Sub-nanoscale Temperature, Magnetic Field and Pressure sensing with Spin Centers in 2D hexagonal Boron Nitride

Spin defects in solid-state materials are strong candidate systems for quantum information technology and sensing applications. Here we explore in details the recently discovered negatively charged boron vacancies ($V_B^-$) in hexagonal boron nitride (hBN) and demonstrate their use as atomic scale sensors for temperature, magnetic fields and externally applied pressure. These applications are possible due to the high-spin triplet ground state and bright spin-dependent photoluminescence (PL) of the $V_B^-$. Specifically, we find that the frequency shift in optically detected magnetic resonance (ODMR) measurements is not only sensitive to static magnetic fields, but also to temperature and pressure changes which we relate to crystal lattice parameters. Our work is important for the future use of spin-rich hBN layers as intrinsic sensors in heterostructures of functionalized 2D materials.

A remedy to these limitations may be provided by recently discovered defects in layered materials. One of the most prominent stackable 2D materials is hexagonal boron nitride (hBN) which hosts a large variety of atomlike defects including single photon emitters [11][12][13][14] . Spin carrying defects have been theoretically predicted and experimentally confirmed in hBN [15][16][17][18][19] . Currently, the most understood defect is the negatively-charged boron vacancy center (V % # ) 20 , which can be readily created by neutron irradiation, ion implantation or femtosecond laser pulses 21,22 . Due to its spin-optical and properties, the V % # center is proving to be a promising candidate system for quantum information and nanoscale quantum sensing applications and has thus expanded the already large suite of unique features displayed by 2D materials 13 .
The recently identified VBin hBN displays a photoluminescence (PL) emission band around 850 nm and has been found to be an electronic spin-triplet (S=1) system with a ground state zero-field splitting (ZFS) '( /ℎ ≅ 3.5 GHz between its spin sublevels ( = 0 and ( = ±1 16 . Here, we study the effect of external stimuli on the defect's properties and demonstrate its suitability for sensing temperature, pressure (as lattice compression) and magnetic fields. Notably, our experiments show that the resolution and range of operation of the hBN V % # center is competitive or exceeding those of similar defect-based sensors 23 .
The results presented in this work were obtained on single-crystal hBN. The V % # centers were generated in the sample via neutron irradiation (≈2.3´10 18 n·cm −2 ), as described elsewhere 16,20 . The hBN single crystalline sample consists of a stack of a few thousand mono layers. The distance between two adjacent layers is ≅ 6.6 Å, while the in-plane distance between two identical atoms is ≅ 2.5 Å (Fig. 1a). As shown by temperature dependent X-ray data 24 , the hBN lattice undergoes highly anisotropic thermal expansion with and changing in opposite directions, i.e., while decreases with cooling, increases, as schematically shown in Fig. 1b. This crystallographic feature can be used to monitor local temperature variations optically, via ODMR, since the temperature-driven compression/expansion of the lattice parameters and causes a direct change in the ZFS parameter '( of the triplet ground state Spin-Hamiltonian 25 . Figure 1c shows continuous wave (cw) ODMR measurements for three different temperatures, with (dark blue) or without (cyan) an external magnetic field applied. At room temperature and in the absence of the external magnetic field, the ODMR spectrum of the V % # shows two resonances ( @ , B ) centered symmetrically around C , which corresponds to the ZFS parameter '( /ℎ = C = 3.48 GHz with the splitting due to the non-zero off-axial ZFS parameter '( /ℎ ≅ 50 MHz. When applying an external static magnetic field , @ and B split further following: Here, h is Planck's constant, g is the Landé factor and is the Bohr magneton. The separation of the two resonances @ and B can clearly be seen in Fig. 1c (dark blue traces). The visible substructure in both ODMR peaks is due to hyperfine coupling of the electron S=1 spin system (negatively charged boron vacancy) with three equivalent nearest nitrogen atoms, each possessing nuclear spin = 1 for the most abundant 14 N isotope (99.63%). In total, seven hyperfine peaks can be resolved, whose relative separations are temperature and magnetic field independent. A closer look at Fig. 1c reveals that cooling down the sample results in a shift of the ODMR peaks to higher frequencies. Thus, the dependencies of the ODMR spectrum on temperature and magnetic fields can provide a basis for the use of the V % # center as a thermometer and magnetometer at the sub-nanoscale.

