Sudoku-inspired high-Shannon-entropy alloys

Current definition of high-entropy alloys (HEAs) is commonly based on the configurational entropy. But this definition depends only on the chemical composition of a HEA and therefore cannot distinguish the information content that is encoded in various local atomic arrangements and measurable by the Shannon entropy in information theory. Here, inspired by the finding that two-dimensional (2D) Sudoku matrices exhibit higher average Shannon entropy than 2D random matrix counterparts, we propose high-Shannon-entropy alloys (HSEA), whose structures are based on 3D Sudoku matrices. Despite the constraints on the atomic arrangements to form a 3D Sudoku 4 x 4 x 4 matrix, we find that, a prototypical HSEA, NbMoTaW has a lower energy than the same HEA with a random structure due to the smaller lattice distortion. We compute the electronic structures and mechanical properties and find that the HSEA exhibits enhanced average Fermi velocity and ductility over the random NbMoTaW HEA. Finally, we evaluate the formation energies of single vacancies and a vacancy cluster in the HSEA, whose configurations mimic Sudoku puzzles with one and several missing numbers, respectively. We find that a range of large energies are required to generate such vacancies, which depend on the location and species of missing atoms. Our work shows that introducing the Shannon entropy to HEAs offers a useful metric to reveal more atomic details of HEAs and that HSEAs represent an uncharted domain in the complicated energy landscape of HEAs which may be endowed with improved properties.