Enhancing the brightness of electrically driven single-photon sources using color centers in silicon carbide

Practical applications of quantum information technologies exploiting the quantum nature of light require efficient and bright true single-photon sources which operate under ambient conditions. Currently, point defects in the crystal lattice of diamond known as color centers have taken the lead in the race for the most promising quantum system for practical non-classical light sources. This work is focused on a different quantum optoelectronic material, namely a color center in silicon carbide, and reveals the physics behind the process of single-photon emission from color centers in SiC under electrical pumping. We show that color centers in silicon carbide can be far superior to any other quantum light emitter under electrical control at room temperature. Using a comprehensive theoretical approach and rigorous numerical simulations, we demonstrate that at room temperature, the photon emission rate from a p–i–n silicon carbide single-photon emitting diode can exceed 5 Gcounts/s, which is higher than what can be achieved with electrically driven color centers in diamond or epitaxial quantum dots. These findings lay the foundation for the development of practical photonic quantum devices which can be produced in a well-developed CMOS compatible process flow. Theoretical simulations show how to improve the performance of silicon carbide single-photon sources by a factor of ten thousand. Atomic-scale defects in silicon carbide can be used to generate single photons, which are an important resource for quantum communication protocols. Dmitry Fedyanin and co-workers from the Moscow Institute of Physics and Technology have developed a numerical model of silicon carbide single-photon emitting diodes. Their calculations accurately reproduced experimental observations on an existing device and allowed them to identify the important factors influencing performance. Based on this knowledge, the authors suggest a new design that should increase the diode brightness by four orders of magnitude. Combined with silicon carbide’s compatibility with existing silicon electronics processes, this makes it a highly competitive candidate for realising a practically useful room temperature, electrically-driven single-photon emitter.


INTRODUCTION
Silicon carbide has been a recognized material for high-power and high-temperature electronics for several decades. 1 At the same time, despite the fact that light emission from semiconductors was for the first time observed from silicon carbide [2][3][4] and 6H-SiC yellow light emitting diodes were serially produced in the USSR in the 1970s, for a long time SiC could not find applications in the optoelectronics of the XXI century. Due to the indirect bandgap and thus low efficiency of light-matter interaction, SiC could not compete with other semiconductor materials. However, recently it attracted great research interest owing to the progress in an entirely different field of research.
After the first studies of color centers in diamond, [5][6][7] it has become clear that point defects in the crystal lattice of dielectrics and wide-bandgap semiconductors can be efficiently used in quantum information technologies. These defects can be created in diverse solid-state structures. At the same time, their optical properties are much closer to the properties of isolated atoms and molecules than those of quantum dots. 8 This gives a unique opportunity to exploit the quantum optics effects at room temperature. 9 Silicon carbide, as well as diamond, can host diverse color centers, such as carbon antisite-carbon vacancy complexes (C Si V C ), [10][11][12] divacancies (V C V Si ), 13,14 silicon vacancies (V Si ), 12,15,16 nitrogen-vacancy (NV) centers, 17,18 silicon antisite-stacking fault complexes (Si C -SF), 19,20 and tiny polytype inclusions. 21 These point defects in the crystal lattice of silicon carbide can be excited optically and emit single photons on demand. [10][11][12]16,20,22 However, the greatest advantage of silicon carbide over diamond and recently emerged 2D materials 23,24 is its semiconductor properties. SiC can be efficiently doped with both donors and acceptors and demonstrate distinct electron and hole conductivity. 1,25 Moreover, SiC devices can be fabricated in a well-developed CMOS compatible process flow. 1 These advantages are very promising for the design and development of practical electrically-driven devices for quantum information and communication systems, particularly for the development of electrically pumped singlephoton sources 26 which can operate at room temperature. Implementation of electrical pumping is essentially important for practical single-photon sources since optically pumped devices are much less energy efficient and are more difficult to be integrated on a chip. 27 Another advantage of silicon carbide is that SiC is an indirect bandgap semiconductor material, which ensures a low background luminescence level due to radiative band-to-band transitions, which are unavoidable in electricallypumped devices at high injection levels. The result of the recent intense research efforts was the demonstration of single-photon emission from color centers in silicon carbide under electrical pumping at room temperature. 19 The measured brightness of these emitters was higher than that of the electrically pumped color centers in diamond [28][29][30] and zinc oxide, 31 as well as quantum dots 32 at room temperature, but lower than that of the optically pumped color centers. 22,33 Further research aimed at developing practical single-photon sources urgently needs an understanding of the physics behind the process of photon emission from a single color center in SiC under electrical pumping and knowledge about how the demonstrated record emission characteristics can be further improved.
