Selection rules in the excitation of the divacancy and the nitrogen-vacancy pair in 4H- and 6H-SiC

In this study, we address the selection rules with respect to the polarization of the optical excitation of two colour centres in 4H-SiC and 6H-SiC with potential for applications in quantum technology, the divacancy and the nitrogen-vacancy pair. We show that the photoluminescence (PL) of the axial configurations of higher symmetry (C3v) than the basal ones (C1h) can be cancelled using any excitation (resonant or non-resonant) with polarization parallel to the crystal axis (EL||c). The polarization selection rules are determined using group-theoretical analysis and simple physical arguments showing that phonon-assisted absorption with EL||c is prohibited despite being formally allowed by group theory. A comparison with the selection rules for the silicon vacancy, another defect with C3v symmetry, is also carried out. Using the selection rules, we demonstrate selective excitation of only one basal divacancy configuration in 4H-SiC, the P3 line and discuss the higher contrast and increased Debye-Waller factor in the selectively excited spectrum.


INTRODUCTION
Since the last few decades, the development of solid-state quantum bits (qubits) based on colour centres in wide-bandgap semiconductors, has been led by the negative nitrogen-vacancy (NV − ) centres in diamond, showing impressive progress in applications to nano-scale sensing and quantum communications [1][2].However, no individual colour centre or material platform is ideal for all different applications.With time, more materials with specific advantages attract attention and join the race [3,4].Among these, silicon carbide (SiC) is a promising one due to that it hosts various colour centres emitting light near and at telecom wavelengths with excellent optical and spin properties and, more importantly, for large-scale applications, it is the only wide-bandgap semiconductor that has industrial wafer-scale productions with wellcontrolled doping, mature CMOS technology, and established nanofabrication techniques [3][4][5][6].
Among various colour centres in SiC, the Si vacancy (VSi) [7] and the neutral divacancy (VCVSi 0 or simply VV) [8], i.e., an uncharged complex of a Si vacancy (VSi) and a nearest C vacancy (VC), are most studied.The divacancy spins have long coherence times, ranging from milliseconds in natural SiC [9] to several seconds in isotope-purified materials [10], and can be optically controlled at room temperature or even up to 550 K [11], making it suitable for broad temperature-range quantum sensing.Having emissions in the near-infrared spectral region around 1100 nm with a high-fidelity spin-to-photon interface [12] and a good control of single nuclear spins [13], the neutral divacancy is also promising for quantum applications.
An analogous defect to the NV − centre in diamond is the negative N-vacancy centre in SiC, i.e., the negatively charged complex between a N shallow donor at a C lattice site and a nearest Si vacancy (NCVSi − , denoted hereafter just NV) [14].The centre has an electronic structure analogical to that of the NV − centre in diamond but emits light near the O-band of telecom wavelengths (~1176-1243 nm) [15] which is more favourable for long-distance quantum communications.
In non-resonant excitation experiments, the phonon-assisted absorption process is also involved.Since this process is also subject to selection rules, some defect configurations with a certain symmetry may not be excited.This is reflected in reported PL spectra, which often show ZPLs with very different intensities depending on the polarization of the excited laser [15,18,19].Weak ZPLs excited with inappropriate polarization may appear in PL spectra of ensembles but can easily be missed in a scan for single emitters.Non-resonant excitation is a common way for activating and stabilizing the bright charge state of colour centres and is needed for obtaining their PL spectra.Identification of the selection rules for excitation including non-resonant excitation of ZPLs of defect centres with different symmetries is important for optimizing the PL or ODMR detection of individual ZPLs.In this work, we investigate the selection rules in non-resonant excitation for the axial (C3v) configurations of VV and NV centres in 4H-and 6H-SiC and compare with the selection rules for the silicon vacancy which also has C3v symmetry.

