Silicon carbide (SiC), which is a broadband semiconductor, contains the family of quartet spin color centers (spin S = 3/2), where a unique mechanism of optically induced alignment (polarization) of populations of spin levels at room temperature and above up to 300°C occurs. The main polytype for applications in quantum technologies is 4H-SiC, where S = 3/2 spin centers with mentioned properties are implemented [1–4].

Optically induced alignment of spins allows the optical detection of magnetic resonance and spin level anticrossing (LAC) signals by the photoluminescence intensity; the microwave power is not required in the latter case [5, 6]. The technique of optical detection of the spin transitions leads to a giant increase in the sensitivity up to the possibility of detecting single spins at room temperature [7, 8]. Level anticrossing spectroscopy of S = 3/2 spin centers in SiC is promising foundation for the development of magnetic field and temperature sensors with the possibility of submicron spatial resolution, as well as magnetic field sensors operating at high temperatures and radiation, in particular, in space [9]. When two energy levels of the spin system cross under the variation of the magnetic field, the physical properties of the quantum system vary in the region of crossing; the anticrossing of levels occurs if two states that should cross in the first approximation are coupled by an additional perturbation. The optical properties of the system change in the region of LAC because the interaction between the spins results in the so-called flip–flop transitions of two or more spins simultaneously with the conservation of the total energy, tending the system of populations of levels to equilibrium. As a result, LACs lead to the appearance of pronounced features on the magnetic field dependence of the photoluminescence intensity of spin centers. Level anticrossing can be considered as the optical detection of magnetic resonance with zero frequency. There is an opinion that all crossings of levels are anticrossings because all states are coupled in a certain order of perturbation theory [10].

In this work, we report the study of color centers with the spin S = 3/2 in the hexagonal 4H-SiC polytype. Centers with the spin S = 3/2 were introduced in the 4H-SiC single crystal with a low nitrogen concentration by exposing it to a 2-MeV electron beam with a fluence of ~1018 cm–2. Color centers with the spin S = 3/2 are conventionally denoted by the corresponding zero-phonon photoluminescence lines; in this work, we experimentally study V2 centers in 4H‑SiC.

Figure 1a presents the scheme of energy levels of the V2 spin center in 4H-SiC, which has the spin S = 3/2 both in the ground state (GS) and in the excited state (ES). The energy levels and corresponding wavefunctions were calculated using the VISUAL EPR [11] and EasySpin [12] software packages; calculations with both packages gave the same results. Nonresonant transitions from the ground state to the excited state occur under the excitation by an infrared laser to the region of phonon repetitions with the subsequent relaxation to lower levels of the excited state 4E. Reverse transitions from the excited state 4E to the ground state 4A can occur through two ways: (i) the radiative transition with spin conservation in the form of phosphorescence, which is shown in the inset of Fig. 1a; the lifetime in the excited state is ~6 ns and (ii) the nonradiative transition without spin conservation from the excited state 4E to an intermediate metastable state (the so-called intersystem crossing) with the subsequent transition from the metastable state to the ground state 4A. In this case, since the transitions are spin-selective, the populations of the spin center are aligned after several cycles of optical excitation (usually, within a submicrosecond time interval). In particular, for V2 in 4H-SiC, lower levels with S = ±1/2 are populated, whereas levels with S = ±3/2 are depleted [8]. Before optical excitation, the populations of spin levels at room temperature satisfied by a Boltzmann distribution and were almost identical because the splitting of the fine structure between S = ±1/2 and S = ±3/2 levels is as small as Δ = 70 MHz.

