Introduction

Silicon is a mature platform for quantum technology, offering multiple advantages for the integration of defect quantum emitters. As compared to other host materials1, high-purity isotopically enriched single crystals of silicon2,3 and integrated photonic devices4,5 are readily available. These characteristics of the host material enable the improvement of optical and spin properties of the quantum defects. Recent experimental advances allowed for the isolation of single quantum emitters in this material, such as G-, W- and T-centres, with a great promise for quantum communication6,7,8,9,10,11,12,13,14,15,16,17. However, to further reduce the transmission losses in optical fibres, it is essential to find an active defect with the emission wavelength in either the L- or the C-band and an optically addressable spin state. This caused a revival of interest in the magneto-optical properties of deep defects in silicon, including those configurations that have been discovered decades ago11,18,19,20,21.

Carbon and oxygen atoms are the most common impurities and therefore are important sources of colour centres in silicon crystals. When both are present in a sample, the oxygen interstitial is able to trap a mobile carbon atom, forming a carbon–oxygen interstitial pair (CiOi)22,23. This stable defect complex was thoroughly characterised by various experimental techniques. Its photoluminescence (PL) signal was associated with the C-line at 789 meV24,25 in the spin singlet, neutral charge state26 and with the Si-G15 electron paramagnetic resonance (EPR) centre in the spin doublet, positive charge state27. Local vibrational modes (LVM) in the PL spectrum were identified at 64.5, 72.6, 138.1, and 145.3 meV energies24,25. The uniaxial stress perturbation effect on the PL line was consistent with C1h point symmetry26. Deep-level transient spectroscopy (DLTS) measurements identify a deep donor level at EVBM + 360 meV position28. Time-resolved PL measurements reveal the excited state lifetime of 2.3 μs at cryogenic temperatures and a second slower decay activated at elevated temperatures with an activation energy of 3.1 meV29. The latter was associated with a reverse intersystem crossing (rISC) from a lower-lying triplet excited state. From the measurements, radiative and ISC lifetimes of 14 and 2.8 μs were obtained, respectively. PL excitation measurements reported a series of excited states, revealing the bound exciton origin of the C-line30. It was assumed, that the triplet state is dark owing to the quenched orbital momentum of the hole by the defect strain field. However, this phosphorescence was detected at low temperatures despite the measured isotropic g-factor in the triplet state31.

The defect structure was tentatively identified32,33 using density functional theory (DFT) calculation within the local density approximation (LDA)34,35. The identification was based on the calculated LVMs and the position of the donor level, where the latter could be compromised by the underestimation of the band gap by DFT-LDA method. The atomic structure of the defect is shown in Fig. 1a. It represents a pair of neighbouring Si–C and Si–O defects, connected into a square-like structure by a mutual displacement from the initial split-interstitial positions towards the opposite directions (up–down configuration)36.

Fig. 1: Geometry and electronic structure of the defect.
Fig. 1: Geometry and electronic structure of the defect.
Full size image

a The atomic structure of the CiOi defect complex, consisting of neighbouring carbon (Ci−Si1) and oxygen (Oi−Si2) split-interstitial defects. b Kohn–Sham level structure of the \({{{{{\rm{C}}}}}_{{{{\rm{i}}}}}{{{{\rm{O}}}}}_{{{{\rm{i}}}}}}^{0}\) defect ground state (\({}^{1}{A}^{{\prime} }\)), singlet (1A″), and triplet (3A″) excited states in the spin-polarised HSE06 calculation. VBM, CBM, and C labels show the valence band maximum, the conduction band minimum and the dangling bond orbital of the carbon atom, respectively.