II. EXPERIMENTAL
The samples used for the studies on the NV pairs are bulk N-doped (in the range of 10 17 cm −3 ) 4H-and 6H-SiC, irradiated and annealed to form NV pairs.The samples containing divacancies are either as-grown bulk high-purity semi-insulating (HPSI) materials, or electron irradiated to fluences in the 10 17 -10 18 cm −3 range HPSI bulk 4H-and 6H-SiC substrates, annealed to about 800 C for 20 -30 min to create divacancies.PL measurements are carried out using a Jobin Yvon HR460 monochromator equipped with 1200 and 300 g/mm gratings, an InGaAs multichannel detector.The excitation laser is a tuneable Ti-sapphire laser.Most spectra presented here are obtained with laser at ~ 930 nm unless stated otherwise.The samples are mounted in a variable temperature closed cycle cryostat and cooled down to 3.8 K.The resolution of ~4 Å with the 300 g/mm grating is sufficient for most of the measurements, albeit a higher resolution of ~1 Å has been needed using the 1200 g/mm for some closely spaced lines in 6H-SiC.A halfwave plate is placed in the path of the laser for rotating the polarization, and a polarizer is used in the emission path.The laser line is filtered using a 1000 nm long-pass filter.

III. THEORETICAL BACKGROUND
The polarization selection rules for the zero-phonon line (ZPL) emission, as derived from the symmetries of the ground and excited states and group-theoretical analysis, have been discussed for many colour centres in previous works.For the divacancy and the VSi such consideration can be found in Refs.[20] and [21], respectively.From a group-theoretical point of view, the selection rules for the divacancy and the NV centre are the same because their ground and excited states have the same symmetry.Furthermore, the same selection rules apply for resonant excitation in the ZPLs since the matrix elements for absorption into and emission in a ZPL are the same.
However, the selection rules for non-resonant excitation of a certain defect are rarely considered, and when there is such consideration, it is usually limited to experimental observations only (e.g., [22,23]).Non-resonant excitation refers to excitation at higher energy than the ZPL into the phonon sideband (PSB) of the defect.Hence, it is phonon assisted with creation of a phonon in the absorption process, and one may assume that absorption will always be allowed if this phonon has a suitable symmetry.In this work, we refute this notion by showing that the phonon-assisted absorption into the excited state of some defects is also subject to strict selection rules.This may lead to configurations of the polarization of the exciting laser light with respect to the c-axis for which the absorption from the ground to the excited state of the defect is strictly prohibited in the case of phonon assisted transitions even though a formal group-theoretical analysis classifies it as allowed.
We examine here two different defects in the hexagonal 4H-and 6H-SiC polytypes: the neutral divacancy (VV 0 , denoted shortly VV) and the negative nitrogen-vacancy pair (NV − , or just NV) and compare with the silicon vacancy VSi which has been treated elsewhere [17,23,24].The presence of inequivalent lattice sites for Si and C, two in 4H-SiC and three in 6H-SiC, yields several defect centres (configurations) for the same type of defect.Thus, the divacancy and the NV pair in 4H-(6H-SiC) have four (six) inequivalent configurations, respectively.The silicon vacancy occupies either hexagonal or cubic Si sites, yielding two inequivalent configurations in 4H-SiC and three in 6H-SiC.All abovementioned defects have distinct ZPLs, and in some cases, the PL centres have been associated with specific configurations.
The configurations of VV and NV are further divided into axial and basal.The two (three) axial configurations possess C3v symmetry, in 4H-(6H-) SiC, respectively.The remaining two (three) configurations in 4H-(6H-) SiC have lower symmetry, C1h.Our study focuses on the high symmetry (C3v) axial configurations.The term "axial" reflects the fact that the orientation of the C3 axis coincides with the crystal axis (c-axis), along which the two constituents of the binary defects are aligned.We will show that the orientation of the exciting photon's polarization with respect to the c-axis determines the probability for photon absorption, with this probability vanishing for polarization parallel to the c-axis (E||c) in the case of both the divacancy and the NV pair in their axial configurations.However, the obtained selection rules are likely to be approximately valid also for the basal configurations with lower C1h symmetry [25,26], but with different orientation of the polarization direction for which the absorption nearly vanishes.We underline that we consider here usual non-resonant excitation with photon energies above the energy of the corresponding ZPL.The selection rules for resonant excitation into the ZPL are simpler and follow the selection rules for the photons emitted in the ZPL.
