Single-electron states of phosphorus-atom arrays in silicon

Maicol A. Ochoa, Keyi Liu, Michał Zieliński, and Garnett Bryant

Phys. Rev. B 109, 205412 – Published 8 May 2024

Abstract

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.

Received 4 March 2024
Revised 25 April 2024
Accepted 29 April 2024

DOI:https://doi.org/10.1103/PhysRevB.109.205412

Physics Subject Headings (PhySH)

Analog computation Quantum engineering Quantum simulation

Doped semiconductors

Electronic structure Hubbard model Tight-binding model

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Maicol A. Ochoa ^1,2,*, Keyi Liu ^1,3, Michał Zieliński ⁴, and Garnett Bryant ^1,3,†

¹National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
²Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA
³Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
⁴Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, ulica Grudziadzka 5, 87-100 Toruń, Poland

^*maicol@nist.gov
^†garnett.bryant@nist.gov

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Issue

Vol. 109, Iss. 20 — 15 May 2024

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Images

Figure 1
Phosphorus dimer overlap analysis. (a) Energy spectrum as a function of P-P separation. For each separation $d$ , we show the lowest 24 dimer states. Black and blue dots correspond to doubly degenerate, spin-conjugate states. Brown dots correspond to fourfold degenerate states. The horizontal gray dashed line indicates the single-electron ground energy in a single phosphorus atom. (b)–(f) Overlap histograms, given by Eq. (2), as a function of the type of single-dopant state and the dimer separation $d$ for (b) first, (c) third, (d) fifth, (e) seventh, and (f) ninth dimer state. The dimer grows along the [100] direction. The blue dashed lines in (a) indicate the dimer states identified with symmetric and antisymmetric combinations of single-phosphorus $A_{1}$ electron states.
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Figure 2
The phosphorus dimer along the [100] direction. (a) Single-electron tunneling energy $t$ , (b) on-site energy shift $λ$ , and (c) single-electron energy spectrum as a function of P-dimer separation $d$ along the [100] direction. In (c), solid and dashed lines show the corresponding symmetric and antisymmetric energies $ɛ_{2 P}^{-}$ and $ɛ_{2 P}^{+}$ for each case of dimer-state pair identified through the overlap analysis.
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Figure 3
The phosphorus dimer along the [110] direction. (a) Single-electron tunneling, (b) on-site energy shift, and (c) single-electron energy spectrum as a function of P-dimer separation $d$ along the [100] direction. In (c), solid and dashed lines show the corresponding symmetric and antisymmetric energies $ɛ_{2 P}^{-}$ and $ɛ_{2 P}^{+}$ for each case of dimer-state pair identified through the overlap analysis.
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Figure 4
The linear phosphorus trimer along the [100] direction. The $d$ dependence for (a) on-site energy shift at the inner impurity, (b) on-site energy shift at the outer impurity, and (c) nearest-neighbor electron tunneling. (d) Single-electron energy spectrum as a function of the separation $d$ of each outer impurity from the inner one. For each case, we used solid, dotted, and dashed lines to show fits corresponding to the energies $ɛ_{3 P}^{+ 1}$ , $ɛ_{3 P}^{0}$ , and $ɛ_{3 P}^{- 1}$ , respectively.
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Figure 5
The square array. (a) Nearest-neighbor tunneling energy, (b) diagonal or next-nearest-neighbor tunneling energy, (c) on-site energy shift, and (d) single-electron energy spectrum as a function of the square side length $d$ . For each case, we used solid, dotted, and dashed lines to show fits corresponding to the energies $ɛ_{2 P \times 2 P}^{+ 1}$ , $ɛ_{2 P \times 2 P}^{0}$ , and $ɛ_{2 P \times 2 P}^{- 1}$ , respectively.
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Figure 6
Confinement potential sections for an electron in a $2 P \times 2 P$ array and a 2P array. We show the potential along the line joining nearest-neighbor (NN) atoms (black, circles), and the diagonal (blue, squares) in the $2 P \times 2 P$ array with $d = 7 \sqrt{2} a_{o}$ . For the 2P array formed by the P atoms in the diagonal, we present the potential energy along the straight line (red, diamonds).
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