Optical crosstalk in single photon avalanche diode arrays: a new complete model. Opt. Express 16, 8381-8394

: One of the main issues of Single Photon Avalanche Diode arrays is optical crosstalk. Since its intensity increases with reducing the distance between devices, this phenomenon limits the density of integration within arrays. In the past optical crosstalk was ascribed essentially to the light propagating from one detector to another through direct optical paths. Accordingly, reﬂecting trenches between devices were proposed to prevent it, but they proved to be not completely effective. In this paper we will present experimental evidence that a signiﬁcant contribution to optical crosstalk comes from light reﬂected internally off the bottom of the chip, thus being impossible to eliminate it completely by means of trenches. We will also propose an optical model to predict the dependence of crosstalk on the distance between devices.


Introduction
Photodetection systems based on Single Photon Avalanche Diode (SPAD) arrays are characterized by both high photon detection efficiency (up to 50-60 % in the visible range) and good timing resolution (about 30 ps).These features make them suitable for being proficiently employed in many different fields, ranging from life science (DNA and single molecules analysis [1]) to electronic industry (non-invasive testing of VLSI circuits [2]), astronomy (adaptive optics [3]) and even military or commercial applications (e.g.quantum cryptography [4]).
A potential drawback affecting SPAD arrays is optical crosstalk: when a device detects a photon, secondary photons are emitted by the SPAD itself due to hot carriers relaxation [5].These photons can be detected by adjacent detectors, thus corrupting the data acquired by the array.The crosstalk intensity increases with reducing the distance between pixels; hence, this phenomenon sets a limit to the array density.Despite its importance, very few studies focused on this subject [6,7,8] and all of them agreed that crosstalk depends only on direct optical paths between devices.For this reason some of these works [7,8] proposed the introduction of trenches coated with metal between detectors in order to make crosstalk vanish, but experimental data showed just a decrease for its magnitude.
In this work we demonstrate that optical crosstalk depends strongly on indirect optical paths, i.e. on photons emitted by the pixels and reflecting off the surfaces of the chip.In particular, we show that the main contribution to crosstalk comes from photons reflected off the bottom of the chip; these photons can by-pass trenches making them partially ineffective.We also present an optical model to predict the dependence of crosstalk on the position of the SPADs within the array.

Tested samples
Basically SPADs are diodes reverse biased above breakdown.Under this condition the electric field at the p-n junction is so strong that even a single carrier crossing the space charge region can trigger a self-sustaining avalanche via impact ionization.If the carrier is photogenerated, the leading edge of the avalanche current marks the absorption of a photon; conversely, thermally generated carriers cause the detector noise.A suitable circuit (AQC, Active Quenching Circuit) quenches the avalanche and restores the initial bias condition to allow a subsequent detection [9].
The SPADs discussed in this work were developed at the Politecnico di Milano and fabricated at the CNR-IMM Silicon facility (Bologna, Italy) by using a double-epitaxy process [10].The cross-section of a single device is schematically sketched in Fig. 1.The active region of the SPAD is the p-n junction formed of the two heavily-doped regions near the top surface.The heavily-doped isolation acts as electrical isolation between the pixels of the array.Furthermore, as we will show later (see section 4), its high doping level make it an effective optical barrier, analogous to the trenches described in [8].
The SPADs were arranged in 7 × 2 arrays: the spacing between the two rows was 365 µm whereas the SPADs in a row were 289 µm distant (see Fig. 2).A set of 10 arrays of our standard fabrication process were tested in order to achieve a statistically significant number of data.

Optical crosstalk
Figure 3 shows a simplified representation of optical crosstalk.Whenever an avalanche is triggered in one pixel of the array, photons are emitted at different wavelengths [11]; these photons can trigger other avalanches in the neighboring pixels.According to this scheme, every time a crosstalk event occurs, we have: • emission from the first device (the emitter) • propagation through the bulk of the array • detection by the second device (the detector) In order to build a detailed model for optical crosstalk, each phase will be carefully considered.
As far as emission is concerned, the two critical points are the spatial and spectral distribution of the emitted light.Propagation is strictly connected to the characteristics of the bulk of the array, both in terms of absorption coefficient and refractive index.Detection, finally, depends on the SPAD PDE (Photon Detection Efficiency), i.e. the probability that a single photon can be absorbed in the active region of the detector and trigger an avalanche.

