Nitrogen-vacancy (NV) complex in diamond is one of the most prominent solid state defects as the negatively charged NV defect (NV ) is a leading contender for quantum technologies. In quantum information processing applications, NV is photoexcited that often leads to photoionization to neutral NV defect, NV , and re-ionization back to NV should occur to control the spin of NV . As a consequence, understanding the photophysics of NV is crucial for controlling NV . Furthermore, recent studies have shown that the electron spin of NV can also be initialized and read out at certain conditions that turns single NV a potential quantum bit. Quantum optics protocols rest on detailed knowledge on the electronic structure of the given system, which is obviously missing for NV in diamond. In this study, we combine the group theory and density functional theory calculations toward exploring the nature of the ground and excited states of NV . We show that the effective three-electron system of NV leads to high correlation effects that make this system very challenging for ab initio simulations.
I. INTRODUCTION
Nitrogen-vacancy (NV) center1 in diamond has been receiving a continuous interest due to its broad applicability for quantum technologies.2–8 Most of these applications utilize its negative charge state (NV ) that belongs to the 637 nm1 ( eV) zero-phonon-line (ZPL) photoluminescence center. The NV color center possesses electronic spin that can be initialized, coherently driven, and read out, thus NV may act as an exemplary qubit operating at room temperature.9–12 An early theoretical work suggested13 that the neutral counterpart, NV , may also act as a qubit. This has been recently realized in experiments,14–17 where NV possess electronic spin in the ground state with the corresponding 575 nm ( eV) ZPL emission (Fig. 1). However, in stark contrast to the extensively studied NV , various magneto-optical properties of NV are vaguely studied and the underlying physics is left unknown.
Electronic structure of the neutral nitrogen-vacancy ( ) center of diamond. (a) Defect levels inside the bandgap of diamond. (b) Single particle (Kohn–Sham) orbitals of defect levels.
Electronic structure of the neutral nitrogen-vacancy ( ) center of diamond. (a) Defect levels inside the bandgap of diamond. (b) Single particle (Kohn–Sham) orbitals of defect levels.
Both 2,18–20 and eV21,22 color centers are active in both absorption (ABS) and photoluminescence (PL) processes. NV defect’s eV ZPL is accompanied19 by a broad phonon sideband both in PL and ABS exhibiting only minor asymmetry [cf., Figs. 2(c) and 2(d)]. The sideband mainly couples with a meV quasilocal vibration and the strength of coupling can be depicted by a Huang–Rhys (HR) factor. Consequently, the ZPL exhibits a Debye–Waller factor of , thus only 3% of total intensity resides in the ZPL in agreement with previous ab initio simulations within23,24 and beyond the HR approach.25
Comparison of theoretical and experimental phonon sidebands for the negative and neutral NV defects. (a) and (c) Absorption line shape of the 2.156 and 1.945-eV peaks of NV and NV , respectively. We scaled the experimental and theoretical line shapes to exhibit the same area under their curves between the ZPL and 100 meV, 200 meV for NV , NV , respectively. (b) and (d) Luminescence line shapes for 2.156-eV NV and 1.945-eV NV . We normalized the experimental (expt.) and theoretical line shapes in their full range. Expt. data are taken from different sources as follows: [a] Fig. 1(b) from Ref. 21, [b] Fig. 2(a) data from Ref. 29 at C anneal, [c] Ref. 32, [d] Fig. 4 in Ref. 33, [e] Fig. 6(b) in Ref. 19, and [f] Fig. 3(b) in Ref. 20.
Comparison of theoretical and experimental phonon sidebands for the negative and neutral NV defects. (a) and (c) Absorption line shape of the 2.156 and 1.945-eV peaks of NV and NV , respectively. We scaled the experimental and theoretical line shapes to exhibit the same area under their curves between the ZPL and 100 meV, 200 meV for NV , NV , respectively. (b) and (d) Luminescence line shapes for 2.156-eV NV and 1.945-eV NV . We normalized the experimental (expt.) and theoretical line shapes in their full range. Expt. data are taken from different sources as follows: [a] Fig. 1(b) from Ref. 21, [b] Fig. 2(a) data from Ref. 29 at C anneal, [c] Ref. 32, [d] Fig. 4 in Ref. 33, [e] Fig. 6(b) in Ref. 19, and [f] Fig. 3(b) in Ref. 20.
