Focusing now on a detailed analysis of the optical band. The PSB of an optical transition is generated by the electron-phonon interactions involved in the transition. In the absence of vibronic interactions, such as the Jahn-Teller effect, the PSB can be described by the linear symmetric mode model [11]. The linear symmetric mode model was applied extensively by [12] in the analysis of diamond color centers. Davies showed how the model could be used to extract the electron-phonon spectral density, referred to as the one-phonon band. Applying critical-point analysis allows this spectral density to be interpreted in terms of structural features of the defect [13]. This is the approach adopted in this section.

Given an emission band *I*_{em}(*ω*), the bandshape function, *I*(*ω*), of a center is proportional to *I*_{em}(*ω*−*ω*_{0})*ω*^{−3}, where *ω*_{0} is the ZPL frequency and *I*_{em} is the emission spectra. The band function is given by

$$I\mathrm{(}\omega \mathrm{)}={e}^{-S}{I}_{0}\mathrm{(}\omega \mathrm{)}\otimes \left[\delta \mathrm{(}\omega \mathrm{)}+{\displaystyle \sum _{n=1}^{\infty}}\frac{{S}^{n}}{n\mathrm{!}}{I}_{n}\mathrm{(}\omega \mathrm{)}\right]$$(4)

where *S* is the total Huang-Rhys factor, *I*_{0} is the ZPL shape, and ⊗ denotes convolution. The function

$${I}_{n}\mathrm{(}\omega \mathrm{)}=\underset{-\infty}{\overset{\infty}{{\displaystyle \int}}}{I}_{n-1}\mathrm{(}\omega -x\mathrm{)}{I}_{1}\text{(}x\text{)d}x$$(5)

is the *n*-phonon band that is constructed by successive convolutions of the one-phonon band. The one-phonon band represents all processes involving the creation and annihilation of a single phonon. Through self-convolutions, it generates the *n*-phonon band *I*_{n}, which describes all *n*-phonon processes with total energy *ħω*.

In theory, one can directly extract the one-phonon band from the band function by applying an inverse Fourier transform to equation (4), rearranging to obtain an expression for *I*_{1}(*t*), and then applying a Fourier transform, $\widehat{\mathcal{F}},$ to obtain

$${I}_{1}\mathrm{(}\omega \mathrm{)}=\frac{1}{S}\widehat{\mathcal{F}}[\mathrm{log}\text{\hspace{0.17em}}\mathrm{(}{\widehat{\mathcal{F}}}^{-1}[I\mathrm{(}\omega \mathrm{)}-{e}^{-S}{I}_{0}\mathrm{(}\omega \mathrm{)}]\mathrm{)}]$$(6)

However, because of the largely featureless PSB of the ST1 center, the large Huang-Rhys factor (*S*=5), and the comparatively low signal-to-noise of our experimental spectra, this direct Fourier deconvolution method is difficult because it is sensitive to numerical and spectral noise. Thus, the direct method is only sufficient to obtain an initial estimate of *I*_{1}(*ω*).

These issues can be overcome by using an iterative deconvolution method developed by Kehayias et al. [13]. We applied this method by first obtaining an approximate one-phonon band from Fourier deconvolution. We then smoothed and tapered this approximate one-phonon band to form our first estimate ${I}_{1}^{0}\mathrm{(}\omega \mathrm{)}$ that was appropriately continuous and restricted to *ω*ϵ[0, Ω], where Ω is the phonon cut-off of diamond. Next, we calculated the normalized PSB components ${I}_{n}^{0}\mathrm{(}\omega \mathrm{)}$ via successive convolutions of ${I}_{1}^{0}\mathrm{(}\omega \mathrm{}\mathrm{)}\mathrm{.}$ We applied the following equation to generate an improved estimate ${I}_{1}^{1}\mathrm{(}\omega \mathrm{)}$ for the one-phonon band:

$${I}_{1}^{k}\mathrm{(}\omega \mathrm{)}={e}^{S}I\mathrm{(}\omega \mathrm{)}-{I}_{0}\mathrm{(}\omega \mathrm{)}-{\displaystyle \sum}_{n=2}^{\infty}\frac{{S}^{n}}{n!}{I}_{0}\otimes {I}_{n}^{k-1}\mathrm{(}\omega \mathrm{)}$$(7)

where *k* is the inductive step index. We then inductively repeated this procedure of calculating the normalized PSB components and generating the next estimate of the one-phonon band until it converged. This processes only required a small number of iterations, and the results are depicted in Figure 5. As can be seen, the generated band function matches the central line of the spectrum very well.

Figure 5: The PSB spectrum from experiment (gray points), a polynomial fit of the spectrum to show its centerline (dashed blue), the calculated PSB (dashed orange), and its constituent *n*-phonon bands.

The sum of *n*-phonon bands equals the calculated PSB.

We now turn to critical point analysis to relate the features of the one-phonon band to structural components of the defect. The key assumption of critical point analysis is that the center does not significantly perturb the phonon modes of pristine diamond. Using this approach features in the one-phonon band corresponds to either frequencies of high mode density and/or where there is strong coupling to the defect orbitals. Figure 6 shows the extracted one-phonon band against the phonon band structure (PBS) and density of states (DOS) of diamond. The absence of spectrally sharp features at frequencies above Ω shows that the optical transition does not couple to local modes; only weak coupling to the continuum modes is present. This validates the key assumption of our application of critical point analysis.

Figure 6: The comparison of the one-phonon band with the DOS and PBS of pristine diamond [14].

The shaded areas connect adjacent points to highlight regions of interest.

As shown in Figure 6, the one-phonon band’s largest feature is a broad peak centered at 60 meV. The prominence of this feature indicates that at this frequency, two things are occurring: a high density of modes and strong coupling to the defect. Indeed, the feature is coincident with the “leveling out” of the transverse phonon bands at the *L*-point, which implies a higher relative density of modes of that phonon type. Furthermore, as the *L*-point lies on the edge of the Brillouin zone, these phonons also result in maximum displacement between equivalent atoms in neighboring unit cells. This implies that for the defect to strongly couple to these modes, its orbitals must be well localized to the nearest-neighbor atoms of a lattice site. This is just like the orbitals of the NV^{−} center in diamond that surround a vacant lattice site.

The plots in Figure 7 show that the one-phonon bands of ST1 and NV^{−} are indeed remarkably similar. Assessment of the critical points of the NV^{−} phonon band by Kehayias et al. [13] showed that it also couples most strongly to phonon modes at the *L*-point. They observed that *L*-point modes would result in the greatest distortion of the electron density localized to the dangling *sp*^{3} orbitals about the center’s vacancy. The strong similarities of the one-phonon bands of ST1 and NV^{−} strongly indicate that the ST1 center contains a vacancy and the orbitals involved its optical transition are highly localized to this vacancy. We use this conclusion to greatly simplify the identification of possible defect structures of the ST1 center.

Figure 7: Comparison of the one-phonon bands of the NV^{−1} [13] and ST1 optical transitions.

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