Experimental results
Before modeling optical crosstalk, we performed a complete experimental characterization of this phenomenon.First we investigated the dependence of crosstalk on the position of the SPADs in the array for each possible couple of devices, then we focused on each step discussed in the previous section.
The dependence of crosstalk on the position of the two SPADs can be characterized by different techniques.We considered coincidence measurements and what we called pseudo-crosstalk measurements.Coincidence measurements are performed evaluating the correlation between the output signals generated.This allows to obtain crosstalk probability directly, but requires long acquisition times in order to achieve an acceptable signal-to-noise ratio (SNR) [12].
In pseudo-crosstalk measurements the count-rate of a detector is evaluated while another device is operating as emitter, the latter being biased above breakdown at a constant current.An analogous measurement was performed by Haitz [6].In this case a value proportional to crosstalk probability and not crosstalk probability itself is obtained.However, since shorter acquisition times than before are required to get the same SNR, we characterized the crosstalk on the array mainly with this kind of measurements.Some coincidence measurements were also performed in order to verify the equivalence of the two approaches.
Pseudo-crosstalk measurements were performed for all the emitter-detector pairs within the arrays, and the acquired data were filtered by using the digital lock-in technique, which is here briefly described.Each measurement consists of a large number of cycles (varying from 100 to 10000) made of two adjacent time intervals.During the first interval the counts of the detector are evaluated with the emitter turned on; in the subsequent interval this procedure is repeated with the emitter turned off.By subtracting the latter contribution from the former and averaging the resulting values for all the performed cycles, it is possible to remove the noise floor contribution (background + detector dark counts).The intervals duration is made sufficiently short (100 ms) so that thermal transients could be neglected.
Considering only direct optical paths we expect a monotonic (decreasing) behavior of the crosstalk with the distance between devices.In particular, even neglecting the absorption from the silicon, the crosstalk should decrease with the distance at least as a 1/R 2 law (due to geometrical attenuation).Including the absorption we expect a more marked decrease of the crosstalk with a dependence from the distance proportional to 1/R 2 • exp(−αR), where α is the (effective) silicon absorption coefficient.
In Fig. 4 we report, for a given detector (in this case SPAD 1, see Fig. 2), the dependence of pseudo-crosstalk on the position of the emitter; Figure 5 shows, instead, its dependence on the distance R from the detector.Surprisingly the crosstalk not only does not follow a 1/R   The values shown represent the mean pseudo-crosstalk over all the couples at the same distance on the array.For comparison 1/R 2 curves are reported (dashed lines).Since crosstalk does not follow this law and it does not even decrease monotonically, also optical paths different from the direct ones have to be considered to describe this phenomenon.
can not be explained considering only direct optical paths between devices, therefore different contributions, namely indirect paths, were investigated.
Firstly we experimentally evaluated whether a fraction of the emitted photons can effectively travel through the thick substrate (500 µm) without being absorbed.For this purpose we performed emission measurements from the bottom of the chip; the light was collected by an optical system and an high-resolution Hamamatsu C4880 Silicon CCD (see Fig. 6).Despite high silicon absorption in the substrate and relatively low Fresnel transmission coefficient at the silicon-air interface we revealed a consistent photon flux escaping from the bottom, only 10 4 times less intense that the light escaping from the top.This experiments proofs that some emitted spectral components are able to propagate though the substrate and reach the bottom of the chip.
We further investigate whether photons reflecting off the bottom could give a contribution to crosstalk performing some pseudo-crosstalk measurements after having manually placed a reflecting metal sheet under the chip (as it is shown in Fig. 7).The purpose of the experiment was to modify the bottom reflected components in order to test whether this condition could change the overall crosstalk.Actually we noticed a remarkable increase in the pseudocrosstalk measurements thus demonstrating that bottom reflected components give a contribution to crosstalk.We could not simulate these experiments in order to exactly predict the expected crosstalk increase since the metal sheet position (distance from the chip and tilting) were not accurately controlled.However considering the geometrical factors and the Fresnel coefficients at the