Both and -eV ZPLs are attributed to transitions between the same and electronic orbitals [Fig. 1(b)] that would imply comparable optical features. Surprisingly, however, there is a significant asymmetry in NV defect’s PL and ABS phonon sidebands as plotted in Figs. 2(a) and 2(b), respectively. HR factor of the PL spectrum was reported for NV defect26 and another study reported at 77 K (Ref. 15). ABS spectra exhibit much broader sideband quantified by an HR factor.21 Interestingly, the PL signal is coupled with phonons with significantly smaller energy –45 meV15,27 than those of NV . The radiative lifetime of the excited state of NV is reported as 13.2(2),28 19(2),29 17(1),30 and 20(1)31 ns in different studies.
The -eV ZPL of NV occurs between the orbitally degenerate ground state to an orbitally non-degenerate confirmed by uniaxial stress measurements.21,34 Furthermore, splits to two Kramers doublets ( and ) due to spin–orbit coupling. In this context, we note that NV defect’s spin–orbit splitting was already reported as GHz35 or GHz36 by various experiments more than a decade ago, yet the for NV has not been reported until recently. The exact spin–orbit splitting value37 is still ambiguous as 14 and 9.8(8)16 GHz were reported by two different research groups. Additionally, we note that electron paramagnetic resonance (EPR) for NV defect’s spin has not yet been reported, to our best knowledge. The lack of EPR signal is attributed to linewidth broadening caused by the Jahn–Teller effect.38,39
On the other hand, under continuous green illumination at 532 nm, an spin associated with the metastable state of NV was indeed observed in EPR38 where early ab initio calculations confirmed the model based on the favorable comparison of the observed and calculated hyperfine signatures between the electron spin and N [ MHz and MHz in experiments] and C nuclear spins.13 The observed near free electron g-factors ( , ) imply very small second-order spin–orbit interaction associated with this state. In this case, the C symmetric axial field causes a zero-field splitting between the and levels at MHz as observed in the EPR spectrum.38
The existence of the metastable state has been confirmed by ab initio calculations with various levels of theory.13,40–42 However, the level position referenced to the ground state scatters between 0.4 and 0.86 eV (see Table I). According to a recent theoretical study with combining selection rules and ionization potentials of NV (Ref. 42), successive two-photon absorption of NV via the excited state could populate the state over the state of the state, which would explain the occupation of the metastable state and the EPR signal of this metastable state under green optical pumping that is enhanced by the population difference in their spin levels. This theory sets the position level over the level at around 0.4 eV.
Electronic excited states for NV0. All energies are in eV units. We note that we optimized the atomic positions of atoms in our present work (p.w.) with HSE06, PBE, LDA functionals. However, the atomic positions were not relaxed within configuration interaction (conf. int.)c and Hubbard modeling (Hubbard)d results.
Method | |4A2⟩ | |2A2⟩ | |2E*⟩ | |2A1⟩ |
expt. | ∼0.4a | 2.156 | n.a. | n.a. |
HSE06 | 0.48p.w. | 2.36p.w. | 2.33p.w. | ∼2.8p.w. |
PBE | 0.58p.w. | 1.20p.w. | 1.24p.w. | n.a |
LDA | 0.86b | 1.22p.w. | 1.26p.w. | n.a |
conf. int. | 0.68c | 1.65c | 2.04c | 2.93c |
Hubbard | 0.68d | 2.64d | 2.88d | 3.29d |
Method | |4A2⟩ | |2A2⟩ | |2E*⟩ | |2A1⟩ |
expt. | ∼0.4a | 2.156 | n.a. | n.a. |
HSE06 | 0.48p.w. | 2.36p.w. | 2.33p.w. | ∼2.8p.w. |
PBE | 0.58p.w. | 1.20p.w. | 1.24p.w. | n.a |
LDA | 0.86b | 1.22p.w. | 1.26p.w. | n.a |
conf. int. | 0.68c | 1.65c | 2.04c | 2.93c |
Hubbard | 0.68d | 2.64d | 2.88d | 3.29d |
Although directly not known, the difference between photoionization thresholds of |3A2⟩ → |2E⟩ and |3E⟩ → |4A2⟩ are 432 and 342 meV, respectively, which hints for ∼0.415 eV energy position for the |4A2⟩ multiplet. However, these photoionization thresholds were recorded at room temperature where phonon-assisted transition can also occur, thus ∼0.415 eV is still an estimate.
Ab initio taken data from Ref. 13.
