Donor-Acceptor Pair Quantum Emitters in Hexagonal Boron Nitride

: Quantum emitters are needed for a myriad of applications ranging from quantum sensing to quantum computing. Hexagonal boron nitride (hBN) quantum emitters are the most promising solid-state platform to date due to its high brightness, stability, and the possibility of spin photon interface. However, the understanding of the physical origins of the single-photon emitters (SPEs) is still limited. Here, we present concrete and conclusive evidence that the dense SPEs in hBN, across entire visible spectrum, can be well explained by donor-acceptor pairs (DAPs). Based on the DAP transition generation mechanism, we have calculated their wavelength fingerprint, matching well with the experimentally observed photoluminescence spectrum. Our work serves as a step forward for the physical understanding of SPEs in hBN and their applications in quantum technologies.


Introduction
Layered van der Waals materials have received much attention not only for their novel optoelectronic properties 1 but also for the capability of hosting a wealth of optically active defects that act as single-photon emissions (SPEs) 2,3 . In particular, SPEs from two-dimensional (2D) hBN exhibit stable, bright, and efficient emission into the zero-phonon line (ZPL) even at ultra-high temperatures [3][4][5][6][7][8][9][10][11] . Furthermore, the spin information of these emitters serves as a promising platform for fundamental research and potential applications in quantum technologies 12,13 . However, the origin and the nature of the h-BN SPEs remain under debate [14][15][16][17] . The emission spectra of both intrinsic and engineered defects in hBN show broadband spectral range, distinct polarization profiles as well as different quantum efficiencies 9,14,15,17 . Several bottomup growth techniques as well as post-processing approaches such as ion implantation or electron irradiation have been aimed at identifying the sources 12,13,[18][19][20] . In addition, multiple theoretical calculations show that different types of atomic defects may exist, including nitrogen or boron vacancy complexes, antisite defects, or substitutional defects with carbon or oxygen 14,15,17,[21][22][23][24] , but a conclusive evidence for these prediction results is still lacking.
As one of the fluorescence mechanisms, the donor-acceptor pair (DAP) transition between the ionized donors and acceptors has been studied in traditional semiconductors, including silicon 25,26 , silicon carbide 27,28 , diamond 29 and other compound semiconductors [30][31][32][33] . Generally, the DAP transition is characterized by a series of sharp PL lines with a broadband wavelength range at low temperatures [26][27][28][29] . In particular, by designing the emitters with a well-defined donor-acceptor distance and orientation in covalently linked organic molecules, it can enable a promising avenue for manipulation and coherent control of spin states 34,35 . Recently, the DAP induced emission has been theoretically predicted in hBN 36 .
Here, we present experimental evidence for the existence of DAP quantum emitters in hBN. At low temperatures, we observe dense sharp photoluminescence (PL) lines over a broad range of wavelengths from blue (~ 440 nm) to red (~ 800 nm) color. Based on the DAP transition mechanism, we used DAP model to calculate the emission lines, matching the experimental results very well. We also give possible origin of DAP by comparing fitting results and density function theory (DFT) calculations. Our work indicates that the DAP transition mechanism may explain the broad distribution of observed quantum emitters in hBN and serve as a foundation for designing scalable quantum devices.

