Materials exhibiting zero thermal expansion (ZTE), also known as the Invar effect, are rare, while those with positive (PTE) or negative (NTE) thermal expansion are common. The Invar effect, where a material shows minimal volumetric change over a temperature range, is highly desirable for applications requiring dimensional stability, particularly in temperature-sensitive devices and high-precision instruments like seismographs and telescopes. Recent advances in nanofabrication have reduced electronic devices to the nanoscale, emphasizing the importance of understanding the thermal expansion of 2D materials, which act as building blocks for such devices. While some 2D materials show PTE (e.g., MoS2) or NTE (e.g., graphene), no 2D Invar materials have been identified. The large number of 2D materials in recently developed databases makes data mining and machine learning approaches valuable for identifying materials with exceptional properties. This study uses these methods to investigate the existence of 2D Invar and anti-Invar materials.

The Invar effect was first discovered in 1897 by Guillaume, leading to his Nobel Prize in Physics in 1920. Since then, Invar materials have been found in various materials classes, including ferromagnetic alloys, ferroelectric relaxors, and silica-based glasses. The origin of the Invar effect is often linked to a cancellation of PTE by NTE contributions within the material's structure. Recently, machine learning has been successfully employed in discovering new bulk Invar alloys with exceptionally low thermal expansion coefficients. This study extends this approach to 2D materials, leveraging the available databases of 2D material properties to identify suitable candidates for machine learning analysis.

The researchers employed a two-pronged approach: classification and symbolic regression. Initially, they used a random forest method and the sure independence screening sparsifying operator (SISSO) method to identify the most crucial descriptors for predicting thermal expansion in 2D materials. From this they found that in-plane tensile stiffness (E2D) and out-of-plane bending stiffness (D) were the most significant descriptors, effectively classifying PTE and NTE materials. A support vector machine (SVM) classifier was trained using these descriptors to effectively categorize the 2D crystals. Next, to predict the linear thermal expansion coefficient (LTEC) at 500K (α500K), a symbolic regression model was developed, trained on a dataset of calculated properties including E2D, D, and isothermal compressibility (KT). The model generated an equation linking these variables to α500K. This model, alongside the SVM classifier, was used for high-throughput screening of potential ZTE and ELTE (extremely large thermal expansion) 2D materials. The predicted candidates were further verified using the quasi-harmonic approximation (QHA) and Grüneisen theory methods to assess their thermal expansion behavior over a temperature range (typically 300-600K or lower depending on the material). DFT calculations were extensively used to determine the necessary input parameters such as E2D, D, and phonon dispersions for the QHA and Grüneisen methods.

The study successfully identified two interpretable mechanical descriptors, E2D and D, effectively distinguishing between PTE and NTE 2D crystals. The SVM classifier showed high accuracy in classifying materials based on these descriptors. The symbolic regression model provided a robust prediction for α500K, showing reasonable accuracy when compared to the QHA calculations. Through high-throughput screening of a large dataset of stable 2D crystals from the C2DB database, two 2D Invar materials, ZrO2 and HfO2, were identified with LTEC values within ±2 × 10⁻⁶ K⁻¹ over a range of temperatures (300-600K). These were 2D transition metal oxides characterized by high E2D and moderate D, leading to a balance between in-plane and out-of-plane deformation modes. Furthermore, three 2D anti-Invar materials exhibiting extremely positive thermal expansion (EPTE) were discovered: SbSe2, HfBi2, and HfSb2. These materials showed significantly high LTEC values (>15 × 10⁻⁶ K⁻¹). Their characteristics include large D and small E2D, indicating the dominance of out-of-plane thermal fluctuations contributing to the large expansion. Finally, three ENTE materials (As2, Ge2, and Sn2) were identified, showing extremely negative thermal expansion at low temperatures. Their properties indicated small D and large E2D.

The findings address the research question by demonstrating the effectiveness of machine learning in discovering 2D materials with exceptional thermal expansion properties. The identification of 2D Invar and anti-Invar materials significantly broadens the scope of available materials for nanoscale applications. The interpretability of the descriptors (E2D and D) provides valuable physical insight into the relationship between mechanical properties and thermal expansion behavior in 2D crystals. The success of the combined SVM and symbolic regression approach highlights the potential of data-driven methods in materials discovery and design. The discovered materials have potential applications in various fields, including temperature-stable nanoscale devices, thermal management systems, and advanced metamaterials.

This study successfully employed machine learning to discover novel 2D Invar and anti-Invar materials. The identified interpretable descriptors offer valuable insights into the relationship between material properties and thermal expansion. The discovered materials expand the possibilities for nanoscale devices requiring precise control over thermal expansion. Future research could focus on exploring the thermal expansion behavior of these materials in different environments, such as under strain or on substrates, and investigating the potential for designing heterostructures to fine-tune thermal expansion properties.

The study primarily focused on free-standing 2D crystals of infinite size. Finite-size effects, substrate effects, and the influence of specific entropy-driven wavy conformations were not fully considered. The accuracy of the symbolic regression model relies on the quality and quantity of the training data, and potential improvements could be achieved with a larger and more diverse dataset. The QHA method, while widely used, has limitations in capturing the full complexity of thermal expansion behavior at low temperatures.