Ab initio data by means of screened configuration interaction on top of LDA (local density approximation of DFT) from Ref. 41.
Ab initio data by means of generalized Hubbard Hamiltonian on top of B3LYP functional from Ref. 40 in a C71H85 diamond molecular cluster.
This example shows that it is crucial to know the lifetime of this metastable state so one can reliably set up quantum optics protocols for this potential qubit state. This is associated with the intersystem crossing (ISC) between and as well as the photoionization cross section from to the ionization bands. To our knowledge, this issue has not yet been resolved for NV as well as the origin on the phonon sidebands of the absorption and luminescence spectra. In this study, we aim to understand the photophysics of NV and the non-radiative decay routes that are all associated with the strong electron–phonon coupling of the system.
The paper is organized as follows. In Sec. II, we describe our ab initio methodology. Then, we depict the electronic structure of NV and establish the nomenclature of our paper in Sec. III. Next, we discuss the interactions that govern the ground state and its optical excitation at 2.156-eV in Sec. IV. After understanding both spin–orbit and electron–phonon couplings of NV , we show and discuss our results on the ISC rates of in Sec. V and the photoionization threshold energies and processes in Sec. VI. Finally, we will conclude our paper in Sec. VII.
II. METHODOLOGY ON ATOMISTIC SIMULATIONS
We apply ab initio calculations with the use of simple cubic 512-atom supercells embedding NV defect within the framework of the spinpolarized density functional theory (DFT) as implemented in the vasp 5.4.1 code.43 The 512-atom model suffices to sample the Brillouin zone of the supercell at the point. We converged the electronic structure by self-consistent cycles with eV convergence threshold. The applied projector-augmentation-wave method (PAW)44,45 on the core electron orbitals are made possible to use a relatively low cutoff (370 eV) plane wave basis for electron wavefunctions. We converged the forces acting on ions below eV/Å.
We applied the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional46,47 that reproduces the experimental bandgap and the charge transition levels of defects in group-IV semiconductors within 0.1 eV accuracy.48,49 We determined the excitation energies for excited states within 0.1 eV accuracy with the SCF method50 that provides accurate ZPL energy and Stokes-shift for the optical excitation spectra of the triplets of the NV center23,24 and group-IV impurity-vacancy centers51,52 of diamond.
We used the HR theory23,53,54 to predict the phonon sidebands near the ZPL emission. We used the Perdew–Burke–Ernzerhof (PBE) semilocal functional55 to determine the vibration frequencies and eigenvectors as PBE is reported to accurately predict these features.23 However, we employed the computationally demanding HSE06 to determine the optical excitation energies by means of the SCF method where the atomic positions upon excitation are relaxed.
We determined the spin–orbit coupling as implemented in the vasp code within the scalar-relativistic approximation.56 During the non-collinear calculations for spin–orbit coupling, we fixed the spin quantization axis along the trigonal symmetry axis of the defect. We determined the spin–orbit coupling parameters directly from matrix elements between single particle Kohn–Sham orbitals that reproduced the observed fine structure of defects within accuracy.24,51,52,57,58
III. PRELIMINARIES: ELECTRONIC STRUCTURE
Here, we define the basic nomenclature of the paper. We note that the orbitals and levels of NV from DFT calculations have been already published in several papers.40,59–66 Furthermore, the respective many-body states and spin–orbit coupling coefficients acting between them were also thoroughly analyzed.67,68 As we will show in this paper, the strong electron–phonon coupling should be explicitly considered for the many-body states of NV that was lacking in previous studies. Therefore, instead of referring toward these papers, we explicitly reiterate the respective wavefunctions and interactions that we use in the entire paper. Therefore, we begin our discussion presenting electronic structure of NV in Sec. III A. We describe the different forms of electronic correlations governing multiconfigurational character , , , multiplets of in Secs. III B–III D. We summarize the multiplet energies within different levels of theory from the literature accompanied by our present results in Table I.
A. Electronic structure of NV0
B. Dynamic correlation in the |2E⟩ ground state
We quantify the effect of electronic correlation in energy units by removing the symmetry constraint on the wavefunctions and enforcing C symmetry only on the atomic positions. We report , , meV energy gain by means of LDA, PBE, HSE06 functionals, respectively, where we enforce symmetrization by putting two electron halves on , orbitals resulting in the electron occupation. We note that we determined the vibration frequencies and modes within this smeared occupation by means of PBE functional to avoid broken symmetry wavefunctions as given in Eq. (9). However, this introduces an artificial self-interaction as the electrons on and repel each other due to exchange integrals present in hybrid functionals such as HSE06. In summary, we highlight that the electronic correlation between the two , multiplets is a significant effect that cannot be neglected. We set the reference point of our electronic total energy (for example, in Fig. 3) to this broken-symmetry solution on which we removed all possible symmetry constraints both on electronic and ionic degrees of freedom with the spinpolarized DFT method.
