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Quantum computing with defects
J. R. Weber, W. F. Koehl, J. B. Varley, A. Janotti, B. B. Buckley, C. G. Van de Walle, D. D. Awschalom
TL;DR
Finding robust solid-state qubits beyond diamond’s NV-1 center requires systematic criteria for candidate defects and hosts. This paper combines physical criteria with electronic-structure analysis to identify and evaluate potential NV-like centers, including in 4H-SiC and other tetrahedral semiconductors.
Problem
Systematically identifying deep-center defects and hosts with NV-1-like qubit properties remains an important research need.
Method
The paper combines defect and host criteria with band-structure analysis and electron counting to evaluate candidate quantum-defect systems.
Results
The analysis identifies defect-level orbital splittings as important quantities for discovering new defect systems for quantum applications.
Takeaways & Limitations
The proposed criteria provide a framework for screening tetrahedrally coordinated semiconductors and their deep centers as potential NV analogs.
Takeaways & Limitations
Accurately computing the energy splittings that contribute to criterion D1 is difficult.
Abstract
from arXiv · showhide
Identifying and designing physical systems for use as qubits, the basic units of quantum information, are critical steps in the development of a quantum computer. Among the possibilities in the solid state, a defect in diamond known as the nitrogen-vacancy (NV-1) center stands out for its robustness - its quantum state can be initialized, manipulated, and measured with high fidelity at room temperature. Here we describe how to systematically identify other deep center defects with similar quantum-mechanical properties. We present a list of physical criteria that these centers and their hosts should meet and explain how these requirements can be used in conjunction with electronic structure theory to intelligently sort through candidate defect systems. To illustrate these points in detail, we compare electronic structure calculations of the NV-1 center in diamond with those of several deep centers in 4H silicon carbide (SiC). We then discuss the proposed criteria for similar defects in other tetrahedrally-coordinated semiconductors.
Defect and Host Criteria for NV-like Systems
NV-like defects require a qubit-compatible bound state, spin-selective optical initialization and readout, host-isolated transitions, and thermally stable levels, while suitable hosts should minimize decoherence. First-principles calculations can screen many of these properties, but non-radiative processes and small spin splittings remain difficult to predict accurately.
- Motivation: NV-1’s long coherence and high-fidelity ambient optical initialization and measurement motivate criteria for identifying comparable deep-center qubits.The NV-1 ground state can have coherence times up to 1.8 ms, while manipulation rates above 200 MHz have been demonstrated.
- Defect criteria: Candidate defects should provide a paramagnetic, long-lived bound state with resolvable spin sublevels, optical polarization, sublevel-dependent luminescence, host-isolated transitions, and thermally separated bound states.These requirements are designated D1–D5 and target qubit manipulation, initialization, measurement, suppression of host interference, and preservation of spin information.
- Host criteria: Suitable hosts should have wide band gaps, weak spin-orbit coupling, high-quality single-crystal availability, and zero-nuclear-spin isotopes for reducing decoherence.The latter three host properties are intended to reduce decoherence in the defect.
- Computational screening: First-principles calculations can predict defect spin, internal optical transitions, dipole-related lifetimes, and defect-level positions relative to each other and host band edges.Constrained density functional theory can calculate optical-transition energies, with excitonic effects included through the Bethe-Salpeter formalism.
- Computational limitations: Accurate prediction remains difficult for spin-sublevel splittings, non-radiative decay paths, and luminescence differences based on small microwave-scale splittings.Nevertheless, defect atomic configurations and electronic structures are correlated, enabling calculations to prioritize vacancy-related candidates in tetrahedrally coordinated semiconductors.
Formation Energies, Defect-Level Diagrams, and Configuration-Coordinate Diagrams
The section uses hybrid-functional calculations to evaluate defect formation energies, defect-level diagrams, and configuration-coordinate diagrams for NV−1-like centers. In diamond and 4H-SiC, these analyses identify stable charge states, spin configurations, and excitation properties relevant to quantum-computing criteria.