Here, we perform a rigorous theoretical and numerical study of single-photon electroluminescence of color centers in a silicon carbide diode and predict the photon emission rate and the correlation between emitted photons. Using a comprehensive computational approach, we perform 2D numerical simulations of the p + -n-n + 4H-SiC single-photon emitting diode. We demonstrate a highly nonlinear dependence of the photon emission rate on the pump current, which can unexpectedly turn into a linear (unsaturated) dependence at high pump currents due to selfheating effects. Our numerical simulations accurately reproduce the triple exponential second-order autocorrelation functions obtained in the experiment in a broad range of injection currents. Finally, we propose and simulate an optimized p-i-n singlephoton emitting diode. This diode can give the possibility to increase the photon emission rate by four orders of magnitude compared to what was observed in the experiments and design a gigahertz single-photon source which does not require cooling and operates under ambient conditions.

RESULTS
Single-photon emitting diode on silicon carbide We focus on the silicon antisite defect (Si C ) in 4H-SiC 34 near or inside the stacking faults (SFs) 35 (hereafter, this defect is referred as the Si C -SF defect). From this defect, bright single-photon emission under electrical pumping was recently experimentally observed. 19,20 The radiative transition in the neutrally charged 36 Si C -SF defect produces a photon polarized in the basal plane of 4H-SiC, 19 which promotes the collection efficiency and gives the possibility to achieve a high brightness of the single-photon source. Despite that most of our calculations and simulations are performed for the Si C -SF defect, we claim that our analysis can be readily used for any electrically pumped color center in silicon carbide since the electrical properties of the system are determined by the SiC crystal rather than by the point defect in the crystal lattice.
Since the process of electroluminescence of the color center essentially involves interaction of the color center with the crystal, namely electron and hole exchange, we start with the electron and hole transport in the p + -n-n + 4H-SiC diode, which contains a color center in the n-type region of the structure (see Fig. 1a). Such a geometry was recently used to design an electrically pumped single-photon source. 19 The n + substrate is doped with nitrogen at a concentration of 7 × 10 19 cm −3 . The concentration of donors in the n-type epilayer is much lower and is equal to 4 × 10 15 cm −3 .
The p-type region obtained by implantation of aluminum in the epilayer features a nonuniform distribution of acceptors, which is quantitatively shown in Fig. 1a. To find densities of electrons and holes, which govern the electron and holes exchange rates between the color center and the SiC crystal, we have performed self-consistent 2D finite difference simulations of carrier transport (details are provided in the Methods section and in Supplementary Methods). Figure 1b, c shows the distribution of electrons and holes in equilibrium and at high injection levels. Whereas in equilibrium there are no holes in the n-type region, the density of holes reaches 10 14 -10 15 cm −3 at high injection current densities at the top contact (hereafter, referred simply as the injection current density). At the same time, electrons start to penetrate deep into the p-type region, as the bias voltage increases (see Fig. 1b). In Fig.  1b, c, we can introduce an additional region. In this transition region, there are no electrons and holes in equilibrium. At the same time, at high injection levels, one can simultaneously create high densities of both electrons and holes in this region. Such a region should always exist in p-n junctions, and its position and width depend on the doping profile. As shown below, the transition region is crucially important for achieving a high rate of single-photon emission. The electron and hole exchange rates between the color center and the semiconductor crystal are determined by the electron and hole capture processes and are therefore governed by the densities of the corresponding charge carriers.
