Magneto-optical spectra of the split nickel-vacancy defect in diamond

Nickel is a common impurity in high-pressure high-temperature diamond and may contaminate chemical vapor deposited diamond used for high-power electronics or quantum technology applications. Magneto-optical fingerprints of nickel have been known since decades, however, no consensus has been reached about the microscopic origins of nickel-related electron paramagnetic resonance, photoluminescence, and optically detected magnetic resonance spectra. The unknown nickel-related defect structures in diamond make it difficult to control them or harness them for a given application. As a consequence, nickel is considered as an impurity in diamond that should be avoided or its concentration should be minimized. Recent advances in the development of ab initio magneto-optical spectroscopy have significantly increased its accuracy and predictive power that can be employed for identification and in-depth characterization of paramagnetic color centers in diamond. In this study, we extend the accuracy of the ab initio magneto-optical spectroscopy tools towards self-consistent calculation of second-order spin-orbit coupling for paramagnetic color centers in solids. We apply the full arsenal of the ab initio magneto-optical spectroscopy tools to characterize the split nickel-vacancy defect in diamond which is one of the most stable nickel-related defect configurations. As a result, electron paramagnetic resonance and optical centers are positively identified in various charge states of the nickel-vacancy defect in diamond. In particular, the 1.40-eV optical center and the NIRIM-2 electron paramagnetic resonance center are identified as the single negative charge state of the split nickel-vacancy center. The defect possesses $S=\frac{1}{2}$ spin state with an orbital doublet ground state. We find that the coherence time of the ground-state spin is about 0.1 ms at cryogenic temperatures which can be optically initialized and readout by a $\mathrm{\Lambda }$-scheme protocol. Since the defect has inversion symmetry the optical signal is insensitive to the stray electric fields, which is an advantage for creating indistinguishable solid-state single-photon sources. We predict that the negatively charged nickel-vacancy defect has similar optical properties to those of the well-known silicon-vacancy defect in diamond but is superior in terms of electron spin coherence times. Our study resolves a few decades controversy about the nickel-related spectroscopy centers in diamond and turns nickel from an impurity to a resource in quantum technology applications.

Figure 1

(a) Kohn-Sham levels of neutral, negative, and doubly negative NiV defects near the band edges of diamond. We denote the ab initio charge transition levels on the left. We also denote the excitation process and the Kohn-Sham levels of optically excited ${\mathrm{NiV}}^{-}$ (1.4 eV) and ${\mathrm{NiV}}^{2-}$ (2.97 eV) centers. Additionally, we show photoionization process for ${\mathrm{NiV}}^0$ defect (1.22 eV), where an electron at VBM (valence band maximum) is excited into the $e_g$ level, thus transforming it into the ${\mathrm{NiV}}^{-}$ in the process. VB and CB correspond to valence band and conduction band, respectively. (b) Visualization of single-particle Kohn-Sham wave functions. The gray ball depicts the impurity atom, while the black balls depict the six first-neighbor carbon atoms. The real ($|e_x\rangle$, $|e_y\rangle$) degenerate orbitals can be transformed to the complex ($|e_{+}\rangle$, $|e_{-}\rangle$) combinations as $|e_{\pm }\rangle =\frac{1}{\sqrt{2}}\left[\right|e_x\rangle \pm i|e_y\rangle ]$ (see details in Ref. [4]) to show the effective $L_z=\pm 1$ orbital moment of the wave functions. We note that the Ni $3d$ atomic orbitals significantly couples to $|e_g\rangle$ while $|e_u\rangle$ shows significant Ni $3p$ and $4p$ characters too. These Ni atomic orbitals are mainly responsible for the spin-orbit coupling in $|e_g\rangle$ and $|e_u\rangle$.

Figure 2

(a) Supercell size ($L$) dependence of ab initio intrinsic spin-orbit coupling ${\lambda }_0$ (black curve) in ${\mathrm{NiV}}^{-}$ (see Sec. 3c2 for details). We use the ${\lambda }_{g/u}$ parameters (black and red curves) to approximate the ${\mathrm{NiV}}^{2-}$ system (see Sec. 3b2 and Appendix pp3). (b) Kohn-Sham orbitals of ${\mathrm{NiV}}^{-}$ system by means of PBE functional and applied spin-orbit coupling.

Figure 3

(a) Ab initio PL spectrum of ${\mathrm{NiV}}^{-}$ optical defect within Huang-Rhys theory with taking into account only the symmetric ($A_{1g}$) and symmetry-breaking ($E_g$) phonons too. ZPL stands for the zero-phonon line where the theoretical ZPL was aligned to the experimental one at 1.4 eV for the sake of clarity in the comparison of the corresponding phonon sidebands. We applied 11.5-meV Gaussian broadening on the theoretical sideband and 4 meV on the ZPL. Experimental data of the 1.4-eV center is taken from Ref. [29]. (b) Sketch of nickel-vacancy defect exhibiting $D_{3d}$ symmetry.