excitation.The results from the analysis are summarized in Table I for the three defects considered here, VSi − , and the axial configurations of VCVSi 0 , and NV − .Table I.Selection rules for the divacancy and the NV pair in their axial configurations, and the silicon vacancy VSi analysed within the single C3v group.A and F denote the allowed and forbidden transitions, respectively.The transition with E||c assisted by phonons of E symmetry (denoted by A * ) is formally allowed by group theory but prohibited due to physical reasons discussed in text.
NV pair or the divacancy (axial, C3v symmetry): We notice that the latter two defects have common symmetry in 4H-and 6H-SiC (C3v) and the same structure of the ground and excited states ( 3 A2 and 3 E, respectively), hence, their excitation is governed by the same selection rules.On the other hand, VSi has 4 A2 ground state in both polytypes.The excited state has two counterparts, 4 A2 and 4 E. In 4H-SiC, the 4 A2 is the lowest in energy excited state for both the cubic and hexagonal inequivalent vacancy configurations.The selection rules obtained within the single group C3v for ZPL emission and resonant excitation are those between two 4 A2 quartets, yielding emission (or resonant excitation in the ZPL) allowed only with E||c polarization.Another ZPL termed V1 appears at higher energy as a companion of the V1 ZPL (associated with VSi at the hexagonal site) at higher temperature, well visible above ~5 K, stemming from the 4 E  4 A2 transition.The selection rules derived within the single group C3v predict E⊥c polarization.We indicate, however, that the selection rules for VSi derived within the single group are only approximate since a proper analysis should be carried out within the double group C ̅ 3v owing to the halfinteger spin of the defect.

A. Emission selection rules
Although considered previously in different works, we summarize here and display in Table I the polarization selection rules for the ZPLs of the three defects considered here.We notice that the polarization of the ZPLs for the divacancy and the NV pair agree with the grouptheoretical analysis (see Table I) and previous publications considering the specific cases of the divacancy in 4H-SiC [20] and the divacancies and NV pairs in 4H-and 6H-SiC [27].
According to the selection rules in C3v symmetry, the polarization of the ZPLs of the axial configurations (PL1 and PL2 in 4H-SiC, QL1, QL2, and QL5 in 6H-SiC) is perpendicular to the axis of the vacancy pair, i.e., to the c-axis (E⊥c).

B. Selection rules with respect to the polarization of non-resonant excitation
We use excitation at 930 nm from a continuous wave Ti-sapphire laser with linearly polarized emission.The polarization of the laser is rotated using /2 plate which preserves the linear polarization for any rotation angle.The samples are mounted so that the c-axis is in the focal plane of the objective or the collecting lens, hence the linear polarization of the incident laser can be chosen to conclude any angle with the c-axis.In two cases, the NV pair in 4H-SiC and the divacancy in 6H-SiC, we had bulky samples with parallelepiped shape allowing weakly focused laser beam (by a lens of focal length ~150 mm) on the sample and the use of macroscopic collecting system consisting of a parabolic mirror and a 200-mm focal length lens for focusing on the monochromator slit.Sometimes with bulky samples we have used also rectangular geometry: the k-vector of the exciting beam is perpendicular to the c-axis and to the detection direction.Thinner substrates (of thickness 350 -500 m) are mounted edge-on and a microscope objective is used to focus the laser on the edge of the sample and collect the emitted PL. recorded with a polarization of the excitation laser EL perpendicular and parallel to the c-axis (EL⊥c and EL||c, respectively).The most notable feature is the complete vanishing of half of the divacancy lines (in both 4H-and 6H-SiC) when excited with EL||c polarization.In complete analogy with the divacancy, half of the ZPLs associated with the NV pair vanish completely when the laser excitation applied to the sample has EL||c polarization.This is illustrated in Fig. 2 (part (a) displays the spectra of 4H-SiC, part (b)in 6H-SiC).
