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Abstract
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
1. Introduction
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 [8], fluorescence medical imaging [9], and STED microscopy [10].
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 [19]. 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.
Fig. 1 (a) SPAD structure, consisting of a silicon rib waveguide end-fire coupled to an input Si3N4 waveguide. (b) SPAD doping profile, with an asymmetric placement of the p-n+ junction. The cross section is constant along the length of the waveguide.
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 [25]. 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:
where ϵr is the relative dielectric permittivity of the semiconductor, ϵ0 is the permittivity of free space, q is the elementary charge, NA and ND are the number of ionized acceptors and donors, respectively, Vbi is the built-in voltage, and VA is the applied bias. The chosen doping levels ensure that the depletion region covers a large part of the waveguide width. We also explore other doping levels in this work and assess their effect on SPAD performance.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)).
Fig. 2 (a),(b) Optical mode profiles at 640 nm for the fundamental (quasi-)TE modes of 600 nm wide silicon and Si3N4 waveguides, respectively. (c) 3D FDTD simulations of end-fire coupling efficiency from the fundamental TE mode of the Si3N4 waveguide to the fundamental TE (top) and other modes (bottom) of the silicon waveguide. In both plots, each curve shows the coupled power for a fixed silicon waveguide width W, normalized to the input power.
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 [29] model with the phosphorus parameter set in conjunction with the Canali [30] 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) [29]. Ionization coefficients are obtained from the Selberherr impact ionization model [31], 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 [32].
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 [37]. Timing jitter arises from the stochastic nature of the impact ionization and avalanche buildup process, and cannot be modeled by deterministic simulators [38]. 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.
Fig. 3 Electric field profile for a silicon rib waveguide (outlined in white), showing the waveguide core of width W = 900 nm and the surrounding regions, at a reverse bias voltage of VB = 21 V. In regions with a lower electric field magnitude, 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.
4.2. Implementation
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 [39]. 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. [38] 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 [40]. 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 [41]:
where Δlq-n is the shortest of the three relevant length scales: the widths of both quasi-neutral regions and the 250 nm height of the silicon waveguide. In other words, at least 50 steps are required to traverse the full width or height of the quasi-neutral region.The time duration of this Brownian motion displacement is
where is the diffusion constant given by the mobility µ, Boltzmann constant k, temperature T, and elementary charge q.The total displacement of each random walk step is then
where the first term describes the Brownian motion with a randomly chosen direction at each step, and the second term describes the displacement under the drift velocity ve(r) or vh(r) (depending on charge carrier type) at the current position r.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
and the energy gain is where the term within the square brackets is a dot product.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]
Table 1. Parameters used in the simulation
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,
respectively, and the corresponding energies for an impact ionization caused by a hole are, The a and b coefficients are listed in 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 [47]:
where vi,x is the component of the drift velocity in the x direction (i.e. across the width of the waveguide). We set a detection current threshold Idet = 0.2 mA as a reasonable discriminator threshold used in experimental SPAD characterization setups [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.
Fig. 4 (a) Photon detection efficiency (PDE) for different waveguide widths W. Error bars show the uncertainty given by the standard deviation. (b) Maps of successful avalanches at different reverse bias voltages VB for W = 900 nm. The plots show the 900×250 nm waveguide core, with each pixel being ≈20×20 nm. Color scale values indicate the number of successful avalanches caused by initial photo-generated charge carriers injected within that pixel, normalised to the maximum value in each plot. The data in each plot consists of ≈20k simulation runs. (c) Map of position-dependent probability of avalanche success for W = 900 nm at VB = 19 V over ≈500k runs. Each ≈ 10 × 10 nm pixel shows the probability of an initial photo-generated electron-hole pair injected within that pixel resulting in a successful avalanche. All devices in this figure have a n+ (p) doping concentration of 1 × 1019 (2×1017) dopants/cm3, and a breakdown voltage of Vbr ≈ 12.4 V.
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.
Fig. 5 (a) Simulated times taken to reach the detection threshold tava for a device of width W=900 nm and n+ (p) doping concentration of 1×1019 (2×1017) dopants/cm3, at a reverse bias voltage of VB = 19 V. Histogram bin size is 0.2 ps. The full-width-half-max (FWHM) and full-width-tenth-max (FWTM) ranges are indicated by dashed lines. Total simulation runs ≈20k. (b) and (c) show the results from the same device, but only for cases with the initial photo-generated charge carriers injected into the high-field region (zones 7 to 8) and the p-side quasi-neutral region (zones 1 to 3), respectively. The zones are as defined in (f). (d),(e) Times taken for intial charge carriers injected into n- and p-side quasi-neutral regions, respectively, to diffuse to the depletion region. Total simulation runs ≈ 500k; only results from successful avalanches are included in the histograms. Histogram bin size is 0.5 ps. (f) Electron ionization coefficients αe within the 900×250 nm waveguide core at different doping levels, with VB signficantly above breakdown. (g) Median avalanche build-up times tbu for charge carriers injected into each zone.
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 [36]. 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:
and consider the dependence of both on the initial injection position. We find a median tinit ≈ 5 ps across all regions in our simulated devices. However, there is a significant variation of tbu across some devices. Regions with high ionization coefficients, also indicative of high electric field, show lower tbu due to more rapid impact ionizations (see 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.
Fig. 6 (a) Median time taken to reach the detection threshold tava, and (b) full-width-half-max (FWHM) and full-width-tenth-max (FWTM) timing jitter for different waveguide widths W. All plots share the same legend. All devices in this figure have a n+ (p) doping concentration of 1×1019 (2×1017) dopants/cm3, and a breakdown voltage of Vbr ≈12.4 V.
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) [52]. 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.
Fig. 7 (a) Photon detection efficiency (PDE), (b) median time taken to reach the detection threshold tava, and (c) full-width-half-max (FWHM) and full-width-tenth-max (FWTM) timing jitter at different doping levels (in units of dopants/cm3) for waveguides of width W = 900 nm. Error bars in (a) show the uncertainty given by the standard deviation. Dashed vertical lines indicate breakdown voltages Vbr. All plots share the same legend.
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.
6. Conclusion
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 [54]. 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.
Funding
National Research Foundation Singapore (NRF) (NRF-CRP14-2014-04); Agency for Science, Technology and Research (A*STAR) (SINGA).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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