The C-centre in silicon exhibits emission in the telecom L-band with a confirmed high electron spin state which has a great potential in quantum communication with the possibility of scalable creation of the quantum emitters in carbon and oxygen-contaminated silicon by CMOS compatible technology (e.g., ref. 12). However, the lack of understanding the nature of the electronic structure and the origin of the emission of the C-centre precludes it from being considered as a qubit. In this work, we aim to significantly advance the magneto-optical characterisation of the \({{{{{\rm{C}}}}}_{{{{\rm{i}}}}}{{{{\rm{O}}}}}_{{{{\rm{i}}}}}}^{0}\) defect by means of high-level DFT calculations, in order to elaborate on the feasible quantum optics protocols for quantum technology applications. To this end, we determine the key zero-field splitting (ZFS) parameters of the defect, taking into account the effects of hyperfine and spin–orbit-interaction. We propose a spin-selective optical cycle for the coherent manipulation and sensitive readout of the triplet electron spin state, suitable for the optically detected magnetic resonance (ODMR) measurements. Hyperfine interaction between the electron and proximate nuclear spins mediates to realise a quantum register with long-living memory as the ground state singlet does not disturb the proximate nuclear spin states.

Results

Electronic structure

The calculated electronic structure of the \({{{{{\rm{C}}}}}_{{{{\rm{i}}}}}{{{{\rm{O}}}}}_{{{{\rm{i}}}}}}^{0}\) defect is shown in Fig. 1b. The only in-gap state is induced by the carbon impurity dangling bond resulting in a deep donor level. The calculated (+/0) charge transition level of EVBM + 408 meV is in good agreement with the DLTS level (360 meV)28. Note that we label the states by their irreducible representations in the C1h point symmetry group. The \({}^{1}{A}^{{\prime} }\) is the ground state with two electrons occupying the deep defect level. Promoting one electron from the deep level to the conduction band minimum in the same or opposite spin channel results in the singlet and triplet excited states, respectively. In the double group notation, the irreducible representations of these semi-localised defect states, formulated in the two-electron representation in the orbital and spin product basis, are as follows

$${| }^{1}\left.{A}^{{\prime} }\right\rangle =\frac{1}{\sqrt{2}}\left\vert {a}^{{{\prime\prime}} }{a}^{{{\prime\prime}} }\right\rangle \otimes \left\vert \uparrow \downarrow -\downarrow \uparrow \right\rangle =\left\vert {{{\Gamma }}}_{1}\right\rangle ,$$
(1)
$${| }^{1}\left.{A}^{{{\prime\prime}} }\right\rangle =\frac{1}{2}\left\vert {a}^{{{\prime\prime}} }{a}^{{\prime} }+{a}^{{\prime} }{a}^{{{\prime\prime}} }\right\rangle \otimes \left\vert \uparrow \downarrow -\downarrow \uparrow \right\rangle =\left\vert {{{\Gamma }}}_{2}\right\rangle ,$$
(2)
$${| }_{0}^{3}\left.{A}^{{{\prime\prime}} }\right\rangle =\frac{1}{2}\left\vert {a}^{{{\prime\prime}} }{a}^{{\prime} }-{a}^{{\prime} }{a}^{{{\prime\prime}} }\right\rangle \otimes \left\vert \uparrow \downarrow +\downarrow \uparrow \right\rangle =\left\vert {{{\Gamma }}}_{2}\right\rangle ,$$
(3)
$${| }_{\pm }^{3}\left.{A}^{{{\prime\prime}} }\right\rangle =\frac{1}{2}\left\vert {a}^{{{\prime\prime}} }{a}^{{\prime} }-{a}^{{\prime} }{a}^{{{\prime\prime}} }\right\rangle \otimes \left\vert \uparrow \uparrow \pm \downarrow \downarrow \right\rangle =\left\vert {{{\Gamma }}}_{1}\right\rangle .$$
(4)