interfaces we conservatively estimated that the bottom reflected components would be increased no more that 20%.We used two different materials (Cu and Al) and in both cases we observed a 10-20 % increase in optical crosstalk for the nearest SPADs, with respect to the measurements performed without the sheet.Despite the roughness of the setup, this increase not only demonstrate that photons reflected on the bottom gives a contribution to crosstalk but also that this contribution must be dominant.In order to establish which spectral components mostly contribute to crosstalk, the value of the absorption coefficient for the different regions were considered.Figure 8 shows the absorption coefficient of silicon as a function of the wavelength for different doping levels [13,14,15].In the visible range the absorption coefficient decreases monotonically; conversely, in the near IR (about 1000-1300 nm), it starts increasing due to the mechanism of free-carrier absorption [16].
Since direct paths travel mainly through the isolation (doping 10 20 cm −3 ) whereas indirect paths travel mainly through the substrate (doping 4 • 10 17 cm −3 ), we focused on these two regions.According to the data shown in Fig. 8, the absorption coefficient for the isolation is greater than about 300 cm −1 for every wavelength, whereas the absorption coefficient for the substrate has a lower minimum around 1100 nm (about 3 cm −1 ).This means that for direct paths the minimum attenuation (computed considering 180 µm of isolation between the emitter and the detector, see Fig. 9(a)) is about 221.Moreover, it is worth noting that the isolation will attenuate much more the whole spectrum since the emitted wavelength are not concentrated to a single wavelength corresponding to the minimum absorption; hence, the isolation can be considered as an optical barrier between devices.Conversely, considering the shortest indirect path traveling almost entirely through the substrate (single reflection off the bottom of the chip, see Fig. 9(b)) the attenuation is only 1.17 at 1100 nm.Thus, indirect paths involving reflections off the bottom of the chip give the most important contribution to crosstalk and the spectral region of interest must be the near-IR (around 1100 nm).On the other hand, direct paths give a small contribution and can be neglected.
To gain a more quantitative understanding on which spectral components give a contribution to crosstalk we further investigated SPAD emission spectrum and quantum efficiency.
The spectra of avalanching silicon p-n junctions found in previous works differ significantly from one another [17,18,19], therefore we decided to perform some of the emission spectrum for our devices.
We assembled an optical system to perform this kind of measurements.The light emitted by a SPAD biased above breakdown was collimated by means of achromatic doublets, the different spectral components were separated by a prism and the beam was then focalized on the previously mentioned high-resolution CCD.The system was calibrated by using a tunable monochromatic light as the input signal and marking out the correspondence between wavelengths and pixels.The monochromatic input was achieved by filtering the light emitted by a broad-band source by means of a monochromator (FWHM of 10 nm).The resulting data were finally scaled by the quantum efficiency of the CCD.
The measured spectrum is reported in Fig. 10: it extends up to 1100 nm and beyond, thus indicating that actually these spectral components mostly contribute to crosstalk.
As just mentioned, we also measured the SPAD PDE (Photon Detection Efficiency), i.e. the probability that a photon reaching a detector can trigger a self-sustaining avalanche current.
Typical silicon SPAD applications require the PDE to be known within the visible range, therefore most of the available experimental setups for PDE measurements and the literature data are relative to this part of the spectrum.According to the previous results, we needed to measure the PDE in the near-IR region, thus we built a specific setup.The light emitted by a broad-band source was collimated and narrow-band filtered (FWHM of 10 nm) around a central wavelength tunable from 400 to 1250 nm.Comparing the optical power incident on a near-IR power-meter with the counts detected by a SPAD, it was possible to calibrate the system and compute the PDE.
Figure 11   the near-IR region (about one order of magnitude, moving from 1100 to 1200 nm), and the corresponding decrease of the SPAD emission spectrum at the same wavelengths (see Fig. 10) the crosstalk would tend to decrease with the increasing of the wavelength.On the other hand, the absorption coefficients rapidly decrease from a very high value in the visible range to an extremely low level in the near-IR (see Fig. 8), thus the crosstalk would tend to increase with the wavelength.A trade off between these two opposite effects takes place and thus the dominant contribution to crosstalk arises from photons in a narrow band centered at a wavelength between 1100 and 1200 nm.