- Formation Energies: Formation energies indicate defect stability, relative configuration and charge-state stability, and likely formation under both equilibrium and non-equilibrium processes.Charge-state stability is determined by formation energy’s dependence on the Fermi level, while formation-energy magnitude remains informative for processes such as ion implantation.
- Defect Centers in SiC: Compared with diamond NV−1, NCVSi−1 has vertical transitions about half as large and relaxation energies more than 75% smaller because SiC’s larger lattice constant reduces dangling-bond overlap.The reduced overlap produces a smaller splitting between the a1(2) and ei levels.
- Defect Centers in SiC: The isolated VSi defect can have a ground-state triplet for four electrons, but VSi−1 forms a spin-3/2 ground-state quartet and therefore lacks spin-conserving triplet excitations.VSi charge states 0, -1, and -2 are stable in n-type material within 0.3 eV of the conduction-band minimum, while proximity of a level to the VBM raises concern for criterion D4.
Discovering NV Analogs in Other Material Systems
The paper proposes screening tetrahedrally coordinated semiconductors for NV analogs by analyzing vacancy-level positions, orbital splittings, impurities, band gaps, and charge-state electron counting.
- Cation Vacancies: For cation vacancies, anion dangling-bond energies and ΔCV are key quantities for positioning t2 levels away from the VBM while satisfying splitting criteria.Jahn–Teller distortions, impurities, and reduced symmetry split t2 levels, but the splitting must remain neither too large nor too small.
- Cation Vacancies: Cation-vacancy complexes should use donor impurities that interact attractively with the vacancy without making δCV large enough to push a1(2) near or below the VBM.Because cation vacancies tend to be negatively charged, suitable donors are elements to the right of the host anion in the periodic table.
- Cation Vacancies: The SA-center in ZnSe, a Zn vacancy–donor complex, has a positive-charge-state ground-state triplet with six electrons, although its full criteria remain unconfirmed.This charge state is expected to be stable in p-type material.
- Anion Vacancies: Anion vacancies are generally less favorable for triplets because cation dangling bonds and ΔAV place t2 levels near the conduction-band minimum; oxygen vacancies in ZnO illustrate this problem.For ZnO, only the a1(1) level lies within the band gap.
- Anion Vacancies: AlN satisfies the anion-vacancy requirements by placing VN t2 levels within the band gap, while donor impurities can both bind the vacancy and supply electrons for the desired occupation.The screening criteria include a sufficiently large band gap, cation dangling-bond orbitals below the CBM, and small ΔAV.
Beyond NV Analogs
Only a small subset of deep centers has been examined, and future work must test other defect classes against the outlined criteria. Relaxing the requirement that spin be a good quantum number may enable relaxed optical selection rules and alternative optical-polarization mechanisms.
- Beyond NV Analogs: Only a small subset of the vast range of deep centers has been discussed in detail, leaving broader exploration necessary.Future work is needed to identify other classes compatible with the proposed defect and host criteria.
- Beyond NV Analogs: No isolated substitutional or interstitial impurity center satisfying criteria D1-D5 has yet been identified.Many such impurities act as deep centers, but their compatibility with all five criteria remains unresolved.
- Beyond NV Analogs: Vacancy-related complexes with D4h symmetry in MgO and CaO show optical spin polarization, but their common features with diamond’s NV-1 center require further study.These octahedrally coordinated hosts provide additional systems for exploration.
- Beyond NV Analogs: Removing the requirement that spin be a good quantum number relaxes optical selection rules and may permit alternative optical-polarization mechanisms.This broadens the classes of deep centers open to investigation.
Methods
The study combines first-principles supercell calculations with defect-formation-energy analysis and constrained DFT to evaluate NV centers in diamond and 4H-SiC. HSE06 calculations provide relaxed electronic structures, while internal defect transitions are treated as more reliable than defect-to-band transitions.
- Defect formation energies: Defect formation energies are computed from total-energy differences, chemical-potential references, charge-state Fermi-level terms, and valence-band alignment.For diamond, the carbon chemical potential is the energy per carbon atom in the crystal, and the Fermi level is referenced to the valence-band maximum.
