Photoionization of negatively charged NV centers in diamond: Theory and ab initio calculations

We present ab initio calculations of photoionization thresholds and cross sections of the negatively charged nitrogen-vacancy (NV) center in diamond. We consider photoionization from the ground ${}^{3}A_2$ and the excited ${}^{3}E$ states. After the ionization from the ${}^{3}E$ level, we show that the NV center transitions into the metastable ${}^{4}A_2$ electronic state of the neutral defect. We reveal how spin polarization of ${\mathrm{NV}}^{-}$ gives rise to spin polarization of the ${}^{4}A_2$ state, explaining electron spin resonance experiments. We obtain smooth curves of photoionization cross sections as a function of energy by employing dense $k$-point meshes for the Brillouin-zone integration together with the band unfolding technique to rectify the distortions of the band structure induced by the artificial periodicity of the supercell approach. Our calculations provide a comprehensive picture of photoionization mechanisms of ${\mathrm{NV}}^{-}$. They will be useful in interpreting and designing experiments on charge-state dynamics at NV centers. In particular, we offer a consistent explanation of recent results of spin-to-charge conversion of NV centers.

Figure 1

(a) Atomic structure of the nitrogen-vacancy center in diamond. (b) Electronic level diagram of ${\mathrm{NV}}^{-}$. Energies of the ZPL between the triplets and the singlets are indicated. Spin splitting of ${}^{3}A_2$ and ${}^{3}E$ states are not to scale.

Figure 2

Band structure of bulk diamond calculated using the HSE density functional (see Sec. 3 for details about computational methods). The shaded area corresponds to the band gap. The photoionization threshold is determined by electrons being excited to the CBM that occurs between $\mathrm{\Gamma }$ and $X$ points.

Figure 3

Photoionization of the ${\mathrm{NV}}^{-}$ center in the single-electron picture. (a) Electronic configuration of the $m_s=1$ spin sublevel of the ${}^{3}A_2$ state. (b) Electronic configuration of the $m_s=1$ spin sublevel of the $E_x$ component of the ${}^{3}E$ manifold. Red arrows show one possibility of photoionization whereby an $e_y$ electron is excited to the conduction band (see the text for a more in-depth discussion).

Figure 4

Photoionization of ${\mathrm{NV}}^{-}$ from ${}^{3}A_2,{}^{3}E$, and ${}^{1}E$ states. Horizontal lines indicate the energy of the entire system: black for ${\mathrm{NV}}^{-}$ and blue for ${\mathrm{NV}}^0$ plus an electron at the CBM. Red arrows indicate possible photoionization mechanisms, and $E_{\mathrm{ZPL}}$ is the ZPL energy of the triplet transition.

Figure 5

Spin physics of the photoionization from the ${}^{3}E$ state. Numbers near arrows show relative probabilities of the transition during photoionization. The $m_s=+1$ ($-1$) spin sublevel transitions into either the $m_s=+3/2(-3/2)$ or the $m_s=+1/2(-1/2)$ sublevel of the ${}^{4}A_2$ manifold with different probabilities. The $m_s=0$ sublevel transitions to the $m_s=\pm 1/2$ sublevels with equal probabilities. Spin sublevels are separated by zero-field splittings $D\left({}^{3}E\right)=1.42\mathrm{GHz}$ [1] and $D\left({}^{4}A_2\right)=1.69\mathrm{GHz}$ [40].

Figure 6

(a) Unfolded band structure of conduction-band states, perturbed by the NV center, along the $\mathrm{\Gamma }–X$ path. The color indicates a relative spectral weight (dark blue is zero); see Ref. [43] for more details. (b) The band structure of bulk diamond folded to the first Brillouin zone of the $4\times 4\times 4$ supercell along the $\mathrm{\Gamma }\text{−}X$ path of the supercell.

Figure 7

Photoionization cross sections from (a) the ${}^{3}A_2$ and (b) the ${}^{3}E$ state of ${\mathrm{NV}}^{-}$. Blue lines: cross sections ${\tilde{\sigma }}_{\mathrm{ph}}\left(\epsilon \right)$ without vibrational broadening [Eq. (3)]; red lines: actual cross sections ${\sigma }_{\mathrm{ph}}\left(\epsilon \right)$ [Eq. (4)]; dashed lines show ${\tilde{\sigma }}_{\mathrm{ph}}$ calculated using a constant momentum matrix element in Eq. (3) and DOS corresponding to a parabolic band (see text). The insets show the spectral function of electron-phonon coupling $A\left(\epsilon \right)$.

Figure 8

Calculated cross section as a function of photon energy. Solid blue: photoionization from the excited-state ${}^{3}E, {\sigma }_{\mathrm{ph}}$; dark red: stimulated emission, ${\sigma }_{\mathrm{st}}$; orange: intradefect absorption, ${\sigma }_{\mathrm{intra}}$; dashed blue: photoionization from the singlet state ${}^{1}E$. Photoionization thresholds from ${}^{3}E$ and ${}^{1}E$ are indicated (estimated error bar 0.1 eV), together with the experimental values of the ZPL energy for ${\mathrm{NV}}^{-}$ and ${\mathrm{NV}}^0$.

Figure 9

The ratio of the photoionization cross section and the cross section for stimulated emission ${\sigma }_{\mathrm{ph}}/{\sigma }_{\mathrm{st}}$ as a function of photon energy.