Fig. 1.
figure 1

(Color online) (a) Scheme of energy levels of the V2 center in 4H-SiC, which has the spin S = 3/2 both in the ground state (GS) and in the excited state (ES). The cycle of the optically induced alignment of populations of spin levels in the ground state consists of excitation with spin conservation, emission with spin conservation, and nonradiative recombination through the intermediate metastable state (ISC) without spin conservation. As a result, lower levels with S = ±1/2 are populated, whereas levels with S = ±3/2 are depleted. (b) Fragments of the block diagram of the setup for the detection of level anticrossing under the variation of the magnetic field B0 (i) in option 1 (Mode 1), where the magnetic field is modulated with a small amplitude at a low frequency and (ii) in option 2 (Mode 2), where the exciting laser radiation intensity is modulated at a low frequency. In both options, a signal is detected at the low frequency with the lock-in detector (Lock-in).

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A laser with a wavelength of 785 nm and a power of 150 mW, which provided a spot ~1 μm in diameter in a scanning confocal microscope, was used to align spin centers and to excite photoluminescence (the photoluminescence spectrum is shown in the inset of Fig. 1a). An LAC signal in the varying magnetic field was detected in two options (modes). Fragments of the block diagram of the setup for the detection of level anticrossing under the variation of the magnetic field B0 in two options are shown in Fig. 1b. In option 1 (Mode 1), where the magnetic field was modulated by a modulation field, which is directed along B0 and had a small amplitude of 0.1 G and a low frequency of 162 Hz under continuous optical excitation, the LAC signal was detected in the form of a derivative. In option 2 (Mode 2), where the exciting laser radiation intensity is modulated at a low frequency of 156 Hz (light-on, light-off) in the presence of only the varying magnetic field B0, the LAC signal was detected in the form of an absorption curve. Under pulsed optical excitation in option 2, a dynamic process occurs with the alignment of populations of spin levels and with the destruction of the alignment in the LAC region in a short time of an optical pulse (submilliseconds in our experiments). We are going to use this method to study transient spin processes at various LAC points; moreover, this method also allows one to detect LAC signals for excited states with S = 3/2. In both options, by signal is detected at the low frequency with the lock-in detector (Lock-in) by a change in the photoluminescence intensity.

The studied spin color centers have axial symmetry along the c axis of the crystal and contain a negatively charged silicon vacancy with the spin S = 3/2. The axial symmetry of a center is due to the presence of excitation in an intrinsic defect along the c axis [13]; in our opinion, this intrinsic defect is a neutral carbon vacancy, which is not covalently bonded to the mentioned silicon vacancy. Experiments were performed on a radio-spectroscopic complex fabricated at the Ioffe Institute in the form of an optically detected magnetic resonance spectrometer based on a confocal optical microscope (NT MDT). The 4H-SiC crystal plate sample with S = 3/2 spin centers and with the hexagonal axis perpendicular to the crystal plane was placed in the confocal microscope. The axial axis of the spin centers are parallel to the c axis of the crystal. The scanning magnetic field B0 and the modulation magnetic field (for option 1) were parallel to the axis of the spin center.

The electron paramagnetic resonance (EPR) spectrum of S = 3/2 spin centers is described by the spin Hamiltonian

$$H = g{{\mu }_{{\text{B}}}}{\mathbf{B}} \cdot {\mathbf{S}} + D[{\mathbf{S}}_{z}^{2} - 1{\text{/}}3S(S + 1)] + A{\mathbf{I}} \cdot {\mathbf{S}},$$
(1)

where B is the magnetic field, g = 2.003/2 is the isotropic g-factor, μB is the Bohr magneton, D is the fine structure parameter in the axial crystal field, A is the isotropic hyperfine interaction with one 29Si (I = 1/2) nucleus located in the second coordination sphere of the silicon vacancy entering the center structure. The fine structure splitting for V2 centers in 4H-SiC is Δ\({v}\) = 2D = 23.4 × 10–4 cm–1 = 70 MHz According to the proposed model of S = 3/2 spin centers, the main role is played by the silicon vacancy with four carbon atoms in the nearest environment. An anisotropic hyperfine structure with one 13C nucleus was optically observed in the tenfold 13C-enriched 6H-SiC crystal in our previous work [14]. The hyperfine structure splitting for the interaction with 29Si in the second coordination sphere of the silicon vacancy (12 SiNNN atoms, where the subscript NNN means the next nearest neighbor) is 9 MHz = 3 × 10–4 cm–1 [15].

