Introduction

Silicon technology is probably the most advanced area of materials processing, and the readily available and low-cost starting materials ensure the enduring success of silicon-based electronics. Defects of the silicon crystal, which can be applied as quantum emitters and/or qubits, are receiving increasing attention recently, especially in view of integration with silicon based photonic devices. The defects suggested so far, like the G-centre (CSi-I-CSi)1,2, the T-centre ([C2]Si + H)3,4, the W-centre (I3)5,6,7, and the C-centre (Ci+Oi)8,9, are all complexes. (Here “I” denotes a silicon self-interstitial, the “Si” subscript a substitutional, while the “i” subscript an interstitial impurity.) The erbium-related centres10 are even more complicated and their exact structure is not even known. Tightly controlled generation of such complexes makes defect engineering very challenging, because formation of complexes requires special annealing procedures to combine the parts of the complex, and those often lead to the formation of unwanted defects at the same time. Therefore, it would be very advantageous to find a simple defect (containing a single impurity) with the appropriate properties. Recently, a defect, with photoluminescence (PL) in the telecom band11,12 could have been created and erased on demand by applying femtosecond laser pulses with varying the dose of irradiation. Based on the observed spectra, the defect was identified as a single carbon interstitial (Ci)13. The PL of Ci is well known from earlier studies5,14,15 and this defect has been amply characterized by deep level transient spectroscopy (DLTS)16,17,18 and electron spin resonance (ESR)18,19 as well, providing definitive information about its charge transition levels and geometrical structure. Unfortunately, to the best of our knowledge, theoretical calculations could not reproduce these properties so far20,21, making the assignment in Ref. 13 somewhat uncertain. It is also an open question whether a single Ci defect can act as a qubit when isolated with the afore-mentioned or other techniques.

Using advanced computational techniques, in this paper we provide a full characterization of the Ci defect, in very good agreement with PL, DLTS, and ESR data, and confirm that the observed emission can really be assigned to its neutral charge state, as a radiative recombination between a bound exciton singlet state and the closed shell localized singlet state. We also find that an optically addressable metastable triplet state also exists, so the defect could be applied as a quantum memory. Comparing it to a known optical detected magnetic resonance centre in silicon (G-centre), we propose that a carbon interstitial could act as a quantum bit and may realize a spin-to-photon interface in complementary metal-oxide semiconductor-compatible platforms.

Results

The standard implementations of first-principles DFT (density functional theory) contain approximate local- or semi-local exchange functionals (LDA and GGA, respectively), which underestimate the band gap and delocalize defect states. Since defects are usually calculated in big supercells of the host crystal, the application of first-principles many-body methods are as yet computationally prohibitive. Quantitatively correct results may be obtained within (generalized) Kohn-Sham DFT by the application of semi-empirical hybrid exchange functionals, which mix semi-local and non-local (Hartee-Fock-type) exchange. We have shown earlier that the two parameters of the Heyd-Scuseria-Ernzerhof (HSE) exchange functional22, i.e., α (for mixing non-local and semi-local exchange) and μ (to describe electronic screening), can be tuned so that the functional mimics the exact DFT exchange functional23,24. This means that it provides the piece-wise linear behaviour of the total energy as a function of the occupation numbers, with a proper derivative discontinuity at integer values25. The latter is equivalent with the reproduction of the exact single-particle band gap. The linearity condition is satisfied, when the generalized Koopmans’ theorem (gKT)26 is fulfilled, i.e., the position of the highest occupied (or lowest unoccupied) Kohn-Sham level matches the ionization energy (or electron affinity), calculated from total energy differences. We have also shown that such optimized HSE(α,μ) functionals can yield very accurate results for defects in semiconductors27,28,29. Particularly in silicon, the original HSE06 = HSE(0.25,0.20) parametrization30 is optimal, providing a (0 K) band gap of 1.16 eV, in excellent agreement with experiment31, and resulting in charge transition levels within 0.1 eV to the measured ones27. Therefore, in this study, we apply the HSE(0.25,0.20) functional to calculate the properties of Ci, as described in detail in the Methods section.

As is well known from ESR signals associated with its positive charge state18,19, the structure of the carbon interstitial, Ci, corresponds to a so-called [001] split interstitial (or dumbbell) configuration, often called (C-Si)Si[001], with the carbon and a silicon atom sharing a lattice site, and giving rise to C2v symmetry. As shown in Fig. 1, the threefold coordination results in sp2 hybridization, with pure p-like dangling bonds on both atoms, perpendicular to each other. The C 2p dangling bond is lower in energy than the Si 3p (due to the higher electronegativity of carbon). Figure1 also shows the occupations in the neutral ground state and in the possible excited states. Removing or adding an electron preserves the C2v symmetry. Table 1 shows the calculated charge transition levels in comparison to experiment. The improved agreement with respect to an earlier HSE calculation21 may be in part due to the larger supercell, and in part due to the use of the self-consistent potential correction method, instead of an a posteriori energy correction (especially for the less localized negative state).