Jahn–Teller effect for the ground state of NV . (a) Adiabatic potential energy surface (APES) of NV defect’s ground state. We relaxed the atomic positions that of Jahn–Teller minima (green dots) and barrier saddle points (blue dots) by means of the HSE06 functional. Additionally, we obtained the case (red dot) by enforcing symmetry constraints only on atomic positions and thus electronic correlation effects that of Eq. (9) can appear. Finally, we interpolated the geometries and calculated the electronic structure along the purple line to ensure the continuous shape of the APES. Additionally, we depict purple route in (b) inset that shows the raw HSE06 data points while (a) depicts the APES defined by Eq. (9), thus and are dimensionless therein. We note that only the green and blue data points that of (a) follow the axis on the top while the purple data points do not.
Jahn–Teller effect for the ground state of NV . (a) Adiabatic potential energy surface (APES) of NV defect’s ground state. We relaxed the atomic positions that of Jahn–Teller minima (green dots) and barrier saddle points (blue dots) by means of the HSE06 functional. Additionally, we obtained the case (red dot) by enforcing symmetry constraints only on atomic positions and thus electronic correlation effects that of Eq. (9) can appear. Finally, we interpolated the geometries and calculated the electronic structure along the purple line to ensure the continuous shape of the APES. Additionally, we depict purple route in (b) inset that shows the raw HSE06 data points while (a) depicts the APES defined by Eq. (9), thus and are dimensionless therein. We note that only the green and blue data points that of (a) follow the axis on the top while the purple data points do not.
C. Correction scheme for the |2A2⟩ excited state
D. Correction scheme for the |2E2*⟩ excited state
IV. NATURE OF THE GROUND STATE
Upon depicting the electronic structure of the ground state, we aim to describe the observed fine structure of the optical transition. We analyze the adjoint Jahn–Teller (Sec. IV A) and spin–orbit coupling (SOC) (Sec. IV B) effects. Additionally, local and external strain around the defect may affect the optical lines. Therefore, we will discuss both effects to approximate the effective SOC parameter that splits the multiplet into two electron–phonon coupled doublets: , . Finally, we will discuss the possible excitation schemes from the ground state in Sec. IV D.
A. The Jahn–Teller effect in the |2E⟩ ground state
Jahn–Teller (JT) coupling parameters for the |2E⟩ ground state by means of various DFT functionals: LDA, PBE, HSE06.
. | LDA73 . | LDAp.w. . | PBEp.w. . | HSE06p.w. . | expt. . | . |
---|---|---|---|---|---|---|
: | 62.5 | 64.5 | 63.5 | 62.5 | meV | |
EJT: | 73.2 | 61.2 | 49.9 | 30.1 | meV | |
δJT: | 10 | 9.9 | 6.0 | 20.7 | meV | |
3Γtun.a: | 21.4 | 25.5 | 29.2 | 14.0 | 13.6b | meV |
p factor: | 0.118c | 0.161 | 0.199 | 0.304 |
. | LDA73 . | LDAp.w. . | PBEp.w. . | HSE06p.w. . | expt. . | . |
---|---|---|---|---|---|---|
: | 62.5 | 64.5 | 63.5 | 62.5 | meV | |
EJT: | 73.2 | 61.2 | 49.9 | 30.1 | meV | |
δJT: | 10 | 9.9 | 6.0 | 20.7 | meV | |
3Γtun.a: | 21.4 | 25.5 | 29.2 | 14.0 | 13.6b | meV |
p factor: | 0.118c | 0.161 | 0.199 | 0.304 |
Energy of first vibronic excited state of JT system depicted by Eq. (14) known as tunneling splitting.71,72
B. Spin–orbit coupling in the |2E⟩ ground state
First, we determined the SOC parameter by constraining a half–half electron into Kohn–Sham orbitals and leave orbitals empty. This way, we completely cancel out the symmetry breaking effect both JT and electronic correlation origins thus the wavefunctions will be complex valued, that is, , where only spin–orbit coupling can split the degeneracy. We approximated the spin–orbit coupling parameters as (318) GHz with HSE06 (PBE) functional depicting negligible difference on the choice of the DFT functional.