The calculated zero-phonon line (ZPL) energy in the singlet-singlet transition is 750 meV after the same finite size correction as for the charge transition level calculation was applied (see Methods). This correction is justified by the excitonic nature of the excited states, where the electrostatic potential of the localised hole is corrected similarly to the ionised ground state37. The calculated ZPL energy is smaller than the calculated donor transition energy \(\left({E}_{{{{\rm{CBM}}}}}-{E}_{(+/0)}=780\,{{{\rm{meV}}}}\right)\), leading to a stable neutral emitter. We obtain the energy of the triplet level to be 2.8 meV below the singlet excited state, in good agreement with the temperature activation energy (3.1 meV) for the rISC process29. We validate our HSE06 DFT results by comparison with G0W0 and Bethe-Salpeter-equation (G0W0+BSE) calculations in 216-atom supercell model38,39,40 (see Supplementary Note 3). To this end, we use the HSE06 relaxed ground state atomic positions as the computation of the quantum mechanical forces at the G0W0 + BSE level has not yet been implemented. Thus, we can compare the ground state electronic structure and the vertical excitation energies in both calculation methods. We obtain energy differences between the HSE06 ΔSCF and G0W0+BSE calculations of 11 and 15 meV for the vertical excitation energy and the excited state singlet-triplet splitting, respectively. The G0W0+BSE exciton wavefunction is dominated by a single Slater determinant in complete agreement with the ΔSCF electronic configuration. This justifies the ΔSCF method for the excitation calculations of the defect.

Spin properties

After obtaining the electronic states, we explore the spin interactions crucial for the tight control of the quantum bit defect. We start with the calculation of the dipolar spin–spin interaction of the electron spin in the triplet excited state, governed by the Hamiltonian operator

$${\hat{H}}_{{{{\rm{SS}}}}}=D\left({\hat{S}}_{z}^{2}-\frac{S\left(S+1\right)}{3}\right)+\frac{E}{2}\left({\hat{S}}_{+}^{2}+{\hat{S}}_{-}^{2}\right).$$
(5)

Our calculations result in axial D = −771 MHz and rhombic E = 94 MHz ZFS parameters. In order to describe the new eigenstates, we introduce the magnetic quantum number mS in the notation. The D parameter lifts the degeneracy of the triplet state to 2Γ1(±) + Γ2(0), while E further splits the ±1 spin states resulting in Γ1(−), Γ1(+), Γ2(0), in ascending energy order.

Spin-orbit coupling is described by the Hamiltonian operator

$${\hat{H}}_{{{{\rm{SO}}}}}={\lambda }_{z}{\hat{L}}_{z}{\hat{S}}_{z}+{\lambda }_{+}{\hat{L}}_{+}{\hat{S}}_{-}+{\lambda }_{-}{\hat{L}}_{-}{\hat{S}}_{+},$$
(6)

which belongs to the totally symmetric Γ1 irreducible representation in the double group. The selection rules for spin–orbit coupling are depicted in Fig. 3a. We further express the two-particle operators in (6) with single-particle ones (\({\hat{l}}_{z}{\hat{s}}_{z}\), \({\hat{l}}_{\pm }{\hat{s}}_{\pm }\)) and expand the matrix elements between the wavefunctions to a combination of matrix elements between one-particle spin-orbitals (see Supplementary Note 1). Furthermore, we substitute the DFT-calculated matrix elements Λij of the spin-orbit Hamiltonian expanded on the ordered basis functions of the one-particle spin–orbitals in the above expansion, namely,

$$\begin{array}{c}\begin{array}{cccc}\left| a^{\prime\prime\uparrow} \right\rangle & \left| a^{\prime\prime\downarrow} \right\rangle & \left| a^{\prime\uparrow} \right\rangle &\left| a^{\prime\downarrow} \right\rangle\end{array}\\ {{\Lambda}}\left[{{\mathrm{meV}}}\right]= \left(\begin{array}{cccc} 0.186 & 0 & 0 & 0.082+0.061i\\ & 0.735 & 0.235 & 0\\ & & -0.010 & 0\\ & & & -0.010.\end{array}\right)\end{array}$$
(7)