Temperature Sensing
The observed shift of the resonances to higher frequency values (Fig. 1c) is independent of the applied magnetic field and is solely due to a reduction of the ZFS parameter '( . Over the temperature range 295 -10 K, '( undergoes a variation ∆ '( ≅ 195 MHz. This is a relatively large change compared to analogous spin systems in 3D materials (≈30-fold). For instance, the NVcenter in diamond exhibits a shift ∆ '( ≅ 7 MHz 25 , while the '( of VSi in SiC is almost constant over the same range. Only more complex spin defects such as Frenkel defects (VSi-Sii) in SiC display a comparably strong effect (∆ '( ≅ 300 MHz) 5 .
To quantify this temperature-induced shift of the ground state triplet energy-levels we combine temperatureand magnetic field-dependent ODMR measurements. Figure 2 summarizes the shift of '( in the ODMR spectrum as a function of temperature both, in the presence (a) and absence (b) of an external magnetic field. In Fig. 2a, an external magnetic field of 8.5 mT is applied. A monotonic, nearly linear increase of the resonance frequencies associated to a change in the zero-field splitting parameter '( can be observed for temperatures down to 50 K. Zero-field ODMR (Fig. 2b) shows the same behavior. From Figs. 2a, b we now extract the ZFS values '( /ℎ and plot them against temperature (Fig. 2c). Both temperature dependencies, represented by dark blue and cyan diamonds, match perfectly and thus confirm that the temperature scaling of '( is indeed independent of the magnetic field.  The temperature dependency can be explained by considering the change in the delocalization of the spindefect wave function delocalization due to temperature-induced structural deformations of the crystal lattice. This is consistent with e.g. the case of the NVcenter in diamond 25,26 . The latter shows a linear behavior for the shift of the ZFS associated to the relative change of the lattice constant is observed 26 : where is the proportionality factor, explicitly written as / and ( ) is the relative change of the lattice parameter. Applying the same concept to hBN with its two lattice parameters and results in the equation: Here, '( /ℎ is the experimentally measured ZFS frequency. '(,B[\] /ℎ = 3.48 GHz is the ZFS at T = 295 K that we choose as reference. ∆ '(, and ∆ '(,Z are the frequency shifts induced by the relative changes in and , ( ) and Z ( ) are relative changes of and , respectively (see also Eqs. (6,7) below). The temperature-dependent lattice parameters ( ) and ( ) for hBN were determined in Ref. 24 and are plotted in Fig. 2c in addition to the ODMR data. The proportionality factors and Z are the significant parameters that connect lattice deformation and ZFS and will be derived from the experimental data in the following. To do so, Eq. (2) is fitted to the experimentally measured ZFS '( , as shown in Fig. 2c. The fit perfectly reproduces the experimental data, which highlights the remarkable linear response of the resonance frequency to changes of the lattice constants in this case due to temperature. Figure 3 shows the relationship between temperature-dependent lattice parameters and ZFS for a magnetic field of = 8.5 mT. By inserting the crystallographic data for the hBN lattice parameters 24 into Eq. (2), we obtain a surface with respective slopes and Z , as shown in Fig. 3d (grey). We extract the values for of (−84 ± 15) GH and (−78.2 ± 8.8) GH or Z of (−24.32 ± 0.59) GH and (−24.6 ± 1.0) GH at B = 0 mT and 8.5 mT, respectively. The out-of-plane Z can be determined more precisely, since the relative change of lattice parameter is one order of magnitude larger. The values coincide within the experimental error and can be combined as: Remarkably, the ratio / Z ≈ 3.3, which means the influence of the lattice distortion on the ZFS in-plane is at least three times stronger than the influence of the interplanar distance on the same, indicating a localization of the V % # spin density in the plane as predicted by the theory 17 .
Finally, we propose a polynomial which allows a direct determination of '( from the sample temperature : where T is the temperature, ℎ is Planck's constant, is an integer, and the polynomial coefficients b are summarized in Table 1 for different temperature ranges. To obtain the coefficients b , the essential step is to determine the relative changes of the lattice parameters from the crystallographic data 24 by using the Equations (6) and (7): Temperature range (K) C (GHz) @ (MHz K -1 ) B (kHz K -2 ) e (Hz K -3 )

Pressure Sensing
The observation that a temperature-induced change in the lattice parameters directly results in a shift of the ZFS '( leads to the consideration of utilizing the V % # center also as a sensor, for externally applied in-plane or out-of-plane pressure. For a first-order estimation we assume an isothermal system without shear strain and derive the perspective sensitivity based on reported elastic moduli for hBN crystals 27,28 . In cartesian coordinates, the pressure vector σ ghi is given by the elastic moduli tensor multiplied with the relative change of the lattice parameters ghi : This can be rewritten to obtain ,Z directly: p = g /u v w @@ + x w @B y = h /u v w @B + x w @@ y (10) Substituting these relationships into Eq. (2) yields the ZFS as a function of the applied pressure: Based on our estimates for ,Z in Eqs. (3,4) and the reported elastic moduli 27 , we obtain the sensitivity to measure the ZFS shifts for each direction of the applied pressure: Consequently, we find that the ZFS shift ∆ '(,ghi is directly associated with external compression of the hBN lattice and therefore V % # can be utilized as a pressure sensor. Remarkably, the out-of-plane sensitivity along the -axis is much higher due to the small ee coefficient. This makes this type of sensor particularly useful to measure vertical indentation in 2D heterostructures.