Figure 2a presents LAC signals for V2 S = 3/2 spin centers in the ground state (GS) detected in the orientation B || c at room temperature by changes in the photoluminescence intensity in the 4H-SiC single crystal with the natural isotopic composition. Line anticrossing signals were detected using the scanning confocal optical microscope in the focused volume (~1 μm3) of the pump laser beam with a wavelength of 785 nm. The blue line corresponds to LAC signals in the form of the derivative of the varying magnetic field detected with the low-frequency modulation of the magnetic field with an amplitude of about 0.1 G (option 1). The red line presents LAC signals recorded with the low-frequency modulation of the pump laser beam intensity (option 2); the modulation of the beam in the form of periodic beam-on and beam-off pulses with the period T = 1/f, where f is the modulation frequency, is shown. Figure 2b shows the scheme of energy levels in the magnetic field for even silicon isotopes having zero nuclear magnetic moment: S = 3/2, I = 0, D = 35 MHz = 11.7 × 10–4 cm–1, and Δ = 2D = 70 MHz = 23.4 × 10–4 cm–1. We took the spin functions in the form of eigenfunctions of the operators \({{\hat {S}}_{Z}}\) and \({{\hat {I}}_{Z}}\), which are correct functions in the case of the magnetic field parallel to the axis of the axial center, except for the LAC regions [16]. The wavefunctions of electrons in the S = 3/2 state with MS = ±1/2 and MS = ±3/2 are denoted as \({\text{|}}{\kern 1pt} \pm {\kern 1pt} 1{\text{/}}2\rangle \) and \({\text{|}}{\kern 1pt} \pm {\kern 1pt} 3{\text{/}}2\rangle \), respectively. Levels corresponding to the lower \({{M}_{S}} = \pm 1{\text{/}}2\) states (\({\text{|}}{\kern 1pt} \pm {\kern 1pt} 1{\text{/}}2\rangle \)) are indicated by thicker lines to emphasize the predominant population of these levels due to optical alignment. The wavefunctions of 29Si nuclei in the I = 1/2 state with \({{m}_{I}} = + 1{\text{/}}2\) and \({{m}_{I}} = - 1{\text{/}}2\) are denoted as \({\text{|}}\alpha \rangle \) and \({\text{|}}\beta \rangle \), respectively. Thus, wavefunctions including the electron and nuclear spins are denoted as \({\text{|}}{{M}_{S}};{{m}_{I}}\rangle \); in the general case, wavefunctions are linear combinations of presented functions. Circles mark LAC points in the ground state: the first point LAC1 with the electron spin projection change \(\Delta {{M}_{S}} = \pm 2\) and the second point LAC2 with the electron spin projection change \(\Delta {{M}_{S}} = \pm 1\). Figure 2c shows the scheme of energy levels in the magnetic field with allowance for hyperfine interactions with one 29Si nucleus located in the second coordination sphere of the silicon vacancy entering the spin center (SiNNN): S = 3/2, I = 1/2, A = 9 MHz = 3 × 10−4 cm−1 = 3.2 G. As in Fig. 2b, levels corresponding to the lower MS = ±1/2 states are indicated by thicker lines to emphasize the predominant population of these levels due to optical alignment. Circles indicate LAC signals, and the circles indicating two LACs, where two levels with MS = +1/2 cross with the MS = –3/2 level, are marked by thick lines. For comparison, dashed lines present energy levels for I = 0 shown in Fig. 2b; crosses mark two LAC points for these levels. Arrows show the c-orrespondence of experimentally observed LAC signals to the anticrossing points in the system of energy levels.