We evaluate Eq. (16) by means of HSE06 functional, thus we used , and GHz that results in GHz that is significantly larger than that of the experimentally derived ones at GHz (Ref. 14) or GHz (Ref. 16). The and factors cannot correspond to this degree of discrepancy. Therefore we conclude that the constrained occupation that we forced upon our SOC calculations is not a reliable method. However, the broken symmetry solution of Eq. (9) cannot be used to this end as and orbitals are mixed in that case, which also brings matrix elements.
Nevertheless, we attempted to determine from the correlated Eq. (9) wavefunction. Therefore, we determine from the SOC matrix elements GHz between unoccupied and occupied orbitals by means of HSE06 (PBE) functionals. However, it still overestimates experimental data as the reduced value is GHz with HSE06 functional. We note that the value might be estimated by the non-correlated orbital in the excited state of NV , which results in GHz with HSE06 functional (Ref. 24). From this estimation the final spin–orbit gap yields GHz. In summary, our results cannot accurately determine the spin–orbit gap from the Kohn–Sham DFT level of theory because of the highly correlated nature of the ground state.
Lastly, we note that we determine as GHz matrix element as calculated from the constrained occupation. We use this value for the ISC transition (see Sec. V for details).
C. Quadrupolar and hyperfine parameters
Theoretical hyperfine and quadrupolar parameters for 14N within the |2 E⟩ ground state.
MHz | A⊥ ˭ −3.2 MHz |
A1 ˭ 0.1 MHz | A2 ˭ −0.1 MHz |
Q ˭ −4.8 MHz | |
Q1 ˭ 9.6 kHz | Q2 ˭ −9.7 kHz |
MHz | A⊥ ˭ −3.2 MHz |
A1 ˭ 0.1 MHz | A2 ˭ −0.1 MHz |
Q ˭ −4.8 MHz | |
Q1 ˭ 9.6 kHz | Q2 ˭ −9.7 kHz |
Atomic positions of the NV defect from two different orientations. We show two different viewing directions: (a) depicts the system viewed along the [111] trigonal symmetry axis, while (b) depicts a viewing angle perpendicular to it. We depict the [111] oriented coordinate system in (i) used within Eqs. (18) and (20) for hyperfine and quadrupolar parameters larger than MHz. Additionally, we show the Cartesian coordinate system of diamond in (ii). We depict the orbital by the blue/red colors for negative/positive parts of the wavefunction.
Atomic positions of the NV defect from two different orientations. We show two different viewing directions: (a) depicts the system viewed along the [111] trigonal symmetry axis, while (b) depicts a viewing angle perpendicular to it. We depict the [111] oriented coordinate system in (i) used within Eqs. (18) and (20) for hyperfine and quadrupolar parameters larger than MHz. Additionally, we show the Cartesian coordinate system of diamond in (ii). We depict the orbital by the blue/red colors for negative/positive parts of the wavefunction.
D. Optical spectra between the doublet states
As mentioned in the Methods section (Sec. II), we determine the optical sidebands of PL and ABS spectra by means of the Huang–Rhys method.23,53 First, we calculated the Huang–Rhys sideband by the geometry differences between the configuration and the correlated ground state as depicted in Eq. (9) with all symmetry constraints lifted. However, the predicted Huang–Rhys factor significantly underestimates the intensity of the PL sideband as plotted in Fig. 2(b).
The absorption spectrum is much broader than the PL spectrum for the NV defect. We attribute this phenomenon to the presence of the second excited state close to the first excited state. Furthermore, the ionization bands emerge at about 2.7 eV (e.g., Ref. 48), which overlaps with the phonon sideband of the related absorption. In Sec. IV E, we analyze the optical transition dipole moments toward the doublet excited states from the ground state.
