Integrated photonics platforms are crucial to the development and implementation of scalable quantum information and networking schemes, but many such devices still rely on external bulk photodetectors. We report the design and simulation of a waveguide-based single-photon avalanche diode (SPAD) for visible wavelengths. The SPAD consists of a p-n junction implemented in a doped silicon waveguide, which is end-fire coupled to an input silicon nitride waveguide. We developed a 2D Monte Carlo model to simulate the avalanche multiplication process of charge carriers following the absorption of an input photon, and calculated the photon detection efficiency (PDE) and timing jitter of the SPAD. We investigated the SPAD performance at a wavelength of 640 nm and temperature of 243K for different device dimensions and device doping configurations. For our simulated parameters, we obtained a maximum PDE of 0.45 at a reverse bias voltage of ~20 V, and full-width-half-max (FWHM) timing jitter values <8 ps.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Quantum communication between nodes in a scalable quantum network is one of the key challenges in developing practical quantum technologies of the future [1, 2]. Recent developments based on CMOS-compatible material platforms highlight the remarkable promise of a compact, low-cost and mass-manufacturable quantum photonics device technology [3–5]. However, state-of-the-art demonstrations still require the coupling of light to external photodetectors. For a truly integrated photonics platform, these quantum devices need to be combined with on-chip single-photon detectors.
Geiger-mode single-photon avalanche diodes (SPADs) are photodetectors operating in reverse bias above the breakdown voltage so as to achieve single-photon sensitivity. In SPAD operation, incident single photons can trigger an avalanche current which is used for detection. Apart from quantum communications, a high detection efficiency, low jitter single-photon detector is desirable in a wide range of photon starved applications, e.g. lidar [6, 7], non-line-of-sight imaging , fluorescence medical imaging , and STED microscopy .
An on-chip waveguide-coupled single-photon detector would be an important component of a scalable integrated photonics platform. However, most modern SPADs are free-space-coupled, large area devices where the photon arrives at normal incidence to the absorbing silicon material [11, 12]. Waveguide-integrated superconducting nanowire single-photon detectors (SNSPDs) have shown high detection efficiency (90%) at a low dark count (few Hertz) and low timing jitter of 20 ps [13, 14]. However, SNSPDs have to be operated at cryogenic temperatures of a few degrees Kelvin, which is below the critical temperature of the superconducting material. Maintenance of cryogenic temperatures is energy intensive, expensive and not amenable to scalable architectures. On the other hand, conventional SPADs typically only require Peltier cooling and can even operate at room temperature [15, 16]; this offers many practical advantages in system implementation.
In this paper, we design a silicon waveguide-coupled Geiger-mode SPAD, on a silicon nitride (Si3N4) on silicon-on-insulator (SOI) platform, suitable for quantum photonics applications at visible wavelengths. Moreover, it uses CMOS-compatible materials and thus can be readily incorporated into more complex silicon photonic integrated circuits, leveraging on mature fabrication techniques and an optimized set of component devices [17–19]. To model the SPAD operation, we first perform a DC electrical simulation to extract the relevant electrical parameters, e.g. electric field, carrier mobilities, ionization coefficients, and breakdown voltage. Subsequently, we use a Monte Carlo technique based on a random path length (RPL) model [20–23] to simulate the avalanche process in the SPAD, and obtain the photon detection efficiency (PDE) and timing jitter. We study a series of devices with variations of waveguide width and doping levels.
2. Waveguide-coupled SPAD design
2.1. Device geometry
The SPAD is based on a SOI platform, and consists of a silicon rib waveguide end-fire coupled to an input Si3N4 rectangular waveguide (see Fig. 1). Silicon is a suitable photodetector material in the visible range due its high absorptivity, while Si3N4 has high transmittivity and a moderately high refractive index . The structure is cladded with 3 µm of silicon dioxide above and below. The length of the silicon waveguide is 16 µm, with an absorption of >99% at 640 nm. For ease of fabrication, we fix the height of the silicon and Si3N4 waveguide layers at 250 nm and the rib height at 125 nm; and vary the width W between 450–900 nm in our study.
2.2. Doping profile
The choice of doping profile is essential in determining the performance characteristics of the SPAD. In surface illuminated configurations, the location and orientation of the diode junction, as well as the presence of a thick intrinsic region, is used to enhance the absorption length and to target specific wavelengths of light. This often comes at the cost of a reduced wavelength detection range or increased timing jitter [12, 24]. In a waveguided configuration, we have no need for such a trade-off since the absorption length is determined by the length of the waveguide. We design our SPAD to consist of a p-n+ diode with a single continuous depletion region along the length of the waveguide (see Fig. 1(b)).
