Temperature dependent spin-phonon coupling of boron-vacancy centers in hexagonal boron nitride
Zhongyuan Liu Ruotian Gong Benchen Huang Yu Jin Xinyi Du Guanghui He Eli Janzen Li Yang Erik Henriksen James Edgar Giulia Galli Chong Zu

























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The negatively charged boron-vacancy center () in hexagonal boron nitride (hBN) has recently emerged as a highly promising quantum sensor. 

Compared to the nitrogen-vacancy (NV) center in diamond, the change with temperature of the spin transition energy of is more than an order of magnitude larger, making it a potential nanoscale thermometer with superior sensitivity.

However, the underlying mechanism of the observed large temperature dependence remains an open question.

In this work, using isotopically purified , we systematically characterize the zero-field splitting, hyperfine interaction, and spin relaxation time of from 10 to 350 K.

We carry out first-principle calculations of the spin-phonon interaction and show that a second-order effect from finite-temperature phonon excitations is responsible for the observed changes in experiments.

By fitting our experimental results to a physically motivated model, we extract the dominant phonon mode which agrees well with our simulations.

Finally, we investigate the dynamic nuclear spin polarization process at cryogenic temperatures.

Our results provide key insights in centers and their utilization as nanoscale thermometers and phonon sensors. 




Temperature Dependent Spin-Phonon Coupling of Boron-Vacancy Centers in Hexagonal Boron Nitride
    
Zhongyuan Liu,^1,* 
Ruotian Gong,^1,* 
Benchen Huang,^2,* 
Yu Jin,^2 
Xinyi Du,^1
Guanghui He,^1

Eli Janzen,^3
Li Yang,^1
Erik Henriksen,^1
James Edgar,^3
Giulia Galli,^2,4,5
Chong Zu^1,6,7,†


^1Department of Physics, Washington University, St. Louis, MO 63130, USA

^2Department of Chemistry, University of Chicago, IL 60637, USA

^3Tim Taylor Department of Chemical Engineering, Kansas State University, Manhattan, KS, 66506, USA

^4Pritzker School of Molecular Engineering, University of Chicago, IL 60637, USA

^5Material Science Division, Argonne National Lab, IL 60439, USA

^6Center for Quantum Leaps, Washington University, St. Louis, MO 63130, USA

^7Institute of Materials Science and Engineering, Washington University, St. Louis, MO 63130, USA

^*These authors contributed equally to this work

^†To whom correspondence should be addressed; E-mail: zu@wustl.edu


    April 29, 2024
==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



Introduction— 

Optically-addressable solid state spin defects are promising platforms for quantum applications <cit.>, and when residing in atomically-thin van der Waals materials they may exhibit properties superior to those of their conterparts in three-dimensional materials <cit.>.

Among a wide-range of contestants, the negatively charged boron-vacancy center, , in hexagonal boron nitride (hBN) has attracted much growing interest <cit.>.

Several interesting properties of the host material, including the large bandgap and excellent stability, as well as the readily controllable spin degree of freedom at room temperature, makes in hBN well-suited as a platform for technologies.

When compared with its three-dimensional counterparts, for example the nitrogen-vacancy (NV) center in diamond, the ground state zero-field splitting (ZFS) of exhibits a ∼25 times larger variation from cryogenic to room temperatures, placing it in an advantageous position for nanoscale thermometry <cit.>.

However, the underlying mechanism of such a large temperature response remains an open question.

In particular, the first-order lattice displacement of hBN with temperature can only account for a negligible fraction of the changes in the measured ZFS of  <cit.>; higher-order spin-phonon interactions should be included to quantitatively capture the experimental results.

In this letter, we experimentally characterize the temperature dependence of the ZFS, hyperfine interaction, and spin relaxation time T_1 of in isotopically purified . 

Compared to conventional hBN with natural abundant ^14N, our offers much narrower spin transitions and substantially improved measurement resolution <cit.>.

We then carry out first-principle calculations of the temperature-induced phonon-mediated interaction of ,  motivated by recent investigations of the NV centers in diamond <cit.>.

Our combined experimental and theoretical study allows for the identification of the main mechanism responsible for the observed temperature dependence of the properties of , namely a second-order spin-phonon coupling with a characteristic phonon mode energy around 18meV.