E. Optical transition dipole moments
The calculated 6.6 ns radiative lifetime overestimates the optical transition’s strength as the experimentally observed excited state lifetime varies around 13–20 ns (Refs. 28–31). This result is surprising as our ab initio method correctly predicts radiative lifetime of NV excited state lifetime [13.7 ns in comparison to the experimentally observed 12.0 ns (Ref. 79)] by determining Debye. We tentatively explain the discrepancy with the complex nature of the electronic excited states. We conclude from our ab initio calculations that the bright excited state of NV does not have a pure character because electron–phonon coupling can mix and excited states with each other as the two energy levels are very close to each other. In such a case if the coupling between them is strong it could also explain why the second ZPL for is missing in ABS as the two electronic excited states are merged into common vibronic bands. The calculation of the electron–phonon coupling, i.e., the pseudo-Jahn–Teller effect between and is beyond the scope of this study as it is extremely difficult to achieve a reliable adiabatic potential energy surfaces and energy spacing for the two excited state doublets at high accuracy.
In summary, we suggest polarization dependent measurements both for the sideband both in ABS and PL to resolve this issue. The optical polarization already been resolved for the ZPL absorption80 that points to a transition, whereas later measurements confirmed34 that the excited state is a multiplet. In particular, an multiplet active through “ ” and “ ” directional dipoles only, whereas additionally active through the dipole pointing toward “ .” However, we propose that the optical sidebands contain not only the character, thus “z” dipole activity should be present toward the multiplet at higher energies in the absorption spectrum. The variation of the observed optical lifetimes of NV with various excitation techniques in bulk diamond might be also explained by the intertwined and character of the excited state.
V. LIFETIME OF THE |4A2⟩ METASTABLE STATE
Intersystem crossing (ISC) from the metastable state of the NV defect. (a) Mechanism of the ISC process. (b) and (c) ISC process within linear and log scale, respectively. We depict the theoretically viable region by green stripes in (c). We take an upper bound for the ISC rate ( ) from EPR measurements (Ref. 38, see Sec. V for details).
Intersystem crossing (ISC) from the metastable state of the NV defect. (a) Mechanism of the ISC process. (b) and (c) ISC process within linear and log scale, respectively. We depict the theoretically viable region by green stripes in (c). We take an upper bound for the ISC rate ( ) from EPR measurements (Ref. 38, see Sec. V for details).
Additionally, we determine the spectral function for ISC by means of the JT theory: , where is the HR spectral function, exhibiting HR factor and / is the eigenspectrum/wavefuncion of JT Hamiltonian of Eq. (14). At zero temperature, there would be no -phonons present within the metastable quartet, thus its electronic plus – –phonon wavefunction would be a vacuum state for vibrations. Any can be expressed as within the Born–Oppenheimer basis, where . Therefore, it can be concluded that the ISC process cannot target JT levels transforming as or representations because their expansion coefficient is zero due to group theory constraints and s are thermally not activated. ISC toward to the lowermost four relaxes with the coefficient and is conserved fully from the initial . However, if the system relaxes to the first vibronic -level ( ) at meV first by coefficient, a very fast relaxation will occur back to or toward the JT tunneling state at meV. We note that here transfers are allowed, and we expect that orbital relaxation84 would be the dominating process in the fs region.85 Nevertheless, the system would relax to from within due to the same orbital relaxation.
Surprisingly, the correction from the JT theory is negligible. We calculated the ISC process fully by means of the HR theory only: , we obtained almost the same transition rate as that of the JT theory [see Fig. 5(c)]. We used two estimates for . First, we used our GHz ab initio result as we discussed in Sec. IV B. However, our previous study about ISC processes for NV indicated that the off diagonal SOC matrix elements may overestimate24,82 the strength of the transition. Therefore, we used the previous result only as an upper bound, and we used , where is the SOC parameter of NV in its excited state as suggested by previous studies.24,79,81,82 Finally, we use the HSE06 correlation parameter as discussed in Sec. III B. We note that HSE06 DFT usually predicts the position non-correlated states within eV precision. Therefore, we estimate the position of level as eV (see Table I). However, we note that configurational interaction wavefunction methods predict at eV, so we use 0.7 eV as an upper bound for . Therefore, all the necessary parameters are known to evaluate Eq. (26). By taking all the broad energy intervals into account, our theory predicts the lifetime of within ns within the interval (see Table I).
