Influence of Second-Order Effects due to Hyperfine Interaction on the Magnitude of the Larmor Frequency ¹⁴N

The negatively charged boron vacancy ( \({\text{V}}_{\text{B}}^{-}\) ${\text{V}}_{\text{B}}^{-}$ ) in hexagonal boron nitride (hBN) is currently considered an intriguing quantum object for testing and developing quantum technologies on two-dimensional van der Waals materials. This article presents results from photoinduced electron spin echo (ESE)-detected electron spin resonance (ESR) and electron–nuclear double-resonance (ENDOR) spectroscopy at the W-band (ν = 94 GHz), focusing on the interactions of the \({\text{V}}_{\text{B}}^{-}\) ${\text{V}}_{\text{B}}^{-}$ electron spin with the three nearest nitrogen nuclei (¹⁴N, I = 1). The lines in the ENDOR spectrum are due to both hyperfine and quadrupole interactions for M_S = ± 1 levels and only quadrupole interactions for M_S = 0 levels. We show that significant hyperfine interaction with the three nearest nitrogen atoms, despite the high magnetic field, results in a mixing of the hyperfine sublevels for M_S = 0. We show that significant hyperfine interaction with the three nearest nitrogen atoms, despite the high magnetic field, results in mixing of the hyperfine sublevels. This mixing shifts the ¹⁴N Larmor frequency from its nominal value defined as \({{\varvec{\nu}}}_{{\varvec{L}}}=\boldsymbol{ }{{\varvec{g}}}_{{\varvec{N}}}{{\varvec{\mu}}}_{{\varvec{N}}}{\varvec{B}}/{\varvec{h}}\) ${\mathit{\nu }}_{\mathit{L}}={\mathit{g}}_{\mathit{N}}{\mathit{\mu }}_{\mathit{N}}\mathit{B}/\mathit{h}$ . This shift observed through ENDOR experiments can be understood using spin-Hamiltonian formalism within the second-order perturbation theory. These findings enhance an understanding of electron–nuclear interactions in hBN.