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Optical thermometry based on level anticrossing in silicon carbide

We report a giant thermal shift of 2.1 MHz/K related to the excited-state zero-field splitting in the silicon vacancy centers in 4H silicon carbide. It is obtained from the indirect observation of the optically detected magnetic resonance in the excited state using the ground state as an ancilla. Alternatively, relative variations of the zero-field splitting for small temperature differences can be detected without application of radiofrequency fields, by simply monitoring the photoluminescence intensity in the vicinity of the level anticrossing. This effect results in an all-optical thermometry technique with temperature sensitivity of 100 mK/Hz1/2 for a detection volume of approximately 10−6 mm3. In contrast, the zero-field splitting in the ground state does not reveal detectable temperature shift. Using these properties, an integrated magnetic field and temperature sensor can be implemented on the same center.

Temperature sensing with high spatial resolution may be helpful for mapping of biochemical processes inside living cells and monitoring of heat dissipation in electronic circuits [1][2][3] . Frequently used contact-less methods exploit temperature-dependent features either in Raman spectra of microfabricated chips 4,5 or in photoluminescence (PL) spectra of nanoprobes such as quantum dots 6 , nanocrystals 7,8 and fluorescent proteins 9 . Typical temperature resolution of these methods is several hundreds of mK or lower.
Using quantum-mechanical properties of the nitrogen-vacancy (NV) in diamond, the temperature sensitivity better than δT = 10 mK/Hz 1/2 is achievable 3,10-12 . It is based on the moderate thermal shift dν 0 /dT = − 74 kHz/K 13,14 of the optically detected magnetic resonance (ODMR) frequency in the NV center (ν 0 = 2.87 GHz at T = 300 K) and the use of the advanced readout protocols, particularly temperature-scanned ODMR 15 or thermal spin echo 10,11 . However, this method is not universally usable, because the application of high-power radiofrequency (RF) fields in the pulsed ODMR technique may alter the temperature at the probe during the measurement. Therefore, the realization of highly-sensitive and RF-free optical thermometry is of broad interest.
Our approach is based on the silicon vacancy (V Si ) centers in silicon carbide (SiC), demonstrating appealing properties for quantum sensing applications [16][17][18] . Particularly, the V Si excited state 19,20 shows a giant thermal shift, exceeding 1 MHz/K 18 . Furthermore, these centers reveal an utterly long spin memory 21 and possess favorable absorption and PL in the near infrared spectral range 22 , characterized by a deep tissue penetration. The concentration of the V Si centers can be precisely controlled over many orders of magnitude down to single defect level 23,24 and they can be incorporated into SiC nanocrystals as well 25 .
We perform proof-of-concept thermometry measurements using 4H-SiC crystals. The 4H-SiC sample under study was grown by the physical vapour transport method. Silicon vacancies were created by irradiation of the crystal with 2 MeV electrons with a fluence of 10 18 cm −2 . The V Si centers possess a half-integer spin state S = 3/2 26 , which is split without external magnetic field in two Kramers degenerate spin sublevels m S = ± 3/2 and m S = ± 1/2. Here, we address the V Si (V2) center 27 with the zero-field splitting (ZFS) in the ground state (GS) 2D G = 70 MHz [ Fig. 1(a)]. The spin states are split further when an external magnetic field B is applied. The spin Hamiltonian of the V Si center in the magnetic field has a complex form 20 and five RF-induced transitions are allowed: ν 1 (− 1/2 ↔ − 3/2), ν 2 (+ 1/2 ↔ + 3/2), ν 3 (+ 1/2 ↔ − 3/2), ν 4 (− 1/2 ↔ + 3/2) and ν 5 (+ 1/2 ↔ − 1/2). In the ODMR experiments, we pump the V Si centers into the m S = ± 1/2 state with a near infrared laser (785 nm or 808 nm with power in the range of several hundreds mW). To decrease the detection volume to approximately 10 −6 mm 3 , we use a near-infrared optimized objective with N.A. = 0.3. The PL is recorded in the spectral range from 850 to 1000 nm, allowing optical readout of the V Si spin state: it is higher for m S = ± 3/2. A detailed ODMR dependence on the magnetic field strength and orientation is presented elsewhere 20,28 .
Due to the relatively short excited state (ES) lifetime of 6 ns in the V Si center 22 , the direct ODMR signal associated with the ES is weak. However, in the ES level anticrossing (LAC) between the m S = − 1/2 and m S = − 3/2 states (ESLAC-1) [magnetic field B E1 in Fig. 1(a)] the optical pumping cycle changes [29][30][31][32] . This results in a reduction of the ODMR contrast of the corresponding GS spin resonance 19,20 .