In silicon, electrons have a higher ionization coefficient than holes; this results in a more efficient avalanche multiplication process for electrons . We hence aim to have photo-generated electrons (rather than holes) drift into the avalanche region of the diode in order to increase the PDE. In a p-n+ junction, the depletion region extends largely into the p-doped side, where the dominant charge carriers are electrons. To maximize the spatial overlap between the large depletion region on the p-doped side and the optical waveguide mode, the p-n+ junction is placed asymmetrically within the waveguide, with p- and n+- doped regions having a width ratio of roughly to 5:1.
We choose a nominal n+ (p) doping concentration of 1×1019 (2 × 1017) dopants/cm3, drawing on experience from previously published literature [11, 24, 26]. The depletion width within a 1D approximation is:
Electrical connections to the device are made via metal contacts deposited on top of heavily-doped p++ and n++ regions.
2.3. Optical coupling
State-of-the-art integrated photodetectors for infrared wavelengths typically use a phase-matched interlayer transition to couple light from the input waveguide to the detector [17, 27, 28]. However, this is difficult to achieve in our SPAD due to the large difference in refractive indices for silicon (n = 3.8) and Si3N4 (n = 2.1). Thus, we choose to end-fire couple the input Si3N4 waveguide to the silicon rib waveguide in the same layer.
The optical modes in the silicon and Si3N4 waveguides have different shapes (see Fig. 2(a) and 2(b)). To investigate the dependence of optical coupling on the widths of the silicon waveguide W and the Si3N4 waveguide WSiN, we perform 3D Finite Difference Time Domain (FDTD) simulations using Lumerical software (see Fig. 2(c)).
For a fundamental (quasi-)TE mode at 640 nm in the input Si3N4 waveguide, near-optimal coupling to the fundamental TE mode of the silicon waveguide can be obtained by choosing W = WSiN, with coupling loss <1 dB for widths below 1 µm. Choosing the two waveguides to be of the same width also simplifies device fabrication. The remaining power is mostly reflected, with minimal excitation of other waveguide modes (< 25 dB for W = WSiN). As such, we only consider the fundamental TE mode of the silicon waveguide at 640 nm for subsequent Monte Carlo simulations.
3. DC electrical analysis
Using the device dimensions and doping profile, we perform a DC electrical analysis of the SPAD (ATLAS device simulator, Silvaco Inc.) to obtain the electric field E and the charge carrier mobilities (µe, µh), drift velocities (ve, vh), and ionization coefficients (αe, αh), where the subscripts e, h denote electron and hole, respectively. These position-dependent parameters are required for the Monte Carlo simulation of the avalanche process. We also obtain the breakdown voltage Vbr by analysing the current-voltage characteristics, and identifying Vbr as the reverse bias voltage at which the current increases sharply.
As the geometry and doping profile are constant along the length of the silicon waveguide, a full 3D simulation is not necessary, and we simplify the calculations by considering only the 2D waveguide cross-section. We use ATLAS’s built-in graduated meshing system with a maximum element size of 5 × 5 nm in the waveguide core region (given by the width W and the 250 nm height) and minimum element size of 1 × 1 nm in a ~200 nm region around the diode junction.
Within ATLAS, we use the Masetti  model with the phosphorus parameter set in conjunction with the Canali  model to obtain doping- and field-dependent mobilities and drift velocities. Amongst the available mobility models in ATLAS, we use the Masetti model as it is built upon the most updated empirical mobility data, and provides more detailed information at high dopant concentrations (we simulate devices with up to 1019 dopants/cm3) . Ionization coefficients are obtained from the Selberherr impact ionization model , which only considers local electric fields; the dead space effect of impact ionization modeling is taken into account within the RPL model in the Monte Carlo simulator. The implementation details of these models can be found in the ATLAS manual .
Conventional SPAD modules are typically Peltier-cooled to suppress dark noise. While we are unable to simulate dark noise, we expect that the small active volume of our waveguide SPAD will translate to lower dark count rates compared to free-space-coupled large-area devices [33,34]. We conservatively assume that our SPAD will be operated at similar temperatures to conventional ones, and perform our analysis at 243 K. We also note that dark count rates also depend significantly on the excess bias voltage and hence a comprehensive experimental study will be needed to make any conclusions on improvements on dark count rate for our device.