In both polytypes, half of the ZPLs for both considered defects correspond to the number of lines due to the axial configurations.We will show that the vanishing of lines with EL||c polarization identifies exactly these configurations and is a consequence of the C3v symmetry and the symmetries of the ground (GS) and the excited (GS) states which are common for these two defects.Moreover, we will demonstrate that photons with EL||c polarization are not absorbed at all by the axial configurations, hence neither the ZPLs nor the associated phonon sidebands can appear in the spectra excited with EL||c.
We refer to Table I, where the selection rules in C3v symmetry for direct and phonon assisted transition between the GS and the ES are listed.Since both defects (divacancy and NV) have integer spin (S = 1), the group-theoretical analysis is restricted to the single group C3v.We see that the direct transitions for the axial configurations are forbidden with E||c polarization and allowed with E⊥c polarization, in agreement with theoretical analysis and experimental data for the divacancy in 4H-SiC [20] and our experimental data.Here E denotes the electric field polarization of the ZPLs, hence the same selection rules apply for resonant excitation (EL||cforbidden, EL⊥callowed).
Let us consider now the phonon assisted transitions for the axial configurations of the divacancy and the NV pair.There exist three possible symmetries for the phonons in C3v, namely, A1, A2, and E. The former two symmetries are one-dimensional, and the atomic displacements are along the c-axis of the crystal, whereas the two-dimensional E representation of C3v describes phonons with displacements in the basal plane, i.e., perpendicular to the caxis.Table I shows that all phonon-assisted transitions with E⊥c polarization are allowed, whereas for E||c only transitions assisted with phonons of E symmetry are allowed according to the formal group theoretical analysis.However, light with E||c polarization cannot interact with phonons of E symmetry because the electric field of the incident photons (E||c) is orthogonal to the atomic displacements of phonons with E symmetry.Consequently, such transitions are physically forbidden.
Thus, we obtain that in the cases of the axial configurations of the divacancy and the NV pair in both 4H-and 6H-SiC, both direct and indirect (phonon assisted) transitions with E||c polarization are prohibited.When photon absorption is considered, i.e., the polarization E represents the polarization of the incident laser EL, this means that absorption of photons by the axial configurations of the considered defects is prohibited for EL||c not only for the direct but also for the indirect (phonon assisted) absorption.This is in complete agreement with the experimental data.
We now discuss the non-axial configurations of the divacancy and NV, a. k. a. basal plane configuration, which have C1h symmetry per se.The ground state has A´´ symmetry compatible with the A2 symmetry of the ground state in C3v.(Here, we denote the two irreducible representations of C1h with A´ and A´´ to distinguish them from the A1 and A2 representations of C3v.)The excited state of E symmetry splits into A´ and A´´ states when the symmetry is lowered to C1h.Trivial analysis shows that from a group-theoretical point of view, all transitions are allowed, i.e., with both E||c and E⊥c polarizations.In addition, for each of the basal configurations, there exist three equivalent orientations of the defect axis (the line connecting the two components of the defect).We notice that if E||c polarization is considered, the polarization orientation is equivalent for all three orientations of a basal configuration.However, if an arbitrary E⊥c polarization is considered, with photons propagating (nearly) along the c-axis, as is the most common experimental geometry for both the impinging laser excitation and the received PL from the c-face of a sample, the orientation of the polarization is different with respect to the axis of each of the three equivalent orientations.This has some interesting consequences, briefly discussed here in a qualitative manner.

C. Experimental implications
Let us consider first PL measurements on an ensemble of divacancies, or NV centres.In the usual backscattering-from-surface geometry of a c-plane sample with the c-axis nearly perpendicular to the sample surface, the impinging photons propagate approximately along the c-axis and, therefore, have perpendicular to c polarization (EL ⊥ c).Whatever the orientation of this polarization with respect to the crystal axis, there will always be three equivalent angles between the axis of the defect and the laser polarization differing by 120.Thus, if we rotate the laser polarization in the basal plane, we do not expect any polarization dependence, i.e., PL intensity for the basal configuration will have more or less isotropic distribution with the angle of rotation of the polarization.
