Origin of structure and zero-phonon-line anomalies of XV centers in diamond (X=Si, Ge, Sn, Pb

Color centers in diamonds have emerged as a promising candidate for quantum information and quantum computing applications. Compared to the well-known and widely studied nitrogen-vacancy ${\mathrm{N}}_{\mathrm{C}}\text{\ensuremath{-}}{\mathrm{V}}_{\mathrm{C}}$ (NV) color center with ${C}_{3v}$ symmetry, the group-IV vacancy color centers ${\mathrm{V}}_{\mathrm{C}}\text{\ensuremath{-}}X\text{\ensuremath{-}}{\mathrm{V}}_{\mathrm{C}}$ (XV, $X=\mathrm{Si}$, Ge, Sn, Pb), exhibit structures with the ${D}_{3d}$ symmetry, which give rise to more stable coherent optical transitions for the zero-phonon line (ZPL) due to its inversion symmetry. Moreover, it is experimentally found that the ZPL peak of XV centers increases from Si to Sn to Ge to Pb, i.e., it does not vary monotonically with the atomic numbers. So far, the physical origin of the unusual local structures and the abnormal trend of ZPL of the XV centers are not well understood. In this paper, based on density-functional theory calculations and symmetry analysis, we demonstrate that the large size of the X atoms plays a dominant role in moving the X atoms away from the substitution site to the bond-center site between the two carbon vacancies to form the ${D}_{3d}$ structure that can effectively reduce the local strain energy. Meanwhile, we find that the abnormal trends of ZPL of the XV centers derive from a competition of the $p\ensuremath{-}p$ coupling and $p\ensuremath{-}d$ coupling between X atoms and the divacancy based on the band-coupling mechanism. Our study, therefore, provides insights into the origin of the abnormal trends of ZPL and the local structure of XV centers in diamonds.