Fig. 1: Electronic structure of Ci (alias (C-Si)Si[001]).
Fig. 1: Electronic structure of Ci (alias (C-Si)Si[001]).
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Schematic representation of the defect states in the neutral charge state, and their occupation in the ground state and in the dark and bright singlet excited states as well as in the metastable triplet state are depicted. The zero-field splitting (ZFS) of the latter is also indicated (with D = 439 MHz and E = 38 MHz), from which intersystem crossing (ISC) may occur. The square of the wavefunction for the C 2p and the Si 3p orbital is displayed at equal isovalues. (As can be seen, the Si 3p state is less localized than the C 2p.) The wave function alternates its sign around the node. The calculated HSE06 values for the zero phonon line (ZPL) are given, but the arrows showing the transitions are not to scale. CBM denotes the conduction band minimum.

Table 1 Calculated and observed charge transition levels, E( + /0) and E(0/-)
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Our calculated formation energy for the neutral charge state (with respect to perfect silicon and diamond) is 3.72 eV, as also found in earlier calculations32. We have reported previously33 the calculated diffusion barrier of Ci, via the reorientation mechanism of Ref. 34 to be 0.70 eV (also in good agreement with the observed values between 0.72 and 0.75 eV18,35), which explains the stability of this colour centre below the annealing temperature at ~50 °C18. (Note that the conversion of Ci to CSi + Sii is endotherm by 1.86 eV so, unless mobile species are created, Ci is stable up to the temperature it begins to move.)

We first study the ESR fingerprints of the defect which provide direct information about the localization of the defect wave functions via hyperfine interaction between the electron spin and the nuclear spins. This comparison both verifies the accuracy of our calculations and provides important information about the feasibility of employing this colour centre as a qubit. Table 2 contains the hyperfine tensor in the positive charge state, showing fair agreement with experimental values obtained on a sample enriched with 13C. This indicates that the applied hybrid functional describes the localization of the defect wave functions well. Hyperfine interaction with 29Si have not been published to our knowledge but we provide the calculated data in Supplementary Table 1. (We note that another ESR centre was associated with the negatively charged (C-Si)Si[001] defect18 but no trace of hyperfine interaction with 13C or 29Si – within 0.16 T ≈ 4 GHz – was found.) Based on the good agreement of the charge transition levels and the hyperfine data on the 13C atom in the positive charge state, we conclude that the (C-Si)Si[001] structure is well established, and at the same time the applied method is validated. Next, we focus on the interpretation of the PL spectrum5,13.

Table 2 Calculated and observed hyperfine tensor of the positively charged carbon interstitial, 13Ci
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The observed zero-phonon line (ZPL) associated with the Ci defect is at 0.856 eV according to the literature (see, e.g., Ref. 15), so the corresponding wave length (1448 nm) falls into the telecom region. We note that the defect is susceptible to strain (see e.g., Ref. 19) and that is likely responsible for the observed variance in the ZPL emissions of the defects in the silicon-on-insulator (SOI) sample13 which produce a strain field towards the defects in silicon. The observed charge transition levels clearly imply that the negative and positive charge states cannot explain this emission as they would immediately be converted to the neutral state by illumination with higher-than-ZPL energy. (If the defect was in the positive charge state then photo-excitation with higher-than-ZPL energy of the Ci center will promote an electron from the valence band to the empty defect level in the gap which turns it to neutral; if the defect was in the negative charge state then the same photo-excitation energy will promote the electron from the occupied defect level to the conduction band which again turns it to neutral.) We continue the discussion of the emission for the neutral charge state. The ground state is a closed-shell singlet which transforms as the trivial A1. The excited state may be constructed by promoting an electron from the carbon dangling bond to the silicon dangling bond (see Fig. 1). The calculated ZPL energy (see Supplementary Note 2 for details) is 0.86 eV, including an exchange correction of 0.29 eV, which brings it close to the experimental value. However, this excited state is dark and transforms as A2 and the optical transition is only allowed by phonon participation which clearly goes against the observed PL spectrum showing a strong ZPL emission. On the other hand, it is intriguing that the donor (+/0) charge transition level, obtained by DLTS is ≈ EC - 0.87 eV (using the 0 K band gap for conversion where EC is the conduction band minimum), which is only 14 meV higher than the ZPL energy. Thus, the defect may have a bound exciton excited state where the hole is located in the carbon dangling bond orbital and the electron sits on a state split from conduction band minimum (CBM) and the binding energy of the exciton is about 14 meV. Our calculations imply that the dark singlet excited state lies at higher energy than the singlet bound excited state does. Indeed, the calculated ZPL of the bound exciton recombination is 0.84 eV, in good agreement with the measured value of 0.856 eV.

