Nuclear Spin-Mediated Relaxation Mechanisms of the V$_{B}^-$ Center in hBN

TL;DR

This work addresses the microscopic origin of $T_1$ relaxation for the $V_B^-$ center in bulk hBN at low temperatures by developing a parameter-free spin-dynamics model based on cluster expansion with an extended Lindbladian. Through a sequence of progressively richer cluster models, the authors demonstrate that an extended central-spin description including the electron spin and the three nearest $^{15}$N nuclei is essential, and that incorporating two additional distant $^{15}$N bath spins yields an accurate, largely exponential $T_1$ consistent with experiments and captures the field-dependent relaxation behavior. The study reveals three magnetic-field regimes (low-field, near GSLAC, high-field) with distinct relaxation mechanisms dominated by electron–nuclear and nuclear–nuclear flip-flops, and highlights the critical role of nuclear-spin entanglement in mediating relaxation. These insights provide a microscopic framework for engineering longer $T_1$ times and inform the design of nuclear-spin-based quantum technologies in hexagonal boron nitride, with implications for quantum sensing and quantum memory implementations.

Abstract

The negatively charged boron vacancy $V_B^-$ defect in hexagonal boron nitride (hBN) has recently emerged as a promising spin qubit for sensing due to its high-temperature spin control and versatile integration into van der Waals structures. While extensive experiments have explored their coherence properties, much less is known about the spin relaxation time $T_1$ and its control-parameter dependence. In this work, we develop a parameter-free spin dynamics model based on the cluster-expansion technique to investigate $T_1$ relaxation mechanisms at low temperature. Our results reveal that the $V_B^-$ center constitutes a strongly coupled electron spin-nuclear spin core, which necessitates the inclusion of the coherent dynamics and derived memory effects of the three nearest-neighbor nitrogen nuclear spins. Using this framework, this work closely reproduces the experimentally observed $T_1$ time at $B = 90\,\mathrm{G}$ and further predicts the $T_1$ dependence on external magnetic field in the $0 \le B \le 2000\,\mathrm{G}$ interval, when the spin relaxation is predominantly driven by electron-nuclear and nuclear-nuclear flip-flop processes mediated by hyperfine and dipolar interactions. This study establishes a reliable and scalable approach for describing $T_1$ relaxation in $V_B^-$ centers and offers microscopic insights to support future developments in nuclear-spin-based quantum technologies.

Figures (6)

• Figure 1: Magnetic field dependence of the fine structure of the ground state spin system of the V$_\text{B}^{-}$ center in hBN. Spacing of the energy levels is dominated by the zero-field splitting and Zeeman terms, except at the ground state level avoided crossing (GSLAC) of the electron-spin sublevels, where hyperfine interaction heavily mixes electron and nuclear spin states.
• Figure 2: Possible interactions of a V$_\text{B}^{-}$ defect in hBN. (a) Spin density of the V$_B^{-}$ defect, highlighting various interactions. The hyperfine interactions (Eq. \ref{['eq:H_SB']}) are represented by gray wavy lines, while the dipole-dipole interactions between nuclear spins, including nitrogen-nitrogen, boron-boron, and nitrogen-boron interactions, are depicted by blue, pink, and green wavy lines, respectively (corresponding to the second term in Eq. \ref{['eq:H_B']}). The interactions of both electron and nuclear spins with an external magnetic field are present but not explicitly visualized in this figure. The spin density illustrates the localized wavefunction, which correlates with the strength of the hyperfine interaction. (b) Rotational invariant perpendicular hyperfine components in decreasing order plotted in a log-log scale. The steps indicate different neighboring shells. The first three nearest-neighbor nitrogen nuclei exhibit an extraordinarily strong hyperfine interaction, indicating a strongly induced spin flip-flop process.
• Figure 3: Spin dynamics models. (a) Model 1: Two-spin system model in V$_\text{B}^{-}$. Each cluster contains an electron spin and a nitrogen nuclear spin. (b) Model 2 and 3: Three/Four-spin system model in V$_\text{B}^{-}$. Blue clusters represent nitrogen clusters, containing an electron spin and three nitrogen nuclei. Pink clusters represent boron clusters, containing an electron spin and two boron nuclei. (c) Model 4: Six-spin system model in V$_\text{B}^{-}$. Each shaded area represents a six-spin cluster, including an electron spin and the closest three nitrogen nuclei in the central spin and two nitrogen nuclei from the bath. Each cluster has the same four-spin central spin and two different environmental spins. Boron nuclei are neglected. Blue and pink atoms depict nitrogen and boron atoms, respectively.
• Figure 4: Spin population of $\boldsymbol{m_s}$ = 0 state of the first cluster at $\boldsymbol{B}$ = 800 G. The decoherence is taken into account based on (a) Model 1, (b) Model 2 with 47 nitrogen and 48 boron atom clusters, respectively, (c) Model 3 T$_2^* = 200$ ns and 47 nitrogen and 48 boron atom clusters, and (d) Model 4 with 28 nitrogen atom clusters. The $T_1$ time is obtained from fitting a$e^{(-(t/T_1)^n)}$ + $y_0$.
• Figure 5: Polarization dynamics at $\boldsymbol{B}$ = 90 G among 28 clusters, each containing six spins based on Model 4. (a) Population of polarization of an electron spin of V$_\text{B}^{-}$ in m$_s$ = 0 state of the first cluster. (b) Population of spin-up polarization of one of the first three nearest $^{15}$N nuclear spins of the first cluster. Note that the behaviour for the other two first nearest nuclear spins remains unchanged. (c) Summation of all 28-cluster populations of spin-up polarization of two $^{15}$N nuclear spins from the bath. Note that the value of the population is shifted by 0.5 for simplicity in analyzing the spin-bath contribution. (d) The zoom-in of the nuclear-spin polarization population in (b). (e) Convergence of the extrapolated $T_1$ time as a function of evolution time.