An another aspect of is that it is EPR active. As we mentioned in the Introduction, it can be occupied with ionizing NV . However, the ISC rate cannot be too fast, otherwise the optically induced EPR could not be observed. Indeed, we fitted Lorentzian peaks on the first top-left experimentally measured EPR spectra from Fig. 1(a) in Ref. 38 recorded at K. We estimated the linewidth of the Lorentzian broadening as 0.068(3) mT that corresponds to 1.8(1) MHz in EPR frequency. Therefore, we can interpret the linewidth as natural line broadening to estimate the spin lifetime of as . However, the broadening may arise from other sources such as hyperfine interaction broadening due to distant C isotopes, for example. Thus, we use this value only as a minimum estimation for [see the green area in Figs. 5(b) and 5(c)]. Our final conclusion is that the lifetime of is in the s regime at cryogenic temperatures. We summarize the level structure of NV and the possible transitions in Fig. 6. It is worthy to consider the route on which the system may end up in . However, it can be seen that the is separated from by 1.6 eV, thus ISC transition is negligible between the states. Therefore, in Sec. VI, we will present how photoionization of NV can populate the metastable spin quartet state, .
Summary of ISC rates and level structure of NV . We take the experimentally observed zero-field splitting value from Ref. 38. Additionally, we expect that spin relaxation time is close to that of of NV because both multiplets are similar orbital singlets.
Summary of ISC rates and level structure of NV . We take the experimentally observed zero-field splitting value from Ref. 38. Additionally, we expect that spin relaxation time is close to that of of NV because both multiplets are similar orbital singlets.
VI. PHOTOIONIZATION OF THE NV DEFECT
The NV defect is often observed under illumination. The most typical photoexcitation of NV defect occurs with green laser (e.g., at 532 nm in wavelength or 2.33 eV in energy), which can lead to charge switching between the negatively charged and neutral NV defects.
Understanding of photoionization of NV defect is difficult because of the possible non-linear optical effects such as successive two-photon absorption processes via real excited states and the presence of long-living in NV and in NV . It is very challenging to calculate the photoionization threshold energies and rates at ab initio level for two reasons: (i) the singlet states in NV cannot be accurately calculated by DFT and (ii) simple ionization and Auger-ionization might compete where the latter is a two-body interaction, an inherently complex task for ab initio methods. Issue (i) can be principally solved by embedding wavefunction techniques that was applied to the ionization of of NV .86 We note that the photoionization cross section in that study was calculated within the -point in a 512-atom supercell, which is not fully convergent because the continuous band region at elevated energies above the ionization threshold energy requires much higher k-point sampling87 and band unfolding methods.42 Nevertheless, the calculated photoionization threshold energy at 2.2 eV implies that direct ionization, , occurs under green illumination. Issue (ii) was dealt with representing the appropriate exciton wavefunctions by combination of the respective Kohn–Sham wavefunctions in the Coulomb-integrals associated with the Auger-ionization rates.88 The method was applied to the of NV ionization to the ground state of NV and resulted in about 1 ns inverse rate.88 This result was later criticized with the argument that the electron density in the applied 512-atom supercell was very high so the Auger-ionization rate was orders of magnitude overestimated, which should depend on the carriers density.42 On the other hand, no other ab initio data have been yet published about the Auger-ionization rates of the NV defect to justify this argument.
In this study, we proved that even the ground state of NV is highly correlated that was not considered in the previous studies in the context of photoionization cross section. Because of the high complexity of the multiplet states of NV and numerical challenges of calculating direct and Auger-ionization rates, we rather show the estimated photoionization threshold energies for the and processes (see Fig. 7). Furthermore, we use the Slater–Condon principles to identify the feasible ionization processes as alluded in Refs. 42 and 89 for and , respectively. Briefly, the Slater–Condon selection rule implies that direct ionization and Auger-ionization only occurs for changing maximum one and two single particle wavefunctions between the initial and final multiplet states, respectively.
Summary of photoionization processes of NV . We round the related transition rates, thus we depict the lifetimes as rough ps, ns, and s estimates.
Summary of photoionization processes of NV . We round the related transition rates, thus we depict the lifetimes as rough ps, ns, and s estimates.
