We spectroscopically investigate micrometer sized multi-layer flakes of hBN in a home-built laser scanning confocal microscope under continuous wave excitation at λ=532 nm. Experiments are performed at room temperature if not indicated otherwise. The commercially available flakes (Graphene Supermarket, Graphene Laboratories Inc., Ronkonkoma, NY, USA) are diluted in a solution (50% water, 50% ethanol) with a concentration of 5.5 mg/ml and put in an ultrasonic bath to break up agglomerates. The solution is drop cast (5–10 μl) onto a silicon wafer with an iridium layer for enhanced photon collection efficiency. The substrate is heated on a hotplate to 70°C to evaporate the liquid. After drop casting, individual flakes can be examined in the confocal microscope.

Figure 1A, D and G show typical spectra of point emitters inside the flakes. Although they differ in their central wavelengths, their spectral shapes are very similar. The spectra are fit with four Lorentzian lines which we will discuss later in closer detail. Saturation measurements in Figure 1B, E and H show typical saturation count rates (≈1–2 Mcts/s) and saturation powers (≈1 mW) of these emitters, in good agreement to previous reports [8], [9], [10]. The red lines are fits according to

Figure 1: Photoluminescence spectra and single photon emission from individual defects in hBN.

(A, D, G) Typical spectra of three defects (E1, E2 and E3) in hBN consisting of four Lorentzian lines; (B, E, F) Saturation measurements on the defects from A, D and G with no significant background contribution; (C, F, I) *g*^{(2)}-intensity correlation measurements on the defects shown in A, D and G. Photons are collected from the spectral regions enclosed in dashed lines in A, D and G, respectively. See main text for details.

$$I\mathrm{(}P\mathrm{)}=\frac{{I}_{\text{sat}}\cdot P}{{P}_{\text{sat}}+P}+{C}_{\text{back}}\cdot P\mathrm{.}$$(1)

Here, *I*_{sat} and *P*_{sat} are the saturation count rates and saturation powers of the emitters, whereas *C*_{back} describes a potential contribution due to linear background emission stemming from the host material. Note that this contribution is negligible in the presented data. This is in accordance with the very clean spectra presented in Figure 1A, D and G, where also no significant background contribution is visible. In approximately one out of 50 flakes background-free emission can be found. Contrary to these findings, Figure 2A) shows a spectrum, which clearly contains additional background emission. This background emission is also visible as a prominent linear increase in a corresponding saturation measurement in Figure 2B. Note that the saturation measurements are always taken by integrating the emission intensity over the ZPL and the two sidebands (cf. the four Lorentzian lines in Figure 1).

Figure 2: Photoluminescence spectra and single photon emission from individual defects in hBN with strong and vanishing background emission, respectively.

(A) Spectrum of an emitter in hBN (E4) with a clear background contribution. (B) Saturation measurement on the emitter in A. The background contribution is visible as prominent linear increase in the emission rate at increasing excitation powers. (C) *g*^{(2)}-function on an hBN emitter (E3) showing typical bunching timescales of several hundreds of microseconds. (D, E, F) Spectrum, saturation measurement and *g*^{(2)}-function of an emitter (E5) with a clean spectrum. Background contribution becomes relevant at about 20×*P*_{sat}. Still, *g*^{(2)}(0) is strongly limited even at almost vanishing excitation powers.

As a last step, we perform *g*^{(2)}-photon correlation measurements (Figure 1C, F and I to get information about the photon statistics. In order to reduce the potential influence of background emission (although not present in the current experiments) on the photon statistics measurements, we restrict the spectral window from which we collect photons for the *g*^{(2)}-measurements to the region of the ZPL (regions enclosed by dashed lines in the corresponding spectra) and take the measurements at excitation powers far lower than the emitters’ saturation powers. Despite vanishing background fluorescence, the *g*^{(2)}-functions do not vanish at all at zero time delay as one would expect for an ideal single photon source. Figure 2D–F further shows an example, where background emission from the host material is present but becomes relevant only at about 20×*P*_{sat}. Nevertheless, for almost vanishing excitation power (*P*=3.5 μW), the value of *g*^{(2)}(0) is still much larger than zero. As we show below also the timing jitter of the photon detectors does not explain the deviation from ideal single photon statistics as the emitter fluorescence lifetime is larger than the jitter. Instead, we have to assume that the asymmetric shape of the ZPL is due to the presence of two independent emission lines.

In the following we develop a model for the photon correlation functions that, besides background emission and the timing jitter of the photon detector, accounts for the presence of a second emitter and prove that this model fully reproduces the measurements. We start with the well-known *g*^{(2)}-function for a three-level system:

$${g}_{i}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau \mathrm{)}=1-\mathrm{(}1+a\mathrm{)}\cdot {e}^{-\frac{\mathrm{|}\tau \mathrm{|}}{{\tau}_{1}}}+a\cdot {e}^{-\frac{\mathrm{|}\tau \mathrm{|}}{{\tau}_{2}}}$$(2)

We now, step by step, include all experimental parameters that influence the shape of the *g*^{(2)}-function: Although negligible in the presented data (but not in general), we start with uncorrelated background emission, that can be extracted from saturation measurements. Including this into the model, the *g*^{(2)}-function reads [28]

$${g}_{p}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau \mathrm{)}=\frac{1}{{p}^{2}}\cdot [{g}_{i}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau \mathrm{)}-\mathrm{(}1-{p}^{2}\mathrm{)}]$$(3)

