Quantum behavior of hydrogen-vacancy complexes in diamond

Hydrogen plays an essential role in the growth process of artificial diamond and can easily form complexes with lattice vacancies. Despite substantial efforts to resolve the electronic structure and the ground-state properties of the hydrogen-vacancy (HV) center, the final remarks are ambiguous, while the complexes of vacancy with two and more hydrogen atoms remain unexplored. In this paper, we used spin-polarized, hybrid density-functional theory method to investigate electronic structure and magneto-optical properties of various hydrogen-vacancy clusters in diamond. Our theoretical results indicate a very strong tendency toward the formation of ${\mathrm{H}}_n\mathrm{V}$ complexes up to four hydrogen atoms that are mostly electrically and optically active centers. One of the investigated defects introduce highly correlated electronic states that pose a challenge for density-functional theory and, therefore, require special treatment when charge- and spin-density-related properties are determined. We introduced an extended Hubbard model Hamiltonian with fully ab initio provided parameters to analyze the complex electronic structure of highly correlated ${\mathrm{H}}_2{\mathrm{V}}^0$ defects. The role of quantum tunneling of hydrogen in HV center and its impact on the hyperfine structure was discussed. We demonstrate that experimentally observed ${\mathrm{HV}}^{1-}$ center is similar to well-known ${\mathrm{NV}}^{1-}$, i.e., I) it possesses triplet ${}^3\mathrm{A}$ ground state and ${}^3\mathrm{E}$ excited state in $C_{3v}$ symmetry; II) the calculated zero-phonon line is 1.71 eV (1.945 eV for ${\mathrm{NV}}^{1-}$). A detailed experimental reinvestigation based on optically detected electron paramagnetic resonance spectroscopy is suggested to verify whether the ${\mathrm{HV}}^{1-}$ center has metastable singlet shelving states between the ground and excited state triplets and, as a result, whether it may exhibit a spin-selective decay to the ground state.

Figure 1

Formation energy of the investigated complexes as a function of Fermi level position in the fundamental band gap of diamond. The Fermi energy level is referenced to the VBM.

Figure 2

The geometric sketch of ${\mathrm{H}}_n\mathrm{V}$ center in diamond. All atoms are numerated in accordance with Table 2.

Figure 3

Adiabatic potential energy surface of the ground and excited states of ${\mathrm{HV}}^0$. The deformation modes of ${\mathrm{HV}}^0$ corresponding to different electronic states are shown as well. The vertical excitation leads to ${}^2\mathrm{A}^{\prime }$ excited state in configuration $Q_3$. A metastable state (nearly degenerate in energy with the ground state) with the same type of distortion was found, which leads to an avoided crossing (the conical intersection) at $C_{3v}$ symmetry point.

Figure 4

Localized Kohn-Sham levels of the investigated complexes in diamond calculated with HSE06 hybrid functional. The light orange shaded area represents the conduction and valence band of an ideal diamond. Spin down (up) channels are indicated by blue (red) triangles, whereas, the filled (unfilled) triangles represent the occupied (empty) states. For the sake of clarity, the corresponding K-S states in different charge states are linked by black dotted lines.

Figure 5

3D representation of the symmetry-broken Kohn-Sham wave functions for highly correlated ${\mathrm{H}}_2{\mathrm{V}}^0$ defect in diamond. The symmetrical $a_1$ and $b_1$ orbitals calculated for the closed-shell singlet ${\mathrm{H}}_2{\mathrm{V}}^{2-}$ are also depicted. The red (blue) lobes indicate the positive (negative) phase of the wave functions with arbitrary selected isosurface value.

Figure 6

HSE06 calculated electronic structure and the lowest energy excitation of highly correlated ${\mathrm{H}}_2{\mathrm{V}}^0$ defect in diamond. I represents the symmetry-broken solution, II represents the self-consistent solution with fixed symmetrical orbitals, III represents the lowest energy optical transition calculated for symmetrical orbitals, and IV represents the non-highly correlated triplet state of ${\mathrm{H}}_2{\mathrm{V}}^0$.