Figure 4

(a) Theoretically predicted PL sideband of the ${\mathrm{NiV}}^{2-}$ by means of Huang-Rhys theory. We aligned the theoretical ZPL energy at 2.87 eV to the observed one at 2.964 eV, for the sake of sideband comparison. (b) Theoretically predicted PLE spectra for the 2.964-eV PL center. We obtained the PLE spectra by the following procedure. First, we convoluted the phonon sideband within Huang-Rhys theory with the imaginary dielectric function. Then, we aligned the mean value of this spectrum to the $\mathrm{\Delta }\mathrm{SCF}$ ZPL energy for the singlet transition at 3.33 eV. (c) Imaginary part of the dielectric function at the ground state of ${\mathrm{NiV}}^{2-}$ by means of DFT PBE calculations with $4\times 4\times 4$ $k$-point sampling within the Monkhorst-Pack scheme [85]. The experimental PL and PLE spectra of 2.97-eV center are taken from Ref. [37]. The theoretical curves (green and blue) are scaled to exhibit the same area under the curve as that of the red colored area of experimental spectra.

Figure 5

(a) Many-electron states of NiV. The excitation energies inside the parentheses are experimental data (see Table 1 and the main text for details and interpretation). (b) Optical absorption spectra of low-nitrogen-concentration diamond taken from Ref. [44].

Figure 6

Proposed quantum optics protocol for ${\mathrm{NiV}}^{-}$. (a) At zero magnetic field the ${|}^2{E}_u\rangle$ ground state is split by $\lambda$ into ${\mathrm{\Gamma }}_4$ and ${\mathrm{\Gamma }}_5\oplus {\mathrm{\Gamma }}_6$ spin-orbit sublevels [89]. (b) If one applies magnetic field along the $z$ [111] axis, the Kramers doublets split. We label the individual levels by their $S_z=\{\uparrow ,\downarrow \}$ spin and $L_z=\pm 1$ orbital moment by $|e_{u\pm }^{\uparrow ,\downarrow }\rangle$ labels. (c) If the magnetic field is misaligned from the $C_3$ rotation axis, the spin projections will disorient from the $\mathit{B}$ field direction according to Eq. (4). That is, the spin-orbit coupling does not allow the spin to fully align towards the $\mathit{B}$ field, thus, optical transitions “D” will be optically active. We note that the $|a_{1g}\rangle$ orbital does not possess $L_z$ angular moment, thus, its spin perfectly follows the magnetic field direction.

Figure 7

Kohn-Sham levels of (a) ${\mathrm{Ni}}_{\text{s}}{\mathrm{N}}_{\text{s}}$, (b) ${\mathrm{Ni}}_{\text{s}}{\left({\mathrm{N}}_{\text{s}}\right)}_2$, (c) ${\mathrm{Ni}}_{\text{s}}{\left({\mathrm{N}}_{\text{s}}\right)}_3$ complexes. We denote the ab initio charge transition levels on the left bar for each figure. We also depict the excitation process for ${\mathrm{Ni}}_{\text{s}}{\left({\mathrm{N}}_{\text{s}}\right)}_3^{-}$ that we associate with the NE4 EPR center and 1.72-eV optical center. We note that we lifted the $C_{3v}$ symmetry constraint for ${\mathrm{Ni}}_{\text{s}}{\mathrm{N}}_{\text{s}}^0$ and ${\mathrm{Ni}}_{\text{s}}{\left({\mathrm{N}}_{\text{s}}\right)}_3^{+}$, therefore, $e_x$ ($e_y$) orbitals become $a^{\prime }$ ($a^{″}$) in the lower $C_{1h}$ symmetry.

Figure 8

Ab initio PL spectrum of ${\mathrm{Ni}}_{\text{s}}{\left({\mathrm{N}}_{\text{s}}\right)}_3^0$ defect within Huang-Rhys theory with taking into account only the symmetric ($A_{1g}$) phonons. ZPL stands for the zero-phonon line where the theoretical ZPL was aligned to the experimental one at 1.717 eV for the sake of clarity in the comparison of the corresponding phonon sidebands. We applied 1-meV Gaussian broadening and its area under curve was fitted to the experimental data. Additionally, we applied 3.5-meV broadening for the theoretical sideband. Experimental data are taken from Fig. 11 in Ref. [50].