Fig. 2.
figure 2

(Color online) (a) Level anticrossing signals detected by a change in the photoluminescence intensity of V2 spin centers (blue line) with the low-frequency modulation of the magnetic field and (red line) with the low-frequency modulation of the intensity of exciting laser radiation. (b) Scheme of energy levels for even silicon isotopes in the magnetic field. Levels corresponding to the MS = ±1/2 states are indicated by thicker lines to emphasize the predominant population of these levels due to optical alignment. Circles mark level anticrossing points in the ground state. (c) Scheme of energy levels in the magnetic field with allowance for hyperfine interactions with one 29Si nucleus in comparison with (dashed lines) energy levels for I = 0; crosses mark two level anticrossing points.

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In Fig. 2c, the LAC marked by circles in the direction of increasing magnetic field correspond to the following transitions: \({\text{|}}{\kern 1pt} + {\kern 1pt} 1{\text{/}}2,\alpha \rangle \leftrightarrow {\text{|}}{\kern 1pt} + {\kern 1pt} 1{\text{/}}2,\beta \rangle \); \({\text{|}}{\kern 1pt} + {\kern 1pt} 1{\text{/}}2,\alpha \rangle \) \( \leftrightarrow \) \({\text{|}}{\kern 1pt} - {\kern 1pt} 1{\text{/}}2,\beta \rangle \); \({\text{|}}{\kern 1pt} + {\kern 1pt} 1{\text{/}}2,\beta \rangle \leftrightarrow {\text{|}}{\kern 1pt} - {\kern 1pt} 1{\text{/}}2,\beta \rangle \); \({\text{|}}{\kern 1pt} + {\kern 1pt} 1{\text{/}}2,\alpha \rangle \leftrightarrow {\text{|}}{\kern 1pt} - {\kern 1pt} 3{\text{/}}2,\beta \rangle \); \({\text{|}}{\kern 1pt} + {\kern 1pt} 1{\text{/}}2,\beta \rangle \, \leftrightarrow \,{\text{|}}{\kern 1pt} - {\kern 1pt} 3{\text{/}}2,\beta \rangle \); \({\text{|}}{\kern 1pt} + {\kern 1pt} 1{\text{/}}2,\alpha \rangle \, \leftrightarrow \,{\text{|}}{\kern 1pt} - {\kern 1pt} 3{\text{/}}2,\alpha \rangle \); \({\text{|}}{\kern 1pt} - {\kern 1pt} 1{\text{/}}2,\beta \rangle \) ↔ \({\text{|}}{\kern 1pt} - {\kern 1pt} 3{\text{/}}2,\beta \rangle \); and \({\text{|}}{\kern 1pt} - {\kern 1pt} 1{\text{/}}2,\alpha \rangle \leftrightarrow {\text{|}}{\kern 1pt} - {\kern 1pt} 3{\text{/}}2,\beta \rangle \). The notation of wavefunctions in the approximation of high magnetic fields is used. There are the allowed electron spin transition with and without nuclear spin flip, the forbidden electron spin transition with and without nuclear spin flip, and the transition with nuclear spin flip without electron spin flip.