Magnetic Field Sensing
As shown in Fig. 2a, the two resonant transitions @,B are equally separated with respect to C over the entire temperature range between 5 K and 350 K. It should be pointed out that the magnetic field sensing is based on the g-factor, which is independent of the lattice parameters. In Figure 4, we demonstrate the principle suitability of a V % # center in hBN for magnetic field sensing, where we show the resonant microwave transitions @ and B over a broad range (0 − 3500 mT) and exemplarily simulated for two distant temperatures, T = 295 K (dark blue) and T = 5 K (light blue). For a magnetic field applied in the -direction of the hBN lattice, the behavior can be described with Eq. (1). Due to the non-zero '( , the Zeeman splitting term µ N leads to a linear regime only for > 3 , when the applied magnetic field is large enough to separate the two otherwise partially overlapping @,B transitions.

Figure 4. Experimental (diamonds) and simulated (dark and light blue traces) resonant frequencies of V % # for different temperatures and magnetic fields. a) ODMR measurements (pink diamonds) at B < 20 mT. b) cw EPR measurement at T = 5 K and microwave frequency of 9.4 GHz (light green) and electron spin-echo measurements at T = 8 K and 94 GHz (dark green). Note the axes are shifted for better visibility and comparability.
To extend the magnetic field range of our measurements beyond the confocal ODMR setup limit of ≈ 20 mT, we applied cw electron paramagnetic resonance (cw EPR) and electron spin-echo detected (ESE) EPR. The advantage is that the cw EPR measurements are performed at a microwave frequency of 9.4 GHz (X-band) (light green diamonds) and the ESE EPR measurements at a microwave frequency of 94 GHz (W-band) (dark green diamonds) which allow extending the magnetic field range to 3500 mT. The multifrequency spin resonance approach enables us to determine the g-factor with extremely high accuracy as = 2.0046 ± 0.0021.

Discussion
As we mentioned, one of the crucial parameters for high-sensitivity sensing is the distance between the sensor and the object to be sensed. In this regard, sensors based on the hBN V % # center are particularly appealing as -based on the presented model of lattice constant variations -the required hardware ideally consists of only three hBN layers, with the intermediate one hosting a V % # center. This corresponds to a minimum distance of the defect to the surface in the sub-nanometer range of »0.33 nm. A further thickness reduction, e.g. to a single monolayer, would indeed eliminate the interlayer contribution ( ), but make the sensing effect completely dependent on the interaction of the V % # wave function with the parameters of the adjacent material. For single layers, '(,B[\] would differ from the theoretical calculation 29 , and a calibration would be required to determine the set of parameters ( '(,B[\ ] , p , Z ) specific for the adjacent material. To substantiate this hypothesis, however, further measurements and calculations must be carried out.
In the following, we benchmark the properties of the hereby proposed sensor based on V % # centers in hBN against other defect-based sensors in silicon carbide and diamond. For this purpose, the general sensitivity is derived, which includes the respective coupling coefficient Γ d,|,N representing the sensitivity of the ODMR frequency shift due to the corresponding external influence, and the general resolution in relative change of frequency in relation to acquisition time and noise level 5 . An overview of all calculated coupling coefficients and resolutions is summarized in Table 2.
In order to facilitate comparison with other color centers in 3D materials, we consider a linear regime (50 -350 K) in our analysis. In this range, a proportionality between ∆ '( and is given by the factor Γ d = −623 ‰Šƒ ‹ . This value is almost one order of magnitude larger than the corresponding factor for NVcenters in diamond (−74 ‰Šƒ ‹ ) 30 . This remarkable (> 8-fold) difference is in particular due to the larger relative change of the lattice parameters as a function of temperature in hBN, while is of the same order of magnitude. For NVcenters, a value of OE• = −14.41 GHz is reported 26 , which is comparable to Z ≈ −24 GHz for V % # in hBN. Note that p can be neglected here, since on the one hand the relative change of the lattice parameter a is negligible and on the other hand it counteracts the effect due to the expansion of the in-plane distance while cooling the sample. The general resolution d B[\] obtained at room temperature is approximately 3.82 ‹ √Šƒ which is of the same order of magnitude as defect-based temperature sensors in SiC (1 ‹ √Šƒ ) using the same cw ODMR setup 5 . It should be noted, however, that at cryogenic temperatures the resolution of the V % # center is enhanced by a factor of ≈20 ( d \C] = 0.19 ‹ √Šƒ ), as both the ODMR contrast ΔPL/PL and the PL intensity increase, which significantly reduces the required measurement time. Despite the smaller coupling coefficient, a temperature sensor based on NV's in diamond is still more sensitive (0.76 •‹ √Šƒ ) 31 , mainly due to stronger PL emission, higher ODMR contrast and an optimized pulsed measurement protocol that exceeds the sensitivity of standard cw ODMR measurements performed here. Analogously, the magnetic field resolution of the V % # can be quantified at room temperature as N B[\] = 85.1 '' √Šƒ ( N \C] = 4.33 '' √Šƒ at T=50 K). This is comparable to VSi in SiC (10 '' √Šƒ ) 5 but lower than for NVs in diamond (3 "' √Šƒ ) 32 . However, spin defects in 3D materials can lose their superior properties if they are close to the surface of the crystalline host 10 , which also leads to an inevitable limitation of the minimum achievable sensor-to-object distance. This can significantly hinder the effectiveness of the spin defects-based sensors in 3D materials, highlighting the potential advantages of the V % # center as a sub-nanometer scale sensor.