4. 2D Monte Carlo simulator
4.1. Initial motivations
Besides PDE, timing jitter is also a critical performance indicator of single-photon detectors for applications such as quantum key distribution (QKD) [35, 36] and lifetime measurements of single-photon sources . Timing jitter arises from the stochastic nature of the impact ionization and avalanche buildup process, and cannot be modeled by deterministic simulators . In contrast, Monte Carlo simulators can capture the probabilistic outcome of individual avalanches, and thus can evaluate the timing performance through statistical analysis of repeated simulation runs.
In our Monte Carlo device simulator, the Random Path Length (RPL) model is used to simulate the avalanche multiplication process [20, 22, 23]. Briefly, an absorbed photon gives rise to an electron-hole pair. Each charge carrier is accelerated by the electric field, and may cause an impact ionization after traveling a certain path length, creating further electron-hole pairs until the device current crosses a detection threshold.
Given the probabilistic nature of the RPL model, charge carriers under the same conditions may or may not cause an impact ionization; if it does, the path length traversed until the impact ionization is also random. Overall, this leads to a distribution of random detection times (i.e. time between the initial photon absorption to when the detection threshold is crossed), from which we obtain the detection timing uncertainty, i.e. the timing jitter. The initial photo-generated charge carriers may also fail to cause sufficient impact ionizations to trigger a self-sustaining avalanche process and reach the detection threshold; we thus regard the PDE as the ratio of successful detection events to the total number of simulation runs.
The most efficient way to simulate the RPL model would be to use a 1D simulator, considering only electric fields normal to the p-n+ junction (Ex), as had been done in previous SPAD simulation work [20, 22, 23]. However, the geometry of the rib waveguide with a shallow height results in significant orthogonal Ey field components (see Fig. 3). As such, minority charge carriers (electrons and holes in the p and n+ regions, respectively) near the device edges are accelerated outwards of the silicon waveguide, potentially leading to their loss from the SPAD. A 1D simulator would not take this into account, and thus likely overestimate the PDE. On the other hand, a full 3D simulation is not necessary, given the symmetry along the length of the silicon waveguide. Therefore, we choose to implement a 2D Monte Carlo simulator.
While the DC electrical analysis considers the silicon waveguide as well as the surrounding material, we restrict the Monte Carlo simulation to only the waveguide core region, as the contribution of charge carriers is negligible outside this area.
Each simulation run models the absorption of a single input photon, the injection of a photo-generated electron-hole pair at that location, and the subsequent avalanche process. The main steps are:
- Defining the depletion and quasi-neutral regions.
- Absorbing an input photon and injecting a photo-generated electron-hole pair.
- Simulating charge carrier diffusion in the quasi-neutral regions.
- Simulating avalanche buildup with the RPL model.
For each device setting (device geometry, doping levels, and bias voltage), we repeat the simulation runs until we obtain sufficient statistics. The following subsections describe the implementation of the simulator in detail.
4.2.1. Defining the depletion and quasi-neutral regions
The SPAD can be divided into the central depletion region, and the p- and n-type quasi-neutral regions at the sides of the silicon waveguide. The different electric field strengths in these regions give rise to different charge carrier transport dynamics. The strong electric fields in the depletion region result in the drift along the electric field being the dominant transport process, with diffusion being negligible . In the quasi-neutral regions, diffusion governed by Brownian motion has to considered alongside the drift force under the weaker electric fields, while impact ionization can be ignored.
Although the transition between these regions would not be abrupt in an actual device, we follow Ref.  and use a threshold electric field to distinguish the high-field depletion region from the low-field quasi-neutral regions. We choose a threshold field of Ethr = 1 × 105 V/cm, on the same order as the breakdown field in silicon . The transport of charge carriers in different regions is simulated differently, as detailed in the following subsections.
4.2.2. Absorbing an input photon and injecting a photo-generated electron-hole pair
The probability of a photon absorption event at any position is proportional to the amplitude of the input optical field. We use the spatial profile of the optical mode as a probability density map to determine the injection point of the photo-generated electron-hole pair for each simulation run.
If the injection point is in the depletion region, the simulation proceeds straight to step 4. For the quasi-neutral regions, we proceed to step 3. These photo-generated charge carriers are injected with zero initial energy.
4.2.3. Simulating charge carrier diffusion in the quasi-neutral regions
For the photo-generated electron-hole pair injected into the quasi-neutral regions, we immediately remove the majority carrier from the simulation as it would travel towards the metal electrodes and not towards the depletion region. Thus, we only consider the movement of the minority charge carrier (electron and hole in p and n+ regions, respectively) of the photo-generated electron-hole pair. We model its Brownian motion as a series of random walk steps, and additionally factor in the drift due to electric fields at each step. Impact ionization is not considered here.
