Abstract
Solidstate defects acting as single photon sources and quantum bits are leading contenders in quantum technologies. Despite great efforts, not all the properties and behaviours of the presently known solidstate defect quantum bits are understood. Furthermore, various quantum technologies require novel solutions, thus new solidstate defect quantum bits should be explored to this end. These issues call to develop ab initio methods which accurately yield the key parameters of solidstate defect quantum bits and vastly accelerate the identification of novel ones for a target quantum technology application. In this review, we describe recent developments in the field including the calculation of excited states with quantum mechanical forces, treatment of spatially extended wavefunctions in supercell models, methods for temperaturedependent Herzberg–Teller fluorescence spectrum and photoionisation thresholds, accurate calculation of magnetooptical parameters of defects consisting of heavy atoms, as well as spinphonon interaction responsible for temperature dependence of the longitudonal spin relaxation T_{1} time and magnetooptical parameters, and finally the calculation of spin dephasing and spinecho times. We highlight breakthroughs including the description of effectivemass like excited states of deep defects and understanding the leading microscopic effect in the spinrelaxation of isolated nitrogenvacancy centre in diamond.
1 Introduction
Quantum information is physical [1]. Solidstate defect spins are a conceivable platform to realize the elementary unit of quantum information, i.e., quantum bits or qubits [2]. Two prototypical representatives are the phosphorus donor (Pdonor) in silicon and the nitrogenvacancy (NV) centre in diamond. From electronic structure point of view, these two defects reside the opposite sides of the spectrum: the Pdonor can be described by the socalled effective mass state with hydrogenlike Rydberg series of excitation energies split from the conduction band of silicon, which are weakly localized wavefunctions, whereas the carbon dangling bonds in NV centre create strongly localized orbitals with deep levels in the fundamental band gap of diamond. Kane proposed to apply Pdonor spins as a qubit [3]. However, the readout of the single spin in a controlled fashion had been great challenge for relatively long time that could be realized in a singleelectron transistor device operating at hundreds of millikelvin temperatures [4]. The readout of the single spin of diamond NV centre has been realised optically, i.e., optically detected magnetic resonance (ODMR), which was the first optically read single defect spin in a solid [5]. In this case, the readout and initialization of the electron spin of diamond NV centre could be readily carried out by optical means at room temperature. Recently, the single electron spin electrical readout via photoionised electrons and holes has been realised for diamond NV centre which is a hybrid scheme: photoexcitation is required for creating spindependent photocurrent from a single NV defect and optical initialisation of the spin, and then the spindependent photocurrent is observed to read the electron spin state [6, 7]. The coherent manipulation of these electron spins were realised electron spin resonance techniques [8, 9]. The coherent control and readout of single defect spins define the underlying defects as quantum defects, and the quantum defect with its host material can be called as a quantumcoherent material.
Since the discovery and realisation of these quantum defects an intense research has begun to seek alternative solidstate defect spins both in the experimental and theoretical fronts, which might have favourable properties for certain quantum technology applications [10, 11]. Recently, data mining techniques with machine learning algorithms have been spread at the theoretical fronts. The data mining can be approached either towards the host materials [12–14] or to extend this with creating defect structures and calculating their key qubit parameters [15, 16]. In these approaches, there are assumptions for selection of materials and defects that might be too restrictive and might lead to overlooking important candidates. For instance, it was assumed that the host materials should have wide band gap with low density of nuclear spins, at least, for the defect qubits alike diamond NV centre [10, 12]. However, certain quantum technology applications such as quantum communication does not require room temperature operation, and smallbandgap silicon has become a promising platform to realize spintophoton interface with quantum memory [11, 17], [18], [19], [20], [21], [22], [23]. One of the most promising platforms to host qubits in twodimensional (2D) materials is hexagonal boron nitride (hBN) which has 100% nuclear spin abundance but the coherence times of defect spins can be well extended with using good control of these nuclear spins [24–27]. Certainly, the selection criteria can be changed based on these recent findings. Nevertheless, quantumcoherent materials can only be interpreted together with their defect qubits, thus selection of host materials should be followed by finding defects for which the electron spin can be initialised and readout with sufficiently long coherence times. Calculation of the coherence times for any hypothetical defects, e.g., Ref. [13], could lead to misleading results in certain cases where the spin density distribution and so the strongest hyperfine interaction between the electron spin and nearest nuclear spins characteristic for the actual defect would strongly affect its electron spin’s coherence time [24]. The automatically generated defects are often selected from the thermodynamically most stable ones [15, 16]. It has been found that this selection may omit very important complex defects realizing qubits, e.g., Gcentre in silicon [28], which is one of most promising qubit candidates in silicon. Furthermore, it has been shown for diamond NVcentre [29] that the strongly coupled electronphonon states are inevitable for understanding the optical spinpolarisation of its electron spin. These polaronic states are also usually ignored in these databases. These examples clearly call for improving the ab initio magnetooptical spectroscopy methods, in order to increase the credibility and prediction power of these databases.
Ab initio methods have significantly contributed to understanding and control of diamond NV qubits and exploration of alternative quantum defects which was summarized in a recent review paper [30]. In that review paper, the diamond NV qubit is thoroughly described including the electronic structure and polaronic solutions and spectra within Jahn–Teller theories, and the desiderata of the target of computations and the developed computational methods are summarized in detail which are not perpetuated here. We assume that the readers are aware of the basic description of diamond NVlike qubits and the previously implemented methods to compute their magnetooptical properties which are the basis of further developments.
For instance, new findings have been reported for the very basic property of this qubit such as the spinlattice relaxation (T_{1}) time [31] which apparently demonstrates that even the most studied diamond NVcentre has been not fully understood to date. It is an immediate quest to further advance ab initio methods, in order to accurately calculate the excited states together with quantum mechanical forces, the electronphonon coupling and the basic magnetic parameters such as the zerofieldsplitting between the electron spin’s levels (ZFS), or understanding the temperature dependence of these parameters and the spinlattice relaxation (T_{1}) time, and the coherence time (T_{2}) also as a function of spin bath around the target defect qubit. This paper provides a comprehensive review about recent years ab initio developments on solidstate defect spin qubits along these directions.
We illustrate the advances of computational methods on various defects in solids: we shall discuss (i) deep defects such as NV centre, siliconvacancy centre and nickelvacancy centre in diamond, divacancy and vanadium centres in silicon carbide, and boronvacancy in twodimensional hexagonal boron nitride; (ii) the shallow excited states of deep defects such as the neutral siliconvacancy centre in diamond and various interstitial defects in silicon; (iii) and shallow donors in silicon.
2 Computational methods
The ab initio investigation of solidstate defect qubits alike diamond NVcentre or silicon Pdonor requires full and accurate description of the host material and the embedded isolated defect. The photostability of the quantum defects depends on the ionisation threshold energies, therefore it is critical that the crystalline bands and the resonant or localized defect states are computed at equal footing. Greenfunction methods are principally ideal to represent the topology of the problem, i.e., the embedded defect in a perfect solid. However, practical implementations of Greenfunction methods suffer from the consistent calculation of quantum mechanical forces which is required to calculate the ionic coordinates of the given point defect. As the geometry of the defect is highly decisive in their magnetooptical properties, the supercell method is most often employed to model the quantum defects which methodology readily offers the calculation of quantum mechanical forces based on the HellmanFeynman theorem [30]. In this review article, we focus on those method developments and implementations which work within this formalism because the supercell formalism guarantees the simultaneously accurate calculation of the ionisation threshold and intradefect optical transition energies. The ground state properties are typically calculated by means of KohnSham density functional theory (KS DFT) [32], which can be a starting point for the calculation of excited states. A natural choice for the basis set for supercell formalism is the planewave basis which is combined by pseudopotential or projector augmentation wave (PAW) methods (see Ref. [33] and references therein). The computation methods of the ground state thermodynamic properties of point defects in solids within this formalism was already described in detail in the literature (e.g., Ref. [34]) that can be applied to the specific quantum defects [35]. In short, the quantum defect’s local properties and associated parameters are computed within plane wave supercell KS DFT methods, and implementation based on these methods or relied on the parameters obtained by these methods will be discussed in the review paper.
3 Method developments and results
In this section, we collect recent developments on ab initio calculation of properties of quantum defects. We start with the treatment of the excited states and geometries, and then we continue with the discussion of the ionisation threshold energies with the photoionisation spectrum, the supercell treatment of shallow defect states, the zerofieldsplitting parameters, hyperfine parameters and gyromagnetic tensor, and the spinlattice relaxation T_{1} times as well as the spin coherence T_{2} times. Although, it is unconventional to start the discussion with the excited states prior the description of finite size effects of supercells we decided to do so because we shall consider the finite size effects of shallow excited states in the section of supercell modelling of defects which requires description about the calculation of excited states. We shall also show that accurate description of the excited states is needed to calculate some ground state related magnetic parameters such as the zerofield splitting parameter.
3.1 Excited states
3.1.1 Following the topology of the problem: DFT + CI multiscale methods
The accurate calculation of the excited states is still under extensive research. The popular manybody perturbation method, GW + BSE (see Ref. [36] and references therein), fails for the highly correlated singlet states of the diamond NV centre [37, 38]. The highly correlated states may be recognized of which manybody electronic wavefunction (Ψ) can be genuinely described by two or more Slaterdeterminants with significant weights. This may be quantified with the function of oneparticle density matrix
as if and only if the Ψ can be described as a single Fockstate when
In KS DFT theory, these manybody states cannot be captured by the known exchangecorrelation potentials (V_{xc}); however, it provides a very good description for the host materials and simple defect states, i.e., for which Λ = 0. In general, the bands of the host crystal can be well described by KS DFT method whereas the strong Coulomb interaction between localized defect orbitals confined to a small place, e.g., around a vacancy, represents a problem for KS DFT method. The topology of the problem implies a method in which the Coulomb interaction between the localized defect orbitals is directly calculated, called configurational interaction (CI), which are in contact with the bath of itinerant electrons that can be treated by KS DFT. This can be considered as a multiscale problem where the interaction between electrons are calculated with different precisions in a single system. As we will see below the challenges are to define the set of orbitals, often called “active space”, for which the precise method should be applied (here, CI) and to find an interface between the levels of different approaches, here DFT and CI, which can produce selfconsistent results. Interfacing different levels of theory in a single system is a common problem for all the multiscale methods.
By following the above mentioned topology of the problem, the manybody Hamiltonian may be described as
where v^{eff} is the partially screened Coulombinteraction
The methodology was first implemented by Bockstedte and coworkers within VASP code [39, 40] which was called DFT + CIcRPA method [41]. Here, CI refers to the configurational interaction between the electrons in the active space whereas constrained random phase approximation was applied to compute
NV  HSE06 [41]  HSE06 [49]  DDH [44]  DDH [44]  GWQP [50]  GWQP [50]  Exp. 