The situation is drastically different if PL measurements on single defects with basal configuration are considered, as studied in Ref. [22] for single NV pairs.If we rotate the polarization of the incident photons in the basal plane, there will be one direction in which the polarization is nearly parallel to the defect axis.Since the absorption for the axial configurations vanishes when the laser polarization is parallel to the c-axis (i.e., to the defect axis), we may anticipate that also for the basal configurations the absorption will have a minimum when the laser polarization is parallel to the defect axis and maximum when it is perpendicular to it.Hence, we expect a strong dependence of the PL on the polarization orientation of the incident light for basal configurations, which has been observed in [22] for a single configuration of the NV pair in 4H-SiC at room temperature.The single defect with this property has been attributed to an axial configuration (the defect is labelled NV-15 in Ref. [22]), but it likely should be attributed to one of the basal configurations.On the contrary, no strong dependence on the incident light polarization which would involve nearly vanishing of the PL for a certain incident light polarization is expected for the axial configurations (single defects or ensemble), because in the backscattering geometry considered here the laser polarization will be always perpendicular to the c-axis (EL⊥c) and the transition between the ground and excited state is always allowed.The defect labelled NV-1 in [22] has this property, the PL varies by a factor of 2 when the exciting laser polarization is rotated in the basal plane but does not vanish.Hence, this defect should be associated with the axial configural, not with the basal one as suggested in [22].
Using the notion that the axial configurations of VV and NV vanish when excited with EL||c as a tool for verifying the configuration, we can compare the experimental results presented in Fig. 1 with theoretical calculations.Theoretical data for the ZPLs of the NV centre in 4H-SiC can be found in [15,16,28] and in 6H-SiC in [18].The theoretical estimates for the ZPL positions of the divacancy are available in [29,30] and [17] for 4H-and 6H-SiC, respectively.The vanishing of the NV2 and NV3 lines when excited with EL||c implies that these two lines are associated with the axial hh and kk configurations in 4H-SiC [cf.Fig. 1(a)].The theoretical calculations, however, yield that the lowest energy ZPLs are associated with the axial configurations, indicating that the precision in the ZPL calculations is still insufficient for comparison with experiment [15,28].Nevertheless, using the D-tensor parameters for identifications of the microscopic NV configurations correctly identifies the ZPL positions of the axial configurations.[16] In [18], the SL2, SL5, and SL6 lines are associated with the axial configurations in 6H-SiC, which is correct according to Fig. 1(b) (notice that Ref. [18] uses different notations for the ZPLs of the NV pair in 6H-SiC).For the divacancy, all theoretical calculations predict that the axial configurations have the lowest energy [29,30].This is confirmed in Fig. 1(c) since PL1 and PL2 vanish for EL||c polarization.However, the assignment of the axial configurations for 6H-SiC in Ref. [17] needs adjustment, since the QL5 line identified in this reference as a basal configuration (hk1) does vanish, while the QL6 line suggested to be axial configuration (k2k1) does not vanish for EL||c polarization.Since, according to Ref. [17], these two configurations have significant overlap of their Gaussian standard deviations (cf.Fig. 2(b) of Ref. [17]), indicating substantial probability that their ordering in energy is reversed compared to theory, it is most likely that the correct association of QL5 and QL6 is with k2k1 and hk1 configurations, respectively.Here we use the notations of Refs.[18,31] for the hexagonal (h) and cubic (k1, k2) lattice sites in 6H-SiC (Ref.[17] uses somewhat different labelling of the lattice sites, hence hk1 is hk2 and k2k1 is k2k2 in Ref. [17]; the order is VSiVC).The corrected assignment for QL5 and QL6 is reflected in Table II.