The DAP transition in hBN
The DAP transition describes a fluorescence mechanism that includes the Coulomb interaction between the ionized donors and acceptors. Considering the nature of deep levels in the wide bandgap of semiconductors (e.g., hBN), the DAP transition process can be simply described as following: the electrons (holes), created by laser excitations, can be trapped by the ionized donor sites (D + ) (or the neutral A 0 ) and acceptor sites (A -) (or the neutral D 0 ) before the radiative recombination and the photon emissions, as shown in Fig. 1(a). In general, two types of DAP transitions exist considering the random distribution of D + and Asites: Type 1, where both donors and acceptors are located at the same atomic element species, as illustrated by the red arrows in Fig. 1(b); and Type 2, where the donors and acceptors occupy different atom sites as indicated by the blue arrows in Fig. 1(b). This radiative recombination of DAPs in semiconductors can generate a series of sharp lines with a broadband distribution in the PL spectra, as reported for other semicondcutors [25][26][27][28][30][31][32][33] .   Fig. S1. Interestingly, we find that the spectrum shows dense sharp emission lines, and covers a wide energy range from 2.80 eV (443 nm) to 1.544 eV (803 nm), confirmed on multiple samples (Fig. S1). Furthermore, these lines also show a narrow linewidth down to ~75 µeV, as shown in Fig. S1, which is close to the narrowest BN emitter linewidth currently reported 8 . Fig. 1(d) and Fig. S2 shows the (") ( ) measurement results of two randomly selected PL lines, which prove that they are single-photon emitters. We also measured the stability of PL lines at different emission wavelengths and found that they are stable and bright, as shown in Fig. S3. Based on the DAP transition mechanism, we use theoretical DAP model below to calculate the emission lines. In this model, the electrons, instead of relaxing from the valance band to donors or acceptors as reported in previous studies, such as in GaP [31][32][33] , relax between defect levels within the large bandgap of hBN. Thus the energy of the emitted photons can be written as 31 : where, EA and ED are the donor and acceptor levels, respectively. Rm is the m-th (m=1, 2, 3, ...) nearest distance between the donors and acceptors as determined by the crystal structure and lattice constant, and m is the shell number, and ' ! ()** " + # describes the Coulomb interaction between the donor and acceptor. The discrete Rm in the Coulomb term is responsible for the observed sharp lines in the emission spectra. For large Rm, the lines are closely located and barely distinguishable, resulting in a broad peak. Fig. 2(a) shows the calculated spectral distributions and the zoom-in result for type 1 DAPs based on five different | $ − % | values (other zoom-in results are shown in Fig. S4 in SI). To further confirm the rationality of the model, we used a series of values that slightly deviate from one | $ − % | to calculate the DAP lines, as shown in Fig. S5, and found that these calculated lines can not match well with the experimental lines, especially for the small m numbers. Moreover, we also found that these calculated DAP lines based on the same | $ − % | values match well with the experimental results from other samples, as shown in Fig. S6. These results indicate that these sharp hBN PL lines here are indeed from the DAP process. Figure 2(b) shows the calculated emitted-photon energies of the DAP as a function of Rm between the donors and acceptors. We found that the calculated results match most of the PL lines. Remarkably, we found that ~70% percent of the theoretically calculated emission wavelengths match precisely the experimental results, as shown in Fig. 2(c), indicating the truth of the DAP mechanism here. The unmatched lines can be attributed to the inhomogeneous distribution of the defects and the multipole term 27 . In addition, the crystal structure of the hBN sample may be deformed with strains, which can also lead to small deviations 32 . Meanwhile, some of the emission lines may originate directly from the color centers without DAP effects. As shown in Fig. S7, our results suggest that both two types of DAP transitions may coexist in our hBN samples due to the random doping. The lifetimes of different DAP lines are also measured as shown in Fig. S8, which become longer as the shell number increases. These results can be understood by considering that the overlap between electrons and holes in DAP is reduced with Rm increase and thus, the lifetime is slightly changed 32 . The weak lifetime dependence on shell number can be understood by considering the deep level nature in hBN, and small varied Rm (0.7 nm to 1.6 nm) here, as well as the relatively large exciton Bohr radius (on the order of a few nanometers) of Wannier exciton type in 2D materials 1 . More detailed discussions can be found in the SI.
a n t i ( E ) energy levels). For example, C , (-) denotes a C impurity substituting for a lattice B atom and half-filled with electrons. Anti-site denotes B . and N , , V , and C / denote boron vacancy and carbon interstitial, respectively. (c) The |ED-EA| of different type1 DAPs from our fitting result (red circle) and DFT calculation (black square).
To understand the origin defect of DAP, we calculated the electronic energy band structures of the bulk hBN and its possible defect states (Fig. 3) based on density function theory (DFT). At 4 K, the electron-phonon coupling is weak (almost no phonon sideband is observed in our spectra) and lattice relaxation hardly affects the calculation of defect energy levels, thus we did not consider the lattice relaxation 14 . The calculated band gap of bulk hBN is 5.98 eV, which is consistent with previous experimental results 37 . We consider all possible transitions based on DFT calculated level of defects, i.e. electron transition from full-filled or half-filled level to half-filled or empty level, and get the transition energy by the energy difference between these levels. Comparing with the fitting results, we list the possible origin of DAPs in the DAP transition mechanism in hBN provides a new way to understand the dense SPEs in layered semiconductors and opens a new way to produce single-photon sources for quantum technology.