It is not practical to simulate every collision event; instead, for simulation efficiency, we choose a nominal step size Δdrw, treating each step as a “macrocollision”, i.e. a group of true collisions :
The time duration of this Brownian motion displacement is
The total displacement of each random walk step is then
The simulation continues until the charge carrier reaches the depletion region edge (as defined by the threshold field), where the simulation of the charge carrier continues under the RPL model described below; or when it is lost from the SPAD through the other edges of the waveguide core, and the simulation run ends.
4.2.4. Simulating avalanche buildup with the RPL model
In the depletion region, charge carriers accumulate energy as they are accelerated by the electric field. Above the ionization energy threshold, they can probabilistically cause an impact ionization, creating a new electron-hole pair. The repetition of this process leads to an avalanche of charge carriers, giving rise to a macroscopic current that indicates a photon detection event.
The simulation is carried out in time steps of duration Δtrpl = 1 fs. In every step, we consider individually the evolution of each charge carrier i having an initial position ri,0 and a current position ri. The displacement under the electric field drift is
The charge carrier traverses a random ionization path length before causing an impact ionization. To simulate this, each charge carrier is assigned a random number Xi from a uniform distribution between 0 and 1. We define the dead space ds,i as the total distance traveled by the charge carrier when its energy ϵi crosses the ionization energy threshold ϵthr,e and ϵthr,h for electrons and holes, respectively (see Table 1). Above the threshold, the charge carrier accumulates an incremental ionization probability along its path with each time step, depending on the ionization coefficients αe(ri) and αh (ri). The cumulative probability that the charge carrier has caused an impact ionization is [22, 42]
Impact ionization occurs when Pion,i > Xi, generating a new electron-hole pair at the current location ri, and the energy ϵi is redistributed among the three charge carriers as follows [43, 44]. For an impact ionization caused by an electron, the electron and hole energies are,Table 1.
For simplicity, we do not consider the effects of random walk within the RPL model, even if the charge carriers travel to the quasi-neutral regions as defined previously. Charge carriers that exit the waveguide core are considered to be lost from the SPAD and removed from the simulation. We also assume that the charge carriers are not lost via recombination, as the carrier lifetime is >10 ns for our doping concentrations, while the transport time across the sub-micron depletion width under high electric fields is below 1 ns [38, 46].
We obtain the device current using Ramo’s theorem from the contributions of each charge carrier :45]. The simulation ends with a successful detection event when I > Idet; otherwise it ends with a failed avalanche when the number of charge carriers drops to zero, or when the simulated time exceeds 1 ns.
4.2.5. Repeating the simulation to obtain sufficient statistics
For each device setting, we perform ≈ 20 k simulation runs to obtain adequate statistics for subsequent analysis. In addition, we also performed a separate set of ≈500 k simulation runs for width W = 900 nm at VB = 19 V reverse bias. To reduce simulation time, this separate set of simulations are performed with a reduced detection current threshold of 0.02 mA. From the regular simulation runs, we do not observe any case where the device current rises above 0.02 mA but then fail to reach 0.2 mA, thus these results will still predict the PDE correctly. While these reduced-threshold simulations do not capture the full avalanche behaviour, they can still provide insight into the diffusion of initial photo-generated charge carriers injected into the quasi-neutral regions, which requires a large number of simulation runs to observe due to low probabilities of occurrence.
5. Simulation results
We first investigate the PDE and temporal behavior for devices with a nominal n+ (p) doping concentration of 1×1019 (2×1017) dopants/cm3, then explore the effects of varying the doping concentrations.
5.1. Photon detection efficiency (PDE)
By treating the success and failure outcomes of the simulation runs as a binomial distribution, we obtain the PDE as the ratio of successful detection events to the total number of simulation runs, with an uncertainty given by the standard deviation. For devices with nominal n+ (p) doping concentration of 1×1019 (2×1017) dopants/cm3, the breakdown voltage Vbr ≈ 12.4 V; PDE increases with reverse bias voltage VB and saturates at about 20 V (see Fig. 4(a)). We observe a maximum PDE of 0.45 for waveguide widths W of 750 nm and 900 nm, with narrower devices being less efficient.
Figure 4(b) illustrates the spatial distribution of successful avalanches. At low VB, only photo-generated charge carriers injected in the central depletion region are sufficiently accelerated to cause enough impact ionizations, which leads to successful avalanches. As VB increases, the depletion region becomes wider, and the impact ionization rate is enhanced. This results in more successful avalanches caused by photo-generated charge carriers injected nearer the waveguide edges, and an overall higher PDE. However, PDE saturates at high VB and does not increase above ~0.45.