Physics of the electroluminescence process Similar to the NV center in diamond, 37 the Si C -SF defect in 4H-SiC has at least two charge states. 19,36 This requirement is essential for electroluminescence since the process of electron and hole exchange is accompanied by the change in the charge of the color center. One of these states is positively charged, and one is neutral (see Fig. 2a). In principle, a negatively charged state can also exist (Fig. 2a), making the process of single-photon electroluminescence more complicated (for details, see Supplementary Methods). However, theoretical studies of charge states of the Si C defect (D I center) in 4H-SiC, 34 which has a similar structure, 36 do not reveal the presence of such a state. 38 Since the photoluminescence measurements do not show a noticeable difference between defects in the n-type and p-type silicon carbide, 19 we can conclude that the defect has a deep-donor-like nature and, consequently, it is not ionized in either n-type or ptype silicon carbide unless the dopant concentration reaches 10 20 cm −3 . Therefore, the observed single-photon photoluminescence spectrum can be attributed only to the transitions between the ground and excited states of the neutrally charged Si C -SF center (see Fig. 2a). Figure 2a shows the schematic diagram of the single-photon electroluminescence for the Si C -SF center. In the steady state, the process is cyclic. Therefore, let us consider the Si C -SF center in the neutral ground state |g 0 > right after photon emission. The color center in the neutral ground state can either capture a hole from the valence band and become positively charged or capture an electron from the conduction band and become negatively charged. Since according to the DFT studies, 38 the negatively charged state does not exist, the Si C -SF center can capture only a hole. Here, we assume that there is only a one energy state |g + > in the positively charged Si C -SF center. The hole capture rate can be found as c p p, where p is the density of holes in the vicinity of the color center. The hole capture rate constant c p is given by c p = σ p υ p , where υ p is the average thermal velocity of holes in silicon carbide, and σ p is the capture cross-section by the neutral Si C -SF center. σ p is estimated to be about 5 × 10 −15 cm 2 . 39, 40 We should note that in principle, a hole can be captured into some excited state and then experience relaxation to the ground state |g + >. However, such nonradiative relaxations are typically orders of magnitude faster than the hole capture rate 41 and can therefore hardly be noticed in the electroluminescence measurements. The electroluminescence studies do not indicate the existence of radiative transitions among excited and ground levels of the positively charged Si C -SF center. 19 Hence, only one energy state | g + > in the positively charged Si C -SF center can be considered.
The positively charged center attracts negatively charged electrons and can capture one of them from the conduction band. This process changes the charge of the Si C -SF center from positive to neutral (|g + > → |e 0 > transition in Fig. 2a). The electron capture rate is equal to the product of the electron capture constant c n and the density n of free electrons in the vicinity of the color center. c n = σ n υ n , where σ n is the electron capture crosssection, and υ n is the average thermal velocity of free electrons. The electron capture cross-section can be found as follows. The attractive potential of the color center forms a set of discrete bound levels, which can be populated by electrons from the conducting band. To be captured, an electron should lose an energy larger than k B T, otherwise, it will be released back to the conduction band by thermal excitation. According to the cascade capture model, 42,43 the energy is lost due to interaction with acoustic phonons. In each act of the electron-photon interaction, the electron loses only a small portion of its kinetic energy, which defines the cascade nature of the process (see Fig. 2b). The electron capture cross-section σ n can be found as a ratio of the volume of the Thomson sphere with radius r T to the energy relaxation length due to electron-phonon interaction: 42 σ n = 4πr T 3 /(3l 0 ). The energy relaxation length l 0 = lk B T/(2m eff s 2 ) is significantly larger than the electron mean free path l since only a small fraction of the electron kinetic energy is dissipated in each act of phonon emission. Here, s is the speed of sound in 4H-SiC and m eff is the effective electron mass. For the transition region in Fig. 1b, c, where we expect the highest photon emission rate from the color center, σ n = 6.1 × 10 −15 cm 2 at room temperature.
In the next stage, the excited state of the neutral center relaxes to the ground state via radiative (|e 0 > → |g 0 >) and nonradiative transitions (|e 0 > → |s 0 > → |g 0 >) (see Fig. 2a for details). After that, the Si C -SF center appears in the neutral ground state |g 0 > and can repeat the cycle of the electroluminescence process.