We also investigate the dynamic nuclear spin polarization of the nearest three ^15N and find that the polarization persists down to 10K.



Experimental system— 

The is an atomic spin defect in the hBN lattice (Fig. <ref>a).

The electronic ground state of exhibits a spin-1 triplet with |m_s = 0⟩ and |m_s = ±1⟩.

In the absence of external fields, the |m_s = ±1⟩ are degenerated and separated from |m_s = 0⟩ by a zero-field splitting (ZFS) D = (2π)× 3.48GHz at room temperature (Fig. <ref>b).

The Hamiltonian governing and the nearest three nitrogen nuclear spins takes the form, 


    ℋ   = D S_z^2
         + γ_e B_z S_z
         + ∑_j=1^3𝐒𝐀^j𝐈^j
         - ∑_j=1^3γ_n B_z I_z^j    ,


where B_z is an external magnetic field along the c-axis of hBN (or ẑ direction), 𝐒 and S_z are the electron spin-1 operators, 𝐈^j and I_z^j are the nuclear spin operators, and
𝐀^j is the hyperfine tensor coupling electronic and nuclear spins.

The factors γ_e= (2π)× 2.8MHz/G and γ_n are electronic and nuclear gyromagnetic ratios.

Throughout our experiment, the external magnetic field is B_z ≲ 760G.

In this case, the energy splittings between |m_s = 0⟩ and |m_s = ±1⟩ sub-levels, D ±γ_e B_z, are much larger than the strength of the hyperfine interaction |𝐀^j|, thus the secular approximation can be adopted: 
∑_j𝐒𝐀^j𝐈^j ≈∑_jA_zz^j S_z I_z^j = A_zz(∑_jI_z^j)S_z, leading to a nuclear spin dependent energy shift of the electronic transition.

This results in 2ℐ+1 transitions between |m_s=0⟩ and |m_s=-1⟩ (or |m_s=+1⟩) sub-levels, where ℐ = ∑_j I^j is the total nuclear spin number accounting for all three nearest nitrogen atoms (Fig. 1b).

The corresponding spin transitions of can be probed via optically detected magnetic resonance (ODMR) spectroscopy: by sweeping the frequency of the applied microwave drive while monitoring the fluorescence signal of , we expect a fluorescence drop whenever the microwave is resonant with an electronic spin transition.

We report measurements for hBN with naturally distributed isotopes (99.6% ^14N with I = 1), and for the isotope-engineered (99.7% ^15N with I=1/2) <cit.>.

For conventional , there are in total seven broad hyperfine spin transitions; while for , we observe four sharp resonances, enabling the measurement of ZFS and hyperfine splitting, A_zz, with substantially improved resolution (Fig. 1cd).

Experiment results—

To investigate the temperature-dependent spin properties of , we load hBN samples with thickness ranging from ∼ 30 - 70 nm into a closed-cycle optical cryostat (Fournine Design) for temperature control from 10-350K.

A small external magnetic field B_z ≈ 90 G is used to lift the degeneracy between |m_s=±1⟩.

Figure <ref>c,d show the ODMR spectra as a function of temperature.

We fit the spectra with two groups of equally spaced Lorentzians, corresponding to |m_s=⟩↔|m_s=±1⟩ transition respectively.

The ZFS, D(T), can be extracted from the average frequency of the two transition groups, while the hyperfine splitting, A_zz(T), is obtained from the fitted spacing between adjacent resonances in each group (Fig. <ref>a).

In addition, we also perform spin relaxation measurements and extract the T_1 timescale of the sample under different temperatures  (Fig. <ref>b).



We find that the ZFS of exhibits a dramatic change of more than 200 MHz from 10 to 350 K, which is around 30 times larger than the change observed for NV centers <cit.>.

We determine the susceptibility of at room temperature, χ(T=300 K), to be -880±65 kHz/K for , and -784±35 kHz/K for .

These results highlight the potential use of as an ultra-sensitive nanoscale thermometer.

Furthermore, within our measured temperature range, we observe a slightly smaller change of ZFS in compared to , which can be ascribed to the fact that the heavier nuclei in  lead to atomic displacements less sensitive to the temperature change.