- Computational setup: First-principles calculations use 64-atom diamond and 96-atom 4H-SiC supercells, finite-size corrections for charged defects, PAW pseudopotentials in VASP, a 400 eV cutoff, and a 2×2×2 k-point mesh.
- Electronic-structure calculations: HSE06 hybrid-functional calculations include atomic relaxations and yield band gaps of 5.36 eV for diamond and 3.17 eV for 4H-SiC.
- Site selection: The calculations select hexagonal sites in 4H-SiC because inequivalent-site energies differ by less than 0.1 eV, and use the shorter-bond-associated nitrogen site for SiC NV defects.
- Excitation energies: Constrained DFT excitation energies remove an electron from an occupied defect level and place it into another occupied defect level.Internal defect-level transitions are likely more accurate than defect-to-band transitions, which are used only qualitatively to assess whether such transitions are suppressed.
Figure Legends
The figure legends depict defect-state stability, spin-resolved electronic levels, optical transitions, and dangling-bond interactions for NV-1 in diamond and related SiC defects.
- Figure Legends: A figure legend identifies the stability of NV-1 in diamond and VSi-2 in SiC.The passage names the two defects in the context of stability.
- Figure Legends: Optical diagrams compare the NV-1 center in diamond with the NcVSi-1 center in SiC, marking absorption, emission, and zero-phonon-line transitions with their energies.The legends identify both systems and indicate the transition types and energies.
- Figure Legends: Atomic sp3 dangling bonds interact to form a1 and t2 levels in an ideal vacancy, with t2 splitting further in vacancy complexes.The legend presents this progression from isolated dangling bonds to vacancy-complex level structure.
Table Legends
The supplied legends identify defect-level energies, NV-1 optical spin transitions, defect formation-energy trends, vacancy-complex level diagrams, triplet-excitation coordinates, and defect-level development in tetrahedral semiconductors. Supporting material also outlines host screening, defect-level splitting, electron counting, and the need for first-principles validation.
- Table 1 reports defect-level energies for vacancy and NV centers in diamond and SiC, with underlining marking occupied levels.
- Figure 1 shows a 1.945 eV spin-conserving transition between NV-1 ground and excited triplets, with spin-selective decay enabling optical spin polarization and measurement.Decay from ms = ±1 excited-state sublevels to an intermediate singlet is much stronger than decay from ms = 0.
- Figures 2–5 present formation-energy plots, vacancy-complex defect-level diagrams, triplet-excitation configuration coordinates, and defect-level development in tetrahedrally coordinated compound semiconductors.
- Supporting information describes screening tetrahedrally coordinated hosts using band-structure parameters and a band-gap threshold above 2.0 eV, while noting that the threshold is arbitrary.Diamond, Si, and GaAs are listed for comparison in Table S1.
- Section S2: Trends in defect level splitting: Defect-level splitting depends on interacting dangling-bond orbitals, with anion electronegativity, orbital localization, and anion separation determining vacancy-level overlap.
- Section S3: Electron counting for defects: Electron counting identifies charge states that produce the occupations and spin states required for spin-conserving triplet excitation, but explicit first-principles calculations remain necessary for broader defect criteria.The NCVSi center in SiC is given as an example with ne = 6 and Q = −1.
Figure Legends
The legends define how defect levels develop in tetrahedrally coordinated semiconductors, describe vacancy-related complexes, and specify the sources and conditions for host-material parameters.
- Figure Legends: Figure S1 depicts sp3 dangling bonds forming a1 and t2 levels in an ideal vacancy, with further t2 splitting in vacancy complexes.The diagram covers the progression from atomic dangling bonds to vacancy-complex defect levels.
- Figure Legends: Figure S2 provides schematic defect-level diagrams for vacancy-related complexes with occupations of six and four electrons.The two occupations are labeled A and B in the legend.
- Table Legends: Table S1 lists host material parameters, with Eg and ΔSO values generally taken at room temperature from Ref. 29 unless otherwise noted.Asterisks, daggers, and double daggers identify data from Supporting Information References 1, 2, and 3, respectively.