Following Fedyanin and Agio 37 and neglecting thermal excitations, we can quantitatively describe the populations of the ground and excited states of the neutral and positively charged Si C -SF centers: In this system of equations, n and p are the electron and hole densities in the vicinity of the color center, respectively, n e , n s , and n g are the populations of the excited, shelving and ground states of the neutral defect, and n + is the population of the positively charged state. 1/τ r , 1/τ nr , 1/τ s are the transition rates between the excited, shelving, and ground states of the neutral center. From the photoluminescence measurements, 19 τ r and τ nr are estimated to be 3.6 ns and 36 ns, respectively. The lifetime of the excited state equals τ e = 1/(1/τ r + 1/τ nr ) = 3.3 ns. The lifetime of the shelving state estimated from the experimentally measured g (2) functions 19 is found to be about 35 ns. For the complete list of (1) in the steady state, we can find the single-photon emission rate: where n ss e denotes the steady-state population of the excited state of the neutral defect. Expression in the square brackets contains two terms. The first term represents electrical pumping, while the second is responsible for the transitions from the excited to the ground state. It can be clearly seen that the long-lived shelving state can substantially affect the photon emission rate, but only at very high injection levels.
Single-photon emission upon electrical injection Using the results of the electron and hole transport simulations, we can quantitatively evaluate the photon emission rate from the Si C -SF center in the silicon carbide diode depicted in Fig. 1a. Figure  3a shows the dependence of the photon emission rate on the position of the Si C -SF defect in the p + -n-n + SiC diode at a high injection level. The emission rate depends strongly on the distance from the edge of the p + region. The simulations demonstrate that only defects in a narrow region can give bright single-photon emission, which was observed in the experiment. Remarkable is that the width of this region is about 150 nm and it coincides with the transition region shown in Fig. 1b, c. As follows from Eq. (2), bright photon emission requires the density of both electrons and holes to be significantly high. This requirement can be satisfied only in the transition region, which is clearly seen in Fig. 3b, which shows the ratio between the electron and hole capture rate. In this figure, one can identify two areas, where either the electron capture (blue region) or hole capture (yellow region) process is slow. In between these areas, the electron and hole capture rates are nearly equal to each other. Figure 3c shows the numerically simulated and experimentally measured dependences of the single-photon emission rate on the pump current. One can see that the plotted curves are nonlinear, and the emission rate is not directly proportional to the pump current. We should emphasize that this nonlinearity is not due to saturation. The photon emission rate is indirectly related to the pump current via the carrier densities. In turn, the carrier densities are not directly proportional to the injection current, as in most semiconductor devices. Since the position of the bright color center in the experiment is not known exactly, 19 we can allow a slight variation to fit the experimental data better (Fig. 3c). At low pump currents, one can see good quantitative agreement between the theoretical (red curve) and experimental (dotted Fig. 3 a Dependence of the single-photon emission rate from the Si C -SF center on its position within the SiC diode at a pump current density of 1.84 A/cm 2 , which corresponds to the maximum current of 14 mA in the experiment. 19 b The ratio of the electron capture rate to the hole capture rate at a pump current density of 1.84 A/cm 2 . c Photon emission rate as a function of the current density at points marked in panel (a). The experimental curve is retrieved from ref. 19 and corrected for the detection efficiency (53%) and collection efficiency (estimated to be 1.5%, which is slightly lower than the value reported in ref. 19 ). (d) g (2) curves retrieved from ref. 19 and numerically simulated for the color center at point 2 in panel a Enhancing the brightness of electrically driven... IA Khramtsov et al. curve) results. However, at high pump currents, the discrepancy cannot be ignored. The theory holds that the emission rate starts to "saturate" at high currents, while in the experiment the emission rate is linearly proportional to the pump current.
