Active Learned Equivariant Neural Networks for Inverse Design of Thermoelectric Materials

Thermoelectric materials are capable of converting waste heat into electrical power utilizing the Seebeck effect, and they find applications in numerous domains. A major expected trait of these materials is to be environmentally sustainable and applicable across a wide temperature range. Copper selenide (Cu2Se) shows exceptional promise in this regard and has been an intensively examined candidate in recent years. The thermoelectric figure of merit (ZT), which determines the potential of a thermoelectric candidate, relies heavily on the thermal and electrical transport properties due to the strong interdependence with the Seebeck coefficient (α), electrical conductivity (σ), absolute temperature (T), electronic thermal conductivity (κe), and lattice thermal conductivity (κl). While first-principles calculations provide precise insights into electronic and thermal (phonon) transport properties, they are computationally expensive and limited in scalability, particularly in systems such as Cu2-xSe, which exhibit complex phase space, electronic structures, and heterogeneity that extend much beyond their unit cell representation. Equivariant neural networks, known to substantially improve data efficiency and generalizability, have been successfully implemented in predicting the electronic structure of near-equilibrium phases in the past. However, their application in exploring the phase space of far-from-equilibrium and disordered materials is limited. On the other hand, equivariant Machine Learning Interatomic Potentials (MLIPs) have proven effective in predicting thermal transport properties with precision. In this work, we adopt an on-the-fly active learning scheme to train an E(3)-equivariant neural network in tandem with an MLIP (MACE: Higher order equivariant message passing neural network) to learn the Hamiltonian with ab initio accuracy for electronic transport calculations and phonons for thermal transport while exploring the phase space of Cu2-xSe. Combining the two allows us to predict candidates with promising ZT values. The trained model not only can help us sample promising candidates but also reveal the role of local structure in dictating the thermoelectric performance of these materials.