cRPA  cRPA  cRPA  bRPA  cRPA  cRPA  
αFDC  HFDC  HFDC  HFDC  HFDC  EDC  
^{1} E  0.49  0.49  0.476  0.561  0.375  0.463  ∼0.40 
^{1} A _{1}  1.41  1.39  1.376  1.759  1.150  1.270  ∼1.55 
^{3} E  2.02  2.06  1.921  2.001  1.324  2.152  ∼2.15 
In any embedding methods it is a crucial question how to select the active space orbitals. The ingap defect states are per se localized, therefore, they are naturally involved in the active space. However, defect states resonant with the valence band or conduction band may exist which are coupled to the Blochstates of the host material, thus they do not appear as sharp resonances but are rather broadened. In practice, the coupling strength and the width of broadening depend on the size of supercell and kpoint sampling of the Brillouinzone for the given supercell. For instance, it was found that whereas the lowest energy spectrum of diamond NV centre may converge relatively well for the minimal active space taken from the ingap defect orbitals but this treatment leads to a serious inaccuracy for the isovalent silicon carbide divacancy defect for which explicit involvement of the resonant states in the valence band is necessary to achieve converged results [41]. This result clearly demonstrated that simplified HubbardU model with using only the ingap defect orbitals (e.g., Ref. [38]) is generally insufficient for accurate description of the manybody electronic structure of quantum defects. In practice, an energy region of about 3 eV around the Fermilevel was used to pickup the states for the active space with 512 and 576atom supercell models in the original study [41]. In later studies, the choice of the active space was further investigated [49, 50]. Muechler and coworkers also implemented the DFT + CIcRPA method to the VASP code but they first constructed Wannier wavefunctions [51] with preserving the position of the ingap defect levels which could maximally localize the defect wavefunctions including those that are resonant with the host bands, and then the Wannier orbitals were applied in Eq. (2). With this treatment the results upon the number of orbitals in the active space can rapidly converge [49]. Another possibility is to pickup the states based on localization of the KS orbitals that can be simply quantified as
Another critical question in the embedding methods is the treatment of double counting correction which has been briefly considered above. In Hartree–Fock methods, the double counting correction can be readily derived and it is applied routinely in quantum chemistry. However, DFT is applied to calculate the electronic structure of the ground state in the supercell modeling of defects, and there is no theoretical rationale to apply HFDC in this case. By applying hybrid DFT functionals, Fockexchange is employed to KS orbitals. By assuming a tiny contribution of the semilocal exchangecorrelation in dc correction, one may apply αFDC meaning that the corresponding fraction of the Fockexchange for the active space orbitals is employed. This heuristic treatment may follow the idea of the DFT + U treatment in which U is an orbitaldependent onsite correction, and that is scaled by α when hybrid DFT is applied [52]. However, hybrid DFT is still a functional of the electron density and not the manybody wavefunction, thus this heuristic treatment strictly cannot be justified.
Recently, Sheng and coworkers derived the dc correction for KS DFT with using the Greenfunction approach [50] which might be motivated by the success of interfacing KS DFT and dynamical mean field theory (DMFT) with the same approach [53]. They assumed that the nondiagonal terms of the selfenergy coupling the active space and the environment is negligible, and static approximation is used for the G_{0} and W_{0}. This results in
where ρ_{
A
} is the density matrix of the electrons within the active space, and G_{0} and W_{0} are computed at the quasiparticle energies (ϵ^{QP}) of these electrons [50]. The exact dc correction is called here EDC in which the quasiparticle (QP) levels are solved selfconsistently with calculating ∑_{xc} selfenergy within GW method as
Sheng and coworkers applied the EDC method also to the neutral siliconvacancy (SiV) defect and their sister groupIV vacancy defects in diamond [50]. The ground state of the defect is ^{3}A_{2g} which is a similar wavefunction to that of diamond NV centre. However, the excited state of the neutral SiV defect is highly complex with leaving a hole in each double degenerate orbital which results in three triplet states (^{3}A_{2u}, ^{3}E_{
u
}, ^{3}A_{1u}). These triplet states are highly correlated and they are also coupled by phonons via product Jahn–Teller interaction [56, 57]. According to that study, the product Jahn–Teller interaction leads to a strong ionic relaxation upon excitation (
Despite the remaining issues with the DFT + CI multiscale methods, this approach is very promising to calculate the highly correlated defect states within supercell formalism. The very important next step is to derive the quantum mechanical forces which is then can be used to reoptimise the geometry of the ground state and the excited states. This is required to calculate the characteristic zerophononline energies and Debye–Waller factors of the defects which are key parameters of quantum emitters. The big challenge here is that the original derivation of the EDC method assumed that the offdiagonal term of the selfenergy coupling the active space and the environment can be neglected but once the forces are considered one has to take into account the change of the character of the underlying KS wavefunction selfconsistently with the selfenergy through the coupling between the active space and the environment. In general, it is a known problem in quantum chemistry CI methods that the final results may depend on the initial single particle wavefunctions if restricted active space is applied and the underlying single particle wavefunctions are fixed at the electronic ground state manifold. The DFT + CI multiscale methods face to the same problem, principally.
3.1.2 Density matrix renormalization group methods: an alternative multiscale method
Density matrix renormalization group (DMRG) was originally developed to describe onedimensional quantum models in solidstate physics with local interactions. The underlying mathematical framework, however, is not restricted to models studied in condensed matter physics or applications to molecular clusters but among many others, it can be also used to study nuclear shell models, particles in confined potential, or problems in the relativistic domain. The success of these developments relies on the efficient factorization of interactions and the optimization of the DMRG network topologies based on concepts of quantum information theory leading to tremendous reduction in computational costs (see Ref. [58] and references therein). In particular, the factorization for the manybody wavefunction Ψ with L spatial orbitals reads as
with
where now the spin state (σ) is explicitly written by arrows and n = (n_{1↑}n_{1↓}n_{2↑}n_{2↓} … n_{L↑}n_{L↓}) where n_{
iσ
} ∈ {0, 1}. The components of the state specific C tensor increase exponentially with system size L scaling as 2^{2L} which becomes untractable for few hundreds of electrons. However, the dimension of the matrix product states A_{
i
} can be optimized in DMRG approach,
In short, application of DMRG method on electronnuclei systems can be considered as a special wavefunction method that can be used to accurately calculate the static correlation between electrons. Barcza and coworkers extended this method to interface with DFT calculations of quantum defects [58]. In this postDFT method, the Coulomb integrals of the KS orbitals are directly taken from the supercell DFT calculation in the electronic ground state manifold which is postprocessed by DMRG algorithms. Despite the advantage of DMRG method, hundreds of KS orbitals cannot be directly treated owing to the computational costs. Therefore, an optimal selection of orbitals with tractable size is needed which is responsible for the strong static correlations. This was carried by the complete active space (CAS) selfconsistent field method which is wellknown in quantum chemistry (see Ref. [58] and references therein). The CAS method classifies the set of orbitals to three categories; that is, the socalled core and virtual orbitals are frozen to the mean field level and filled with two and zero electrons, respectively. The third class comprises of the socalled active space orbitals which are populated with the rest of electrons minimizing the energy. Accordingly, the virtual orbitals does not play any role in the corresponding CAS Hamiltonian, whereas the core electrons affect the electrons of the active space through the Coulomb interactions, that is, the Hamiltonian of the active space reads
The oneelectron integrals of the CAS space,
to treat the Coulombic effects of the frozen electrons on the active space orbitals. In other words, this is a multiscale method where the interface between the active space and the environment can be well managed within Hartree–Fock level. Here, the summation runs only on the indices of the core orbitals (c). Finally, the additional energy contribution of the core electrons is summed up in term E^{core}, that is
In practice, the active space is restricted to the most important orbitals featuring strong correlation as was discussed for DFT + CI methods. Even though the method has limitations to provide correct description of dynamic correlations using the relatively small active space, it captures static correlations with high accuracy providing valuable insights into the lowlying energy spectrum and the essential structure and symmetry properties of the corresponding electronic eigenstates. Note also that, contrary to DFT + CI methods, the CAS Hamiltonian in Eq. (7) does not include the KS energies explicitly but only the KS orbitals by construction. Also, the absolute energies of the states computed from the CAS Hamiltonian are not trivially comparable with counterparts obtained on the DFT level of theory due to the different description of the exchange and correlation effects.
The methodology was first applied to the negatively charged boronvacancy defect in hexagonal boron nitride (hBN) [59]. Previous ab initio calculations predicted this defect in hBN as a qubit [60] of which ODMR signal was confirmed in experiment [61] and has become a leading quantum defect in twodimensional (2D) materials. This defect consists of three nitrogen dangling bonds which introduce four closeby levels in the fundamental band gap. Since the vacancy lobes are closely spaced with strong Coulomb interaction they form highly correlated states. The order of the manybody levels and the nature of the states were determined by DMRG method [62]. In the calculations about 50 KS orbitals were used in the active space selected by localization criterion. It was shown that the flake model with about 80 B and N atoms with hydrogen termination of the edge of the flake provides the same electronic structure as the periodic supercell model at DMRG level [58]. This makes it realistic to calculate the electronic structure of defects within the flake model by means of traditional quantum chemistry wavefunction methods [63, 64]. We note that the validity of the flake (molecular cluster) model could be specific to (planar) defects in hBN and may be not applied in general for defects in other 2D materials.
DMRG method was also applied to magnesiumvacancy (MgV) centre in diamond [65]. MgV centre was created by Mg implantation to diamond which has a unique photostability [66] which makes it an interesting quantum emitter. The PL signal was associated with the negatively charged MgV defect which has a similar electronic structure to that of SiV centre [65] with a caveat that the resonant a_{2u} level lies very close to the top of the valence band and the two degenerate e_{ u } and e_{ g } levels lie in the fundamental band gap. These states are localized on the vacancy lobes which create highly correlated manybody states. In particular, the neutral MgV has overwhelmly complicated electronic structure which needs wavefunction method such as DMRG.
Beside the optimization of the selection of the active space orbitals, a critical issue to take the dynamical correlation effects into account. The present Hartree–Fock treatment of the core orbitals [58] could be insufficient for many materials to achieve accurate electronic structure. In order to achieve an extremely accurate total energy of crystals, a coupledclustersingledouble with perturbative triple [CCSD(T)] wavefunction approach is required [67]. It is likely that accurate lowenergy excitation spectrum around the Fermilevel may be achieved at much lower complexity of wavefunction approaches than CCSD(T). One possible route of development is the implementation of more complex levels of wavefunction approaches than Hartree–Fock core and full CI active space model in the DMRG multiscale method, in order to converge towards highly accurate lowenergy excitation spectrum. Another important issue, similarly to the DFT + CI methods, is to compute quantum mechanical forces acting on ions. The concept of the force calculation does exist for DMRG method that was already implemented for quantum chemistry codes (see Ref. [68] and references therein).
3.1.3 Spinflipping timedependent density functional theory and BSE methods
Timedependent DFT (TDDFT) based on KSDFT in the kernel [69, 70] can be principally applied to calculate the low energy excitation spectrum of quantum defects [71, 72]. In order to achieve accurate results, the proper choice of the DFT functional is essential [71]. Unlike the present implementations of GW + BSE and postDFT multiscale methods, TDDFT framework and implementations exist to calculate the quantum mechanical forces acting on ions in the electronic excited state. In a pioneer work it was shown (see Supplementary Materials in Ref. [73]) that the observed Stokesshift of NV centre can be well reproduced by TDDFT calculation when the NV centre is embedded in the core of 1.4nm sized nanodiamond. The optimized geometry by TDDFT method well reproduced the optimized geometry by the ΔSCF method [74] for diamond NV centre. This issue has been recently investigated for the supercell model of diamond NV centre and silicon carbide divacancy centres [75]. It was found that the optimized geometries, the adiabatic potential energy surfaces (APES) of the ^{3}E state and the zerophononline (ZPL) energies are very close to each other as obtained ΔSCF method and TDDFT method based on DDH functional (see Figure 4 in Ref. [75]). The reason behind this good agreement is that Λ ≈ 0 for ^{3}E so it is a very good approximation to describe ^{3}E state as promotion a single electron from a_{1} ingap defect level to the e ingap defect level in the spin minority channel which is exactly constructed by ΔSCF method.
In the recent years, spinflipping TDDFT (sfTDDFT) theory has been developed and applied to molecules [76, 77] which is based on the Casida equations [70] but the original equations are modified in order to calculate the states and energies associated with Δm_{ s } = ±1 spin transition. In principally, the supercell implementation of the sfTDDFT can be applied to diamond NV centre to obtain the singlet ^{1}A_{1} and ^{1}E states with geometry optimization in these states that has not yet been achieved by other means. In the DFT + CIcRPA calculations [41], it was already recognized for diamond NV centre that the ^{1}E state contains Slaterdeterminants associated with symmetry breaking solutions, thus ^{1}E state is dynamically distorted from the C_{3v} high symmetry geometry unlike the ^{1}A_{1} state which stays in C_{3v} symmetry. Insights from group theory, ΔSCF and DFT + CIcRPA calculations with electronphonon Hamiltonian models made it possible to construct the absorption and emission spectra between the ^{1}A_{1} and ^{1}E states for NV centre [29]. Nevertheless, the construction of the absorption spectrum was not accurate as principally it could not use the true APES of ^{1}E and ^{1}A_{1} states. As a consequence, the sharp resonance in the absorption band at 170 meV above the diamond phonon bands was missing in the constructed absorption spectrum.
