Electron spin configurations in phosphorus arrays in silicon

Dopant arrays in silicon are ideal for realizing analog quantum simulations of the Fermi-Hubbard model with quantum control on the parameter space ensured by the dopant number, distribution, and energy modulation from external electric and magnetic potentials. In this study, we analyzed the spin configuration and electron charge distribution in multi-electron states bound to triangular, quadrilateral, and hexagonal phosphorus arrays in silicon, which we obtained by carrying out atomistic tight-binding Hartree-Fock and configuration interaction calculations. By mapping the electron energies into Fermi-Hubbard models and identifying tunneling, on-site and coulomb-repulsion energies, we rationalize the observed spin and charge distributions, classifying them between normal and frustrated magnetic states. Our results reveal the effects of the weak spin-orbit coupling, valley degeneracy, and the dopant lattice distribution in the ground state electron spin configuration. These results are crucial in interpreting quantum simulations and readout from magnetic measurements in these devices.