally from the range of pairwise interactions.

The author concludes that the range of interlayer interactions is a key factor modulating the excess energy scaling. While the study does not claim to replace first-principles methods for specific material prediction, it provides a minimal analytical model that captures the essential physics behind the scaling law breakdown. The work suggests that long-range pair potentials are a necessary condition for the formation of stable non-vdW 2D materials within this simplified framework, offering a theoretical basis for understanding why certain non-layered bulk materials can exfoliate into stable 2D monolayers. The author explicitly states that effects beyond pair potentials (many-body interactions) and environmental factors (substrates, strain) remain subjects for future investigation using density-functional theory.

Technical Summary: Two-Dimensional Structures Formed by Pair Potentials

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