This way, the spin–orbit couplings to each sublevel are obtained as follows

$${\lambda }_{z}=\frac{1}{2}\left({{{\Lambda }}}_{11}-{{{\Lambda }}}_{22}-{{{\Lambda }}}_{33}+{{{\Lambda }}}_{44}\right)=-0.275\,{{{\rm{meV}}}},$$
(8)
$${\lambda }_{-}=\frac{1}{\sqrt{2}}\left({{{\Lambda }}}_{23}+{{{\Lambda }}}_{14}\right)=\left(0.223+0.043i\right)\,{{{\rm{meV}}}},$$
(9)
$${\lambda }_{+}=\frac{1}{\sqrt{2}}\left({{{\Lambda }}}_{23}-{{{\Lambda }}}_{14}\right)=\left(0.108-0.043i\right)\,{{{\rm{meV}}}}.$$
(10)

The spin–orbit coupling of the singlet and triplet states reshapes the relative energies of the spin sublevels within the excited triplet state. It provides a large stabilisation for the mS = 0 level, but is negligible for the other two sublevels, owing to the differences in the energy gaps. Thus, the axial ZFS parameter is modified to D = 5.7 GHz, which is in reasonable agreement with the ~3 GHz readout from the broad peaks in the magneto-PL spectrum31. The re-normalisation also leads to a strong mixing of the excited state singlet and triplet mS = 0 levels, manifesting as a 1% contribution of the triplet state to the excited singlet state.

In addition, we calculate the hyperfine interaction

$${\hat{H}}_{{{{\rm{hyp}}}}}={A}_{xx}{\hat{I}}_{x}{\hat{S}}_{x}+{A}_{yy}{\hat{I}}_{y}{\hat{S}}_{y}+{A}_{zz}{\hat{I}}_{z}{\hat{S}}_{z}$$
(11)

at 13C, 17O, and 29Si isotopes surrounding the defect (see Supplementary Note 2). We conclude that the hyperfine interaction in the neutral triplet excited state of the defect is very similar to that of the positive doublet ground state, only incorporating a scaling factor of 2 of the two spin quantum numbers in the neutral and positive charge states. It can be explained by the delocalised nature of the excitonic excited state, so the similar hole spin density governs the interaction as for the positively charged defect. We find good agreement for the calculated positively charged defect 13C hyperfine parameters of Axx = 5.9 MHz, Ayy = 5.2 MHz and Azz = 153.9 MHz with the reported values of the Si-G15 EPR centre27. The corresponding values in the neutral triplet state are Axx = 3.1 MHz, Ayy = 2.7 MHz and Azz = 77.2 MHz. Additionally, we report the hyperfine couplings of nearest-neighbour (NN) and next-NN (NNN) silicon nuclei [see Supplementary Table I and Supplementary Fig. 1a].

Optical properties

Having characterised the spin properties of the defect, we turn to the detailed description of the optical transitions. First, we calculate the phonon spectrum and simulate the PL spectrum [see Fig. 2a] according to the Franck–Condon principle. We obtain a sideband with LVMs at 63, 70 and 134 meV energies closely resembling the experimental spectrum24,25,31. We also observe an LVM at 87 meV, giving rise to a respective weak experimental signal [see Fig. 2b]. The calculated Huang–Rhys factor is 2.1, corresponding to a Debye–Waller factor of 12%. The bound exciton nature of the excitation results in a relatively delocalised electron orbital, accounting for the observed long optical lifetime in experiments29.

Fig. 2: Simulated photoluminescence spectrum.
Fig. 2: Simulated photoluminescence spectrum.
Full size image

a Calculated PL spectrum of \({{{{{\rm{C}}}}}_{{{{\rm{i}}}}}{{{{\rm{O}}}}}_{{{{\rm{i}}}}}}^{0}\) defect, showing Huang-Rhys factor of S = 2.1. b Calculated PL sideband relative to the ZPL energy and the Si partial Huang–Rhys factors. Our results are compared to the experimental PL sideband reported by Ishikawa et al.31.