The isotope effect is further evidenced by the measured T_1 timescales, limited by spin-phonon interaction <cit.>:  for heavier nuclei and thus weaker spin-phonon coupling strength, the T_1 of in is further extended at lower temperatures.

We also note that while it remains a challenge to resolve the hyperfine interaction strength from the broad ODMR spectra of conventional  <cit.>, the substantially better resolved resonances of enable an accurate characterization of A_zz(T) (see Supplementary Materials). 

The measured amplitude of A_zz(T) displays a noticeable increase from 64MHz to 67MHz with decreasing temperature, whereas the hyperfine interaction of NV centers to the nearby nuclear spins has been measured to be nearly constant with temperature <cit.>.

Finally, we observe that the ODMR contrast of in both hBN samples reaches a maximum around 210K and persists to low temperatures, without any substantial quenching. 

This behaviour is again in opposite to that of the NV center in diamond, whose contrast suffers from a significant drop below 90K <cit.>, hindering its sensing performance at cryogenic temperatures.



Theoretical model— To understand the observed temperature dependence of spin properties, we adopt a theoretical model which was originally developed to study NV centers in diamond <cit.>.

In particular, we write the electronic and nuclear spin transitions as the sum of two terms,


    ν = ν_0(a(T)) + 1/2∑_i ∂^2 ν/∂ q_i^2ħ/M_i ω_i(1/e^ħω_i/k_B T - 1 + 1/2),


where ν is the transition frequency (e.g., ZFS and hyperfine interaction) and a(T) is the lattice constant of hBN as a function of temperature <cit.>, q_i, M_i, ω_i are the normal mode of the crystal, mode-specific effective mass, and frequency, respectively.

Here the first-order term, ν_0(a(T)), corresponds to the thermal expansion of the hBN lattice, while the second-order term represents vibrational contributions caused by finite-temperature phonon excitations near the equilibrium geometry.

We carry out first-principle simulations using density functional theory (DFT) <cit.> with the Perdew-Burke-Ernzerhof (PBE) functional <cit.> to compute the temperature-dependent hyperfine interaction, A_zz(T), between and the nearest three nuclei. 

Interestingly, we find that the variation of the lattice constant with temperature only results in an extremely small change in the magnitude of A_zz(T).

On the other hand, the second-order term accounts for the effect of temperature on the value of A_zz(T) observed experimentally.

Figure <ref>a shows the calculated second-order derivative of the energy of the crystal for each phonon mode.

We find a phonon mode with the largest amplitude at energy around ħω_th≈16 meV, corresponding to an out-of-plane vibration shown in the inset.

The computed temperature dependence of A_zz(T) agrees well with our experimental results (Fig. <ref>b).

For a direct comparison with the NV center, we also investigate the phonon mode associated with the hyperfine coupling between the NV center and its first shell ^13C nuclear spin <cit.>.

In the case of the NV center, the dominant phonon-mode energy is ≃ 70meV <cit.>, requiring a much higher activation temperature than .

Hence the variation of A_zz with temperature is much larger for the center  than that observed  for the NV center within the range (10-350 K) investigated in our experiment.



The dominant phonon mode found theoretically at ħω_th≈16meV can also be independently determined by fitting the experimentally measured spin properties to a physically motivated model <cit.>,


    ν(T) = ν_0 + c_ν(1/e^ħω/k T - 1 + 1/2),


where ν(T=0) = ν_0+1/2c_ν is the transition energy (e.g. ZFS D and hyperfine A_zz) at 0 K and n = (e^ħω/k T - 1)^-1 is the occupation number of the phonon mode.

We apply this model to the measured ZFS and hyperfine interaction of .

For ZFS, the observed temperature dependence of D(T) for both and samples can be accurately fitted using our model, from which we extract the characteristic phonon energy to be ħω_exp = 18.4±1.0meV and 18.8±2.8meV respectively (Fig. <ref>a and Table <ref>), in agreement with our theoretical results.

The measured hyperfine interaction, A_zz(T), shows a temperature dependence that is also consistent with the model with phonon energy set to 18.4 meV (Fig. <ref>a Inset).

Interestingly, the agreement between ħω_exp and the calculated ħω_th suggests that the temperature dependence of ZFS and of the hyperfine interaction originate from the coupling of the electronic spin to the same vibrational phonon mode.