To gain further insight into the saturation behavior, we consider how the probability of generating successful avalanches depends on the injection coordinate of the initial photo-generated electron-hole pair (see Fig. 4(c)); this shows the SPAD behavior independent of the optical mode.
In general, as the initial electron-hole pair is injected futher away from the juntion towards the p-doped side, the avalanche success probability increases as the charge carriers can travel a larger distance over which impact ionizations can occur. However, this probability saturates at ~ 0.7 near the middle of the waveguide, and does not increase even as the initial injection occurs further towards the p-doped side. This indicates that most of the impact ionizations occur near the p-n+ junction, where the electric fields and ionization coefficients are very high (see Fig. 3 and Fig. 5(f)). This high-field region does not become wider with further increases in VB or W; in fact, the peak electric field magnitude |E| decreases with increasing VB.
Injections near the top and bottom edges in the middle section of the waveguide do not lead to successful avalanches. These regions correspond to areas where the electric field has a significant Ey component (see Fig. 3), thus we infer that charge carriers are pushed out of the waveguide and are lost from the device. However, within the quasi-neutral regions, it is possible for the random Brownian motion to overcome the Ey field and bring the charge carrier to the depletion region, thus contributing to the overall device PDE. A larger waveguide height may decrease the effect of the Ey field components, but preliminary simulation results suggest that the PDE does not change significantly.
A large dead space could also suppress the device PDE by limiting the available space for charge carriers to impact ionize. However, the dead space travelled by the initial injected charge carriers is ~0.05W across all widths, which is a relatively small fraction of the device dimensions.
The highest electric fields are concentrated at the junction separating the p- and n+-doped regions. Alternative doping profiles, e.g. ‘L’- or ‘U’-shaped p-n junctions, may be a feasible way of enhancing the device PDE by increasing the overall junction length, and thus the overlap between the optical mode and the high-field regions [48–50].
5.2. Temporal behavior
The distribution of simulated times tava taken to reach the detection threshold Idet is generally asymmetric, and a typical example is shown in Fig. 5(a). From these timing histograms (bin size 0.2 ps), we extract both the full-width-half-maximum (FWHM) and full-width-tenth-maximum (FWTM) timing jitter. We ignore the effect of finite group velocity of the input light along the input waveguide; the propagation time along the full 16 µm waveguide is ~0.1 ps, and the actual effect on the timing jitter is expected to be much less significant.
To further investigate the asymmetric distribution, we consider avalanches caused by photo-generated carriers injected into the p-side quasi-neutral region, as well as those injected into the high field region within the depletion width (Fig. 5(b) and 5(c)). In both cases we still observe an asymmetry, which we attribute to the nature of the impact ionization process. However, the avalanches from the p-side quasi-neutral region show a ~3 ps longer median tava and has a larger contribution to the long tail.
Long tails in the timing distribution can be problematic for certain applications, e.g. high-repetition-rate QKD . Such features with long characteristic timescales (~ns) have been reported in other SPAD work [38, 45, 51], and were attributed to the slow diffusion of charge carriers within the quasi-neutral regions. In our device, the time taken for charge carriers injected into the quasi-neutral regions to diffuse to the depletion region is much faster (~10 ps, see Fig. 5(d) and 5(e)), due to the smaller size of the quasi-neutral regions in our waveguide device compared to conventional free-space-coupled SPADs. We also note that the diffusion in the p-side quasi-neutral region is much faster despite its wider width compared to the n-side, due to the higher mobilities of electrons in the p-side region.
To better understand the dynamics of the avalanche process, we divide tava into an initiation time tinit (where the total number of charge carriers <100) and a rapid avalanche buildup time tbu:Fig. 5(f) and Fig. 5(g)).
In general, tava increases with the waveguide width W and the reverse bias voltage VB (see Fig. 6(a)). As shown in Fig. 4(b), at higher VB, the depletion region becomes wider, and more avalanches are caused by carriers injected nearer the waveguide edges. The slower avalanche build-up for these charge carriers result in the overall longer tava, but does not significantly alter the timing jitter (see Fig. 6(b)). Wider waveguides have a larger spatial distribution of photo-generated charge carriers, which accounts for the longer avalanche times and larger jitter.