Owing to the spin–orbit coupling, ISC and phosphorescence processes can also connect the singlet ground and excited states in a spin-selective optical cycle. The strong mixing with λz spin–orbit coupling in the excited state enables a triplet signal with considerable intensity. According to our spin–orbit calculations, the intensity ratio of the triplet phosphorescence is 1% for the mS = 0 state in the triplet manifold, in good agreement with the experimental findings31. This results in about 1.4 ms radiative lifetime for this state. On the other hand, allowing the intersystem crossing to the ground state can be formulated as a first-order perturbation for the mS = −1 and mS = +1 branches of the triplet (see Methods). Our calculations yield weak transition dipoles of 0.004D and 0.002D for the mS = − 1 and mS = + 1 phosphorescence branches, respectively, which have very long (>10 ms) radiative lifetimes.

The optical properties of the C-centre can be compared to other promising quantum defects in silicon. The G- and T-centres emit in the telecommunication O-band9,16,17 and the W-centre emits near the short wavelength edge of this band11. The emission of the Er+3 ion lies in the C-band41, whereas the C-centre exhibits ZPL in the most favourable L-band24,25, regarding the optical fibre transmission loss. However, the former shows a very long radiative lifetime of about 1−2 ms42,43, which can be decreased 10-fold with integration inside silicon waveguides44. Moreover, emission enhancement up to 70-fold was demonstrated by embedding erbium ions in a Fabry-Perot resonator45. The radiative lifetime of ensemble G-centres was reported to be 5.9 ns46, and a ~40 times enhancement in the PL intensity was achieved by resonant coupling to a dielectric meta-surface47. The emission of the excitonic excited states is much dimmer, as reported for the T-centre with 940 ns excited state lifetime16; however, Purcell enhancement was demonstrated for up to ~58-fold increase in the emission intensity17. Since the nature of the C-centre optical excitation is reminiscent of that of the W- and T-centres, we can assume that the integration in optical resonators could lead to a large Purcell-enhancement, as was already demonstrated for the G- and T-centres, making the relatively dim PL and phosphorescence of the C-centre significantly brighter.

Quantum protocols

We propose a quantum optical protocol to utilise the metastable triplet state and its phosphorescence for ODMR and its hyperfine coupling to nearby nuclear spins for quantum register applications. Optical initialisation of the triplet spin state is feasible owing to this spin-selective ISC to mS = 0 state. The mS = ±1 states are dark in the phosphorescence, suggested by their calculated very long lifetime. We predict an ODMR contrast detected in the phosphorescence signal, where the spin rotation between the bright \(\left\vert 0\right\rangle\) and the dark \(\left\vert \pm 1\right\rangle\) states decrease the phosphorescence intensity [see Fig. 3a]. The ODMR contrast diminishes at elevated temperatures together with the phosphorescence signal itself, owing to the activated rISC rate.

Fig. 3: Proposed quantum protocols.
Fig. 3: Proposed quantum protocols.
Full size image

a Quantum protocol for ODMR in C-centre. The red and blue arrows show fluorescence and phosphorescence, respectively. The blue dashed arrow highlights the spin-selective ISC leading to the main contribution to phosphorescence. The dotted lines represent the weak ISC to the ground state. Orange lines correspond to coherent spin control using microwave excitation. The spin–spin interaction incorporates both dipolar spin coupling and the re-normalisation effect from the spin-orbit interaction. The spin-selective ISC initialises the bright \(\left\vert 0\right\rangle\) spin state. Applying spin rotation leads to a contrast in the phosphorescence emission. b Magnetic field dependence of the triplet electronic and doublet nuclear spin levels labelled by \(\left\vert {m}_{S},{m}_{I}\right\rangle\). Procedure for the Landau-Zener transition is highlighted for negative hyperfine coupling parameters. The initial superposition (green and red) of electronic states \(\alpha \left\vert 0,+\frac{1}{2}\right\rangle +\beta \left\vert -1,+\frac{1}{2}\right\rangle\) is transferred to the coherent superposition of nuclear states \(\alpha \left\vert 0,+\frac{1}{2}\right\rangle +\beta \left\vert 0,-\frac{1}{2}\right\rangle\). The degenerate \(\left\vert 0,\pm \frac{1}{2}\right\rangle\) hyperfine levels are separated for the sake of clarity.