In a seminal work by Jin and coworkers [78], the singlet states of diamond NV centre were directly calculated by sfTDDT including quantum mechanical forces. The ab initio APES could be calculated both for the ^{1}E and ^{1}A_{1}. Although, the sfTDDFT excitation does not involve the double excitation electronic configurations in ^{1}E and ^{1}A_{1} states but their contributions may influence their energy levels – higher than that by a DFT + CI method [50, 78] – but not significantly their optimized geometries. The previously developed model Hamiltonians based on ΔSCF calculations [29] have been confirmed by sfTDDFT calculations [78]: the ^{1}E state is strongly anharmonic whereas the ^{1}A_{1} state shows an almost perfect parabolic APES but the effective phonon frequencies are higher than that for ^{3}A_{2} ground state. On the other hand, the ^{1}A_{1} state shows a slight anharmonicity due to its phonon coupling to the ^{1}E state by the symmetry breaking e phonons. Taking this correction into account, the calculated absorption spectrum shows a perfect agreement with the observed absorption spectrum including the sharp vibration resonance at 170 meV [78]. It was found that the e phonons dominantly contribute to the absorption spectrum, in stark contrast to the optical spectra between the triplet states of the defect. The luminescence spectrum between singlets was not calculated in this study [78] which is a Herzberg–Teller optical transition [29], and the highly accurate calculation of the shape of the phonon sideband would need to solve the multimode Jahn–Teller problem [79].
Spinflip BSE (sfBSE) method can be basically also applied to calculate Λ ≫ 0 states. Here the original idea is that Ψ_{GS}⟩ of the system could be a close shell singlet but the lowenergy singlet excited state might be a
3.1.4 Temperature broadening of optical excitation spectrum with BSE method
The manybody perturbation theory of electronphonon coupled optical transition with nonequlibrium Greenfunctions has been developed in Ref. [84] which was further derived in Ref. [85]. They considered the adiabatic limit for the dipole matrix elements, while they retained dynamical effects only in transition energies [85]. In terms of the phonondependent optical dipole transition moments, it goes beyond the Huang–Rhys theory. The central equation is
where Π_{
λ
} are the exciton (λ) dipole matrix elements within BSE theory, and
In the usual implementation of BSE, it does not take into account the polaron shift. In other words, the nophonon or zerophonon line energy is calculated at the fix coordinate of the ground state from which the GW + BSE calculation starts. The theory was first applied to bulk hexagonal boron nitride which has indirect band gap, so
The observed fluorescence comes from a firstorder forbidden transition between the ^{3}E′′ excited state and ^{3}
Libbi and coworkers rather applied G_{0}W_{0} method on PBE DFT calculations [86]. Then they calculated the nonequlibrium BSE optical spectrum based on Eq. (10). Because of the forbidden nature of the optical transition, the equilibrium BSE results in exactly zero optical dipole moment for the lowest energy transition in agreement with the previous HSE06 DFT result [62]. However, it becomes visible by applying nonequilibrium BSE method (
for electrons and holes, respectively. Here ϵ_{ n k } is the quasiparticle energy of the state {nk} and μ_{el} (μ_{ho}) the chemical potential for electrons (holes). The chemical potential was set for the electrons in such a way that a whole electron is promoted to the excited state manifold as usual in ΔSCF procedure which guarantees a neutral excitation. The chemical potential of the holes is tuned in such a way that the number of holes coincides with that of the electrons excited to the empty states. The nonequilibrium (NEQ) occupations induce a renormalization of the quasiparticle energy levels. It is calculated as
where the first term is the quasiparticle energy determined at the G_{0}W_{0} level of theory using the equilibrium occupations, while the second and the third terms represent the quasiparticle energy at the COHSEX level of theory calculated using the nonequilibrium and equilibrium occupations, respectively.
The calculated I^{BSE}(ω, T) spectrum indicates that the phonons associated with the inplane Jahn–Teller distortion has a very small optical dipole transition moments but the outofplane phonons significantly amplify the optical dipole transition moments [86]. As a consequence, the shape of the PL spectrum is governed by the outofplane phonons and not the inplane phonons. We note that the combination of the ^{3}E′′ electronic state with
3.2 Photoionisation thresholds: manybody and temperature effects
Photoionisation threshold energies and crosssections are key properties for such quantum defects for which the qubits are initialised and read out optically. Photoionisation may promote an electron from the filled ingap state to the conduction band (positive ionisation) or an electron from the valence band to the empty ingap state (negative ionisation). For a given defect in semiconductors and insulators, both process may occur depending on the illumination wavelength. Similar to the neutral photoexcitation processes, phonons may assist the photoionisation processes with yielding temperaturedependent photoionisation thresholds. Ab initio simulations using the usual BornOppenheimer approximation with separating the problem of motion of ions and electrons will result in the lowtemperature photoionisation thresholds that may be not accurate at elevated temperatures.
Photoionisation may occur by simply single photon absorption which basially goes the same way as neutral photoexcitation just the initial Ψ_{
i
}⟩ or the final state Ψ_{
f
}⟩ is the host band in the photoionisation process, i.e.,
Another important consideration about describing the photoionisation is to distinguish the observable manybody picture and the auxiliary singleparticle picture or band structure (see Figure 1). The band structure diagramme is an effective single particle picture including the localised defect states inside the band gap. In other words, band structure virtually plots the singleparticle levels of which singleparticle wavefunctions build up the manybody Slaterdeterminant solution. In the case of shallow donor state such as phosphorus donor (Pdonor) in Si, the defect level is occupied by a single electron in the band gap, and that electron has a relatively weak exchangecorrelation interaction towards the valence band states. Thus, the single electron orbital represents well the manybody total energy with respect to the total energy of the system when the electron is promoted to a higher energy effective mass state. As a consequence, the occupied donor defect level with respect to the conduction band minimum (CBM) can be considered as the photoionisation energy (when the relaxation of the ions upon photoionisation is neglected or minor). However, if the donor level is occupied by two electrons, e.g., sulfur donor in Si, then the two electrons have considerable Coloumb interaction with each other. As a consequence, once an electron is removed from the doubly occupied donor level then we break the Coulomb repulsion between the two electrons and their contribution to the total energy of the system before ionisation which is missing for the single electron left in the donor level after ionisation. Therefore, the difference betwen the doubly occupied donor level and singly occupied donor level does not correspond to the manybody total energy difference of the neutral and singly ionised sulfur defects. For deep defects with multiple electrons localised on the defect this effect is severe: it results in a strong Coulomb and exchange interactions among the localised electrons. Therefore, the manybody ^{3}A_{2} level, of which state is Ψ_{GS} Slaterdeterminant for the diamond NV centre, should not be drawn to the band structure of perfect diamond as ^{3}A_{2} state already contains all the diamond bands. The false level diagramme with mixing the single particle and the manybody pictures will lead to a false impression about the ionisation energies. For instance, if the ^{3}A_{2} level is drawn at the position of the ionisation energy (about 2.65 eV at room temperature [87, 88]) w.r.t. CBM then the ^{3}E level is usually drawn above the ^{3}A_{2} level with the ZPL energy at 1.945 eV. According to this false consideration, the ionisation energy of ^{3}E level w.r.t. CBM would be that of the ^{3}A_{2} minus the ZPL energy which yields about 0.7 eV. The problem with this consideration is that it completely neglects the exchangecorrelation effects between the electrons or other words the manybody effects as will be explained below.
In the example of diamond NV(−) → NV(0) ionisation process, if we start from the ground state ^{3}A_{2} electronic configuration of NV(−) (Figure 2(b)) then a single electron from the e level is promoted to the CBM which results in the ^{2}E ground state of NV(0). Starting the ionisation from ^{3}E state of NV(−), we first have to consider the ^{3}E state which can be well described as a hole is left in the a_{1} level and the double degenerate e level is occupied by three electrons (see left panel in Figure 2(b)). Two ionisation processes are viable at this point: (i) direct process with an electron promoted from the e level to the conduction band which may leave the defect [89]; (ii) Augerprocess occurs after absorption of the second photon where the promoted electron in the conduction band recombines with the a_{1} hole, and the energy gain of this process is used to simultaneously promote another electron from the e level to a high energy state in the conduction band which finalizes the ionisation process (see Refs. [90, 91] and Figure 2(c)). According to the Slater–Condon rules [92, 93], only one spin orbital can change in the manybody wavefunction upon direct ionisation process described by onebody operator, therefore ^{3}E of NV(−) arrives at the metastable ^{4}A_{2} of NV(0) plus an electron in the conduction band by photoexcitation. Since the shelving ^{4}A_{2} level lies above the ground state ^{2}E level of NV(0) the ionisation threshold energy of this process starting from ^{3}E of NV(−) will be higher by about 0.48 eV according to HSE06 calculations [89] then the ionisation threshold energy starting from the ^{3}A_{2} ground state level minus the ZPL energy of the ^{3}E ↔ ^{3}A_{2} optical transition, which finally yields about 1.2 eV ionisation threshold energy [89]. In the alternative Augerprocess, the twobody operator nature of the process makes it possible to arrive at the ^{2}E ground state of NV(0) from ^{3}E NV(−). If the electron is excited to high energy above the CBM then phonons can very quickly (within picoseconds) cool it down to the CBM which is might be bound by the weakly attractive potential of NV(0) towards the CBM electrons which makes this process viable with about 1 nanosecond rate [90]. In the twophoton absorption process of NV(−), the typical excitation energy is about 4.66 eV (green light at 532nm wavelength) which subsequently goes through the ^{3}E level with phonon cooling upon absorption of the first photon and probably a phonon cooling process after absorbing the second photon as explained above. From energetics point of view, it is feasible to arrive at ^{2}E ground state of NV(0) from the ^{3}A_{2} ground state of NV(−). A recent experimental study concludes that the twophoton absorption based charge conversion of NV(−) → NV(0) can be well explained by a dominating Augerprocess. In the rate modelling, a singlephoton optical transition should also occur from the shelving ^{1}E state to a higher lying state [91]. A strong and broad transition around 2.58 eV from ^{1}E to ^{1}E′ has been observed in numerical simulations providing a possible candidate for such a mechanism [41]. ^{1}E′ can decay to the triplet excited states via an intersystem crossing [30, 94].
Nevertheless, such spintochargeconversion (SCC) protocols exist for diamond NV centre which apply lowpower excitation to avoid twophoton absorption processes but dual wavelengths of excitation in which the green illumination is used for the optical cycle between ^{3}A_{2} and ^{3}E states whereas longerwavelengththanZPL illumination is applied to ionise it from the ^{3}E level towards the conduction band [95, 96]. In this case, direct ionisation may occur from the ^{3}E level. In the seminal work of Razinkovas and coworkers [89], the absolute ionisation cross section from the ^{3}E state together with the induced emission was calculated for NV centre as a function of the excitation energy (Figure 2(a)). The calculated ratio of the photoionisation cross section and the cross section for stimulated emission is greater than 2 for the energy around 1.2 eV and 1.93 eV photoexcitation energies, which agree well with the applied photoionisation energies in the SCC experiments [95, 96].
We note that understanding the reionisation process, NV(0) → NV(−), is critical for stabilising the diamond NV(−) qubit state. Typically, the reionisation automatically occurs by the applied green illumination to drive NV(−) by twophoton absorption process which is also subsequential process going through the ^{2}A_{2} excited state: the first photon absorption brings ^{2}E to ^{2}A_{2} and the second photon is then absorbed in the ^{2}A_{2} excited state. The ^{2}A_{2} state consists of a hole left in the a_{1} level and two electrons in the e level. The photoexcitation of the ^{2}A_{2} excited state of NV(0) may also occur by either direct process (promoting an electron from the valence band to the empty a_{1} defect level in the gap) or Augerprocess (occupying the ingap a_{1} hole by an electron from the ingap e level and then promoting an electron from the valence band to the empty e defect level in the gap). Both processes leave a hole in the valence band. In the direct process, the system arrives at the ground state of NV(−) because of the alluded Slater–Condon principle [91]. On the other hand, the Augerprocess enables to arrive at the ^{1}A_{1} and ^{1}E singlet states of NV(−) too, beside the ^{3}A_{2} ground state. The energy cost of these processes varies with the final state. The calculated adiabatic acceptor charge transition level of the NV defect is at about 2.75 eV from the conduction band edge [97, 98], whereas the calculated energy gap between the ^{3}A_{2} ground state and ^{1}A_{1} state is at about 1.6 eV (see Ref. [30] and references therein). The total energy cost to convert the NV(0) ground state to the ^{1}A_{1} NV(−) excited state is then about 4.3 eV which coincides with twice the ZPL energy of NV(0) [91]. This means that a special excited state of ^{1}A_{1} of NV(−) binding a hole resonant with the valence band maximum (VBM) develops. This hole is Coulombically bound which is a special bound exciton state or Rydberg state which has been observed for the SiV defect [11] and has been recently implied and modelled for the ^{3}A_{2} plus bound hole system for the NV(−) defect [99, 100]. The bound hole is weakly localised with following the effective mass theory. By even taking into account the possible relaxation energy of the ions caused by the change in the electronic states, we may claim that a green laser excitation can reach the ^{1}A_{1} plus bound hole state of NV(−) by twophoton excitation of NV(0). Scattering rather to ^{1}E and ^{3}A_{2} states of NV(−) via Augerprocess leaves a hole deep in the valence band at around 1.2 eV and 1.6 eV from the VBM, respectively. According to the calculations [91], a resonant a_{1} state broadened by the diamond bands occurs in this energy region which is originated from the dangling bond orbitals of the carbon and nitrogen atoms near the vacant site. The resonant state is weakly localized unlike the usual diamond bands that are completely delocalized. This should lead to a larger direct and Augerionization rates of NV(0) than those of NV(−) because no such a highenergy resonant state sharing the same spin state with that of the ground state exists in the conduction band, critical in the photoionisation of NV(−). Interestingly, the Augerprocess should lead to a preferential occupation of the m_{ s } = 0 state via spinselective intersystem crossings between the ^{1}E state and the ^{3}A_{2} spin states; however, direct ionisation would result in 1/3:2/3 relative population of the m_{ s } = 0:m_{ s } = ±1 states of ^{3}A_{2} [91]. The ab initio calculation of the rates of the direct and Augerprocesses requires an accurate computation of the excited states of NV(0) which has not yet been solved as we briefly discussed it in Section 3.1.