The results are summarized in Table II for the two defects considered, VV and NV, in 4H-and 6H-SiC.In this table and in Figs. 1 and 2, we use the 'SL' notations of Ref. [27] to denote the NV lines in 6H-SiC because of the following.Firstly, we do not use the 'NV' notations of [18] to avoid confusion with the 'NV' in 4H-SiC and, secondly, in [18] the lines are enumerated in order of ascending wavelength, whereas we prefer to enumerate all the lines for the divacancy and the nitrogen-vacancy pair in order of ascending energies.
Table II.Summary of the line positions and their microscopic identification for the divacancy and the nitrogen-vacancy pair in 4H-and 6H-SiC.The non-resonant (phonon assisted) excitation selection rules stipulate that EL||c polarization cannot excite all configurations denoted as axial.The basal configurations can be excited with both EL||c and EL⊥c polarization.The order of the sites is VCVSi for VV and NCVSi for NV.The notations for the lattice sites h, k1 and k2 for 6H-SiC follow Refs.[18,31] (different from Ref. [17]).[9,23].Note that [9] uses the order VCVSi for labelling.b From Ref. [17].c The QL5 and QL6 lines are reassigned in this work, as explained in text.d According to Ref. [32].e From Ref. [18]

D. Comparison with the silicon vacancy VSi
It is instructive to compare the excitation selection rules for the axial divacancy and NV-pair configurations with those for the silicon vacancy VSi which also possesses C3v symmetry but has half-integer spin (S = 3/2).The GS and the ES of the latter defect are both quartets of A2 symmetry, 4 A2.The selection rules deduced within the single group C3v stipulate that transitions with E⊥c are forbidden for the ZPLs of both configurations, hexagonal and cubic, while the E||c transitions are allowed [21,23].This is also what is approximately observed experimentally (see [33,23]), albeit most experiments can register a weak contribution from the "forbidden" perpendicular to the c-axis polarization.However, the notion that the single-group selection rules are approximately valid also in the description of the silicon vacancy can be misleading, because a proper treatment of the selection rules must be done within the double group C ̅ 3v .In fact, we will show below that the observed dominating polarization of the V1 and V2 lines 4H-SiC is most likely associated with the well-known spin polarization of VSi under optical excitation and not with the "approximate validity of the single-group selection rules".In the case of half-integer spin, all substates of the ground and excited states transform as one of the extra representations of the double group, E1/2, 1 E3/2, or 2 E3/2 [34].The former two-dimensional representation describes the transformation properties of a wavefunction with spin component Sz = ±1/2, and the latter two one-dimensional representations are subject to time-reversal (Kramer's) degeneracy and their direct sum E3/2 = 1 E3/2⨁ 2 E3/2 represents the transformation properties of the wavefunctions with Sz = ±3/2.The selection rules for the ZPL (direct transitions) can be expressed with the following three statements (see, e.g., [35]): E1/2  E1/2allowed with E||c and E⊥c, ( E1/2  E3/2allowed with E⊥c, and E3/2  E3/2allowed with E||c. Here E denotes the emitted photon polarization if ZPL emission is considered, or E  EL if resonant absorption is considered.The selection rules for phonon assisted absorption remain unchanged for EL||c (only phonons of A1 and A2 symmetry can assist the absorption process).
On the other hand, phonon assisted absorption with EL⊥c can only involve phonons of E symmetry, and in this case all phonon assisted transitions are allowed (E1/2  E1/2, E1/2  E3/2, and E3/2  E3/2).Notice that both quartets, 4 A2 (GS) and 4 A2 (ES), contain both substates, Sz = ±1/2 transforming as E1/2, and Sz = ±3/2 transforming as E3/2.Since the zero-field splitting (ZFS) between the substates comprising the GS and the ES is in the tens of MHz range, it is unresolvable in optical experiments and in the ZPLs we expect to see contribution from both polarizations, E⊥c and E||c.