A long-lived quantum memory can be achieved in the defect by transferring the coherent electron spin state to the nuclear spin manifold. This can be achieved by Landau–Zener transitions48,49 at the triplet state level anti-crossing (LAC) condition

$$\alpha \left\vert 0,+\frac{1}{2}\right\rangle +\beta \left\vert -1,+\frac{1}{2}\right\rangle \mathop{\to }\limits^{{{{\rm{LZ}}}}}\alpha \left\vert 0,+\frac{1}{2}\right\rangle +\beta \left\vert 0,-\frac{1}{2}\right\rangle ,$$
(12)

where we note the electron and nuclear spin magnetic levels as \(\left\vert {m}_{S},{m}_{I}\right\rangle\). The nuclear spin flip-flop process is feasible utilising the perpendicular hyperfine couplings (Axx, Ayy) to nearby nuclei. We calculate such couplings to be larger than 10MHz for several positions of the 29Si nuclei (see Supplementary Table 1), allowing for short ramp time of the magnetic field sweep across the LAC [see Fig. 3b], however, decreasing the electron spin coherence time. As a compromise between operation speed and decoherence due to the hyperfine interaction itself, we suggest further lying atoms with hyperfine couplings in the order of 1 MHz as the quantum registers [see Supplementary Fig. 1b]. Alternatively, a weak magnetic field perpendicular to the nuclear spin quantisation axis (set by Azz) and a resonant microwave excitation pulse can also be utilised to map the electron spin superposition onto the nuclear spin as demonstrated for NV centre and a single 13C in diamond in ref. 50. The former allows for Larmor precession in the degenerate nuclear spin manifold of the mS = 0 state, while the latter is tuned to the hyperfine resonance at \(D-E-\frac{{A}_{zz}}{2}\) energy.

$$\alpha \left\vert 0,+\frac{1}{2}\right\rangle +\beta \left\vert -1,+\frac{1}{2}\right\rangle \mathop{\to }\limits^{{\pi }_{{{{\rm{L}}}}}}\alpha \left\vert 0,-\frac{1}{2}\right\rangle +\beta \left\vert -1,+\frac{1}{2}\right\rangle \mathop{\to }\limits^{{\pi }_{{{{\rm{MW}}}}}}\alpha \left\vert 0,-\frac{1}{2}\right\rangle +\beta \left\vert 0,+\frac{1}{2}\right\rangle ,$$
(13)

where πL and πMW are π pulses in the nuclear and electron spin manifolds, respectively. The readout of the nuclear spin memory into the electron spin manifold utilises the same magnetic field sweep as the storage process, however, it is the inverse process for the above pulse sequence. The initialisation of the nuclear spin is feasible with subsequent memory readout and optical pumping processes. We note that there are methods for the fast manipulation of multi-qubit registers with a combination of microwave and radio frequency pulses as well51,52. Based on the above initialisation, storage, and readout protocols, the individual control of the electron spin of the C-centre and a nearby single nuclear spin is feasible for the realisation of long-living quantum memory. Additionally, as the excited triplet state relaxes to the singlet ground state of the defect, further decoherence by the hyperfine coupling to the electron spin is eliminated, increasing the coherence time of the nuclear spin register. We note that isotope engineering of silicon is viable to find the optimum operation conditions, similar to the recent results for divacancy qubits in silicon carbide53.

There are quantum memories already investigated in silicon, the spin doublet T-centre54 and the Er+3 ion41, showing favourable spin coherence properties. For the former one, ODMR was demonstrated using spin selective resonant optical excitation (hyperpolarization method) reporting T1 > 16 s and Hahn-echo T2 times exceeding a millisecond and a second for the electron and hydrogen nuclear spins, respectively16. The erbium ion shows shorter coherence times of T1 ~ 1ms and T2 = 7.5 μs55. However, the advantage of the singlet ground state for the increased nuclear spin coherence was proposed for the ST1 and TR12 centres in diamond as well56,57. These successful applications call for the need for an in-depth experimental study on the spin dynamics in the C-centre as well for a direct comparison to the above contenders.