Here, *p* is the fraction of measured photons stemming from the emitter compared to the measured total count rate. Note, that one should also consider dark counts of the detector in the description. In our case, these dark counts (≈100–200 cts/s) are negligible compared to the signal from the emitters. Second, we include the timing jitter σ of the counting electronics. This jitter is an uncertainty in the time between the arrival and the detection of a single photon and has been measured via ultra-fast laser pulses (σ≈490 ps). It is included via the convolution of Eq. (3) with the Gaussian shape of the instrument response function IRF(t).

$${g}_{p\mathrm{,}j}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau \mathrm{)}=\text{IRF}\mathrm{(}\tau \mathrm{)}\ast {g}_{p}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau \mathrm{)}={\displaystyle {\int}_{-\infty}^{\infty}}\text{IRF}\mathrm{(}\tau \mathrm{)}\cdot {g}_{p}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau -t\mathrm{)}dt$$(4)

Equation (4) is the final description for the case that we collect emission from exactly one single emitter. The blue solid lines in Figure 1C, F and I are fits to the data according to this model. It strikes the eye that this function is not able to reproduce the data. In particular, the model demands a much lower value for *g*^{(2)}(0) than it is provided by the data. We want to stress that we also can reproduce the data by taking the signal to background ratio *p* as a fit parameter. This, however, strongly contradicts our findings of vanishing background in the spectrum and the saturation measurement.

Therefore, as a last step, we also take into account the influence of a second emitter in the detection focal volume. Let *I*_{tot}=*I*_{1}+*I*_{2} be the total detected emission with *I*_{1}=*z*·*I*_{tot} and *I*_{2}=(1–*z*)·*I*_{tot} being the relative fractions of the emission of emitter 1 and emitter 2, respectively. This leads to

$\begin{array}{l}{g}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau \mathrm{)}=\frac{\u3008{I}_{\text{tot}}\mathrm{(}t\mathrm{)}{I}_{\text{tot}}\mathrm{(}t+\tau \mathrm{)}\u3009}{{\u3008{I}_{\text{tot}}\mathrm{(}t\mathrm{)}\u3009}^{2}}\\ ={z}^{2}\cdot {g}_{1}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau \mathrm{)}+\mathrm{(}1-z{\mathrm{)}}^{2}\cdot {g}_{2}^{\text{(2)}}\mathrm{(}\tau \mathrm{)}+\underset{{g}_{\text{mix}}^{\text{(2)}}}{\underbrace{\frac{\u3008{I}_{2}\mathrm{(}t\mathrm{)}{I}_{1}\mathrm{(}t+\tau \mathrm{)}\u3009}{{\u3008{I}_{\text{ges}}\mathrm{(}t\mathrm{)}\u3009}^{2}}+\frac{\u3008{I}_{1}\mathrm{(}t\mathrm{)}{I}_{2}\mathrm{(}t+\tau \mathrm{)}\u3009}{{\u3008{I}_{\text{ges}}\mathrm{(}t\mathrm{)}\u3009}^{2}}}}\mathrm{.}\end{array}$

In order to reduce the number of fit parameters, we assume ${g}_{1}^{\text{(2)}}\mathrm{(}\tau \mathrm{)}={g}_{2}^{\text{(2)}}\mathrm{(}\tau \mathrm{}\mathrm{)}\mathrm{.}$ Because of the independence of *I*_{1} and *I*_{2}, the two mixing terms will be constant for all τ and by making the assumption that ${g}_{1}^{\text{(2)}}\mathrm{(}0\mathrm{)}={g}_{2}^{\text{(2)}}\mathrm{(}0\mathrm{)}=0$ (corresponding to the observed vanishing background), we find ${g}_{\text{mix}}^{\text{(2)}}=2z\mathrm{(}1-z\mathrm{}\mathrm{)}\mathrm{.}$ We eventually arrive at

$${g}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau \mathrm{)}=\mathrm{(}1-2z\mathrm{(}1-z\mathrm{)}\mathrm{)}{g}_{p\mathrm{,}j}^{\mathrm{(}2\mathrm{)}}\mathrm{(}\tau \mathrm{)}+2z\mathrm{(}1-z\mathrm{)}\mathrm{.}$$(5)

In contrast to reports in literature, where the asymmetry of the ZPL is attributed to phonon interaction [8], we here fully reproduce the lineshape by fitting two Lorentzian lines representing two independent electronic transitions. By calculating the areas under the individual Lorentzians, we get information about the relative oscillator strengths of both emitters, corresponding to the parameter *z* in Eq. (5) (numbers also given in the spectra in Figures 1 and 2). By taking into account the double emission spectrum within the model (Eq. (5)) for the *g*^{(2)}-function, we are able to perfectly describe the measured photon correlation data (solid red lines in Figures 1C, F, I and 2C and F).

Interestingly, our photon correlation measurements correspond perfectly to reports in literature in terms of bunching dynamics and dips in the *g*^{(2)}-function at zero time delay [8], [9], [10]. Non-vanishing values of *g*^{(2)}(0) in these reports were always attributed to residual background fluorescence which, however, is not further defined or shown. To our knowledge, the full set of information needed to accurately describe the situation has never been reported [8], [10], [29], [30]. Furthermore, we want to point out that most of the emitters measured in this work show very strong bunching on a timescale of several hundreds of microseconds up to milliseconds as it has been shown in previous work (see, for example, Figure 2C) [9], [10]. Therefore, a proper normalization of the *g*^{(2)}-function to the constant number of events for long time delays τ or to the recorded photon count rates is imperative. The absence of satisfactorily explanations in literature and the excellent agreement of measured photon correlation functions with the double defect model suggest that most probably the majority of *single* emitters in literature are indeed double defects.

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