By considering a singlet bound exciton state, the radiative recombination would be allowed in first order. Indeed, the calculated radiative lifetime is 2.83 μs (see Methods). The estimated radiative lifetime is relatively long because the excited state is delocalized whereas the ground state is localized, which results in a relatively weak optical transition dipole moment (0.96 D). Nevertheless, the spectrum does show a significant ZPL emission according to our simulations (see Fig. 2). The phonon sideband in the PL spectrum can be well explained by the ion relaxations going from the equilibrium geometry of the positively charged defect to that of the neutral defect. The features can be identified as the phonon modes of the Si crystal as the positively charged defect produces a larger tensile strain than the neutral one does. The calculated Huang-Rhys (HR) factor is 2.88. Based on these results, we establish the origin of the PL signal as an optical transition from a bound exciton state to the closed-shell singlet ground state of the neutral Ci defect.

Fig. 2: The calculated and the observed14 photoluminescence (PL) spectra, including the phonon sidebands.
Fig. 2: The calculated and the observed14 photoluminescence (PL) spectra, including the phonon sidebands.
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Here we use the neutral and the positive charge state’s geometries as ground and excited states, respectively, to generate the phonon sideband of the PL spectrum. The simulated spectrum is aligned to the experimental one at the zero-phonon line for direct comparison of the experimental and simulated phonon sidebands. We note that the experimental spectrum sits on a tail of a broad background that is not present in the simulated spectrum.

We now discuss the presence of optically inactive or dark states and their roles. We assume that the dark singlet level may fall above the ionisation energy of the neutral defect. In the opposite scenario, the dark state would significantly reduce the brightness of the defect where the bright state may be activated by elevating the temperature to occupy the bright singlet state. However, it was observed in a recent study13 that the emission intensity is decreasing with raising the temperature which clearly indicates that the dark state’s level should lie above the bright state’s level. The temperature dependence of the PL intensity can be explained by the thermal ionisation of the bound exciton excited state.

We have also explored a triplet manifold which is a metastable state of the defect. This may be generated by promoting an electron from the carbon dangling bond orbital to the silicon dangling bond orbital with spin flip. The corresponding 3A2 level lies 0.29 eV above the ground state. The triplet state keeps the C2v symmetry. The spin levels split by the orthorhombic crystal field with D = 439.3 MHz and E = 37.9 MHz. The 13C and 29Si hyperfine couplings are characteristic to the defect as the spin density is mostly localized on both dangling bonds (see Supplementary Table 3).

Discussion

Silicon is a very promising platform for hosting near-infrared single photon emitters and quantum bits realized by fluorescent point defects12. Stable defects with a single impurity may be preferred over complexes where the latter often require difficult defect engineering protocols to create them even with relatively low yield. The recently demonstrated programmable creation of carbon interstitial defects13 gives hope that telecom wave length single photon emitters can be generated on demand in a given density with good scalability.

The calculated radiative lifetime is 2.83 μs whereas the observed PL lifetime upon 532-nm pulsed laser excitation is between 3 and 8 ns13. The PL lifetime inherits both radiative and non-radiative components where various non-radiative processes may occur such as internal conversion, intersystem crossing or ionisation. Since the excitation energy of a 532-nm laser (2.33 eV) is much larger than the band gap of Si and stable positive and negative charge states of the Ci defect exist, it is likely that ionisation and re-ionisation processes occur depending on the excitation power. Indeed, the PL intensity depends non-linearly on the laser power13 which is a clear signature that photo-ionisation processes take place in the Ci defect. Our calculated radiative rate (2.83 µs), in relation to the experimentally observed total rate (between 3 and 8 ns) implies that the quantum yield of excitation is about 0.1% at the given experimental conditions in Ref. 13 In our estimation, the Debye–Waller factor of this defect emitter is 0.056 which results in 5.6% coherent no-phonon emission of the total emission. As a consequence, observation of single defect emission requires a photonics structure at the ZPL wavelength in order to significantly enhance the PL intensity. This concept was already demonstrated for the G-centre36 and the T-centre37, so it is viable towards Ci emitters too. We predict that the polarization of the emitted coherent photons is perpendicular to the [001] symmetry axis based on the symmetry of the excited state and the ground state.