Indeed, such a behavior is observed in our experiments. Figure 1(b) shows the magnetic field dependence of the ODMR spectrum in the vicinity of the ESLAC-1 at room temperature. The ν 1 and ν 2 lines shift linearly with magnetic field applied parallel to the symmetry axis (B||c) as ν µ 2 0 denoting the g-factor. The transition with Δ m S = ± 2 are also allowed, but corresponding ν 3 and ν 4 lines appear at different frequencies and have lower ODMR contrast 20 . The ν 5 line is not resolved because of the same population of the m S = − 1/2 and m S = + 1/2 states under optical pumping at room temperature 26 . At B E1 = 15.7 mT, the ν 1 contrast drops to nearly zero and according to Fig. 1(a) the ES ZFS can be determined as Simultaneously, the GS ZFS is directly measured as 2D G = (ν 2 − ν 1 )/2. We repeat the above experiment at lower temperature T = 200 K [ Fig. 1(c)]. One can clearly see that the magnetic field associated with the ESLAC-1 is shifted towards higher values B E1 = 21.8 mT, while the splitting between the ν 1 and ν 2 ODMR lines remains the same. In addition, another spin resonance with negative contrast becomes visible ν µ = g B h / B 5 . We ascribe the appearance of the ν 5 line with lowering temperature with different transition rates to the m S = − 1/2 and m S = + 1/2 states. This may occur due to the either temperature-dependent interaction with phonons or some magnetic field misalignment, which in turn leads to the modification of the intersystem crossing as well as of the optical pumping cycle. The detailed analysis is beyond the scope of this work.
The tendency continues with lowering temperature down to T = 60 K [ Fig. 1(d)]. Namely, we observe that the magnetic field associated with the ESLAC-1 is shifted to B E1 = 36.5 mT, indicating a further increase of D E . The splitting between the ν 1 and ν 2 ODMR lines remains unchanged, suggesting D G is nearly temperature independent. These findings are summarized in Fig. 2. The ES ZFS is well fitted to denoting the ZFS in the limit T → 0 and β = − 2.1 ± 0.1 MHz/K being the thermal shift. The latter is by more than one order of magnitude larger than that for the NV defect in diamond 13 and by a factor of two larger than previously reported for 6H-SiC 18 . In following, we use this giant thermal shift for all-optical temperature sensing.
The idea is to exploit the variation of the PL intensity in the vicinity of LAC, occurring even without RF fields. This method has been initially implemented for all-optical magnetometry in SiC 20 , and later extended to the NV centers in diamond 33 . Figure 3 presents lock-in detection of the PL variation Δ PL/PL as a function of the dc magnetic field B z , recorded at different temperatures. The modulation of PL is caused by the application of an additional weak oscillating magnetic field B, i.e.,  Fig. 1(a). A broader resonance at the double magnetic field of 2.5 mT relates to the LAC between the spin sublevels m S = − 3/2 and m S = − 1/2 (Δ m S = 1) , i.e., GSLAC-1. The magnetic fields corresponding to the LACs in the GS (B G1 and B G2 ) are temperature independent, which is in agreement with our ODMR experiments of Fig. 1.
In addition to that, the experimental data of Fig. 3 reveal another resonance at the magnetic field B E2 . It corresponds to the LAC with Δ m S = 2 in the ES (ESLAC-2), as graphically explained in Fig. 1(a). Due to the strong reduction of the ES ZFS with growing temperature, this resonance shifts rapidly following Eq. (1) as . We recall that the lifetime in the ES is about 6 ns 22 . In order to observe ODMR signal associated with a spin state possessing such a short lifetime, one needs a RF field of about 2 mT. This alternating magnetic field, being in resonance with the spin transition, without strong impact on the temperature of the object under measurement is difficult to achieve.  We now discuss how small variations of the magnetic field Δ B and temperature Δ T can be measured. The in-phase lock-in voltage U X at the bias field B G2 can be written as (left inset of Fig. 3 11 12 Using calibration from our earlier experiments 20 , we obtain L 11 = − 39 mV/mT. Because B G2 is temperature independent and the variation of the signal amplitude for |Δ T| < 10 K is negligible, L 12 ≈ 0 mV/K is a good approximation. The linear dependence of Eq. (2) holds for |Δ B| < 100 mT. The same can be written for U X at the bias field B E2 (right inset of Fig. 3) X E2 21 22 and we find L 21 = 1.8 mV/mT and L 22 = 23 mV/K. From the factors L ij , it can be clearly seen that the magnetic field and temperature can be separately measured using GSLAC-2 and ESLAC-2. Particularly, the temperature sensing can be done in two steps. First, the bias field B G2 is applied and one measures U X G2 to determine the actual magnetic field, accounting for Δ B in Eq. (3). Then, after applying B E2 and reading out U X E2 , the magnetic noise can be excluded from the thermometry signal using The dynamic temperature range of such thermometry is |Δ T| < 10 K. A broad range thermometry can be realized (with lower sensitivity) by scanning the magnetic field from 5 mT to 20 mT and determining B E2 , which can be then converted to temperature using µ in combination with Eq. (1). We measure the in-phase and quadrature lock-in signals as a function of time to determine the upper limit of the noise level δU at a given modulation frequency (0.33 kHz). Then using the calibrated values for the L-matrix, we recalculate the noise level into the temperature sensitivity δT = δU/L 22 . It is estimated to be δT ≈ 100 mK/ Hz 1/2 within a detection volume of approximately 10 −6 mm 3 . By improving the excitation/collection efficiency and increasing the PL intensity (the V Si concentration), the temperature sensitivity better than δT ≈ 1 mK/Hz 1/2 is feasible with a sensor volume of 1 mm 3 . The suggested all-optical thermometry can be realized using various color centers in different SiC polytypes 34,35 . Furthermore, because color centers in SiC can be electrically driven 36 even on single defect level 37 , an intriguing perspective is the implementation of a LAC-based thermometry with electrical readout using photoionization of the ES 38 .