The aforementioned ab initio calculations are based on the global energy minimum of the APES in the appropriate electronic structures and charge states which correspond to the zero kelvin solution. However, the photoionisation tresholds of diamond NV centre are often observed at room temperature. Principally, the effective ionisation threshold energies may change as a function of temperature. A very characteristic example is the silicon carbide (SiC) divacancy defects. In particular, four divacancy configurations with similar electronic structures occur in the socalled 4H polytype of SiC [101–103], some of them well observable at room temperature [104, 105]. We note that 4H SiC exhibits a band gap of about 3.3 eV that can host visible and nearinfrared colour centres acting as qubits [11, 106, 107]. The neutral divacancy defects in 4H SiC have isovalent electronic structure to diamond NV centre as depicted in Figure 1 but the energy gaps between the levels are about twice as smaller, thus they produce nearinrared emission. It was found that upon photoexcitation of the defect it falls to a “dark” shelving state due to twophoton absorption or other complex processes. This shelving state has been finally identified as the negative charge state of the defect [108–110]. Close to cryogenic temperatures, a shorter wavelength laser beam was applied to drive the divacancy back to the neutral qubit state by promoting an electron from the e level to the CBM [110]. It was later found that at elevated temperature the quenching of the fluorescence of 4H SiC divacancy (V2) defects does not occur and they remain optically stable [111]. This can be interpreted as the photoionisation threshold energy associated with the V2(−) → V2(0) process is decreasing upon raising the temperature.
The modelling of temperaturedependent photoionisation processes requires the following considerations: (i) the CBM and VBM of the host material may shift with temperature, (ii) the formation enthalpy so the charge transition level of the defect with respect to the band edges may shift with temperature via the vibration entropy, and (iii) the phonon assisted ionisation process may be activated by raising the temperature. The first effect is basically the temperaturedependent electronphonon renormalization of the bands. This can be computed by the manybody perturbation theory on the electronphonon coupling [112–114] that was applied to 4H SiC [115]. The usual case is that the CBM and VBM shifts down and up with raising the temperature, respectively, leading to an effective decrease of the fundamental band gap. It was found by ab initio calculations that the CBM of 4H SiC shifts down by about 5 meV from zero kelvin to room temperature [115]. In other materials with low Debyetemperature, this effect could be significantly enhanced. The second effect assumes an ab initio treatment of the thermodynamic properties of solidstate defects that was thouroughly discussed in Ref. [34]. Here, the key effect is the vibration entropy correction to the formation energy of the defect,
where ℏ and k_{B} are the reduced Planck constant and Boltzmann constant, respectively, and ω_{
i
} is the frequency of the ith phonon mode in charge state q of the defect at the given T temperature. The first term in Eq. (14) is the zeropoint energy. The actual values of F^{
q
}(T) could differ for a given defect in various charge states (ΔF(T) = F^{−}(T) − F^{0}(T)) which results in a shift of charge transition level, here with respect to CBM,
The value of
3.3 Supercell modelling of defects: extrapolation to dilute limits
In the supercell modelling of point defects the goal is to describe isolated point defects. In practice, the size of the supercell is limited up to about 10,000 atoms for KS DFT calculations due to computational resources. By applying accurate hybrid KS DFT functionals the size of the supercell is further reduced to about 1000 atoms. This size of the supercell suffices to obtain accurate results for deep defects because the defect induced wavefunctions and strain fields decay relatively fast from the core of the defect. However, shallow states with weakly localised character of defect wavefunctions such as the case of Pdonor in Si, require extrapolation to dilute limits. For deep defects, certain properties also call for special treatments in the supercell formalism that we nonexhaustively list here: charge correction for charged defective supercells, acoustic phonon couplings to electron orbitals and spins, excitation and ionisation towards electronic bands of the host.
3.3.1 Charge correction schemes: a recent breakthrough
In our previous review paper [30], the charge correction for charged defective supercells was thouroughly discussed. To sketch here the problem and the possible solution we note that the introduced defect charge is neutralized by a compensating jellium charge in the supercell. The charge of the defect and the jellium background charge interact with their periodic images that goes with the leading point charge Coulomb interaction of which energy scales with L^{−1}, where L is the edge of the simple cubic supercell. This theory already gives a hint about the expected scaling property in the correction of the total energy to the isolated defect or dilute limit. Indeed, the total energy of the charged defective supercells converges notoriously slow with the supercell size as indicated by the simple theory above. By applying charge corrections to the total energy, the convergence can be well accelerated. In the previous years (see Refs. [30, 34, 117] and references therein), many a posteriori schemes have been developed for the correction of the total energy of charged defective supercells in 3D and 2D models. In those schemes, the possible change in the character of the wavefunction due to the charge correction was not taken into account that might lead to qualitatively wrong results in notorious cases. This problem is in particular severe for slab models of crystal surfaces with negative electron affinities. Recently, a selfconsistent potential charge correction (SCPC) method was developed to heal this issue which goes beyond the previous a posteriori total energy corrections of the charged defects, and they derive the KS potential associated with the charge correction and selfconsistently solve the constructed KS DFT equations [117].
The SCPC method yield the corrective potential (V_{cor}) in an iterative manner: (i) the distribution of the extra charge in the supercell (δρ) is determined, (ii) the corresponding periodic electrostatic potential (V_{per}) is calculated, (iii) the potential for the same but isolated charge distribution (V_{iso}) is determined by using open (Dirichlet) boundary conditions, and finally, (iv) V_{per} and V_{iso} are used to determine the corrective potential V_{cor}, which is added to the total electronic potential. It should be noted that SCPC always aligns the final potential, considering the difference between the electrostatic potentials of the charged and the reference system far away from the defect position (ΔV).
The method was originally built in the VASP code but an interface has been developed to Quantum Espresso code too. The SCPC method was applied to diamond slab model with NV centre where the (100) diamond surface was terminated by hydrogen. It was shown that without SCPC method the negative charge of the defect artificially pull down the bands of the surface states, socalled image states, which results in a false electronic structure even in large supercells [117]. Here, the selfconsistent correction of the potential is essential. We note that the selfconsistent correction is not a must for many defects in 3D solids and a posteriori charge correction schemes can provide qualitatively good results. Despite the selfconsistent nature of the correction, the SCPC method does show supercell size dependence which comes from the fact that the character of the underlying (localized) defect wavefunctions may change with supercell size due to the defectinduced strain fields and other factors, so supercell size scaling is still necessary with this method but converges faster than the a posteriori total energy correction schemes [117].
3.3.2 Embedding of long wavelength phonons in the finite size of the supercell
The longwavelength acoustic phonons could contribute to the phonon sideband of the optical spectra of defects. Audrius Alkauskas and coworkers developed an embedding method to include the electronphonon coupling in the optical transition [116] that they applied to the optical transition between the triplet states of diamond NV centre [79, 116] which has been also implemented in the sfTDDFT study for the absorption spectrum of the singlet states [78]. Similar treatment is advisable for the temperaturedependent photoionisation spectrum [111].
3.3.3 Reconstruction of the deepenergy valence and highenergy conduction bands
Accurate absorption cross section calculation of excitation from/to deep levels to/from solidstate bands within supercell formalism requires special attention due to band folding in the Brillouinzone [118]. This was recognized by Razinkovas and coworkers when they studied the absorption cross section for the photoionisation of diamond NV centre [89]: they found minigaps in the conduction band which affected the calculated absorption cross section at the given high energy excitation. They used the following technique to circumvent this problem: (i) they identified the folded band in the Brillouinzone of the primitive diamond cell as described in Ref. [118], (ii) then they averaged out the calculated absorption cross section values closest to the corresponding kspace within the energy region of about 0.08 eV. The resulted photoionisation absorption spectrum is then converged well.
3.3.4 Treatment of spatially extended defect wavefunctions: beyond effective mass theory
KS DFT calculation of the properties of shallow donors in Si is computationally very challenging because of the spatial extension of the donor wavefunction. Accurate calculation requires hybrid HSE06 functional, e.g., Ref. [119], which becomes extremely difficult to carry out for sufficiently large supercell sizes. As was done for the total energy correction of charged defective supercells (see Refs. [30, 34, 117] and references therein), one can apply a strategy by studying theoretically and numerically the scaling of the given property as a function of the supercell size, and then extrapolate the result to the dilute (isolated defect) limit. In many cases, the exact scaling properties are unknown or they are too much complex because different effects (electron density and charge distribution, strain field distribution) are intertwinned and they often depend on the local electron density distribution of the defect that might change by increasing the size of the supercell.
In practice, numerical KS DFT investigations could lead to converged results which requires sufficiently large number of sampling points for defining the scaling law, i.e., ideally up to supercell size with about 10,000 atoms. This is prohibited by the accurate HSE06 functional so a typical strategy is to calculate the scaling by the affordable PBE functional and for a limited range of supercell sizes to compare the scaling with HSE06 and PBE functionals to verify the scaling function.
Swift and coworkers studied the shallow groupV dopants of Si by supercell KS DFT calculations [119]. These dopant introduces a singly occupied electron the state split from the CBM. In semiconductor physics, these states are described by the socalled effective mass theory from Kohn and Luttinger which treats the dopant potential as a positively charged Coulomb potential which binds an electron of which state can be described the linear combination of the CBM valleys. The solution of this system results in a Rydberg or hydrogenic series of excitation energies until it converges to the ionisation level. The ionisation or binding energy of the electron (E_{ b }) can be defined as the lowest energy level ϵ^{donor} (1s like envelop function) with respect to CBM of the perfect crystal (ϵ^{CBM}). The validity of this approximation from manybody electronphonon point of view was briefly discussed in Section 3.2. It is known in experiments (see Ref. [119] and references therein) that the ionisation energies of various groupV dopants in Si differ. Therefore, a socalled “central cell correction” was introduced to the effective mass theory which assumes that the 1s ground state wavefunction has the largest overlap with the dopant ion’s attractive potential which will pull down its energy level with respect to the purely hydrogenic solution (enlarging the donor ionisation energy). As the potential of the dopant ion is characteristic to the dopant within short range around the dopant ion, thus the “central cell correction” will be dopant dependent. The central cell correction is merely semiempirical correction to the effective mass theory of shallow donors and acceptors, where the acceptor levels are measured with respect to VBM. In KS DFT calculations, the ionisation energies can be calculated at ab initio level free from any assumptions on the nature of the potential induced by the dopants. It can be estimated from the effective mass theory that the 1s donor wavefunction in Si will decay at around 55 Å from the position of the dopant which would require a supercell of about 64,000 atoms to accommodate the donor wavefunction without significant overlap. This definitely calls to apply a scaling procedure even for PBE DFT functional.
Swift and coworkers [119] applied scaling method for calculating E_{ b } of arsenic (As) and bismuth (Bi) donors in Si by PBE and HSE06 functionals. The calculations were carried for supercells from 64 to 2744 atoms at PBE level and for supercells from 64 to 1000 atoms at HSE06 level. They calculated E_{ b } at a given size of the supercell as
where ΔV potential alignment between the defective and perfect supercells appears the charge correction of defects in Section 3.3.1. Interestingly, the ΔV does not show a clear monotonous decay with increasing size of the supercell which was not explained [119]. The scaling of the HSE06 data was carried out as
where
The neutral donor defects introduce S = 1/2 spin that can interact with the nuclear spins of the dopant or the proximate ^{29}Si I = 1/2 nuclear spins which is called hyperfine interaction. Actually, the interplay between the electron and nuclear spins could represent qubit states in these systems, therefore, understanding this interaction as a function of electric field and strain is highly important (see Ref. [120] and references therein). This hyperfine interaction can be generally written as
where A is the hyperfine tensor and
where
where A_{0} is the value in the absence of strain, K coupling is responsible for the hydrostatic strain, L coupling and N coupling describes the uniaxial and shear strain effects, respectively. Swift and coworkers only focused to the hydrostatic strain in their ab initio study: they found that K scales the same in HSE06 and PBE functionals in relatively small supercells, thus the error in PBE in the difference of absolute values of hyperfine constants as a function of strain is cancelled. As a consequence, the PBE scaling can be applied for larger supercell for extrapolation to the dilute limit. They obtained K = 20.2 which is close to the experimental data at K = 19.1 (see Refs. [119, 120]).
In the aforementioned examples, the defect wavefunction is spatially extended in the electronic ground state. Perhaps, it is not a common knowledge among scientists coming from the quantum optics field that similarly extended wavefunctions could exist in the electronic excited states. Again, the best example is the most studied small band gap material, silicon. As silicon has a band gap of 1.215 eV at cryogenic temperatures, there is a little room to introduce multiple levels by deep defects. Defects may introduce only a single occupied deep level in the fundamental band gap of Si where the electron could be promoted from the ingap defect level to CBM. In this case, the defect can be described as a positively charged centre which Coulombically binds an electron split from CBM. This definitely shows a similarity to the shallow donor states in Si. For example, the photoluminescence Ccentre in Si shows a sharp ZPL at 789 meV for which photoluminescence excitation (PLE) measurements revealed a hydrogenic or Rydberg series of excited states [121]. Later it was shown that this type of PLE features is common for other deep optical centres of Si that was called “pseudodonor” model [122, 123]. Recently, the pseudodonor model of Ccentre in Si has been confirmed by HSE06 calculations [124]. The neutral C_{ i }O_{ i } defect associated with this optical centre indeed produces a deep level in the fundamental band gap (see Figure 3), and the calculated ZPL energy at 750 meV by ΔSCF method agreed well with the experiment. In this calculation, 512atom supercell was employed with the same correction in the total energy of the excited state of the defect (56 meV) as for the positively charged defect. The reasoning behind this method was the following. The excited state involves a spatially extended wavefunction. The scaling property of the total energy of the excited state was assumed to go similarly to that of the positively charged defect within the accuracy of about 50 meV because the extended electron occupying the state split from CBM could behave similarly to all the crystalline valence bands of the system which leaves a positively charged core; in other words, the pseudodonor electron does not “shield” the positively charged core. There is a further note here about the accuracy of ΔSCF method. In a 216atom supercell, the results from ΔSCF method and GW + BSE method were compared. It was found that GW + BSE confirmed the composition of the exciton as the deep hole state and an electon state split from CBM and the vertical excitation energies were within 11 meV. Thus, ΔSCF method can be applied for the bound exciton excitation too which is important to calculate the Stokesshift upon excitation as no quantum mechanical force calculation has been yet implemented to GW + BSE methods. For the case of shallow donors in Si, the geometry change upon ionisation was neglected. However, this cannot be neglected for the fluorescence spectrum of deep defects. Indeed, the sharp features in the phonon sideband of the PL spectrum could be well reproduced by applying the Franck–Condon theory [124]. According to the calculations, Ccentre is a potential building block of quantum repeaters in the telecom Lband [124].
