Single-electron states of phosphorus-atom arrays in silicon

We characterize the single-electron energies and the wave-function structure of arrays with two, three, and four phosphorus atoms in silicon by implementing atomistic tight-binding calculations and analyzing wave-function overlaps to identify the single-dopant states that hybridize to make the array states. The energy spectrum and wave-function overlap variation as a function of dopant separation for these arrays shows that hybridization mostly occurs between single-dopant states of the same type, with some cross hybridization between ${A}₁$ and $E$ states occurring at short separations. We also observe energy crossings between hybrid states of different types as a function of impurity separation. We then extract tunneling rates for electrons in different dopants by mapping the state energies into hopping Hamiltonians in the site representation. Significantly, we find that diagonal and nearest-neighbor tunneling rates are similar in magnitude in a square array. Our analysis also accounts for the shift of the on-site energy at each phosphorus atom resulting from the nuclear potential of the other dopants. This approach constitutes a solid protocol to map the electron energies and wave-function structure into Fermi-Hubbard Hamiltonians needed to implement and validate analog quantum simulations in these devices.