Vibrational modes of negatively charged silicon-vacancy centers in diamond from ab initio calculations

Silicon-vacancy (SiV) center in diamond is a photoluminescence (PL) center with a characteristic zero-phonon line energy at 1.681 eV that acts as a solid-state single-photon source and, potentially, as a quantum bit. The majority of the luminescence intensity appears in the zero-phonon line; nevertheless, about 30% of the intensity manifests in the phonon sideband. Since phonons play an essential role in the operation of this system, it is of importance to understand the vibrational properties of the SiV center in detail. To this end, we carry out density functional theory calculations of dilute SiV centers by embedding the defect in supercells of a size of a few thousand atoms. We find that there exist two well-pronounced quasilocal vibrational modes (resonances) with $A_{2u}$ and $E_u$ symmetries, corresponding to the vibration of the Si atom along and perpendicular to the defect symmetry axis, respectively. Isotopic shifts of these modes explain the isotopic shifts of prominent vibronic features in the experimental SiV PL spectrum. Moreover, calculations show that the vibrational frequency of the $A_{2u}$ mode increases by about 30% in the excited state with respect to the ground state, while the frequency of the $E_u$ mode increases by about 5%. These changes explain experimentally observed isotopic shifts of the zero-phonon-line energy. We also emphasize possible dangers of extracting isotopic shifts of vibrational resonances from finite-size supercell calculations, and instead propose a method to do this correctly.

Figure 1

The geometry sketch of the SiV center. The localized vibration modes $A_{2u}$ (motion of Si atom parallel to $\langle 111\rangle$ axis, left figure) and $E_u$ (motion of Si atom perpendicular to $\langle 111\rangle$ axis, right figure) are shown. These quasilocal vibration modes have significant localization on the six neighbor carbon atoms that move in the opposite direction to that of Si atom in the corresponding vibrations.

Figure 2

Inverse participation ratios [Eq. (1)] of $A_{1g}$ symmetry modes for different supercell sizes as a function of energy. Systems with $N=3,4,5,6,7,8,9$ (size of a cubic supercell, a number next to each graph) correspond to bulk supercells containing $M=8N^3$ atoms. There are no quasilocal modes of $A_{1g}$ symmetry.

Figure 3

(a) Inverse participation ratios [Eq. (1)] for $A_{2u}$ symmetry modes for different supercell sizes as a function of energy. (b) Same for $E_u$ modes. Supercells the same as in Fig. 2.

Figure 4

(a) Localization ratios [Eq. (3)] for $A_{2u}$ symmetry modes for different supercell sizes as a function of energy. Localization ratios are calculated from the data in Fig. 3, and results from different supercells are superimposed on top of one another. The dashed line and shaded region serve as guides to the eye. (b) Same for $E_u$ modes. Note that the $y$ axis has logarithmic scale.

Figure 5

Localization ratios [Eq. (3)] for the $A_{2u}$ symmetry quasilocal modes around energy 43 meV for three Si isotopes. Calculations performed for the $5\times 5\times 5$ supercell.

Figure 6

Frequencies of $A_{2u}$ and $E_u$ vibrational resonances [Eq. (4)] as a function of supercell size $N$. “Bare value” (black circles) is the frequency for a given supercell. “$M$ average” (red squares) and “$N$ average” (blue triangles) are weighted averages, as described in the main text. Horizontal dashed line corresponds to the extracted energy of the resonances.

Figure 7

(a) Localization ratios [Eq. (3)] for $A_{2u}$ (a) and $E_{2u}$ (b) symmetry modes on the excited state for different supercell sizes as a function of energy. The dashed line and shaded region serve as guides to the eye. The $y$ axis has logarithmic scale.