Another deep optical centre in Si, the Wcentre, has been recently isolated as a single quantum emitter with ZPL wavelength close to the telecom region at 1218 nm (1.018 eV) (see Ref. [20] and references therein). The defect contains a complex of three selfinterstitial silicon atoms. The most stable configuration, so called I_{3}V configuration with C_{3} symmetry, has been recently identified by HSE06 calculations where the relative stability changes with respect to PBE calculations [20], which we call here I_{3} for the sake of simplicity. The electronic structure of the neutral I_{3} is very interesting: it shows a single resonant a level at 73 meV below VBM (see Figure 4). At first glance, this defect may be considered as electrically and optically inactive. However, after ionisation, the unoccupied defect level emerges inside the bandgap, and the (+/0) charge transition level is at 55 meV above VBM after applying charge correction in the total energy of the positively charged defect. As the stability of the positive charge state is confirmed, the positively charged defect may Coulombically bind an electron with the state split from CBM, alas, the neutral excitation of I_{3} is a bound exciton with a strongly localized hole on the defect and a loosely bound electron [20]. The pseudodonor nature of the defect was confirmed by HSE06 ΔSCF calculation in 512atom supercell. The estimation of the ZPL energy was based on the full geometry relaxation of the electronic ground and excited states with scaling of supercell sizes,
where L is the side of the simple cubic supercell, A, B, C are fitting constants, where C value corresponds to the dilute limit. L was varied between the supercells of 216atom and 8000atom for PBE calculations and up to 1000atom for HSE06 calculations. It was found that the 216atom supercell results do not fit to the trend and should be ignored. The idea of the formula in Eq. (21) is that the excited state requires charge correction. Since too few data points could be calculated at HSE06 level, it was assumed that the PBE results well reproduce the electrostatics of the problem, and the A and B fit results can be used for HSE06 data points. This procedure finally yields C = 1.102 ± 0.003 eV which is within 0.1 eV when compared to experimental data.
The pseudodonor or bound exciton excitation can occur in wide band gap semiconductors too. A very nice example is the socalled D_{ I } centre in 4H SiC [125, 126]. The optical activity of the defect is identified as the silicon antisite [127, 128] which is an isovalent centre with producing a deep donor level in the fundemental band gap. The defect can be positively charged and then it can Coulombically bind an electron from CBM with producing Rydberg series of excited states [125]. It can be expected that similar bound exciton excited states may be found in diamond.
Indeed, a recent study has identified Rydberg series in the optical spectrum of the diamond neutral siliconvacancy [SiV(0)] defect in a joint experimental and theoretical study [11]. Interestingly, optical spinpolarisation and ODMR signals could be also observed through their bound exciton states [11] which makes the analysis of these states highly important as this defect can be isolated in diamond as a nearinfrared single photon emitter [129].
The electronic structure of SiV(0) in diamond was already briefly described in Section 3.1.1 that we extend here before we proceed to the discussion of the bound exciton states. In SiV(0) defect, Si atom sits in the inversion centre of diamond so the defect can be rather described as a V_{2} defect with six carbon danglings bonds whereas the “dopant” Si ion resides in the empty space of V_{2} with the farthest distance from these six carbon atoms (see Figure 5). The six carbon dangling bonds create a double degenerate e_{ u } level resonant with the valence band and a double degenerate ingap e_{ g } level occupied by two eletrons. This forms the ^{3}A_{2g} ground state. The usual optical activity is associated with promoting an electron from the e_{ u } level to the e_{ g } level for which the optically allowed ^{3}E_{ u } → ^{3}A_{2g} transition yields the ZPL energy at 1.311 eV (see Figure 6(c)). This energy is much smaller than the ionisation threshold energy at 1.53 eV which corresponds to the neutral to the negative charge transition (see Ref. [11] and references therein). It is important to notice the selection rules of optical centre with inversion symmetry that the optical transition is only allowed by changing the parity of the participating wavefunctions.
By increasing the excitation energy above the ZPL at 1.311 eV but below the ionisation threshold energy at 1.53 eV, one can excite the hole from the VBM which results in a SiV(−) defect plus a loosely bound hole, i.e., a bound exciton state of SiV(0). Generally, analysing the hole bound exciton spectrum has an increasing complexity over that of electron bound exciton spectrum because of the orbital degeneracy of VBM at the Γpoint which results in an effective spin–orbit interaction. A detailed description is beyond the scope of the present review paper. We rather defer the readers to the supplemental material of Ref. [11] which is now very briefly summarized here. The threefold degenerate VBM of diamond sligthly splits due to the defect potential resulting in a_{1g} and e_{ g } bands, where a_{1g} band lies above e_{ g } band. The VBM splitting also affected by the spin–orbit coupling which has similar energy as the crystal field splitting induced by the defect potential [11]. The spin–orbit coupling creates lighthole, heavyhole and a splitoff hole in the VBM, where the heavyhole has the shortest Bohrradius effective mass state orbitals [11]. According to the theory from Thiering and Gali [11], the 1s, 2s, … effective mass states will transform as A_{1g}, whereas 2p, 3p, … effective mass states will transform as A_{2u} and E_{ u }, and 3d, 4d, … effective mass states will transform as A_{1g} and E_{ g }. As a consequence, only the ptype effective mass states can be optically excited from the A_{2g} ground state. The lowest energy 1s effective mass state may be observed with the contribution of utype of phonons as phononassisted optical transition. Since the translation motion of Si ion in the void of V_{2} transforms with u oddparity [130] the 1s, 2s, … as well as 3d, 4d, … effective mass states could be optically excited via the A_{2u} quasilocal phonon mode of the Si ion which is about 43.4 meV according the PBE calculations [11]. Indeed, the 1s effective mass state was not observable in the PLE spectrum but well detectable in the optical spinpolarisation spectrum mediated by the Siion vibrations [11] (see Figure 6(a)).
By applying Rydberg scaling to the experimental data (Figure 6(b)), one can find that the expected binding energy of the 1s level is about Δ_{1} − 0.04 = 0.19 eV deeper when the experimental data is corrected with the phonon energy of the Si ion vibration.
An important observation is that the central cell correction makes the 1s level shallower (i.e., its binding energy becomes smaller) than the value of the effective mass theory, in stark contrast to the case of shallow donor and acceptor dopants in semiconductors. The central cell correction energy can reach hundreds of meV for deep defects in diamond.
The qualitative explanation behind this observation can be drawn from Figure 5: the majority of the 1s effective mass state are localized in the core region of the defect in which the localized orbitals are confined. As a consequence, the electron cloud of the localized orbitals will shield the effective attractive potential of the defect and repel the 1s effective mass state which finally shifts its energy level closer to the ionisation threshold energy. The quantative prediction of the 1s energy level calls for ab initio calculations. Unlike the case of the deep defects in Si with bound exciton excited states, the relatively short Bohrdiameter of the VBM hole 1s state makes it possible to embed the excited state wavefunction in a fewthousand atom supercell, viable at KS DFT level. From a remote distance, the SiV(−) + hole system looks completely neutral which is the case of a giant supercell by completely embeding the 1s wavefunction. In smaller sizes of supercells, the systems looks like a negatively charged defect as the hole wavefunction is completely delocalized within the applied supercells, and a total energy correction should be applied similar to the SiV(−) defect. However, the total energy correction for the SiV(−) + hole system should not be exactly the same with the SiV(−) defect as the size of the supercell is increased because the bound hole provides a screening towards the SiV(−) defect core. Thiering and Gali suggested the following formula [11],
where D is the screening length which effectively screens the monopole charge induced by the defect, the stype spherical potential. The quadropole term B may also incorporate the strain field effects too, and C is the value of the dilute limit. It is a critical issue how large could be the screening length. This can be illustrated by numerical modelling of a hydrogen atom in the simple cubic supercell with lattice constants (L) which can be calculated at Hartree–Fock level within VASP with using a soft PAW potential. The results are shown in Figure 7(d). One can clearly see that the Coulombic scaling (−1/L) deviates at sufficiently large supercells. At sufficiently large supercell size (L > 4 Å), the total energy of the system converges exponentially to a constant energy. The dilute limit (C) is found by
where the Coulomb interaction (A/L) is screened by
One can conclude from the results of this simple model that the screening length is multiple times longer than the Bohrradius of the effective mass state. This result could explain the need of simplified computational approaches on silicon defects described above because the Bohrradius itself is already too long to be accommodated by 10,000 atom supercell, so the screening radius cannot be computed at ab initio level. In those cases, Eq. (21) was applied [20] which overcorrects because of the neglect of the screening effect. In diamond, the Bohrradius of the effective mass hole states is short enough to observe the deviation from the formula in Eq. (21) due to the screening effect in few thousand atom supercell calculations. Thiering and Gali in Ref. [11] applied PBE functional to yield the 1s total energy by ΔSCF method including 8000atom supercells, and the resulting screening length was fixed in the fit to HSE06 ΔSCF energies as a function of the supercell size where the maximum size was 1000atom supercell (Figure 7(b)). The scalings of the charge transition level (Figure 7(a)) and the excitation energy of the 1s state show similarities in the range of small supercell size but a clear deviation can be observed for supercell size with
These recent findings cannot be found in the textbooks about semiconductor physics and can be considered as an extension of effective mass theory towards the excited states of deep defects.
The bound exciton states may be formed not just upon optical excitation but by capturing carriers by the deep defects. The capture rate can be, in particular, effective if the defect Coulombically attract the carriers. In the previous example, it can be imagined that the SiV(−) defect in its ground state captures a hole from VBM, which turns SiV(−) to SiV(0) plus a bound hole excited state, and that will decay either radiatively or nonradiatively to the ground state of SiV(0). This effect was first considered in the electroluminescence of single NV defect in diamond (see Supplemental Material in Ref. [99]). In this case, the negatively charged NV defect binds a hole, creating NV(−) plus bound hole from the VBM. That was calculated by HSE06 ΔSCF method without studying the convergence of the excitation energy [99]. In a later study [100], the interaction of two individual NV defects was investigated when the photoionisation of one NV leads to the emission of holes toward the neighbour NV(−) defect which can capture that hole. According to the interpretation of the measurements, a giant σ_{ h } ≈ 3 × 10^{−3} μm^{2} hole capture rate was derived for NV(−). This was rationalised by involving the formation of a bound exciton state featuring an electron localized at the NV(−) plus a bound hole from VBM. Flick in Ref. [100] calculated the total energy of this bound exciton excited state by following the recipe in Ref. [11] with a little modification of Eq. (22) to apply the screening effect also on the quadrupole term. It was found that the binding energy of the exciton is about 40 meV for the 1s state [100]. That is definitely a stable state at room temperature.
3.4 Computation of zerofield splitting for highspin defects and the gtensor for defects with heavy ions
The computation of magnetooptic parameters of defect qubits such as zerofield splitting (ZFS) or gtensor is of high importance not only because that they act as a fingerprint for identification of defect qubits with unknown microscopic structure but they can govern the type of interaction with external magnetic, electric, and strain fields as well as temperature. The importance of interaction between the hyperfine tensor with the strain field was already illustrated in Section 3.3.4. Here, we briefly list the advance in the calculation of ZFS and gtensor for defect qubits.
The highspin (S ≥ 1) defects with axial or lower symmetry may experience the electron spinelectron spin dipoledipole interaction that may be expressed as
where r_{
ij
} = r_{
i
} − r_{
j
}. The 3 × 3 Dtensor can be diagonalized to find the spectrum and spin eigenstates. The D tensor is associated with the twoparticle spin density matrix, n_{2}(r_{1}, r_{2}), which can be approximated by using the Slaterdeterminant of the KS wave functions ϕ of the considered system, so that
where
Each of these m_{
s
} = S − 1 configurations can be obtained by changing the occupation of one of the halffilled KS orbitals from spin up to spin down and subsequently performing the selfconsistent field calculation. It was found that for divacancy defects in 4H SiC that the calculated Dconstants are at around 1.6 GHz but
The ZFS may have other contribution for S ≥ 1 systems than electron spinelectron spin dipoledipole (D_{SS}) interaction as given in Eq. (25). As an example, we mention here the neutral nickelvacancy (NiV) defect in diamond which has the same structure as SiV(0) defect discussed above in this review paper. NiV(0) has also ^{3}A_{2g} ground state similar to that of SiV(0) with six carbon dangling bonds which constitute of the ground state electron wavefuntion. The calculated D_{SS} = 967 MHz = 0.004 meV which is typical for the third neighbour distance of dangling bonds in diamond (see Table 2 below). However, this value is very far from the observed D = 170 GHz = 0.703 meV (see Ref. [133] and references therein). As ^{3}A_{2g} is an orbital singlet, thus firstorder spin–orbit interaction does not enter here. However, secondorder spin–orbit interaction between the ground state triplet and excited state singlet states may play a role which can be selective towards the m_{
s
} = 0 state of the triplet, and it results in an effective energy shift of the m_{
s
} = 0 level and opening the gap between the m_{
s
} = 0 and m_{
s
} = ±1 levels. To illustrate this using the firstorder perturbation theory, we consider the interaction of the ^{3}A_{2g} and ^{1}A_{1g} state that are linked by the parallel component of the spin–orbit operator
Defect  D _{SO}  D _{SS}  D _{SO+SS}  Experiment 