To understand the nature of the difference between theory and experiment, we use additional information provided by the measurements of the correlation between photons. 44 Figure 3d shows the simulated g (2) curves (see Methods section) at moderate and high pump currents, which are plotted next to the experimental data points. First, we note that the theoretical curves ideally reproduce the triple exponential dependences, which were observed in the single-photon electroluminescence measurements. 19 Lohrmann et al. 19 demonstrated that the presence of the fourth high-energy level can explain such a triple exponential dependence. Here, we show that this fourth state, in fact, corresponds to the ground state of the positively charged Si C -SF center rather than to any excited state of the neutral color center. At the same time, we have the same number of transitions among different states of the color center in the proposed electroluminescence model as in the conventional four-level model. The pair of the electron and hole capture processes can be considered as a two-step process that brings the color center from the neutrally charged ground state to the neutrally charged excited state (see Fig. 2a). In contrast, the one-step excitation process in the case of optical pumping stipulates the double exponential dependence in the photoluminescence measurements of Si C -SF centers. 19 Second, at a pump current density of 0.7 A/cm 2 (as well as at lower currents), the agreement between the theoretical and experimental g (2) curves is as good as between the corresponding emission rates. However, at a current of 1.8 A/ cm 2 , one can observe a significant difference in the g (2) traces, which is even higher than the difference in the emission rates (see Fig. 3c).
Self-heating of the single-photon emitting diode We attribute the observed discrepancy in the photon emission rates and g (2) curves between theory and experiment to the change in the properties of the single-photon emitting diode at high currents since our simulations accurately reproduce the experimentally measured data at low injection levels. As the pump current increases, the power dissipated by the diode increases, which gives rise to the device temperature. This was indirectly confirmed in the experiment by the observed displacement (probably caused by thermal expansion) of the color center at high pump currents. 38 Therefore, we proceed with the evaluation of the impact of self-heating on the photon emission rate and the dynamics of single-photon emission. Heat is generated due to the nonradiative recombination in the bulk of the diode and at the top and bottom contacts. 45 Due to the high thermal conductivity of SiC, 39 the temperature distribution across the sample should be homogeneous. The net heating power is approximately equal to the power JV supplied to the diode. In turn, the temperature rise is proportional to the heating power. At high injection levels, the diode operates well above the turn-on voltage 19 and the heating power increases proportionally to the current squared. The temperature rise significantly increases the electron and hole densities due to the relatively high activation energies of donors (nitrogen) and acceptors (aluminum) in SiC. The hole capture rate constant c p also increases due to the increase in the hole thermal velocity. At the same time, the temperature dependence of the electron capture rate constant c n is more complicated. Whereas the electron thermal velocity increases as T 1/2 , the electron capture cross-section is inversely proportional to~T 2 . Thus, c n decreases with temperature increase. Nevertheless, our numerical simulations in a wide temperature range show that the temperature rise increases the brightness of the single-photon emitting diode (Fig. 4a), which is also confirmed by the reduction of the characteristic time of the g (2) function (Fig. 4b). The results are consistent with our assumption that the temperature rise is proportional to the pump current squared: at a current density of 1.2 A/cm 2 the heating power is twice lower than at 1.84 A/cm 2 , so is the temperature rise with respect to the ambient temperature. The numerically simulated curves (Fig. 4) nearly ideally reproduce the experimentally measured data demonstrating that the developed theoretical model accurately describes the photophysics of electrically pumped Si C -SF centers.
Towards an electrically driven gigahertz single-photon source The experimentally demonstrated characteristics of the SiC singlephoton emitting diode are, to the best of our knowledge, superior to any electrically driven system at room temperature. However, they still should be improved to meet the requirements of modern and future quantum technology applications. Particularly, for practical secure quantum communications, the photon emission rate should be in the gigahertz range. Can this be achieved with electrical control?