Figure 3 presents the experimental LAC spectrum in the 4H-SiC crystal with the natural isotopic composition in the orientation B || c at a temperature of 300 K (see Fig. 2a) and simulated EPR spectra for a frequency of 0.1 MHz, which should nearly coincide with the LAC signals because these signal lines are wider than 0.1 MHz The scheme of energy levels calculated for S = 3/2 and for I = (solid lines) 1/2 and (dashed lines) 0 with the parameters same as in Fig. 2 is shown in the bottom panel of Fig. 3, where filled circles indicate EPR transition points for a frequency of 0.1 MHz; the simulated EPR lines for these transitions are shown in the top panel. It is seen that EPR transitions only partially coincide with experimental LAC spectra, i.e., LAC gives a wider set of transitions. The colors of the energy lines provide information on the level crossing points where the spin wavefunctions are mixed, resulting in LAC signals; i.e., this is an intrinsic property of the system, whereas EPR signals reflect nonzero matrix elements of transitions between levels due to an external action (resonant microwave radiation). For simplification, identical colors are used in Fig. 2 for each of the energy levels \({\text{|}}{{M}_{S}};{{m}_{I}}\rangle \) in the entire magnetic field range. In LAC, spin sublevels are mixed by static perturbation, whereas in EPR and optically detected magnetic resonance, the external action depends on the time. This difference is the most pronounced in the system of levels with I = 0, where LAC1 is expectedly not manifested in the EPR spectrum because LAC1 corresponds to the transition with ΔMS = ±2 forbidden in EPR. The wavefunctions for energy levels presented in Figs. 2 and 3 are linear combinations of the corresponding spin states: a linear combination of four \({\text{|}}{{M}_{S}}\rangle \) states for levels with I = 0 and a linear combination of eight \({\text{|}}{{M}_{S}},{{m}_{I}}\rangle \) states for levels with the hyperfine interaction with one 29Si (I = 1/2) nucleus. It is noteworthy that changes in the photoluminescence intensity in the LAC region with and without nuclear spin flip give information on the polarization of nuclear spins, including effects of dynamic polarization of nuclei, and potentially provide a tool to manipulate nuclear spins by applying optical and microwave excitation.

Fig. 3.
figure 3

(Color online) (Top panel) Experimental level anticrossing spectrum in the 4H-SiC crystal with the natural isotopic composition and simulated EPR spectra for a frequency of 0.1 MHz. (Bottom panel) Scheme of energy levels calculated for S = 3/2 and for I = (solid lines) 1/2 and (dashed lines) 0 with the parameters same as in Fig. 2. Filled circles indicate EPR transition points for a frequency of 0.1 MHz; the simulated EPR lines for these transitions are shown in the top panel. The colors of the energy lines provide information on the level crossing points where the spin wavefunctions are mixed.

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The classification of wavefunctions for S = 3/2, I = 1/2, D = 35 MHz, and A = 9 MHz calculated with the software package from [11] has the form |1〉 = \(\underline {{\text{|}}{\kern 1pt} + {\kern 1pt} 3{\text{/}}2,{{\alpha }_{n}}\rangle } \); \(\underline {{\text{|}}8\rangle \;\, = \;\,{\text{|}}{\kern 1pt} - {\kern 1pt} 3{\text{/}}2,{{\beta }_{n}}\rangle } \); \({\text{|5}}\rangle \;\, = \;\,{\text{|}}{\kern 1pt} + {\kern 1pt} 3{\text{/}}2,{{\beta }_{n}}\rangle \); |2〉 = \({\text{|}}{\kern 1pt} + {\kern 1pt} 1{\text{/}}2,{{\alpha }_{n}}\rangle \); \({\text{|}}6\rangle \,\; = \;\,{\text{|}}{\kern 1pt} + {\kern 1pt} 1{\text{/}}2,{{\beta }_{n}}\rangle \); \({\text{|}}7\rangle \;\, = \;\,{\text{|}}{\kern 1pt} - {\kern 1pt} 1{\text{/}}2,{{\beta }_{n}}\rangle \); |3〉 = \({\text{|}}{\kern 1pt} - {\kern 1pt} 1{\text{/}}2,{{\alpha }_{n}}\rangle \); \({\text{|}}4\rangle = {\text{|}}{\kern 1pt} - {\kern 1pt} 3{\text{/}}2,{{\alpha }_{n}}\rangle \).

In this notation, the wavefunctions for levels in Figs. 2с and 3 in zero magnetic field have the form (from bottom to top) \({\text{|I}}\rangle = - 0.7071{\text{|}}3\rangle + 0.7071{\text{|}}6\rangle \); |II〉 = \(0.9922{\text{|}}2) - 0.1244{\text{|}}5\rangle \); |III〉 = \( - 0.1244{\text{|}}4\rangle + 0.9922{\text{|}}7\rangle \); \({\text{|IV}}\rangle \, = \,0.7071{\text{|}}3\rangle \, + \,0.7071{\text{|}}6\rangle \); |V〉 = 0.9922|4〉 + \(0.1244{\text{|}}7\rangle \); \({\text{|VI}}\rangle = 0.1244{\text{|}}2\rangle + 0.9922{\text{|}}5\rangle \); \(\underline {{\text{|VII}}\rangle = 1.0000{\text{|}}1\rangle } \); and \(\underline {{\text{|VIII}}\rangle = 1.0000{\text{|}}8\rangle } \).