Nevertheless, both V1 and V2 manifest dominating E||c polarization which may be due to the polarization of the ground (and, possibly, excited) states under optical excitation.It has been shown that the E3/2 sublevel of the GS has dominant population (above 90 % [36]) under optical excitation [37].The same dominant population of the E3/2 sublevel of the ES can be anticipated, because if the E1/2 sublevel of the ES also had substantial population one would expect to see significant contribution in the ZPL from E⊥c polarization from recombination from E1/2 to both counterparts E1/2 and E3/2, in accord with Eqs.(1a) and (1b).Such contribution is present but nearly negligible, compared to the E||c component of the ZPLs V1 and V2 [33].Thus, if both the GS and the ES have predominantly radiative recombination with E||c, it is the recombination between the two E3/2 counterparts of the ES and GS.Similar hypothesis can be stated for V1´ line.If the ES in this transition has dominant E1/2 population, and the recombination to E3/2 has larger probability than that to E1/2, then we can understand the observed dominant E⊥c polarization in the V1´ line.However, for the moment the only experimental evidence in favour of this hypothesis is that the final state in most of the radiative transitions is E3/2, as demonstrated by the spin polarization [36].Further theoretical work is needed to investigate the probabilities for population of the sublevels in the ES, for both 4 A2 and 4 E, hence the provided possible explanation of the ZPLs' polarizations should be considered hypothetical for the moment.
Experimental data illustrating the presence of E⊥c polarization in the V1 and V2 lines, and weak E||c component in the V1´ line is given in [33].
E. Selective excitation of the PL3 line of the divacancy in 4H-SiC LTPL measurements on an ensemble without special regard for the polarization of the exciting laser usually display all the four ZPL lines PL1 -PL4 if the laser energy is higher than the energy of PL4 (the highest-energy ZPL).As discussed earlier, if excitation with EL||c polarization is used the axial configurations with ZPLs PL1 and PL2 together with their pertaining phonon side bands (PSB) are not excited at all.If, in addition, the laser energy is chosen between the energy positions of PL3 and PL4, then only the PL3 line will be excited.Thus, a single ZPL together with its PSB for the divacancy in 4H-SiC can be isolated in the PL spectrum of divacancies ensemble containing all configurations.The situation is like that of observation of the PL from single defect (in this case, PL3 corresponding to the hk configuration of VSiVC in 4H-SiC), but with the benefit of the much stronger signal associated with the ensemble.
This selective excitation of the PL3 line in an ensemble is depicted in Fig. 3(a).The upper spectrum is obtained with excitation at 1090 nm (1.1371 eV) in between the PL4 (1.1493 eV) and PL3 (1.1191 eV) lines.The laser is filtered using two sharp edge long pass filters with a cutoff at 1100 nm.The bottom spectrum obtained with a common excitation of 930 nm (1.333 eV) above the energy of all four ZPLs is given for comparison.Fig. 3(a) shows two ODMR spectra obtained with the same microwave antenna configuration but different excitation wavelengths, non-selective excitation with laser at 1020 nm (lighter thicker curve) and selective excitation with 1090 nm (darker thinner curve).In both cases EL||c is used, so that the former ODMR spectrum contains contribution from only PL3 and PL4 (PL1 and PL2 are not excited), while the selectively excited spectrum contains the pure PL3 contribution displayed in the top spectrum of Fig. 3(a).We point out two features of the experiment with selective excitation of PL3.Firstly, the ODMR contrast is improved by at least a factor of two as seen from the inset (Fig. 3), which is anticipated because the background contribution from PL4 and its phonon sideband to the ODMR signal of PL3 is removed.The use of excitation with EL not parallel to the c-axis will further deteriorate the contrast due to added contribution from PL1 and PL2 to the background.The second feature concerns the Debye-Waller (DW) factor of the PL3 line which shows a significant increase compared to the spectrum obtained with common excitation at 930 nm.The origin of this improvement is not understood at present and a thorough investigation of the factors influencing the DW factor is beyond the scope of this work.Nevertheless, the possibility of dealing with a single selectively excited ZPL with intensity typical for an ensemble may prove beneficial in future sensor applications.