The photon emission rate is limited by three factors: (1) the lifetime of the excited state, (2) the quantum efficiency and the lifetime of the shelving state, and (3) the electron and hole capture rates. The radiative lifetime of the excited state of the neutrally charged Si C -SF center, the lifetime of the shelving state and quantum efficiency virtually limit the maximum emission rate to 0.1 Gcounts/s as follows from Eq. (2). However, the radiative lifetime τ r can be significantly reduced using an optical Fig. 4 a Photon emission rate versus pump current for the color center at point 2 in Fig. 3a. The red curve is simulated taking into account the self-heating effect, while the grey curves are isothermal curves at 300, 320, 340, 360, and 380 K (the grey arrow shows the direction of temperature increase). The experimental data are the same as in Fig. 3c. b g (2) curves retrieved from ref. 19 and numerically simulated for the color center at point 2 in Fig. 3a Enhancing the brightness of electrically driven... IA Khramtsov et al. nanoantenna, [46][47][48][49][50] which increases the rate of radiative transitions due to the Purcell effect. In addition, the nanoantenna approach can give the possibility to improve the collection efficiency by more than an order of magnitude. 16 The Purcell factor of about 50 can be relatively easily achieved in most nanoantenna geometries, which reduces the radiative lifetime of the excited state to 70 ps. Therefore, we only need to increase the carrier capture rates to (1-10) × 10 9 s −1 , which, however, cannot be done straightforwardly.
Silicon carbide is a wide-bandgap semiconductor, and consequently, the activation energies of donor (~0.05 eV for nitrogen on hexagonal sites and~0.10 eV on cubic sites) 51 and acceptors (~0.19 eV for aluminum) 52 are much higher than in silicon or gallium arsenide. For this reason, the densities of electrons and holes are about 30 times lower than the concentration of the doping impurities. Moreover, in the active region of p-n and p-i-n diodes, where it is possible to create a high density of both electrons and holes, the densities of carriers are 100-10,000 times lower than in the p-and n-type regions. The higher the doping level, the higher the difference. Our simulations show that in the studied geometry (see Fig. 1a), the photon emission rate is limited to about 70 Mcounts/s at current densities below 5 A/cm 2 at room temperature.
To significantly increase the photon emission rate, we need to change the design of the single-photon emitting diode. First, the doping level should be increased. In this case, the density of free carriers in the p-and n-type regions increase not only due to the increased dopant concentrations but also due to the reduction in the activation energies of dopants. 52,53 Second, the size of the lightly doped region should be decreased, since the density of holes rapidly decreases with the distance from the p-type region due to recombination in the active region and the potential barrier at the p-i-n junction. Similarly, the density of electrons decreases with the distance from the n-type region. Therefore, the p-n structure is preferred over the p-i-n structure. However, in this case, bright photon emission can be obtained only from a color center placed in a very narrow region at the p-n junction. To avoid this limitation, we propose a p-i-n diode with a graded doping profile in the narrow i-region, which can be achieved by both CVD growth and ion implantation (Fig. 5a). Figure 5b shows the dependence of the maximum photon emission from the Si C -SF center in the proposed diode configuration. One can see that at a current density of 5 A/cm 2 , the carrier capture processes limit the photon emission rate to about 13 Gcount/s, which is above the limit imposed by the radiative lifetime τ r of the excited state in a bulk material. However, using the Purcell effect in optical nanoantennas, τ r can be reduced by a factor of [46][47][48][49]54 50-200 and a photon emission rate of about 5-10 Gcounts/s is practically achievable. Here, we should emphasize that the operating temperature is approximately equal to 300 K, since the heating power cannot noticeably increase the device temperature if the size of the contact is less than 20 μm. 45 Finally, we note that our theoretical studies demonstrate that the photon emission rate might be further increased to above 15 Gcount/s using a properly designed donor doping profile. However, in this case, one needs novel nanoantennas with record characteristics, which can reduce the radiative lifetime of the excited state of the Si C -SF center by a factor of 1000.