NV(−)  6  2722  2728  2878 
SiV(0)  480  570  1050  929 
GeV(0)  1469  630  2099  2248 
SnV(0)  10,763  630  11,393  
PbV(0)  144,860  660  145,520 
The energy gap between ^{3}A_{2g} and ^{1}A_{1g} is Θ before applying
and it will be dominant over D_{SS}. According to HSE06 calculations (see details in Ref. [133]), Θ ≈ 0.68 eV and λ_{0} = 23.2 meV which results in D_{SO} = 0.79 meV. This is much closer to the experimental data at 0.703 meV.
One can go beyond the firstorder perturbation theory and consider the change in the wavefunction due to spin–orbit interaction (secondorder perturbation),
which may result in a more accurate result than that by firstorder perturbation theory.
In general, the problem can be rephrased by considering the total energy of the system as a function of the spin quantisation direction,
The selfconsistent HSE06 D_{SO} = 0.73 meV which is 0.06 meV deeper than the firstorder perturbation theory value at 0.79 meV, and it brings the result closer to the experimental data at 0.703 meV. This shows that selfconsistent spin–orbit calculation needed for obtaining accurate ZFS for defects consisting of heavy ions. It is interesting to note that selfconsistent spin–orbit PBE calculations results in D_{SO} = 1.35 meV which is significantly larger than the HSE06 and experimental values. We note that Θ ≈ 0.25 eV with PBE which explains the too large D^{SO} with PBE as firstorder perturbation theory showed that D^{SO} scales inversely between the gap of the triplet and singlet levels [133].
These results clearly demonstrate [133] that the energy gap between triplet (highspin) and singlet (lowspin) levels are highly critical in obtaining an accurate ZFS for defects which consist of heavy ions.
The disasvantage of the selfconsistent spin–orbit calculations is that it can principally work for sufficiently large spin–orbit energies, usually created by heavy atoms, so that it does not fall below the numerical noise. For defects with light atoms, one has to rely on the firstorder perturbation theory which was previously sketched for a special case (diamond NiV defect) as an introduction to the problem.
Biktagirov and coworkers [134] implemented the perturbation theory based method to the GIPAWtree of the Quantum Espresso package. They apply collinear spin polarisation approximation with direction a = x, y, z. Then the SO coupling in direction a
Here the sum runs over the spin channels s and s′ and the occupied states o ∈ s. Thereby,
Biktagirov and coworkers applied their method to diamond NV centre and groupIV–vacancy defects [134]. The results are listed in Table 2. As can be seen the defect with the lightest atom exhibits the smallest D_{SO} whereas it increases orders of magnitude with going to heavier atoms. In the groupIV–vacancy defects D_{SS} increases slightly as the heavier atoms push the neighbour carbon atoms farther from each other but the vast contribution comes from D_{SO} except for SiV(0) and partially for GV(0) where the two contributions are similar. We note that PbV(0) shows about D_{SO} = 145 GHz for which selfconsistent D_{SO} calculation would result in a lower value.
Although, the calculated D_{SO} is only 6 MHz for diamond NV centre but it couples directly to the electric field unlike D_{SS} which couples to the electric field only indirectly via changing the electron cloud so the spin density (e.g., Ref. [135]). As a consequence, the field applied along the defect’s symmetry axis, the D_{SO} part dominantly drives the predicted Stark coefficient, 0.034 GHz Å/V, into the experimentally observed confidence interval of 0.035 ± 0.002 GHz Å/V (see Ref. [134]). The simulation was carried in a (111) diamond slab where the electric field was switched on during the calculations of D_{SS} and D_{SO}.
Previously, we discussed the spin level structure in the absense of external magnetic field. Nevertheless, it is highly important to understand the coupling of defect spins to external magnetic fields. The external magnetic fields could be intentionally turned on for manipulation of the qubits on one hand, and on the other hand, randomly distributed electron or nuclear spins proximate to the defect qubits could influence their longitudonal relaxation and coherence times. Here, we discuss the issue of a constant macroscopic external magnetic field interacting with the defect’s electron spin which can be generally described as
where
This issue is illustrated on the neutral vanadium defect substituting the Sisite in 4H SiC which has become a very promising spintophoton interface with a quantum memory and optical emission at the telecom wavelength (see Refs. [136, 137] and references therein). The dorbital of the vanadium ion splits due to the C_{3v} symmetric crystal field of 4H SiC and then a double degenerate eorbital occurs in the gap localized on the dorbital of vanadium, occupied by a single electron. Because of the double degenerate dorbital, one can expect an effective spin–orbit coupling between the orbital and the electron spin, where the low symmetry crystal field will reduce the effective angular momentum of the orbital called Stevens reduction factor (r) (e.g., see the origin of this effect in more detail for groupIV–vacancy defects in Ref. [138]). However, it is also known that this is an E ⊗ e Jahn–Teller system [139, 140] which can also effectively reduce the angular momentum of the electron orbit so the effective spin–orbit splitting known as Ham reduction factor (p). As a consequence,
where the contributions of
where expectation values of the ladder magnetic dipole moment operators are used (μ_{±}) to express g_{⊥}. In Eq. (34) S_{
z
} and
We note that the dorbitals may require a special attention for accurate calculation, and indeed, HSE06 DFT functional overlocalises the d state that should be corrected [142]. For the heavyatom defects one may assume that the vast majority of the spin–orbit coupling, so the effective angular momentum, comes from the single heavyatom. In this case, the analysis of the dorbitals and the actual DFT wavefunctions can reveal the deviation of the dorbitals from the spherical symmetry (see Table 3), so the r can be computed. As an example, Ψ_{1} and Ψ_{4} as well as Ψ_{2} and Ψ_{3} can be coupled in Eq. (34) where the corresponding wavefunction coefficients can be extracted for the given spin–orbit state in the KS DFT calculation. It was found that the vanadium at one site of 4H SiC feels isotropic environment with small effective angular momentum which is an order of magnitude larger in a truly axiallike environment at the other site of 4H SiC. After solving the Jahn–Teller Hamiltonian (see Refs. [30, 141]), the typical Ham reduction factor is at about 0.6 which is significant, so the electronphonon coupling effectively reduce the interaction between the defect’s pseudospin and the external magnetic field parallel to the symmetry axis of the defect. The final typical values of g_{‖} are around 1.9.
Labels  Orbitals  Irreps.  m _{ j }  