We start with the simplest photoionization scheme, i.e., the single photon absorption from the initial ground state to the final ground state. This yields and where the first process was observed at room temperature90 for which the K DFT calculation predicted 2.7 eV.48,87 These energies fall in the blue wavelength region. We note that the ground state is a correlated multiplet [see Eq. (10)] of 71% character, so the direct ionization process is partially quenched as the Slater–Condon selection rule prohibits the direct ionization toward with 28% contribution to the ground state of NV that we depict by dashed arrows in Fig. 7. On the other hand, the shelving state of NV , , can be ionized by green light as where no quenching occurs due to the correlated nature of . We find that the shelving state of NV , behaves very similarly as , which is fully allowed by the direct ionization process. Finally, we note that ionization from is unlikely due to its very short lifetime of 100 ps.91
As was also pointed out in Ref. 42, intersystem crossing is strongly hindered between and because of selection rules and a large energy spacing between these levels so occupation of the shelving could occur via photoionization from NV . Direct ionization from yields an ultraviolet threshold energy for this process, but then it would immediately convert it back to NV . However, it was found in an EPR measurement38 that green illumination for NV ensemble diamond samples can effectively pump the system into . This may be explained two-photon absorption from to via . In this process, the first photon is absorbed to go from to (usual neutral excitation) and the second photon is quickly absorbed before it decays within NV . In Ref. 42, they argued that the Slater–Condon rules dictate that direct ionization occurs only toward with a threshold energy at 1.2 eV, which explains the enhancement of ionization at this energy in a two-color excitation experiment.92 Furthermore, it could also explain a spin polarization of the toward over .42 Indeed, this phenomenon can be observed in Fig. 1(b) of Ref. 38 where the transition (1st spectrum) exhibits induced emission (positive Lorentzian), whereas the transition (4th spectrum) can be identified as absorption (negative Lorentzian). Our analysis shows that the lifetime of can be indeed s long in dark once the state has been occupied. However, green illumination can photoionize it back to the ground state of NV according to our results. Thus, this is a rather complex process where the average lifetime of depends on many factors, e.g., the laser power and the presence of other defects that may induce additional carriers (electrons and holes) upon illumination.
We analyze further the two-photon ionization processes. Our calculations on NV revealed that direct ionization channel exists between and because of the 28% character in the latter. Thus, an additional direct ionization process occurs from 0.7 eV from beside the already considered process in Ref. 42. The Auger-ionization process from is fully active toward . Thus, the sum of direct ionization and Auger-ionization rates will give the total ionization rate. Next, we consider the two-photon ionization processes via . The process is very similar to the reverse ionization process where this process is partially quenched by the electron–phonon related mixture of into similar to that of the quenched radiative lifetime of (see Sec. IV E). Additionally, the process is also forbidden at linear order at the Slater–Condon level. However, it becomes partially allowed due the electron–phonon related mixture of or the correlated nature of (see Ref. 82). We note again that the Auger-ionization process is fully allowed between these two states. Auger-ionization also can connect the and multiplets, and direct ionization is only allowed in the second order since it requires character to be mixed in that is only possible through simultaneous electron–phonon interaction and electronic correlation effects within . We argued in a previous study89 that the Auger-ionization rate can be substantial because of the presence of a deep resonant defect state of NV in the valence band, thus we argue that indeed occurs.89 Finally, ionization may occur from toward upon blue illumination starting at 2.6 eV, thus this route is blocked at the typical green illumination of the NV defect.
VII. SUMMARY
The electronic structure of NV was determined by DFT HSE06 calculations where the correlation of orbitals is taken into account with the guide of the group theory, in order to induce correction in the total energy and the geometry in the excited states. We find that the electron–phonon coupling is very strong and it affects both the fine structure of the ground state as well as the radiative lifetime of the excited state. In particular, we find that the and excited states are intertwined by symmetry breaking phonons, and the optical properties can be only explained by invoking strong electron–phonon coupling between the doublet excited states. We also analyzed the lifetime of the metastable state and provided a detailed analysis on the intersystem crossing between the metastable quartet state and the doublet ground state. Finally, the calculated electronic structure of NV yields new insights into the photoionization processes of the NV defect.
ACKNOWLEDGMENTS
We thank L. Razinkovas, R. Ulbricht, and C. Linderälv for fruitful discussions. Support by the National Research, Development and Innovation Office (NKFIH) as well as by the Ministry of Culture and Innovation within the Quantum Information National Laboratory of Hungary (Grant No. 2022-2.1.1-NL-2022-00004) is much appreciated. A.G. acknowledges the high-performance computational resources provided by KIFÜ (Governmental Agency for IT Development) Institute of Hungary, the European Commission for the projects QuMicro (Grant No. 101046911) and SPINUS (Grant No. 101135699), and the QuantERA II project Maestro (NKFIH Grant No. 2019-2.1.7-ERA-NET-2022-00045). G.T. was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Gergő Thiering: Conceptualization (equal); Investigation (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (equal). Adam Gali: Conceptualization (equal); Funding acquisition (lead); Investigation (equal); Project administration (lead); Supervision (lead); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.