Discussion

The \({{{{{\rm{C}}}}}_{{{{\rm{i}}}}}{{{{\rm{O}}}}}_{{{{\rm{i}}}}}}^{0}\) defect in silicon, identified as the C-centre, was investigated for quantum technology applications. We obtain electronic structures in-line with experimental observations. We explore the spin interactions in the defect and report rhombic ZFS values crucial for the coherent manipulation of the electron spin qubit. We determine the symmetry-allowed spin–orbit couplings, revealing ISC transition with λz coupling between the excited states, and with different λ± coupling strength from the mS = ±1 substates of the triplet excited state is coupled to the ground state. These phosphorescence transitions correspond to the observed triplet radiative transition at cryogenic temperatures31. The lack of phosphorescence at elevated temperatures can be explained by a thermally activated rISC process, thermalising the excited states. On the other hand, population trapping in the triplet state is feasible at cryogenic temperatures, increasing the relative intensity of the phosphorescence assisted by the spin–orbit mixing with the close-lying excited singlet state. We conclude that C-centre in silicon is not only a promising single photon source for quantum communication applications via telecommunication optical fibres, but optical readout of the spin state is feasible in its triplet excited state. This opens the door for several quantum technology applications utilising this defect, including cryogenic temperature nanoscale sensing and quantum communication applications.

Methods

Details on DFT calculations

The atomic structure of the defect is modelled in a 512-atom silicon supercell. To this end, we used the Heyd–Scuseria–Ernzerhof (HSE06) functional58 and Γ-point sampling of the Brillouin zone, as implemented in the VASP plane wave-based code59,60,61,62. For the description of core electrons, projector augmented wave pseudo-potentials are used63,64. Excited states were calculated with the ΔSCF method65. The finite size correction of the charged supercells is obtained from self-consistent potential correction method66 to be 56 meV. The spin–spin contribution to the ZFS parameters was calculated by the approach of Bodrog and Gali 67 using the VASP implementation of Martijn Marsman. The Perdew–Burke–Ernzerhof68 functional was employed to compute the phonon spectrum as well as in the non-collinear DFT calculations in order to manage the computational cost.

Details on GW calculations

G0W038 and BSE39,40 calculations were performed using VASP in a 216-atom supercell model, keeping the Γ-point sampling. The orbital basis was calculated with DFT HSE06 functional, with a number of unoccupied orbitals more than 10 times the occupied ones. The energy cutoff for the calculation of the response function was limited to 150 eV to manage the computational workload. BSE was calculated beyond the Tamm–Dancoff approximation69, including 100 occupied and 100 unoccupied orbitals.

Calculation of phosphorescence rates

The transition dipole of the phosphorescence process between the triplet excited and singlet ground state can be calculated from perturbation theory as70

$$\left\langle^{1}A^{\prime}\right\vert e{{\mathbf{r}}}|^{3}\left.A^{\prime\prime} \right\rangle ={{{\mathrm{\Delta}}}}\mu \frac{\left\langle^{1}A^{\prime}\right\vert \lambda{{\hat{L}}}{{\hat{S}}} |^{3}\left.A^{\prime\prime} \right\rangle }{E(^{1}A^{\prime})-E(^{3}A^{\prime\prime})}$$
(14)

where \({{{\mathrm{\Delta}}}}\mu=\left\langle^{3}A^{\prime\prime}\right\vert e{{\mathbf{r}}} |^{3}\left.A^{\prime\prime}\right\rangle - \left\langle^{1}A^{\prime}\right\vert e{{\mathbf{r}}} |^{1}\left.A^{\prime} \right\rangle =13.7\,{{\mathrm{D}}}\) is the change of the macroscopic dipole moment in the transition.