Single  Double  
Ψ_{1} 

^{2} E 


Ψ_{2} 

^{2} E 


Ψ_{3} 

^{2} E 


Ψ_{4} 

^{2} E 


Ψ_{5} 

^{2} A _{1} 

For the calculation of g_{⊥} (Eq. (35)), the ladder magnetic dipole operator was considered,
The same method was applied to the nickel defects in diamond where the NiV(−) was identified by first principles calculations as an excellent qubit candidate analogous to the groupIVvacancy qubits in diamond [133] which has an optical emission at about 1.4 eV and a highly anisotropic gtensor. In the literature, the NE4/AB1 EPR centre with S = 1/2 spin and relatively isotropic gtensor with g_{‖} = 2.0027(2) and g_{⊥} = 2.0923(2) was previously associated with NiV(−) which is linked to the 1.72eV optical centre (see Ref. [143] and references therein). Clearly, the NE4/AB1 centre should be associated with another nickelrelated defect in diamond. Thiering and Gali tentatively assigned Ni_{s}(N_{s})_{3}(0) defect to this centre which has an unpaired electron on the a_{1} orbital strongly localized on Ni 3d orbitals. In this case, the gtensor is modified from the free electron value because of the orbital moment of the Ni 3d states as explained above for vanadium defects in 4H SiC. This justifies to calculate the total orbital moment
3.5 Spinphonon coupling: temperaturedependent longitudonal spin relaxation time and magnetooptical parameters
3.5.1 Longitudonal spin relaxation time
A key parameter of qubits is the longitudonal spin relaxation time which is the characteristic time of flipping the spin, and it is labeled by T_{1} in the literature. This sets the absolute limit of the spin coherence time, i.e., the characteristic quantum information processing operation time of the qubit. It is of high importance to understand the underlying microscopic processes. In nuclear spin physics, the origin of spin flipping was identified as the interaction between phonons and the spin which is manifested as a highly temperaturedependent phenomenon; therefore, it is also often called spinlattice relaxation time. As T_{1} often exponentially decay with elevating the temperature it is imperative to characterise T_{1} as a function of temperature, in order to explore the applicability of qubits as sensor probes in biology which requires ambient conditions. Our review paper only focusses to the recent advances on defect qubits, in particular, on S ≥ 1 defect qubits.
Spin flipping processes require such interaction Hamilton operator which contains spin shift operators. It can be easily recognized that the spinspin dipoledipole interaction in Eq. (24) contains single and double spin shift operators, e.g.,
Ivády applied the clustercorrelation expansion (CCE) [144, 145] to model the interaction of the central diamond NV centre’s electron spin with other electron spins such as the environmental NV centres, nitrogen donor spins (labelled as P1 EPR centre), and the ^{13}C nuclear spins also as a function of the external magnetic field and strain [62]. The CCE approach will be shortly discussed in the next chapter.
We note that another study only restricted this investigation to the bath of ^{13}C nuclear spins but taking only the dipolar interactions into account with far ^{13}C nuclear sites [146]. However, the bath of ^{13}C with Fermicontact hyperfine terms cannot be ignored for accurate simulations which calls for ab initio simulations.
Ivády calculated the hyperfine tensors for ^{13}C isotopes by HSE06 DFT method in a 1728atom simple cubic supercell [62]. Since the hyperfine tensors should be determined at large distances from the defect site this required a special approach in order to avoid finite size effect problems. Ivády utilised a real space grid combined with the PAW method to calculate hyperfine tensors. The Fermi contact term, dipoledipole interaction within the PAW sphere, and core polarisation corrections are calculated within the PAW formalism from the convergent spin density. The dipolar hyperfine contribution from spin density localized outside the PAW sphere is calculated by using a uniform real space grid. This procedure enabled to obtain hyperfine coupling tensors excluding effects from periodic replicas of the spin density due to the periodic boundary condition. Additionally, hyperfine tensors for atomic sites outside the boundaries of the supercell were calculated by neglecting Fermi contact interactions with achieving a smooth transition in the hyperfine constants at the boundary of the two approaches.
Ivády used the extended Lindbladian equation in order to simulate the spin dynamics of the central spin and its relaxation rate 1/T_{1} in materials where the electron spin density cannot be ignored beside the nuclear spin bath [62]. In this model the total spin relaxation rate (1/
where “NVbasal” and “NVpar” label such NV centres in the environment which have other and parallel symmetry axis with that of the central NV centre. Finally, it was found that the environmental NV centres have a dominant effect on the spin relaxation rate [62]. At special setting of the magnetic fields, either ground state level anticrossing (GSLAC) or excited state level anticrossing (ESLAC), the relaxation rate is accelerated because the P1 centres and the nuclear spins can easily induce spin flipflop processes that were otherwise protected by the energy gap between the electron spin levels of the NV centre. At GSLAC, the spinpolarisation of the electron spin and the coupled nuclear spin also changes that can be observed by the change of the PL intensity as the external magnetic field is swept around the GSLAC region [147]. This modelling also rationalised the photoelectric readout process of the single ^{14}N nuclear spin of the NV centre at ESLAC condition [148].
This theory was also applied to the divacancy qubit in 4H SiC by considering other divacancy spins (S = 1), negatively charged Sivacancy spins (S = 3/2), nitrogen donor spins (S = 1/2), as well as ^{13}C and ^{29}Si I = 1/2 nuclear spins in the environment, also as a function of the external magnetic field [150]. It was found that the crossrelaxation accelerate spin flipflop rates again in the region of GLSAC and ESLAC magnetic fields for each considered environmental spins. At zero magnetic field a simple relation was found for the interaction between Ndonor and the central divacancy spin,
where β = 1.6 × 10^{−35} Hz/cm^{−6} and C is the concentration of the Ndonor. It is noted that nitrogen implanted samples the distribution of nitrogen donor is not homogeneous, and then the “concentration” should be considered near the target divacancy spin which was created as a result of the implantation [150]. The theory was also employed to the Sivacancy S = 3/2 qubit in 4H SiC [151]. In this case, BulanceaLindvall and coworkers considered the interaction the Sivacancy qubit spin with S = 1/2 defects, e.g., Ndonors. Sivacancy in 4H SiC has minor ZFS, thus at a given external magnetic field with similar Zeeman shifts the two spin systems can be effectively coupled by dipoledipole interaction unlike the case of divacancy with S = 1 spin and high ZFS (
High quality materials with single defect spins and low concentration of nuclear spins do not experience spin flipflop events with electron spins in the environment, and the flipflop processes caused by nuclear spins are only observable at special conditions (e.g., external magnetic field is set to GSLAC condition). In this case, the spinphonon coupling is responsible for T_{1} and it becomes strongly temperature dependent. In the case of molecules, vibrations are indeed responsible behind the spin flipping process. In a recent review paper [152], the ab initio theory and its application to molecules have been presented in detail. The formulas and basic equations apply to defect qubits too which are not reiterated here in detail.
Regarding the temperature dependence of T_{1} = 1/Γ of defect qubits, the most studied one is the diamond NV centre [31, 153], [154], [155], and the first ab initio results have been reported for this defect qubit because the theories could be well tested on the accurately recorded experimental data. In recent studies [31, 156], the m_{
s
} = ±1 levels of ^{3}A_{2} ground state was split by a small external magnetic field aligned parallel to the symmetry axis of the defect (
The spin relaxation rate may be expressed as
where the superscript refers to the order of the spinphonon interaction (i.e., terms with superscript 1 or 2 are linear or quadratic in the atom displacements respectively) and the subscript refers to the order in perturbation theory. Γ_{0} is a sampledependent constant term arising from spinspin interactions that was discussed above.
In order to calculate
where the coefficients in Eq. (39) were extracted from VASP PBE calculations as implemented by Thiering and Gali [156]. In order to evaluate the secondorder derivatives, only the diagonal terms were considered which satisfy i = j and distort the C_{3v} symmetric atomic positions by all degenerate e_{
x
}, e_{
y
} phonon modes of the supercell by
where R_{
i
}, X_{
l
} and Y_{
l
} are the dimensionless coordinates (not normal coordinates) for the phonon modes at energy ℏω_{
i
} or ℏω_{
l
}. We note that while the index l only covers the e modes once, the index i covers all a_{1}, a_{2}, e_{
x
}, e_{
y
} modes and thus runs over the e modes twice. Eq. (40) can be employed to transform it into the spinphonon interaction
where
As a next step for defining the equations for the rates, one can apply RPA for these processes so it is assumed that the consequtively absorbed phonons are not coherent. Furthermore, one can further simplify the equations by considering the fact that the ZFS energy of the diamond NV centre is much smaller than the typical phonon energies coupled to the spin. The key matrix elements are the firstorder spinphonon coupling coefficients
Finally, the appropriate spin relaxation rates are
and
respectively. The temperature dependence enter via the Bose–Einstein occupation function (n_{B}) of the phonons at ω, ω′ energies. In the current implementation [156], ℏω′ = ℏω constraint was employed in Eq. (43) that also enforces l = l′ and so
The numerical ab initio calculations provided very slow rates for
The calculated secondorder spinphonon coupling coefficients are depicted in Figure 8. Two broad peaks can be observed at certain phonon frequencies that are associated with the motion of the carbon dangling bonds so the spin density (see Ref. [156] and references therein). As the frequencies of the effective phonons are much higher than the thermal energy of the measurement temperature this will scale as Orbachprocess. Ab initio simulation revealed that two effective phonon frequencies exist, thus Γ = 1/T_{1} can be described as a double Orbachprocess, where the higher effective frequency plays a role at elevated temperatures [156]. The theory also well describes the doubleflip transition and the appropriate rate equations, and both processes are double Orbachprocesses. The double Orbachprocess parameters could be well fitted to the experimental data with providing 68.2 meV and 167 meV effective phonon frequencies (gray lines in Figure 9) which agree well with the features of the calculated spinphonon spectral functions. Insights from theory provided a physically well motivated model for the spinlattice relaxation times of diamond NV centre. The calculated rates are depicted in Figure 9. The agreement between theory and experiment for γ is very good whereas a larger discrepancy is observed for Ω. It was hypothesized that the discrepancy in the predicted Ω is due to the exclusion of combinations of modes for which l ≠ l′, as combinations of modes with different symmetries likely account for significant matrix elements associated with pairs of different spin operators, which correspond to the singlequantum transitions. It was also discussed that
3.5.2 Temperature shifts of magnetooptical parameters
Understanding the temperature shifts of magnetooptical parameters of defect qubits is of high importance in various aspects. One of the most obvious issues is the temperature sensing with defect qubits at the nanoscale which requires temperature characterisation of the basic magnetic parameters. Again, the diamond NV centre is the most investigated defect qubits in this regard (see Ref. [30] and references therein). As an example, the temperature dependence of the ZFS of the diamond NV centre was modelled by the thermal expansion [158] which results in an increase in the distance between carbon dangling bonds so the decrease in the ZFS parameter (Dconstant). However, the obtained coupling coefficient was much lower than the experimental data. Recently, Tang and coworkers pointed out [159] that the thermal expansion model covers a “third order” effect as the measured magnetooptical entity ν will be a statistical average of the phonon mode distribution as