5.3. Varying the doping concentrations
Lower doping concentrations have been shown to increase the multiplication gain in avalanche photodiodes, albeit at a cost of higher breakdown voltages Vbr (and thus higher operational voltages) . However, for our device the PDE still saturates at about 0.45 as we decrease the doping levels (see Fig. 7). The time taken to reach detection threshold tava also shows a clear increase with decreased doping, and timing jitter becomes worse. Thus, on all three counts, we do not observe any performance advantage in lowering the doping concentrations.
We note that our results do not capture the sharp change in PDE near Vbr for n+ (p) doping concentrations of 4×1018 (4×1016) dopants/cm3; we did not run simulations below Vbr ≈ 40.6 V obtained from the DC electrical analysis, but perhaps our simulator would show a significant PDE even below this value.
From our observations, it may be inferred that increasing the doping concentrations (and, in turn, the electric field) further could still yield a high PDE with an even lower operating voltage. However, Zener breakdown becomes dominant at very high electric fields [40, 53], which would then impede the performance of the SPAD.
In conclusion, we have designed and simulated a CMOS-compatible silicon waveguide-based SPAD for visible wavelengths. Our simulated devices show a maximum PDE of 0.45 at ~20 V reverse bias, which is slighly lower but still competitive compared to commercial large-area free-space SPAD modules with PDEs of up to ~0.7 . However, typical FWHM timing jitter in our devices is <8 ps, significantly better than high-timing-resolution free-space SPADs (jitter ~35 ps). Our 2D Monte Carlo simulator can also be adopted to study other device geometries or doping profiles, where device performance can potentially be further improved.
National Research Foundation Singapore (NRF) (NRF-CRP14-2014-04); Agency for Science, Technology and Research (A*STAR) (SINGA).
The authors declare that there are no conflicts of interest related to this article.
References and links
3. J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, “Qubit entanglement between ring-resonator photon-pair sources on a silicon chip,” Nat. Commun. 6, 7948 (2015). [CrossRef] [PubMed]
4. J. Wang, D. Bonneau, M. Villa, J. W. Silverstone, R. Santagati, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. O’Brien, and M. G. Thompson, “Chip-to-chip quantum photonic interconnect by path-polarization interconversion,” Optica. 3, 407–413 (2016). [CrossRef]
5. P. Sibson, C. Erven, M. Godfrey, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. O’Brien, and M. G. Thompson, “Chip-based quantum key distribution,” Nat. Commun. 8, 13984 (2017). [CrossRef] [PubMed]
6. M. Ren, X. Gu, Y. Liang, W. Kong, E. Wu, G. Wu, and H. Zeng, “Laser ranging at 1550 nm with 1-ghz sine-wave gated InGaAs/InP APD single-photon detector,” Opt. Express 19, 13497–13502 (2011). [CrossRef] [PubMed]
7. A. Swatantran, H. Tang, T. Barrett, P. DeCola, and R. Dubayah, “Rapid, high-resolution forest structure and terrain mapping over large areas using single photon lidar,” Sci. Rep. 6, 28277 (2016). [CrossRef] [PubMed]
10. I. C. Hernández, M. Buttafava, G. Boso, A. Diaspro, A. Tosi, and G. Vicidomini, “Gated STED microscopy with time-gated single-photon avalanche diode,” Biomed. Opt. Express 6, 2258–2267 (2015). [CrossRef] [PubMed]
11. M.-J. Lee, P. Sun, and E. Charbon, “A first single-photon avalanche diode fabricated in standard SOI CMOS technology with a full characterization of the device,” Opt. Express 23, 13200–13209 (2015). [CrossRef] [PubMed]
12. K. Zang, X. Jiang, Y. Huo, X. Ding, M. Morea, X. Chen, C.-Y. Lu, J. Ma, M. Zhou, Z. Xia, Z. Yu, T. I. Kamins, Q. Zhang, and J. S. Harris, “Silicon single-photon avalanche diodes with nano-structured light trapping,” Nat. Commun. 8, 628 (2017). [CrossRef] [PubMed]
13. W. H. Pernice, C. Schuck, O. Minaeva, M. Li, G. Goltsman, A. Sergienko, and H. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat. Commun. 3, 1325 (2012). [CrossRef] [PubMed]