Conclusion
In this work, we have analyzed the spin properties of V % # lattice defects in van der Waals hBN crystals in terms of their sensitivity to external perturbations and evaluated their advantages and disadvantages for possible applications as a nanoscale quantum sensor. The advantages include the simple intrinsic nature of the defect basically consisting of a missing boron atom, but also the potentially accessible very small distance between the sensor and the object to be sensed. In particular, we focused on the influence of temperature on the ground-state zero-field splitting, which can be directly measured by cw ODMR and is explained by the temperature-dependent lattice compression/expansion. Externally applied pressure can also induce lattice deformations and therefore be mapped onto the defect ODMR spectrum of the V B − . However, temperature and pressure measurements exclude each other and need to be performed isobar or isothermal, respectively. Nevertheless, V % # can be used for simultaneous magnetic field measurements with high sensitivity, due to the invariability of its g-factor with respect to temperature and pressure. By comparing three spin defect hosting solid systems, diamond, SiC and hBN, we showed that the V % # defect has comparable and, in some cases, even superior properties compared to 3D hosts. The coupling coefficient between zero-field splitting and temperature is eight times larger than the corresponding factor for NV centers in diamond. For completeness, a temperature sensor based on NV centers in diamond is still more sensitive, mainly due to stronger PL emission, higher ODMR contrast and an optimized pulsed measurement protocol that exceeds the sensitivity of cw ODMR measurements. The resolution of V % # to external magnetic fields is comparable to that of silicon vacancies in SiC, but lower than that of NV centers in diamond. However, we believe that the recent demonstration of coherent control of V % # spins in hBN together with the overcoming of inhomogeneous ODMR line broadening by multifrequency spectroscopy 20 will stimulate the development of advanced pulse protocols 33 and lead to a further increase in the resolution of this sensor. In addition, during preparation of our manuscript, a similar work on V % # in hBN was submitted 34 reporting an ODMR contrast of almost 10% and its high temperature stability up to 600 K. Finally, the unique feature of hBN is its non-disturbing chemical and crystallographic compatibility with many different 2D materials, which gains a new fundamental functionality with the embedded spin centers and allows sensing in heterostructures with high sensitivity serving as a boundary itself.

Methods:
ODMR: The low-field ODMR measurements are performed with a lab-built confocal microscope setup. A 532nm laser (Cobolt Samba 100) is coupled into a 50-µm fiber and focused on the sample with a 10× objective (Olympus LMPLN10XIR), which excites an area on the sample with a diameter of about 10 μm. The photoluminescence is separated from the laser by a dichroic mirror and the remaining laser light is rejected by a 532-nm long pass filter. The photoluminescence is then coupled into a 600 µm fiber and directed onto an avalanche photodiode (Thorlabs APD440A). A 0.5-mm wide copper strip-line is used to apply microwaves to the hBN sample placed on top. Microwaves from a signal generator (Stanford Research Systems SG384) are amplified by a Mini Circuits ZVE-3W-83+ amplifier. Lock-in detection is used (Signal Recovery 7230) by on-off modulation of the microwaves. For an external magnetic field, a permanent magnet is placed below the sample.