Multiconfigurational study of the negatively charged nitrogen-vacancy center in diamond

Deep defects in wide band gap semiconductors have emerged as leading qubit candidates for realizing quantum sensing and information applications. Due to the spatial localization of the defect states, these deep defects can be considered as artificial atoms/molecules in a solid state matrix. Here we show that unlike single-particle treatments, the multiconfigurational quantum chemistry methods, traditionally reserved for atoms/molecules, accurately describe the many-body characteristics of the electronic states of these defect centers and correctly predict properties that single-particle treatments fail to obtain. We choose the negatively charged nitrogen-vacancy (${\mathrm{NV}}^{-}$) center in diamond as the prototype defect to study with these techniques due to its importance for quantum information applications and because its properties are well known, which makes it an ideal benchmark system. By properly accounting for electron correlations and including spin-orbit coupling and dipolar spin-spin coupling in the quantum chemistry calculations, for the ${\mathrm{NV}}^{-}$ center in diamond clusters, we are able to: (i) show the correct splitting of the ground (first-excited) spin-triplet state into two levels (four levels), (ii) calculate zero-field splitting values of the ground and excited spin-triplet states, in good agreement with experiment, (iii) determine many-body configurations of the spin-singlet states, and (iv) calculate the energy differences between the ground and exited spin-triplet and spin-singlet states, as well as their ordering, which are also found to be in good agreement with recent experimental data. The numerical procedure we have developed is general, and it can screen other color centers whose properties are not well known but promising for applications.

Figure 1

(a) Side view and (b) top view of the ${\mathrm{NV}}^{-}$ center defect in a 70-atom diamond cluster with $C_{3v}$ symmetry. (c) Top view of the ${\mathrm{NV}}^{-}$ center defect in a 162-atom cluster with $C_{3v}$ symmetry. The color scheme is as follows: carbon (cyan), nitrogen (blue), vacancy (gray), hydrogen (pink). Carbon, nitrogen, vacancy, and hydrogen are denoted by C, N, V, and H, respectively. The rotation axis of the threefold symmetry ($C_3$) and the coordinate axes are shown. Here ${\sigma }_1, {\sigma }_2$, and ${\sigma }_3$ indicate vertical mirror planes passing through the carbons nearest to the vacancy with broken dangling bonds (labeled by 1, 2, and 3), the vacancy, and the $z$ axis.

Figure 2

(a) Top view of six defect orbitals (belonging to two $A_1$ and four $E$ IRRep) in the active space with an isosurface value of 0.06 for the CASSCF(6,6) calculation of the 70-atom cluster. The similar six active orbitals are identified for the 162-atom cluster. The LUSCUS program [56] is used for visualization. (b) Nominal distribution of six electrons over the six active orbitals in the ground spin-triplet (${}^3{A}_2$) state. The actual occupation numbers of the $a_1^N, a_1^C, e_x, e_y, e_x^{\prime }, e_y^{\prime }$, are found to be 1.9986, 1.3753, 0.9883, 0.9883, 0.3248, and 0.3248, respectively, from the CASSCF(6,6) calculation. (c) Schematic flowchart of our computational procedure in which the italicized step is discussed in detail in the Appendix.

Figure 3

Schematic level diagrams of the spin-triplet and spin-singlet states for (a) the 70-atom and (b) the 162-atom diamond clusters obtained using the quantum-chemistry method without SOC or SSC. Here full electron correlation within the six molecular orbitals [Figs. 2 and 2] are considered. The experimental values [31, 35, 36, 37, 42] are shown inside parentheses. The experimental zero-phonon absorption energies are marked with $*$. All energy values are given in units of eV.

Figure 4

Comparison of our calculated spin-triplet and spin-singlet energies to the previous many-body theoretical studies [19, 25, 38, 40, 43, 44] as well as the experimental zero-phonon lines (ZPL) and vertical excitations (VE) [31, 35, 36, 37, 42]. The experimental VE are energies of the maximum-intensity peak of the broad phonon side-band absorption spectra. The experimental ZPL and VE energies of the ${}^{1}E$ and ${}^1{A}_1$ states (relative to the ground state) are taken from the midpoint of the experimental range [36] of the separation between the ${}^{3}E$ and ${}^1{A}_1$ states, while keeping the ${}^1{A}_1{–}^{1}E$ energy difference fixed as the experimental value of 1.190 eV [31].

Figure 5

Schematic diagram of our calculated energy level splitting of (a) the first-excited triplet ${}^{3}E$ state and (b) the ground ${}^3{A}_2$ state due to SOC and SSC in units of GHz (for 70-atom cluster). The experimental values [68] are shown inside parentheses. States ${\mathrm{\Psi }}_{1,...,9}$ are defined in Table 4.

Figure 6

Schematic diagram of our practical procedure to identify two extra unoccupied defect orbitals and to preform the CASSCF(6,6) calculations of an ${\mathrm{NV}}^{-}$ center defect in the hydrogen-passivated 70-atom and 162-atom diamond clusters, using openmolcas. Here initial orbitals in the active space are listed within brackets, where nominal occupation numbers for the spin-triplet ground state are shown inside parentheses. The nominal occupation numbers differ from the actual occupation numbers. The orbitals inside double brackets are final converged orbitals. libmsysm is an orbital-symmetrization program [73] and the function of supersymmetry keyword is defined in the text of the Appendix.