ML Discovers 2D Materials with Unique Thermal Expansion

By Soham NandiReviewed by Susha Cheriyedath, M.Sc.Aug 21 2024

In an article published in the journal Nature, researchers identified mechanical descriptors— in-plane tensile stiffness and out-of-plane bending stiffness—that classified two-dimensional (2D) crystals as exhibiting positive thermal expansion (PTE), negative thermal expansion (NTE), or zero thermal expansion (ZTE).

Using high-throughput calculations and symbolic regression, they helped discover 2D materials with extremely low or high thermal expansion coefficients, including ZTE and 2D anti-Invar monolayers.

Background

Thermal expansion describes how materials change dimensions with temperature variations, characterized by the linear coefficient of thermal expansion (LCTE). While most materials show PTE, some exhibit NTE, and even fewer demonstrate ZTE, known as the Invar effect. The Invar effect, discovered in 1897, has been found in various materials, including ferromagnetic alloys and silica-based glasses, and is crucial for high-precision applications like mechanical and optical instruments.

Recent advances in nanofabrication have led to the development of 2D materials, which are essential for next-generation electronics and devices at nanometer or Ångstrom (Å) scales. Although PTE and NTE 2D crystals have been reported, no 2D material with ZTE has been discovered. Existing databases like computational 2D materials database (C2DB) and 2DMatPedia provide extensive 2D material data, yet identifying materials with ZTE or extreme thermal expansion remains a challenge.

This paper addressed these gaps by using machine learning and symbolic regression to explore the thermal expansion properties of 2D crystals. It identified in-plane tensile and out-of-plane bending stiffness as key descriptors for classifying PTE, NTE, and ZTE 2D crystals. The study predicted the existence of ZTE 2D materials and extreme thermal expansion cases, enhancing the understanding and application of 2D materials in advanced technology.

Computational Methods for Thermal Expansion Analysis

The researchers utilized density functional theory (DFT) with the Vienna ab initio simulation package (VASP) to compute the thermal expansion properties of 2D materials. The calculations employed the projector augmented wave (PAW) method with Perdew-Burke-Ernzerhof (PBE) and local density approximation (LDA) functionals.

Optimizations were performed with an energy convergence of 10^-8 electron volts (eV) and a force convergence criterion of 10^-6 eV/Å. A plane wave cut-off energy of 600 eV and a 17×17×1 k-point mesh were used. To avoid interactions between neighboring layers, a 20 Å vacuum was included.

Thermal expansion was calculated using the quasi-harmonic approximation (QHA) method, with phonon dispersions obtained by the finite difference method using PHONOPY. A 6×6×1 supercell and 4×4×1 k-point sampling were used. Small compressive strains were applied to handle soft out-of-plane transverse acoustic modes. The LTEC was derived from the equilibrium lattice parameters and compared with theoretical and experimental data for validation.

In-plane stiffness and compressibility were calculated from the elastic constants obtained by applying small strains to the equilibrium configuration, and the impact of van der Waals dispersion corrections was assessed. Results were consistent with existing theoretical and experimental data, confirming the reliability of the calculations.

Results and Analysis

In 2D materials, both in-plane and out-of-plane deformation modes influenced thermal expansion behavior. At finite temperatures, a free-standing 2D crystal exhibited these deformation modes, with in-plane expansion caused by bond stretching and out-of-plane rippling due to thermal fluctuations. This interplay resulted in varied thermal expansion characteristics, graphene demonstrated NTE, whereas molybdenum disulfide (MoS2) showed PTE.

To classify 2D crystals based on thermal expansion, a data-driven approach using features like in-plane stiffness (E2D) and bending stiffness (D) was employed. A random forest model identified E2D and D as key features, and a support vector machine (SVM) classifier was used to separate PTE and NTE materials.

The sure independence screening sparsifying operator (SISSO) method provided descriptors that correlated well with observed thermal expansion trends. For instance, materials like tungsten disulfide (WSe2) and tin dioxide (SnO2) were classified based on their E2D and D values, revealing that high D and low E2D corresponded to extreme NTE (ENTE) while high E2D and low D correspond to extreme PTE (EPTE).

Further analysis identified zirconium dioxide (ZrO2) and hafnium dioxide (HfO2) as ZTE 2D materials, maintaining minimal expansion over a broad temperature range. Additionally, materials with high D and low E2D exhibited EPTE behavior, while ENTE materials were typically single-layer with high E2D and low D. The authors integrated mechanical descriptors with thermal expansion predictions to enhance material discovery and classification.

Conclusion

In conclusion, the researchers effectively utilized in-plane tensile stiffness and out-of-plane bending stiffness as critical descriptors to classify 2D materials based on their thermal expansion properties. By leveraging high-throughput calculations and symbolic regression, the research identified materials with diverse thermal expansion behaviors, including ZTE and extreme cases.

The findings demonstrated that both in-plane and out-of-plane deformation modes played crucial roles in determining thermal expansion characteristics. This approach not only enhanced the understanding of 2D material behavior but also laid the groundwork for designing materials with tailored thermal properties for advanced applications.