Transition metal impurities in silicon: computational search for a semiconductor qubit

The study addresses the challenge of realizing scalable, higher-temperature semiconductor qubits by seeking optically addressable, spin-triplet defect centers in silicon analogous to NV centers in diamond. Semiconductor qubits—gate-defined nanostructures, shallow donors, and optically addressable defects—each face limitations in scalability, operating temperature, or photonic interfacing. While NV centers operate at room temperature and offer spin-photon interfaces, their emission is poorly transmitted in silica fibers and diamond is difficult to scale. Silicon benefits from mature manufacturing but lacks spin-photon interfaces and typically requires cryogenic temperatures for long coherence. The research question is whether transition metal impurities in crystalline Si can host deep, optically allowed triplet–triplet transitions within the Si band gap, enabling optical initialization and readout, potentially improving operating temperatures for quantum sensing and enabling spin-photon interfaces, particularly in the mid-infrared.

Transition metal impurities in silicon have long been studied via EPR and first-principles methods. The Ludwig–Woodbury model explains many EPR observations via tetrahedral crystal fields and Hund’s rules. Beeler et al. (1990) performed LDA Green’s function calculations for most 3d TMs in Si (unrelaxed structures), qualitatively matching experiments but deviating from Ludwig–Woodbury and indicating Hund’s rule breakdown for early interstitial and late substitutional TMs. That work was limited by LDA band-edge inaccuracies and lack of relaxation. Hybrid DFT (HSE06) improves Si band gap predictions but overlocalizes TM d states due to homogeneous screening; approaches to address this include enforcing generalized Koopmans’ condition (gKC), e.g., via occupation-dependent potentials (Ivady et al.). Previous focused studies (e.g., Fe interstitial) show improved agreement when gKC is considered, but a comprehensive, gKC-aware survey across the 3d series in Si with structural relaxation and optical properties has been lacking. On the experimental side, few Si color centers are known due to Si’s small band gap; reported centers include Se+, Er3+, and irradiation-induced centers such as the T- and G-centers, with recent telecom-band single-photon emission (e.g., G-center ~1.27 µm; T-center ~1326 nm) but lacking triplet–triplet transitions analogous to NV centers. This motivates a systematic computational search for TM-based deep centers in Si supporting triplet–triplet optical transitions.

The authors performed a systematic defect screening of all 3d and selected 4d/5d transition metals in crystalline silicon, considering interstitial, substitutional, and vacancy–TM complex configurations. The workflow: (1) compute defect formation energies and thermodynamic charge transition levels (CTLs) using spin-polarized hybrid DFT (HSE06) with structural relaxation, (2) evaluate deviation from generalized Koopmans’ condition (non-Koopmans’ energy, E_NK) and apply an occupation-dependent potential (HSE06+U) to TM d orbitals when |E_NK| > 0.2 eV, targeting E_NK < 0.1 eV where applied, (3) construct single-particle defect-level diagrams (DLDs), (4) compute optical absorption spectra to confirm allowed in-gap transitions, and (5) apply selection criteria requiring at least two CTLs in the gap, a spin-triplet ground state, a spin-triplet excited state, and optically allowed transitions within the Si band gap. Computational details: VASP 5.4.4 (GPU-enabled) with PAW potentials; HSE06 with standard mixing parameter α=0.25 yields Si indirect band gap ~1.16 eV and lattice constant ~5.43 Å. Supercells of 216 atoms with Γ-point sampling; plane-wave cutoff 340 eV. Defect formation energies include corrections for finite-size effects, potential alignment, and band-edge issues following Lany–Zunger, implemented per Goyal et al.; static dielectric constant ε=11.11 for image charge corrections. CTLs are extracted from formation energy crossovers. Non-Koopmans’ energy E_NK = ε_N − (E_N − E_{N−1}) is used to assess gKC; when necessary, a Dudarev-type U term is self-consistently determined and applied only to TM d orbitals (leaving Si sp3 states described by HSE06). Optical absorption coefficients are calculated from the complex dielectric function via linear-response using HSE06 Kohn–Sham states, Kramers–Kronig with complex shift 0.009, energy resolution 0.01 eV, and ~800 empty bands for the ~864-electron supercell, targeting the low-energy (mid-IR) spectral range. Additional practical screening considered diffusion coefficients (mobility), nuclear spins of TM isotopes, and potential for ion implantation from the literature. Electron-counting arguments were used to rationalize which TM columns and charge states can host triplet ground/excited states: substitutional candidates tend to appear near the Zn column where two or four in-gap electrons can form triplets; interstitial candidates are favored among early TMs (left of V column) where outer s/d electrons populate states near the gap without Si dangling bonds.