V. SUMMARY
In conclusion, we have demonstrated theoretically and experimentally that the high-symmetry (axial) configurations of the divacancy and the NV pair in both 4H and 6H SiC are highly sensitive to the polarization of the exciting laser.No absorption and hence no excitation of the axial configurations occurs if the polarization of the excitation EL||c.This is shown to be due to the fact that only phonons of E symmetry may participate in allowed transitions between the ground and excited states for EL||c, according to the group theoretical selection rules.However, phonons with in-plane displacements cannot interact with photons of EL||c polarization since the electric field of the electromagnetic wave is orthogonal to the displacement of phonons of E symmetry.A comparison with the properties of the silicon vacancy VSi which also has GS of orbital A2 symmetry and ES of E symmetry for some configurations, but possesses halfinteger spin, shows that excitation of the corresponding configurations is not prohibited.This is attributed to the necessity of analysing the selection rules within the double group C ̅ 3v .The energy ordering of the divacancy ZPLs in 4H-SiC allows selective excitation of only one of the four configurations (PL3 line) using suitable laser energy and EL||c polarization which can be useful for using the PL3 line in sensor applications with an ensemble of divacancies.

Fig. 1
Fig. 1 displays the spectra of the divacancy in 4H-SiC [Fig.1(a)] and 6H-SiC [Fig.1(b)]recorded with a polarization of the excitation laser EL perpendicular and parallel to the c-axis (EL⊥c and EL||c, respectively).The most notable feature is the complete vanishing of half of the divacancy lines (in both 4H-and 6H-SiC) when excited with EL||c polarization.In complete analogy with the divacancy, half of the ZPLs associated with the NV pair vanish completely when the laser excitation applied to the sample has EL||c polarization.This is illustrated in Fig.2(part (a) displays the spectra of 4H-SiC, part (b)in 6H-SiC).

Fig. 1 .
Fig. 1.Low-temperature PL spectra of the NV pair [panels (a), (b)] and the divacancy [panels (c), (d)] in 4H-SiC [panels (a), (c)] and 6H-SiC [panels (b), (d)].The ZPLs due to the axial configurations with C3v symmetry all vanish for polarization of the excitation EL||c.The insets in panels (b) and (d) show spectra recorded with higher resolution; the latter zooms on the nearly degenerated QL5 and QL6 lines, while the former shows the angular dependence of the SL4 and SL5 lines recorded with 15 steps between EL⊥c (topmost curve) and EL||c (bottom curve).NV4´, SL1´, and SL4´ denote PL lines originating from closely spaced counterparts of the excited state detectable at temperatures ~ 4 K.For the NV ZPLs in 6H-SiC we use the 'SL' notations of Ref.[27] to avoid confusion with the NV lines in 4H-SiC.

Fig. 2 .
Fig. 2. Angular dependence of the ZPL intensity on the angle  between the laser polarization and the c-axis for the divacancy and the nitrogen vacancy pair in 4H and 6H polytypes, as denoted for each polar plot.The experimental data is fitted with an expression of the form () = (1 +  cos 2) , where  is the ZPL intensity and  and  are fitting constants specific for each ZPL. = 1 for the axial configurations.The laser polarization is EL⊥c at  = 0° and EL||c at  = 90°.

Fig. 3 .
Fig. 3. Selective excitation of the PL3 line in ensemble of divacancies in 4H-SiC.(a)Selectively excited spectrum of PL3 line (top curve) using excitation with EL||c polarization and energy lower than the PL4 line (laser at 1090 nm, or 1.137 eV).PL1 and PL2 are not excited because there is no absorption for these configurations with EL||c polarization.The bottom spectrum showing all four divacancy lines and their phonon sidebands is provided for comparison (excitation with EL⊥c at 930 nm).Note the enhanced Debye-Waller factor with selective excitation of PL3.(b) ODMR spectra obtained using the selective excitation of PL3 only (dark thinner curve) and of PL3 and PL4 using higher laser energy (light thicker curve), illustrating the contrast improvement for PL3 when the background emission of PL4 is removed, as discussed in text.