In order to turn the single photon emitter into a quantum repeater unit, a quantum memory should be associated with the defect. We found a metastable triplet state which can act as a quantum memory. Decay from the metastable triplet state to the singlet ground state occurs via intersystem crossing (ISC). According to group theory, the |3A2, mS = 0> state is linked to the ground state 1A1, whereas the ISC is forbidden from |3A2, mS = ±1> in the first order. This means that the lifetime of |3A2, mS = 0> is shorter than that of |3A2, mS = ±1 > . This electronic structure and selection rules of ISC strongly resembles those of the G-centre in Si (Ref. 2) for which optically detected magnetic resonance (ODMR) of the electron spin was already demonstrated38. We provide the hyperfine coupling parameters in Supplementary Note 3 that should be assessed in future experiments for identification of the centre. By applying a microwave pulse resonant with the |3A2, mS = 0> and |3A2, mS = +1> or |3A2, mS = -1> levels (see Fig. 1), a contrast in the PL emission of the Ci defect is expected upon resonant microwave transition. Proximate nuclear spins may be applied as quantum registers to store the quantum information which was encoded in the electron spin. On the other hand, too strong hyperfine interaction between the electron spin and nuclear spins has a detrimental effect on the coherence of the electron spin. According to our calculations, the hyperfine constants of 29Si with I = 1/2 nuclear spin in the second and third neighbour shells are all in the order of 10 MHz which could significantly shorten the coherence time of the electron spin. Therefore, it is critical that the defect should be created in isotope engineered silicon with significant reduction of 29Si content (to about 0.5% from the natural abundant 4.5%) so that the probability of finding one 29Si around the defect up to the third neighbour shell could be reduced below 1%. The farther situated 29Si nuclear spins can be safely applied as quantum memories where the quantum information from the electron spin to the nuclear spin can be realised by the Landau-Zener effect or other techniques.

Conclusion

In conclusion, our findings on the electronic structure and magneto-optical properties of a simple carbon interstitial defect in silicon establish a spin-to-photon interface with ZPL emission at the wave length compatible with the fibre optics based communication in the most mature optoelectronics platform. Our analysis revealed that it is critical to engineer this colour centre into a photonics structure for observation of single defect emission, and the ODMR experiments and quantum memory operation can be realised in 28Si isotope enriched silicon host.

Methods

HSE(0.25,0.20) calculations were carried out with the Vienna ab-initio simulation package VASP 5.4, using the projector augmented wave method39,40,41, and a plane wave cutoff of 420 (840) eV for the wave function (charge density). Defects were modelled in a 512-atom supercell (4×4×4 multiple of the conventional Bravais-cell), and all atoms were allowed to relax in a constant volume till the forces were below 0.01 eV/Å. The Γ-point approximation was used for Brillouin-zone sampling. The lattice constant was taken from our earlier HSE06 work27 to be 5.4307 Å (in good agreement with experiment). Total energies of charged systems have been calculated by applying the self-consistent potential correction (SCPC) method for charge correction42. Charge correction was also used in the case of the bound exciton (where the electron wave function is delocalized), as described in Section 3.3.4 of Ref. 43. The zero phonon line (ZPL) was obtained as the energy difference of the relaxed ground and the relaxed excited state, the latter calculated with constrained occupation or ΔSCF method. To obtain the correct ZPL for the singlet-to-singlet transition, an exchange correction in the excited state was applied44 in order to obtain the total energy in the spin singlet eigenstate (see also Supplementary Material in Ref. 45). In case of the bound exciton, a band-filling correction was also used46. (The Γ-point in the reduced Brillouin-zone of the supercell represents the [0.75,0,0] primitive k-point explicitly, while the real conduction band minimum is at [0.83,0,0]. The difference, 50 meV, was subtracted from the calculated transition energy.) The spectrum of the phonon replicas were computed by the generating function method47, based on vibration calculations using the Perdew-Burke-Ernzerhof (PBE)48 functional. The radiative lifetime was calculated with the inverse of \({\Gamma }_{{{{\rm{rad}}}}}=\,\frac{{n}_{D}{E}_{{{{\rm{ZPL}}}}}^{3}{\mu }^{2}}{3\pi {\varepsilon }_{0}{c}^{3}{\hslash }^{4}}\) (see Ref. 49), where the optical transition dipole moment μ was calculated by taking the respective Kohn-Sham wave functions representing the electronic ground and excited states, and nD = 3.485 is the refractive index of silicon at 1.45 μm or ≈0.85 eV (see Ref. 50), EZPL = 0.856 eV is the observed energy of the ZPL, ε0 is the dielectric permittivity of vacuum, c is the speed of light and \(\hslash\) is the reduced Planck-constant.

The hyperfine tensors and zero-field splitting tensor were calculated as implemented in VASP51,52 where the zero-field splitting tensor algorithm was implemented by Martijn Marsman.