14. F. Najafi, J. Mower, N. C. Harris, F. Bellei, A. Dane, C. Lee, X. Hu, P. Kharel, F. Marsili, S. Assefa, K. K. Berggren, and D. Englund, “On-chip detection of non-classical light by scalable integration of single-photon detectors,” Nat. Commun. 6, 5873 (2015). [CrossRef] [PubMed]
15. R. E. Warburton, M. Itzler, and G. S. Buller, “Free-running, room temperature operation of an InGaAs/InP single-photon avalanche diode,” Appl. Phys. Lett. 94, 071116 (2009). [CrossRef]
16. Y. Liang, Y. Chen, Z. Huang, G. Bai, M. Yu, and H. Zeng, “Room-temperature single-photon detection with 1.5-ghz gated InGaAs/InP avalanche photodiode,” IEEE Photon. Technol. Lett. 29, 142–145 (2017). [CrossRef]
17. T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: Monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010). [CrossRef]
18. Y. Huang, J. Song, X. Luo, T.-Y. Liow, and G.-Q. Lo, “CMOS compatible monolithic multi-layer Si3N4-on-SOI platform for low-loss high performance silicon photonics dense integration,” Opt. Express 22, 21859–21865 (2014). [CrossRef] [PubMed]
19. W. D. Sacher, Y. Huang, G.-Q. Lo, and J. K. Poon, “Multilayer silicon nitride-on-silicon integrated photonic platforms and devices,” J. Light. Technol. 33, 901–910 (2015). [CrossRef]
20. D. S. Ong, K. F. Li, G. J. Rees, J. P. David, and P. N. Robson, “A simple model to determine multiplication and noise in avalanche photodiodes,” J. Appl. Phys. 83, 3426–3428 (1998). [CrossRef]
21. S. L. Tan, D. S. Ong, and H. K. Yow, “Theoretical analysis of breakdown probabilities and jitter in single-photon avalanche diodes,” J. Appl. Phys. 102, 044506 (2007). [CrossRef]
22. A. Ingargiola, M. Assanelli, A. Gallivanoni, I. Rech, M. Ghioni, and S. Cova, “Avalanche buildup and propagation effects on photon-timing jitter in Si-SPAD with non-uniform electric field,” Proc. SPIE 7320, 73200K (2009). [CrossRef]
23. J. Ma, M. Zhou, Z. Yu, X. Jiang, Y. Huo, K. Zang, J. Zhang, J. S. Harris, G. Jin, Q. Zhang, and J.-W. Pan, “Simulation of a high-efficiency and low-jitter nanostructured silicon single-photon avalanche diode,” Optica 2, 974 (2015). [CrossRef]
24. S. Mandai, M. W. Fishburn, Y. Maruyama, and E. Charbon, “A wide spectral range single-photon avalanche diode fabricated in an advanced 180 nm CMOS technology,” Opt. Express 20, 5849–5857 (2012). [CrossRef] [PubMed]
25. W. G. Oldham, R. R. Samuelson, and P. Antognetti, “Tiggering phenomena in avalanche diodes,” IEEE Trans. Electron Devices 19, 1056 (1972). [CrossRef]
26. J. J. Ackert, A. S. Karar, D. J. Paez, P. E. Jessop, J. C. Cartledge, and A. P. Knights, “10 gbps silicon waveguide-integrated infrared avalanche photodiode,” Opt. Express 21, 19530–19537 (2013). [CrossRef] [PubMed]
27. C. T. DeRose, D. C. Trotter, W. A. Zortman, A. L. Starbuck, M. Fisher, M. R. Watts, and P. S. Davids, “Ultra compact 45 ghz CMOS compatible germanium waveguide photodiode with low dark current,” Opt. Express 19, 24897–24904 (2011). [CrossRef]
29. G. Masetti, M. Severi, and S. Solmi, “Modeling of carrier mobility against carrier concentration in arsenic-, phosphorus-, and boron-doped silicon,” IEEE Trans. Electron Devices 30, 764–769 (1983). [CrossRef]
30. C. Canali, G. Majni, R. Minder, and G. Ottaviani, “Electron and hole drift velocity measurements in silicon and their empirical relation to electric field and temperature,” IEEE Trans. Electron Devices 22, 1045–1047 (1975). [CrossRef]
31. S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer-Verlag, 1984). [CrossRef]
32. Silvaco, Inc., “Atlas user’s manual,” (2006).
33. J. A. Richardson, L. A. Grant, and R. K. Henderson, “Low dark count single-photon avalanche diode structure compatible with standard nanometer scale cmos technology,” IEEE Photonics Technol. Lett. 21, 1020–1022 (2009). [CrossRef]
34. E. A. Webster, J. A. Richardson, L. A. Grant, and R. K. Henderson, “Single-photon avalanche diodes in 90 nm cmos imaging technology with sub-1 hz median dark count rate,” in Proceedings of the International Image Sensor Workshop, (2011), pp. 262–265.
