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Superexchange coupling of donor qubits in silicon

Mushita M. Munia, Serajum Monir, Edyta N. Osika, Michelle Y. Simmons, and Rajib Rahman

Phys. Rev. Applied 21, 014038 – Published 22 January 2024

Abstract

Atomic engineering in a solid-state material has the potential to functionalize the host with novel phenomena. STM-based lithographic techniques have enabled the placement of individual phosphorus atoms at selective lattice sites of silicon with atomic precision. Here we show that by placing four phosphorus donors spaced 10–15 nm apart from their neighbors in a linear chain, one can realize coherent spin coupling between the end dopants of the chain, analogous to the superexchange interaction in magnetic materials. Since phosphorus atoms are a promising building block of a silicon quantum computer, this enables spin coupling between their bound electrons beyond nearest neighbors, allowing the qubits to be separated by 30–45 nm. The added flexibility in architecture brought about by this long-range coupling not only reduces gate densities but can also reduce correlated noise between qubits from local noise sources that are detrimental to error-correction codes. We base our calculations on a full-configuration-interaction technique in the atomistic tight-binding basis, solving the four-electron problem exactly, over a domain of a million silicon atoms. Our calculations show that superexchange can be tuned electrically through gate voltages where it is less sensitive to charge noise and donor-placement errors.

Received 11 September 2023
Accepted 28 November 2023

DOI:https://doi.org/10.1103/PhysRevApplied.21.014038

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Electronic structure Exchange interaction Quantum information with solid state qubits

Doped semiconductors Quantum dots

Spin

Configuration interaction Exact solutions for many-body systems

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Mushita M. Munia ^1,2,*, Serajum Monir ^1,2, Edyta N. Osika ^1,2, Michelle Y. Simmons ^1,2, and Rajib Rahman ^1,2,†

¹School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
²Silicon Quantum Computing Pty Ltd., Level 2, Newton Building, University of New South Wales, Kensington, New South Wales 2052, Australia

^*[email protected]
^†[email protected]

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Vol. 21, Iss. 1 — January 2024