- Identified seven transition metal impurity defects in Si (three substitutional, four interstitial) that exhibit optically allowed triplet–triplet electronic transitions within the Si band gap, analogous in operating principle to NV centers but emitting in the mid-infrared. - Charge transition levels predicted by HSE06(+U) generally agree well with available experimental data (within ~0.1 eV uncertainty from band edges and residual gKC deviations), while also providing predictions for less-studied defects and challenging some prior CTL assignments. - The optical transition energies fall in the mid-IR, aligning with atmospheric transmission windows and suggesting potential for free-space quantum communication; fiber transmission in silica would be strongly attenuated. - Electron-counting rules explain the positions of viable candidates on the TM periodic table: substitutional candidates cluster near the Zn column where in-gap electron counts of two or four favor triplet ground states; interstitial candidates are favored among early TMs (up to V, possibly Cr with higher charge states) due to their outer s/d electrons contributing to in-gap states without Si dangling bonds. - Estimated smallest in-gap level separations for candidates are ~0.1–0.2 eV, implying limited but non-negligible thermal excitation at room temperature (Boltzmann estimate: ~0.3% excitation at 0.15 eV versus ~0% at 0.6 eV for NV), suggesting that thermal promotion among defect levels may not be the dominant fidelity limiter in Si TMs. - Practical considerations: diffusion data indicate some TMs (Co, Cu) are highly mobile at room temperature and less suitable for high-temperature operation; V and Zn are fairly mobile at high temperature; Sc and Zr are essentially immobile after cooling. Among screened candidates, Zr (including Zr^{+2}) emerges as particularly promising from a mobility standpoint. - Conceptual insight: TM defect behavior and color-center viability may be portable across hosts with similar lattice symmetry and bond lengths; the relative positioning of TM s/d levels to host band edges is critical for forming deep in-gap triplet manifolds.

The work directly addresses the need for optically addressable spin defects in silicon by computationally identifying TM impurities that support in-gap triplet–triplet transitions and allowed optical selection rules. This provides a plausible route to optically initialize and read out spin states in Si, enabling a spin–photon interface and potentially higher operating temperatures for sensing compared to shallow donors. Emission in the mid-IR suggests use in free-space quantum links through atmospheric windows, though not in standard silica fibers. The electron-counting framework and periodic trends rationalize which TM species and charge states can host triplet manifolds in Si, guiding targeted experimental efforts. Agreement of predicted CTLs with experiments where available supports the methodology (HSE06 with gKC-informed +U corrections). Remaining steps—quantifying zero-field splitting for room-temperature spin addressability, computing ZPL energies and Debye–Waller factors for indistinguishable photon generation, and modeling intersystem crossing rates—are identified to translate these candidates into functional Si-based qubits and spin–photon interfaces. The results also suggest broader portability of TM-based color centers across materials with compatible symmetries and bonding, informing searches beyond Si.

This study provides a comprehensive, gKC-aware hybrid-DFT survey of transition metal impurities in silicon, identifying seven defects (three substitutional, four interstitial) with optically allowed triplet–triplet transitions within the Si band gap. The findings open a pathway toward Si-embedded spin–photon interfaces and higher-temperature quantum sensing relative to shallow donors, with mid-IR emission suitable for free-space links. The work reconciles and extends experimental CTL data for TM defects in Si and offers electron-counting rules that rationalize candidate placement in the periodic table. Future work should (i) compute zero-field splitting to ensure magnetic-field-free spin sublevel addressability, (ii) determine ZPL energies and Debye–Waller factors for photonic applications, (iii) model intersystem crossing via first-principles phonon coupling to enable optical initialization, and (iv) assess isotope selection, defect mobility, and complex formation (e.g., with H) to guide fabrication. Zr-based centers, given low diffusion, appear particularly promising for experimental realization.

- The study is computational and constitutes an initial screening; experimental validation of optical spectra, charge states, and spin properties is required. - The generalized Koopmans’ condition is enforced selectively (|E_NK| > 0.2 eV threshold), not exhaustively across all transitions; residual deviations and band-edge uncertainties (~0.1 eV) remain. - Optical analyses are limited to absorption selection rules and energies; zero-phonon lines, Debye–Waller factors, and intersystem crossing rates are not computed. - Spin Hamiltonian parameters (e.g., zero-field splitting) and spin coherence properties are not evaluated. - Defect complexes with common impurities/dopants (e.g., hydrogen) are not considered and could alter CTLs or stability. - Mobility and diffusion-related constraints are discussed qualitatively from literature; detailed kinetic modeling in device-relevant conditions is not performed. - Mid-IR emission limits use in standard silica fibers, affecting long-distance fiber-based quantum communication applicability.