Experimental and Computational Investigation of Thermal Conductivity in Porous Silicon: Doping-Dependent Performance of Molecular Dynamics and Physics- Informed Neural Networks

Silicon is the silent backbone of modern electronics, the material that powers everything from smartphones to solar panels. In its solid, crystal form, silicon is an excellent conductor of heat, which is often a problem for engineers trying to keep tiny computer chips from overheating. However, when silicon is transformed into a sponge-like material filled with tiny holes, its ability to conduct heat drops dramatically. This porous silicon becomes a thermal insulator, a property that makes it incredibly valuable for creating more efficient energy-harvesting devices and for keeping heat where it is needed. The challenge for scientists has always been predicting exactly how well this material will work. The answer depends on a complex mix of factors: how many holes are in the material, how big those holes are, and how the silicon atoms are arranged around them. Because the structure can vary wildly depending on how the material is made, a simple formula often fails to capture the full picture, leaving engineers to guess rather than calculate.

A team of researchers at Taras Shevchenko National University of Kyiv has tackled this uncertainty by building a comprehensive bridge between the microscopic world of atoms and the macroscopic world of heat flow. They did not rely on just one method to solve the puzzle. Instead, they combined three distinct approaches: real-world experiments, computer simulations that track individual atoms, and a new type of artificial intelligence designed to learn the laws of physics. The researchers started by creating their own samples of porous silicon. They took standard silicon wafers and used a chemical bath with an electric current to etch away material, leaving behind a sponge-like structure. By carefully adjusting the strength of the electric current, they could control the density of the holes, creating layers with porosity ranging from 48 percent to 75 percent. To see what they had made, they looked at the samples under powerful microscopes, revealing that the holes formed distinct columns running through the material, a shape that is characteristic of the heavily doped silicon they started with.

To measure how well these materials conducted heat, the team used a technique called photoacoustic measurement. They shone a modulated laser beam onto the silicon, causing the surface to heat up and cool down rapidly. This rhythmic heating created sound waves in the air above the sample, which a sensitive microphone could detect. By analyzing the strength and timing of these sound waves, the researchers could calculate the exact thermal conductivity of the porous layer without touching it. They found that the heat transport was not just about the number of holes, but was fundamentally governed by the initial doping of the silicon wafer. The doping level determined whether the internal structure formed a random, sponge-like network or a more organized, columnar one. This structural difference meant that the same amount of porosity could result in very different thermal behaviors depending on the starting material.

To understand the physics behind these measurements, the researchers turned to two different computational models. First, they used molecular dynamics simulations, which act like a high-speed movie of atoms moving and colliding. This method is excellent for seeing how heat travels through tiny, specific structures where the boundaries of the holes scatter the heat-carrying vibrations, known as phonons. The simulations showed that for the heavily doped, columnar samples, this atom-level scattering was the dominant factor reducing heat flow. However, this method is too slow to model large, complex networks. For that, the team employed Physics-Informed Neural Networks, a form of artificial intelligence that learns to solve heat equations by being taught the fundamental laws of thermodynamics. This approach allowed them to model the complex, random pathways of heat through the porous sponge, capturing how heat flows through the entire network rather than just a tiny slice of it.

The results of these combined efforts revealed a clear picture of why previous studies sometimes disagreed. The researchers found that the thermal properties were dictated by the carrier concentration of the original silicon crystal, which controlled the shape of the pores. In lightly doped silicon, the pores formed a random, labyrinthine structure that made it very hard for heat to find a path through, leading to a rapid drop in conductivity. In contrast, the heavily doped silicon formed column-like structures that acted as thermal highways, allowing heat to move more easily despite having many holes. The molecular dynamics simulations matched the behavior of these columnar, heavily doped samples, while the neural network models excelled at predicting the behavior of the random, sponge-like structures found in lighter doping. By merging their experimental data with these simulations and existing literature, the team used a mathematical technique called symbolic regression to derive a single, unified formula. This formula accurately predicts the thermal conductivity of porous silicon based on its porosity and temperature, covering a wide range from room temperature up to 900 Kelvin.

This work does more than just provide a new number for engineers to use; it resolves long-standing confusion in the field by showing that the material's history matters. The same porosity can yield different results depending on how the material was grown. The team's hybrid approach, which connects the movement of individual atoms to the flow of heat across a whole material, offers a reliable tool for designing better thermal insulation and thermoelectric devices. Their final formula, which works across a broad range of temperatures and hole densities, allows for precise predictions without the need for expensive and time-consuming experiments for every new design. By clarifying how structure and doping interact to control heat, this research provides a solid foundation for the next generation of silicon-based technologies, ensuring that heat is managed exactly where it is needed.