The wavefunctions in a high magnetic field of 50.5 G have the form (from bottom to top) |I〉 = \(0.9954{\text{|}}4\rangle - 0.0955{\text{|}}7\rangle \); \({\text{|II}}\rangle = 1.0000{\text{|8}}\rangle \); |III〉 = \(0.9980{\text{|}}3\rangle - 0.0631{\text{|}}6\rangle \); \({\text{|IV}}\rangle = 0.0955{\text{|}}4\rangle + 0.9954{\text{|}}7\rangle \); |V〉 = \(0.0631{\text{|}}3\rangle + 0.9980{\text{|}}6\rangle \); |VI〉 = 0.9993|2〉 – \(0.0383{\text{|}}5\rangle \); \({\text{|VII}}\rangle = 0.0383{\text{|}}2\rangle + 0.9993{\text{|}}5\rangle \); |VIII〉 = \(1.0000{\text{|}}1\rangle \). The underlined states are pure in the entire magnetic field range. With an increase in the magnetic field by an order of magnitude, all wavefunctions become single-state with a weight of about 0.9999, where the total weight of all other states is less than 0.01. Transformations of wavefunctions in LAC regions are manifested in the color notation of levels in Fig. 3 and are easily calculated in the applied program. As a result, electron and nuclear spin projections change; we emphasize that a microwave power is not necessary in this case, in contrast to optically detected magnetic resonance. This is particularly important for the detection of short-lived excited states because a high microwave power is required to excite a transition between levels in the mentioned case.

CONCLUSIONS

To summarize, the fully optical spectroscopy of hyperfine interactions in a SiC crystal with the natural isotope composition has been demonstrated under ambient conditions. All possible spin level anticrossings in the ground state in the system of hyperfine interactions with one 29Si nucleus located in the second coordination sphere of the silicon vacancy entering the structure of quartet spin center have been detected. The effect of purely nuclear transitions on the photoluminescence intensity of quartet spin centers has been identified. The reported study opens the possibility for the fully optical spectroscopy of hyperfine interactions in numerous quartet spin centers in other SiC polytypes; more than ten such objects have been already revealed in the 6H, 15R, and 21R polytypes [8]. There are serious reasons for the model of quartet spin centers, where the optically induced alignment of spins occurs, in the form of a structure consisting of a negatively charged silicon vacancy with the spin S = 3/2 and an intrinsic defect, which is located in the nearest environment of this vacancy in the position along the c axis and is not covalently bonded to the silicon vacancy. We suggest that this defect is a neutral carbon vacancy although some theoretical calculations show that this defect can be an antisite defect of silicon in the carbon site [17]. The optically induced alignment of quartet spin sublevels does not occur for the negatively charged silicon vacancy in the ideal environment [18, 19].

The possibility of detecting LAC under pulsed optical excitation, which results in the dynamic alignment of populations of spin levels and the destruction of alignment in the LAC region in a short time (submilliseconds) of the optical pulse, has been demonstrated. We are going to use this method to study transient spin processes at various LAC points, including LAC signals for excited states.

We proposed to use quartet spin centers in SiC to develop fully optical methods for measuring magnetic fields [5, 6] and temperature [20] with the micron and submicron spatial resolution. The reported results open possibilities for fully optical manipulation of nuclear spins, including the detection of nuclear polarization effects. A new unique research field, which is based on spectroscopy of hyperfine interactions, has been developed to fabricate sensors with fully optical vector magnetometry with the micron and submicron spatial resolution; the results will be reported elsewhere.