Charge state equilibration of nitrogen-vacancy center ensembles in diamond: The role of electron tunneling

TL;DR

This work demonstrates that charge-state equilibration in NV center ensembles is dominated by multiphonon-assisted electron tunneling between NV^0 and nearby neutral nitrogen donors, rather than thermally activated processes. A combined theoretical framework—density-functional theory for electron–phonon coupling, a zero-range-potential model for inter-defect tunneling, and a configurational-coordinate treatment of the vibrational modes—along with pump–probe experiments, shows that tunneling rates depend strongly on NV–N separation and donor density, while thermally driven channels are negligible. The authors derive analytical transfer integrals, benchmark them against finite-well models, and develop an empirical, tunable description that reproduces the observed seconds-to-hours equilibration timescales across 10–180 ppm nitrogen concentrations. The results quantitatively connect defect energetics, lattice relaxation, and tunneling dynamics, offering a predictive framework for charge-state stability in NV centers and wide-bandgap defects relevant to phosphors and scintillators. This approach can guide strategies to control charge states in quantum sensing and solid-state defect engineering.

Abstract

The charge state stability of nitrogen-vacancy (NV) centers critically affects their application as quantum sensors and qubits. Understanding charge state conversion and equilibration is critical not only for NV centers in diamond but also for defects and impurities in wide-bandgap materials in general. The mechanisms by which these centers change charge state upon optical or electronic excitation without the presence of mobile carriers remain unclear, potentially affecting the performance of applications ranging from phosphors to power electronics. Here, we elucidate this issue for the case of photoionization of NV center ensembles. Using pump-probe spectroscopy, we ionize negatively charged NV centers and monitor the recovery of $\NVm$ on timescales of up to several seconds. We find that the recovery rate depends strongly on the concentration of surrounding nitrogen donors. Remarkably, the equilibration dynamics exhibit no discernible dependence on temperature, ruling out thermally activated processes. The multiphonon-assisted electron tunneling model, supported by density-functional calculations, explains the measurements and identifies tunneling as the equilibration mechanism.

Figures (13)

• Figure 1: Phonon-assisted electron tunneling from $\text{N}^{0}$ to NV$^{0}$. (a) Exothermic charge transfer: the ionization threshold for $\text{N}^0$ is 1.7 eV, and the energy released when the electron is captured by $\text{NV}^0$ is about 2.7 eV. (b) Left: first-principles configuration coordinate (CC) potential energy surfaces of the $\text{N}$ center. Inset: atomic structure of N0 with antibonding orbital isosurfaces in neutral state; arrows show atomic motion along the positive CC $Q$ direction. Right: Optical spectral functions for the tunneling process at $T = 0$ K and 300 K, where the value at $E = 0$ corresponds to the vibrational factor $A(\Delta E; 0)$ in Eq. \ref{['eq:rate']}.
• Figure 2: (a) Schematic of the measurement setup. Abbreviations: OPA -- optical parametric amplifier, PBS -- polarizing beam splitter, BPF -- bandpass filter, PD -- photodiode, DAQ -- Data acquisition card; (b) Excitation scheme illustrating pump and probe dynamics; Normalized probe transmission dynamics for both samples at temperatures of c) 10 K and d) 290 K, with bi-exponential fits to the data. The inset plots display graphs for a 10 s time window.
• Figure 3: (a) Simultaneous fit of the empirical tunneling model (thin solid lines) to the experimental decay curves at $T=10$ K (thick blurry lines) for 10 ppm and 180 ppm nitrogen concentrations. Both datasets are normalized such that the pre-pump population is one and the value at 9 s is zero. Inset: Same data plotted on an absolute scale, with experimental curves rescaled to match the predicted population difference. (b) Calculated tunneling lifetime $\tau = 1/\Gamma$ as a function of $\text{NV}$--$\text{N}$ separation at $T = 0$ K. (c) Time evolution of the relative $\text{NV}^{0}$ population $p(t)$ at $T = 0$ K for different $\text{N}$ concentrations (1--100 ppm), following a single-pulse ionization.
• Figure S1: Ab initio characterization of the electron--phonon interaction in the $\text{N}$ defect. (a) Atomic structure of the neutral $\text{N}$ center, with isosurfaces showing the antibonding localized orbital occupied in the neutral state; arrows indicate atomic motion along the positive CC $Q$ direction. (b) and (c) Configuration coordinate (CC) potential energy surfaces for $\text{N}^{0}$ and $\text{N}^{+}$, with dots representing DFT calculations. Numerical solutions of nuclear wavefunctions are shown with baselines shifted according to vibrational state energies.
• Figure S2: Comparison of $W^{2}$ calculated using the full treatment (including overlaps) and the zero-overlap approximation. The bottom panel shows the ratio of the two models.