Ab initio supercell calculations on nitrogen-vacancy center in diamond: Electronic structure and hyperfine tensors

The nitrogen-vacancy center in diamond is a promising candidate for realizing the spin qubits concept in quantum information. Even though this defect has been known for a long time, its electronic structure and other properties have not yet been explored in detail. We study the properties of the nitrogen-vacancy center in diamond through density functional theory within the local spin density approximation by using supercell calculations. While this theory is strictly applicable for ground state properties, we are able to give an estimate for the energy sequence of the excited states of this defect. We also calculate the hyperfine tensors in the ground state. The results clearly show that (i) the spin density and the appropriate hyperfine constants are spread along a plane and unevenly distributed around the core of the defect and (ii) the measurable hyperfine constants can be found within about $7\mathrm{\AA }$ from the vacancy site. These results have important implications on the decoherence of the electron spin which is crucial in realizing the spin qubits in diamond.

Figure 1

(Color online) The NV center viewed in perspective (left) and along the ⟨111⟩ direction (right); this direction is the $C_3$ symmetry axis of the $C_{3v}$ symmetry group of the defect. The vacant site is indicated by a small pink circle and the neighboring C and N atoms by gray and cyan balls, respectively. The first neighbor atoms around the vacant site have $s{p}^3$ dangling bonds pointing toward the vacant site, which are labeled ${\sigma }_i$ $(i=1,4)$, as in the scheme used in the group theory analysis.

Figure 2

(Color online) The calculated spin-resolved single-electron levels with respect to the valence band maximum (VBM) in the ground state of the NV defect. Valence and conduction bands of the host crystal are shown as blue and pink shaded regions, respectively. The levels are labeled as in Eq. (1) and their occupation is given for a negatively charged defect (a total of six electrons).

Figure 3

(Color online) Isosurfaces of the calculated wave functions of the $a_1\left(1\right)$, $a_1\left(2\right)$, $e_x$, $e_y$ defect states, shown in side (left) and top (right) views relative to the ⟨111⟩ axis. Blue (red) isosurfaces correspond to negative (positive) values of the wave function. The small pink circle represents the position of the vacant site, while the gray and cyan balls show the C and N atoms, respectively. We show the atoms up to the second neighbor from the vacant site. Results are from the SIESTA calculations.

Figure 4

(Color online) Calculated spin density isosurfaces in the $M_S=1$ state for the ${}^{3}A_2$ (left) and the ${}^{3}E$ state (right). The vacant site is depicted by a small pink sphere at the center of each plot. Results are from the VASP calculations.

Figure 5

Energetics of photoluminescence absorption: VI is the vertical ionization energy, ZPL is the zero-phonon line transition, $E$ is the total energy, and $Q$ is the configuration coordinate.

Figure 6

(Color online) (a) The 512-atom cubic supercell with all C atoms shown (atoms within a 64-atom supercell are shown in red). The supercell size in angstroms and in units of the conventional cubic cell lattice constant $a_0$ is indicated on the side of the cube. (b) Perspective view of the calculated spin density isosurfaces in the $M_S=1$ state. Only atoms up to the second nearest neighbor of the vacant site are shown. (c) View along the (111) direction indicating the $C_{3v}$ symmetry of the spin density, which is given in colored contours. The black lines denote the size of the 512-atom supercell, while the red dotted ones show the boundaries of the 64-atom supercell.

Figure 7

(Color online) Variation of the principal values of the hyperfine tensors $A_{11}$ and $A_{33}$ as a function of the distance from the vacant site $\left(R_{\mathrm{vac}}\right)$ for the set of three symmetrically equivalent C atoms (3C). In the inset, we show changes farther from the vacant site on a finer scale. The variation for the set of six symmetrically equivalent C atoms closely follows the one shown for 3C.