35. R. J. Collins, R. Hadfield, V. Fernandez, S. W. Nam, and G. S. Buller, “Low timing jitter detector for gigahertz quantum key distribution,” Electron. Lett. 43, 180–182 (2007). [CrossRef]
36. J. Bienfang, A. Restelli, D. Rogers, A. Mink, B. Hershman, A. Nakassis, X. Tang, L. Ma, H. Xu, D. Su, C. W. Clark, and C. J. Williams, “High-repetition rate quantum key distribution,” Proc. SPIE 6780, 67800C (2007). [CrossRef]
37. M. J. Stevens, R. H. Hadfield, R. E. Schwall, S. W. Nam, R. P. Mirin, and J. A. Gupta, “Fast lifetime measurements of infrared emitters using a low-jitter superconducting single-photon detector,” Appl. Phys. Lett. 89, 031109 (2006). [CrossRef]
38. A. Gulinatti, I. Rech, S. Fumagalli, M. Assanelli, M. Ghioni, and S. D. Cova, “Modeling photon detection efficiency and temporal response of single photon avalanche diodes,” Proc. SPIE 7355, 73550X (2009). [CrossRef]
39. C. A. Lee, R. L. Batdorf, W. Wiegmann, and G. Kaminsky, “Time dependence of avalanche processes in silicon,” J. Appl. Phys. 38, 2787–2796 (1967). [CrossRef]
40. I. Wêgrzecka, M. Wêgrzecki, M. Grynglas, J. Bar, A. Uszyñski, R. Grodecki, P. Grabiec, S. Krzemiñski, and T. Budzyñski, “Design and properties of silicon avalanche photodiodes,” Opto-Electronics Rev. 12, 95–104 (2004).
41. G. Ripamonti and S. Cova, “Carrier diffusion effects in the time-response of a fast photodiode,” Solid State Electron. 28, 925–931 (1985). [CrossRef]
42. A. Spinelli and A. L. Lacaita, “Mean gain of avalanche photodiodes in a dead space model,” IEEE Trans. Electron Devices 43, 23–30 (1996). [CrossRef]
43. Y. Kamakura, H. Mizuno, M. Yamaji, M. Morifuji, K. Taniguchi, C. Hamaguchi, T. Kunikiyo, and M. Takenaka, “Impact ionization model for full band Monte Carlo simulation,” J. Appl. Phys. 75, 3500–3506 (1994). [CrossRef]
44. T. Kunikiyo, M. Takenaka, M. Morifuji, K. Taniguchi, and C. Hamaguchi, “A model of impact ionization due to the primary hole in silicon for a full band Monte Carlo simulation,” J. Appl. Phys. 79, 7718–7725 (1996). [CrossRef]
45. A. Spinelli and A. Lacaita, “Physics and numerical simulation of single photon avalanche diodes,” IEEE Trans. Electron Devices 44, 1931–1943 (1997). [CrossRef]
46. D. Schroder, “Carrier lifetimes in silicon,” IEEE Trans. Electron Devices 44, 160–170 (1997). [CrossRef]
47. S. Ramo, “Currents induced by electron motion,” Proc. IRE 27, 584–585 (1939). [CrossRef]
49. Z. Yong, W. D. Sacher, Y. Huang, J. C. Mikkelsen, Y. Yang, X. Luo, P. Dumais, D. Goodwill, H. Bahrami, P. G.-Q. Lo, E. Bernier, and J. K. S. Poon, “U-shaped pn junctions for efficient silicon Mach–Zehnder and microring modulators in the o-band,” Opt. Express 25, 8425–8439 (2017). [CrossRef] [PubMed]
50. T. Y. Ang, C. E. Png, S. T. Lim, and J. R. Ong, “Silicon modulators with optimized vertical pn junctions for high-modulation-efficiency in the o-band,” in Silicon Photonics XIII, vol. 10537 (International Society for Optics and Photonics, 2018), p. 105370H.
51. M. Ghioni, A. Gulinatti, I. Rech, P. Maccagnani, and S. Cova, “Large-area low-jitter silicon single photon avalanche diodes,” Proc. SPIE 6900, 69001D (2008). [CrossRef]
52. T. Bendib, L. Pancheri, F. Djeffal, and G. Dalla Betta, “Impact of temperature and doping concentration on avalanche photodiode characteristics,” Proc. World Congr. Eng. 1, 5–8 (2014).
53. M. S. Tyagi, “Zener and avalanche breakdown in silicon alloyed p-n junctions —I,” Solid-State Electron. 11, 99–115 (1968). [CrossRef]
54. M. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Rev. Sci. Instr. 82, 071101 (2011). [CrossRef]