Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Single-electron states of phosphorus-atom arrays in silicon



Maicol Ochoa, Keyi Liu, Michał Zieliński, Garnett W. Bryant


We characterize the single-electron energies and the wavefunction structure of arrays with two, three, and four phosphorus atoms in silicon by implementing atomistic tight-binding calculations and analyzing wavefunction overlaps to identify the single-dopant states that hybridize to make the array states. The energy spectrum and wavefunction 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 A1 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 wavefunction structure into Fermi-Hubbard Hamiltonians needed to implement and validate analog quantum simulations in these devices.
Physical Review B


Ochoa, M. , Liu, K. , Zieliński, M. and Bryant, G. (2024), Single-electron states of phosphorus-atom arrays in silicon, Physical Review B, [online],, (Accessed May 26, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created May 8, 2024, Updated May 14, 2024