Temperature dependence of the DAP lines
Next we investigate the behavior of DAP lines at higher temperatures. These dense SPE lines become weaker at elevated temperatures, as shown in Fig. 4(a). When the temperature increases to around 160 K, some of the dense peaks vanish and cannot be individually resolved from each other. This result may be explained by considering the thermal effects on the fine structures and the binding energies of the ground states of the donor/acceptor. At a higher temperature, phonons broaden the luminescent lines, i.e. when the energy difference between two peaks is smaller than the linewidth induced by thermal perturbation at the corresponding temperatures, they cannot be distinguished. On one hand, the thermal energy of the crystal may free the originally paired donor and acceptor with a large Rm and reduce the number of peaks at higher temperatures. On the other hand, with temperature increasing, the thermal energy is big enough to approach the binding energies of the DAP. As a result, the probability of DAP transitions is greatly reduced and other mechanisms such as the transition between the excited states and the ground states of defects dominate the fluorescence process.
We fit one representative PL peak from 4 K to 160 K, and plotted the temperature dependence of the extracted linewidths and ZPL positions, as shown in Fig. 4(b). Because of the phonon effects in higher temperatures, the lineshape shows a change from Gaussian shape to Lorentzian shape, consistent with our following analysis. At low temperatures, the PL ZPL is usually dominated by inhomogeneous broadening induced by spectral diffusion, exhibiting a Gaussian shape; while as the temperature increases, homogeneous broadening induced by phonons starts to dominate leading to Lorentzian shape at higher temperatures 38,39 . From 4 K to 160 K, the linewidths show a T 3 temperature dependence, consistent with the previous results 9, [40][41][42] . We find most of these DAP PL lines follow such a temperature dependence, as shown in Fig. S11.
Furthermore, most of the ZPL positions can be well described by using the O'Donnell equation 43 . It suggests the electron-phonon coupling plays an important role in the broadening and red-shift of the DAP lines. Here Eg(0), S and <ℏω> is around 1.99 eV, 0.20 and 24.61 meV, respectively.

Conclusions
In conclusion, we demonstrated that the DAP transition mechanism could be one of the origins for the wide spectral viability of the SPEs in hBN. These results indicate that SPE on demand can be obtained by filtering the PL lines induced by DAP transition or designing the distance of the donor-acceptor pair. Our work provides a fundamentally new understanding of the hBN SPEs and opens a door to achieve supercontinuum SPS on one monolithic hBN sample. Besides, it also hints that DAP mechanism should be an alternative way to produce quantum emitters in other semiconductors with efficient DAP emission, such as silicon, silicon carbide, diamond and other compound semiconductors.

Experiment Methods:
Photoluminescence (PL) measurement: PL spectra measurements were undertaken in backscattering geometry with a Jobin-Yvon HR800 system equipped with a liquid-nitrogen-cooled charge-coupled detector. The samples are cooled by the Montana cryostat system. A 50x long-working-distance objective lens (NA=0.5) and both 600 and 2400 lines mm -1 gratings were used for the PL measurements at low temperatures. The highest resolution for our system with 2400 lines mm -1 grating is around 40 meV.
The second-order correlation function measurement: The second-order correlation function measurement is carried out by using a home-built Hanbury-Brown-Twiss (HBT) setup. Two silicon Avalanche Photo Diodes (APD) are used to count photons. The 590 nm bandpass and 690 nm bandpass filters combined with a grating monochromator (1200 nm lines mm -1 ) are used during the autocorrelation measurements.