DISCUSSION
In this work, we have presented a theoretical description of emission properties of electrically-pumped Si C -SF color centers in silicon carbide. The combined study of transitions between states of the Si C -SF center and the electron and hole transport in silicon carbide have revealed the photophysics of the color center electroluminescence. It has been shown that the photon emission process is mostly determined by the electron and hole capture processes. Therefore the obtained results can be applied with no change to any color center in 4H-SiC which has neutral and positively charged states, since the electron capture rate by the positively charged defect is determined by the properties of the semiconductor material (sound velocity, electron mobility, and density of free electrons) rather than by the internal structure of the defect. At the same time, the hole capture rate by the neutral defect depends on the defect structure. Nevertheless, for most defects, the capture cross-section is roughly equal to the lattice constant of silicon carbide. 39,40 We also extended our model to color centers with three charge states (negative, neutral, and positive), see Supplementary Methods for details. With minimal changes, this model can be applied to most color center in silicon carbide, such as the C Si V C center. Here we should emphasize that multiple charge states, such as (+2), (+3) or (−2) states, cannot be efficiently excited electrically. The problem is that the singlycharged defect is an attractive center for one type of carriers and is a repulsive center for the other type of carriers. For example, the positively charged defect attracts electrons and repels holes. The capture cross-section for electrons is in the range from 10 −15 to 10 −13 cm 2 , while the capture cross-section for holes is roughly 4-5 orders of magnitude lower. Hence, the photon emission rate from a multiple charge state is expected to be much lower than from a single-charged/neutral state. Nevertheless, silicon carbide can be easily doped with both donors and acceptors, and high densities (>10 18 cm −3 ) of free electron and holes can be created, 52,53 which provides favorable conditions for electrical pumping of even multiple charge states. This property gives silicon carbide a distinct advantage over diamond (a well-recognized host material for single color centers). Since it is not possible to create a high density of both electrons and holes in diamond under ambient conditions, 37,55 color centers in diamond cannot be as bright as in silicon carbide. Our numerical simulations based on the developed theoretical model accurately reproduce the experimental data and explain the recently observed features of silicon carbide single-photon emitting diodes, which include the distinct triple exponential behavior of the second-order autocorrelation function and the linear (unsaturated) dependence of the emission rate on the pump current at high injection levels. We found that the latter is caused by diode self-heating, which cannot be ignored at high pump currents. Similar to diamond, 37 due to the high activation energy of acceptors in silicon carbide, the temperature rise increases the density of free carriers and consequently the photon emission rate. However, the effect is not that strong as in diamond and heating cannot be used to increase the brightness by orders of magnitude. At the same time, self-heating can ensure a linear (unsaturated) growth of the emission rate in a broad range of pump currents despite that the dependence of the carrier densities, which govern photon emission, on the pump current is highly nonlinear. Excellent quantitative agreement between theoretical and experimental curves demonstrates that the developed theoretical approach can be efficiently used in the development of novel single-photon sources. Our theory predicts that as soon as the lifetime of the excited state is decreased below a few tens of picoseconds (which can be achieved with an optical cavity or a nanoantenna), the photon emission rate is governed only by the electron and hole capture processes by the color center. We demonstrate that by using a properly designed p-i-n diode structure and optical cavities, the brightness can be increased by more than four orders of magnitude compared to what was observed in the experiments. Our theoretical findings show that the photon emission rate can be as high as 10 Gcounts/ s at room temperature, which makes silicon carbide single-photon emitting diodes highly advantageous over other single-photon devices for diverse practical quantum optics applications, such as unconditionally secure data communication based on quantum cryptography.

Simulation of the single-photon emitting diode
The 2D numerical simulations of the silicon carbide diode were performed using the self-consistent steady-state model, which comprises Poisson's equation for the electric field and carrier densities, semiconductor driftdiffusion equations for electrons, and holes and carrier continuity equations. These differential equations supplemented with the appropriate boundary conditions at the top and bottom ohmic contacts were solved using the nextnano++ software (for details, see Supplementary Methods). The single-photon emission rate and the second-order autocorrelation function were calculated using the obtained carrier density distributions and the developed theoretical framework.
Evaluation of the g (2) function of the electrically pumped color center The second-order autocorrelation function g (2) (τ) of the Si C -SF center in 4H-SiC was calculated by solving Eq. (1) with the initial conditions n g = 1, n e = n s = n + = 0. 44 The obtained temporal dynamics of the Si C -SF center is related to the g (2) function as g ð2Þ ðτÞ ¼ n e ðτÞ=n ss e ; where n ss e is the steadystate population of the excited state of the neutrally charged center.

Code availability
The nextnano++ software used in the simulations is available at http:// www.nextnano.de/customer/download.php.
Data availability