where {Q_{
i
}} are the normal coordinates of the phonons wihtin quasiharmonic approximation. In the BornOppenheimer approximation, the global energy minimum results in ⟨Q_{
i
}⟩ = 0 (forces are zero) and it becomes nonzero because of violating of the harmonic approximation, i.e., anharmonicity of the phonons. This can be taken into account in the thermal expansion of the lattice. However, the second term is then expected to be dominating. As
where M_{ i } is the modespecific effective mass and ν_{0} is calculated at the lattice constant a which corresponds to the thermal expansion at the given temperature. The spectral function was defined as
The second derivative was calculated numerically as implemented in VASP in a 128atom face centred cubic diamond supercell with 3 × 3 Monkhorst–Pack kpoint sampling and DFT PBE functional [159]. The theory was applied the Dconstant, the hyperfine constant A_{
zz
} and the quadrupole moment Q_{
zz
} of the ^{14}N, and the ZPL energy of the diamond NV centre [159]. It was found that temperature dependence of the Dconstant could be well reproduced where the dynamical effects play a major role, although, the termal expansion effect cannot be neglected. The ZPL energy shifts were well reproduced too by this theory. However, it is unexpected that the calculated spectral function for Dconstant does not show up a peak at around 70 meV phonon energy in the study of Tang and coworkers [159], which is quite visible in the
3.6 Ab initio theory of coherence of defect spins in solids
The coherence time of the defect qubits’ spin corresponds to a decoherence of the transverse nuclear spin magnetisation which is generally labelled by T_{2}. It is also called spinspin relaxation time as usually the interaction of the defect qubit’s electron spin with the nuclear spin bath limits its value in high quality (small electron spin bath) materials. These nuclear spins do not precess with the same frequency in real materials which can lead to a distribution of resonance frequencies around the ideal. Over time, this distribution can lead to a dispersion of the tight distribution of magnetic spin vectors, and loss of signal. This is called the dephasing time, labelled as
The calculation of the hard limit of T_{2} at a given temperature and electon/nuclear spin bath requires the calculation of T_{1} as described in the previous chapter. For S = 1 defect qubit’s spin, the ±1⟩ spin states may split due to the lowsymmetry of the defect or an external small constant magnetic field. In recent studies [31, 156] it has been found for the diamond NV centre that the doubleflip transition (γ rate) is even faster than singleflip transition (Ω rate) induced by phonons. However, γ was neglected in previous studies (see Ref. [156] and references therein), thus T_{2} have likely been overestimates which should be rewritten as
for a superposition in the {0⟩, ±1⟩} singleflip subspace and
for a superposition in the {−1⟩, +1⟩} doubleflip subspace [161, 162].
If spinlattice relaxation does not interfere then the central spin (qubit’s electron spin) and the (nuclear) spin bath interaction and their dynamics should be simulated for obtaining the spin dephasing and decoherence times where the simulations should consider how the qubit’s spin is controlled and driven for yielding
For defect qubits, the CCE approach has been successfully applied that was originally invented by Yang and Liu [144, 145]. It was originally applied to calculating the pure dephasing of the diamond NV centre’s electron spin in the large detuning regime. However, the central spin flip must be considered when the energy relaxation of the diamond NV centre is involved in the nearlyresonant regime, i.e., GSLAC condition [146]. Often, this is called generalised CCE or gCCE approach. To briefly sketch the problem and the neccessity of approximations, an open system
where the Hamiltonian H_{0} can be written as
where h_{0} is the Hamiltonian of the central spin, h_{
i
} is the Hamiltonian of the coupled spin s_{
i
}, and h_{0i} describes the coupling of the central spin and the bath spin s_{
i
}. The last term on the righthand side of Eq. (49) accounts for environmental effects that are not included in
The size of the problem, i.e., the dimension of the Hilbert space, increases exponentially with n, which makes an exact solution unfeasible for large n. To model the dynamics of
added to the master equation of each subsystem, where and C_{0l} and C_{ il } are Lindblad operators. We consider C_{0l} and C_{ il } operators that describe solely spin flipflop transitions of the central spin. Here b_{ il } are timedependent rates determined from the flipflops occurring in subsystem other than i. The Lindbladian formalism ensures that all the spin flipflops occurring in the different subsystems is carried out in all subsystems. This way the central qubit replicas evolve identically in all subsystems. Due to the extended Lindbladian, spin momentum is conserved no longer in the subsystems but in the whole cluster approximation. This way the cluster approximation together with the Lindbladian coupling describes the dynamics of the whole qubitspin bath system approximately. Considering the dynamics, the main approximation of the method is the neglect of the intra spin bath coupling and entanglement that may affect the dynamics of the central qubit through spin diffusion as well as constructive and destructive interference that can give rise to echo signals and dark states, respectively. These limitations can, however, be systematically lifted by including more and more environmental spins in the subsystems of the cluster approximation.
In the CCE approach, the corresponding Hilbertspace can be significantly truncated that are coupled to each other in which the density matrix of the central spin can be consistently calculated. As we increase the order of expansion, the results should converge to the theoretical limit, in good analog to the CI expansion method for approaching the accurate correlation energy of the manyelectron system. For instance, Yang and coworkers found for the diamond NV centre in the nuclear spin bath of remote ^{13}C nuclear spins [146] that gCCE4 and gCCE5 results agree, thus gCCE4 can considered as absolutely convergent for this particular system.
Seo and coworkers [163] applied CCE method for diamond NV centre and divacancy qubits in 4H SiC for understanding the spin dynamics between the qubit’s electron spin and the nuclear spin bath with assuming natural abundance of ^{13}C at 1.1% and ^{29}Si at 4.5%. The authors ignored the Fermicontact term in the hyperfine interaction between the electron spin and the nuclear spins. A constant magnetic field was applied in the simulation. We note that because not the gCCE method was applied in Ref. [163], therefore, these simulations could not well describe the spin dynamics for the magnetic fields at GSLAC and ESLAC conditions of the systems, which results in a rapid decrease of the coherence times. This was later done by gCCE method for divacancy qubits in 4H SiC [164]. Seo and coworkers found that CCE2 level of theory well converges with the aforementioned conditions and the radius of the spin bath at around 50 Å from the defect qubit’s spin provides convergent results. At the CCE2 level, the distance between interacting nuclear spins was set to 8 Å which converged well [163]. It was found that ensemble averages over 50 samples are good enough to produce numerically converged results. For the magnetic fields above 30 mT they found a simple relation between the
In a subsequent publication, the spin dephasing times were calculated for hypothetical defect spins with no Fermicontact hyperfine interaction with moderate magnetic fields (0.1–0.5 mT) in 2D materials [166] by applying the same method. The subject of this study was later extended to 12,000 materials in which both spin dephasing (freeinduction decay) and spin coherence (Hahnecho) times were considered at large constant magnetic fields (e.g., 5 T) [13]. With these simulation conditions, Seo and coworkers [166] found for hBN and molybdenumdisuplhide (MoS_{2}) materials that the spin dephasing time in bulk hBN should be around 18 μs for natural abundance of ^{11}B and ^{10}B isotopes whereas it is about 1.18 ms in MoS_{2}. They attributed the orders of magnitude difference to partially to the variant of nuclear spin density in the two materials and partially to the relatively small gyromagnetic constant of Mo isotopes [166]. By replacing all ^{11}B by ^{10}B should result in
Although, the results on spin dephasing and coherence times with hypothetical defects are somewhat indicative for classifying materials, the defect qubit’s spin relaxation properties may crucially depend on the local environment induced by the defect in terms of ZFS, strain fields and spin density distribution. The last entity is in particular important for materials with dense nuclear spin bath. In that case, the Fermicontact term in the hyperfine tensor is dominant, and such an effect cannot be fully neglected even in diamond or SiC with relatively dilute nuclear spin densities.
For S = 1 defects the case of GSLAC or ESLAC condition was already mentioned where a simple external parameter, magnitude and direction of the constant magnetic field, may drastically change the coherence properties of the defect qubits, e.g., the interplay between the actual Dconstant of the defect spin and the strength of the constant magnetic field [62, 147, 148, 164]. Another interesting example is the socalled clocktransition quantum optics protocol which may be realized by lowsymmetry defect spins, e.g., basal divacancy defects in 4H SiC, where the +1_{
z
}⟩ and −1_{
z
}⟩ levels naturally splits (see Refs. [164, 167] and references therein). Combining the E = 18.4 MHz splitting with a strong longitudinal splitting (D = 1.334 GHz), the ZFS tensor leads to an avoided crossing of electron spin levels at zero magnetic field from which a clock transition emerges. The qubit levels at the clock transition correspond to
Another important defect spin is the negatively charged boronvacancy (
At moderate external magnetic fields, it is challenging to observe the Rabioscillation of the
In pulsed electron spin resonance measurements, the 4spin nature of
4 Summary
In this paper, we reviewed the recent advances on ab initio theory on defect qubits. A strong emphasis was put on the calculation of excited states, photoionisation thresholds and optical excitation spectra also as a function of temperature. A novel theory has been developed on the effective mass states of the excited states of deep defects. Major breakthroughs have been presented on the calculation spin dynamics of the defect qubits which converted the phenomenological description of the spin relaxation times to fully ab initio solution.
Funding source: European Commission
Award Identifier / Grant number: 862721
Funding source: European Commission
Award Identifier / Grant number: 101046911
Funding source: National Research, Development and Innovation Office of Hungary
Award Identifier / Grant number: KKP129866
Funding source: National Research, Development and Innovation Office of Hungary
Award Identifier / Grant number: 20222.1.1NL202200004
Funding source: European Commission and National Research, Development, and Innovation Office of Hungary
Award Identifier / Grant number: QuantERA II MAESTRO

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Research funding: A.G. acknowledges the Hungarian NKFIH grants No. KKP129866 of the National Excellence Program of Quantumcoherent materials project, the EU QuantERA II MAESTRO project, EU HE FETOPEN project QuanTELCO (Grant No. 862721), the EU HE EIC Pathfinder project QuMicro (Grant No. 101046911) as well as the Quantum Information National Laboratory sponsored by the Innovation and Cultural Ministry (Grant No. 20222.1.1NL202200004).

Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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