We now turn to investigate the temperature dependence of the spin relaxation time T_1.

Within the temperature range (10-350 K) considered here, T_1 of is dominated by Raman scattering processes, where the energy difference between phonon absorption/emission is equal to the ZFS of , leading to spin depolarization (see Supplementary Materials) <cit.>.

Therefore the spin relaxation rate of can be approximated as <cit.>


    Γ(T) = 1/T_1 ≈ A n (n + 1) + A_S,


where n = (e^ħω/k T - 1)^-1 and A are the phonon occupation number and coupling coefficient associated with the effective mode, and A_S is a sample-related constant that might vary slightly between experiments.

Using the phonon energy ħω_exp = 18.4  (18.8)meV determined above, the model of Eq. <ref> faithfully reproduces the experimentally measured spin relaxation rates from 50-350K (Fig. <ref>b).

The small discrepancy below 50K may be due to the activation of lower-energy phonon modes as well as processes beyond Raman scattering. 

We note that a prior theoretical study has predicted a power-law scaling of T_1∼ T^-2 <cit.>, which is consistent with our experimental results in the high temperature regime (Fig. <ref>b).

Dynamic nuclear polarization at low temperature—

With the temperature-dependent spin properties in hand, we now turn to investigate the dynamical polarization of the three nearest-neighbor nuclear spins at low temperature.

Nuclear spins feature exceptional isolation from noisy environments, making them ideal candidates for quantum storage applications <cit.>.

At room temperature, several prior works have demonstrated the polarization of proximate ^14N or ^15N with an upper limit of ∼ 30 %; however how to further improve the polarization fidelity remains an open question  <cit.>.

Low temperature offers two potential advantages: ∼ 100x longer electronic spin lifetime T_1 and stronger hyperfine interaction strength.

To enable resonant spin exchange between electronic and nuclear spins (known as electronic spin level anti-crossing, esLAC) <cit.>, we apply an external magnetic field B_z ≈ 760G under which the energy levels corresponding to excited state |m_s = 0⟩ and |m_s = -1⟩ are nearly degenerate (see Supplementary Materials).

We fit the measured hyperfine resonances between |m_s=0⟩↔|m_s=+1⟩ in the ODMR spectra as a function of the laser power to extract the nuclear spin polarization fidelity (Fig. <ref>a).

We find that the nuclear spin polarization at both 10 K and 300K saturate around 30% with similar laser powers (∼10mW) (Fig. <ref>b), suggesting that the spin lifetime T_1 and hyperfine interaction strength are not the limiting sources for nuclear polarization. 



The  spin polarization may be limited from the spin non-conserving terms in the hyperfine interaction (see Supplementary Materials), which are temperature insensitive. 

In addition,  the observed nuclear spin polarization of persists down to 10K, in sharp contrast to the case of NV centers where the dynamic nuclear polarization process is  substantially suppressed at T≲50K, due to the lack of thermal averaging between NV excited states  originating from different orbitals <cit.>. 

For , the lifting of the double degeneracy of excited states leads to a sufficiently large energy gap between the states, which renders the system insensitive to dynamic Jahn–Teller distortions <cit.>.

Conclusion and Outlook—

In summary, our research offers a comprehensive investigation of the temperature-dependent spin-phonon interaction of centers in isotopically purified .

Our results will be of critical importance in designing experiments to make use of centers as nanoscale thermometers and phonon sensors.

Moreover, although we focus on , the methodologies developed here can be readily applied to a broad family of spin defects in two-dimensional materials <cit.>.




Acknowledgements:

We gratefully acknowledge Vincent Jacques, Tongcang Li, Shankar Mukherji, Erdong Song, Xi Wang and Norman Yao for helpful discussions.

This work is supported by NSF ExpandQISE 2328837 and the Center for Quantum Leaps of Washington University.

B. Huang, Y. Jin and G. Galli acknowledge the support of Air Force Office of Scientific Research (AFOSR) under Contract No. FA9550-22-1-0370, the Next Generation Quantum Science and Engineering (QNEXT) hub supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers and resources of the University of Chicago Research Computing Center.

Support for hBN crystal growth (E. Janzen and J. H. Edgar) was provided by the Office of Naval Research, award no. N00014-22-1-2582.




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