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Figure 1
Schematic representation of a simulation using four single phosphorus donors each with an electron. (a) Four phosphorus donors placed in the silicon crystal (gray), where the electrons localized in the two middle donors (blue) are mediators ( $M_{1}$ and $M_{2}$ ) for the electron spins localized in the end donors ( $Q_{1}$ and $Q_{2}$ ) (red). The separation of the blue mediator spins is $r_{M}$ and their exchange coupling is $j_{M}$ . The first qubit $Q_{1}$ is separated from the first mediator $M_{1}$ by $r_{1}$ (approximately $10$ nm) and the second qubit $Q_{2}$ is separated from the second mediator $M_{2}$ by $r_{2}$ (approximately $10$ nm). The corresponding exchange coupling between them is $j_{1}$ and $j_{2}$ , respectively. The total separation of qubits $Q_{1}$ and $Q_{2}$ is $R$ (approximately $30$ nm). The probability density of the lowest single-electron valley-orbital (b) bonding and (c) antibonding state on a logarithmic scale calculated with an atomistic tight-binding method.
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Figure 2
Comparison of indirect and direct exchange coupling of 1 $P$ donors in silicon calculated with use of atomistic FCI. (a) Comparison of indirect exchange (4 $P$ chain) of the end spins (red dots) with nearest-neighbor exchange coupling (1 $P$ -1 $P$ ) (blue dots, replicated from Ref. [35]) along the $[100]$ direction. Dashed lines provide linear fits to both datasets. Here we see that the indirect exchange is greater than the nearest-neighbor exchange, with the exponential dependence on separation being less steep for indirect coupling. (b) Same as (a) but along the $[110]$ direction. Here we see a similar trend in terms of the comparison between direct and indirect exchange as for the $[100]$ direction. However, we also observe oscillations in the exchange coupling along this crystalline orientation due to valley quantum interference.
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Figure 3
Estimate of exchange-coupling strength from the spin Hamiltonian. (a) Energies of four-electron eigenstates calculated with the effective Hamiltonian $H_{eff}$ as a function of $j_{1, 2} / j_{M}$ . The two-lowest states (spanned by $| ↑ S ↓ ⟩$ and $| ↓ S ↑ ⟩$ ) are well separated from the higher states when $j_{1, 2} / j_{M}$ is smaller. The separation of these two energy levels is the superexchange $Δ E$ . The labels on the right refer to the dominant contribution of the corresponding energy states when $j_{1, 2} / j_{M} \approx 0$ . (b) Comparison of the exchange coupling, $Δ E$ , calculated from the effective spin Hamiltonian and the Schrieffer-Wolff Hamiltonian shows that for $j_{1, 2} / J_{M}$ up to approximately $0.3$ , the superexchange from $H_{eff}$ and $H_{SW}$ is the same since the Schrieffer-Wolff Hamiltonian is a valid approximation of the spin Hamiltonian. Beyond this regime, their behavior diverges significantly because the Schrieffer-Wolff transformation no longer holds, indicating the development of an admixture in the singletlike state of the middle-two spins. (c) Contribution of the inner-singlet and inner-triplet basis states to the ground state of $H_{eff}$ as a function of $j_{1, 2} / j_{M}$ . Here we see that as $j_{1, 2} / j_{M}$ increases, the singlet contribution from the inner mediator spins decreases, and the triplet contribution increases. At $j_{1, 2} / j_{M} = 1$ , the ground state has equal contributions from singlet and triplet mediator spin states. The contribution of the inner singlet to the ground state is more than 90% for $j_{1, 2} / J_{M} < 0.3$ (shaded region).
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Figure 4
Exchange coupling as a function of different middle-donor separations, $r_{M}$ . The exchange coupling $Δ E$ is calculated with use of atomistic FCI for different separations of the middle donors $r_{M}$ in (a) the [100] direction and (b) the [110] direction while the total outer-qubit separation is kept fixed ( $R = 29.3274$ nm for the [100] direction and $R = 27.648$ nm for the [110] direction). The system is symmetric ( $r_{1} = r_{2}$ ) to the end donors. The results show that the dependence of exchange energy changes on either side of the equidistant separation (dashed gray line) for all donors. The dashed line indicates where $r_{1} = r_{2} = r_{M}$ , and the left region of this line, where $r_{1} = r_{2} > r_{M}$ , is where the two middle donors form a singlet state and the exchange coupling between the end donors is termed “superexchange.” We compare our FCI results with the effective spin Hamiltonian, $H_{eff}$ of Eq. (1) with $j_{1, 2}$ and $j_{M}$ extracted from Ref. [35]. For both cases, the results show reasonable agreement with each other. We attribute any differences mainly to the fact that the total confinement potential of four donors is deeper than that of two donors, which results in a slightly greater exchange coupling $Δ E$ in our atomistic FCI calculations.
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Figure 5
Modulation of superexchange using voltage biases on phosphorus-doped top gates above the donor plane. (a) Electrostatic potential of the device in the schematic representation of the simulation domain. The qubit donor atoms ( $Q_{1}$ and $Q_{2}$ ) and the mediator donor atoms ( $M_{1}$ and $M_{2}$ ) are placed in the plane between the source ( $S$ ) and drain ( $D$ ) electrodes. Three phosphorus-doped top gates ( $G_{1}$ , $G_{2}$ , and $G_{M}$ ) are placed 30 nm above the donor plane. (b) Total potential of the device in the $x$ - $z$ plane (cut taken at the donor location). Here the voltages of gates $G_{1}$ and $G_{2}$ are $V_{1} = V_{2} = - 0.1 V$ , while the middle gate $G_{M}$ has voltage $V_{M} = 0.5 V$ and the source and drain are grounded. The red and blue dots represent the location of the donors and are not to scale. (c)–(e) One-dimensional potentials with cuts taken at the donor location when $V_{1} = V_{2} = 0$ V and (c) $V_{M} = 0$ V, (d) $V_{M} = 0.3$ V, and (e) $V_{M} = 0.5$ V, respectively, as we deplete the barrier between the donors. The dashed lines represent the potential from the gates and the solid lines represent the individual donor potentials combined with the electrostatic potential from the gates. As the voltage of the top gate increases, the curvature of the potential also increases. (f)–(h) Same as (c)–(e) but when $V_{M} = 0.5 V$ and (f) $V_{1} = V_{2} = - 0.1$ V, (g) $V_{1} = V_{2} = - 0.3$ V, and (h) $V_{1} = V_{2} = - 0.5$ V, respectively. (i) Superexchange as a function of $V_{M}$ when $V_{1} = V_{2} = 0$ V. We see that the magnitude of the superexchange increases as a positive bias is applied to $G_{M}$ . The increased curvature of the applied potential of $V_{M} = 0.5 V$ [comparison of (c),(e)] gives rise to an increase of the magnitude of superexchange by a factor of 10. (j) Superexchange as a function of $V_{1} = V_{2}$ when $V_{M} = 0.5$ V. Here we see the superexchange decreases as more-negative bias is applied as a result of decreased potential curvature [comparison of (f),(h)].
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Figure 6
Effect of donor placement on the magnitude of superexchange calculated with use of atomistic FCI along the [100] direction. Keeping the separation of the middle donors fixed ( $r_{M} = 5.431 nm$ ), we change $r_{1}$ and $r_{2}$ to account for any deviation in donor placement. The superexchange $Δ E$ is maximum when $r_{1} = r_{2}$ . We also see that $Δ E$ remains constant when $r_{1} + r_{2}$ is constant; see the dashed lines.
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Physical Review Applied

Superexchange coupling of donor qubits in silicon

Mushita M. Munia, Serajum Monir, Edyta N. Osika, Michelle Y. Simmons, and Rajib Rahman

Phys. Rev. Applied 21, 014038 – Published 22 January 2024

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