Optical Model
According to the previous experimental results we developed a 3-D optical model in order to perform numerical simulations of optical crosstalk (see Fig. 12).
The key elements of the model are: • the emitters • the detector • the isolation

• the substrate
The entire chip is treated as a piece of uniform material, with the same characteristics (abcoefficients and refractive index) of the substrate.The slight variations (less than 2 %) in refractive index passing from the substrate to the epi-layer [20] was neglected since it is uninfluent regarding the propagation in the substrate.The chip surfaces are treated as smooth silicon-air interfaces.
The emitters are represented by thin cylindrical emitting volumes, approximately coincident with the SPADs high-field regions (thickness of about 0.5 µm and diameter of 20 µm).
According to what we said in section 4, direct optical paths are completely absorbed from the heavily-doped isolation region, thus we inserted a set of perfectly absorbing rings surrounding each SPAD in order to modelize the isolation region.We performed different numerical simulations in the 900-1300 nm wavelengths range for each possible emitting SPAD.The simulations give us the intensity pattern generated by the emitting SPAD on the top chip surface (see Fig. 13).We then calculated the fraction of photons detected by each SPAD integrating over an area corresponding to each active region.The results were scaled with the SPAD quantum efficiency obtaining the final result to be directly compared with the pseudo-crosstalk measurements.
Observing the intensity pattern on the top chip surface (Fig. 13) we note an abrupt increase at a distance d 1 from the emitter (a little bit further than the nearest SPAD), corresponding to the onset of the total internal reflection for photons performing a single reflection off the bottom [21].In fact, photons reaching the bottom silicon-air interface with an angle smaller than the critical angle are partially transmitted, while photons reaching the interface with an angle greater than the critical angle are totally reflected giving rise to stronger contribution to the crosstalk.The condition is showed in Fig. 14(a).
A second weaker local maximum is observed at a distance d 2 = 2 d 1 where the condition of total internal reflection is reached by photons performing a double reflection off the bottom (see Fig. 15.Simulated (solid line) and measured (dashed line) pseudo-crosstalk for the situation of Fig. 4. As before, the two plots refer to the two rows of the array; the abscissas indicate the numbering of the SPAD as reported Fig. 2; the detector (in this case SPAD 1) is indicated by an arrow.The numerical data are in very good agreement with the experimental results, thus indicating the developed model represents a helpful tool for the prediction of the dependence of crosstalk on the position of the devices within the array.Fig. 14(b)).Furthermore, an increase of the top optical intensity can be observed near the edges of the chip close to the emitter, since our model keeps into account reflections off the lateral surfaces.
In our model, using the absorption coefficients reported in curve (b) of Fig. 8, we obtained a maximum contribution to crosstalk for a wavelength of 1100 nm corresponding to an absorption coefficient around 3 cm −1 .The wavelength range that gives a significant contribution to crosstalk was estimated to be lesser that 100 nm.These results confirm our hypothesis that crosstalk is due to a narrow range of wavelengths centered between 1100 and 1200 nm.
A direct comparison of the simulations with the experimental data (see Fig. 15) shows that our model accurately predict the crosstalk dependence with the position of the SPADs for all the couples of devices.The model allows to explain the "bell-shape" of crosstalk shown in Fig. 4 and Fig. 15 from a physical point of view, ascribing it to the total internal reflection phenomenon.We have showed in section 4 that the crosstalk (for the farther SPADs) decreases with the distance much more slower than 1/R 2 (a dependence that we would expect if the crosstalk would be due only to direct optical paths).This effect is also correctly predicted by our model and can be explained considering the indirect optical paths.In fact the indirect optical paths lengths grow "slower" than the direct one with the distance between devices, and so does the corresponding attenuation.

Conclusion
Optical crosstalk represents a potential drawback of SPAD arrays.In the present work this problem was deeply investigated from both the experimental and the numerical point of view.It was demonstrated that one of the main contributions to crosstalk comes from a narrow-band (less than 100 nm) of near-IR spectral components (between 1100 and 1200 nm) reflecting off the bottom of the chip (especially the ones reflecting with an angle of incidence greater than the critical angle, since total internal reflection takes place).

Fig. 1 .
Fig. 1.Cross-section of the planar SPAD structure used in our tests.

Fig. 2 .
Fig. 2. Layout of the tested SPAD arrays.The numbers from 1 to 7 and from 8 to 14 are used to identify the position of the single devices within the array.

Fig. 3 .
Fig. 3. Schematic representation of optical crosstalk between two devices A and B. When a primary signal photon triggers an avalanche in the SPAD A, secondary photons are emitted by the SPAD itself.These photons propagate through the bulk of the array and finally they are detected by the SPAD B.

Fig. 4 .
Fig. 4. Measured pseudo-crosstalk as a function of the position of the emitter and the detector.The two plots refer to the two rows of the array; the abscissas indicate the numbering of the SPAD as reported Fig. 2; the detector (in this case SPAD 1) is indicated by an arrow.

Fig. 5 .
Fig.5.Dependence of the pseudo-crosstalk on the distance R between the emitter and the detector.The values shown represent the mean pseudo-crosstalk over all the couples at the same distance on the array.For comparison 1/R 2 curves are reported (dashed lines).Since crosstalk does not follow this law and it does not even decrease monotonically, also optical paths different from the direct ones have to be considered to describe this phenomenon.

Fig. 6 .
Fig.6.Measurement of the light escaping from the bottom of the chip.The light is collected by a Hamamatsu C4880 Silicon CCD.Also the light escaping from the top surface was measured, in order to perform a comparison of the two contributions.

Fig. 7 .
Fig. 7. Increase in the total optical intensity reaching the detector due to a metal sheet manually placed under the chip.

Fig. 12 .
Fig. 12. Model developed to perform numerical simulations of optical crosstalk.The red dots correspond to the active regions of the SPADs and represent the possible emitting devices (as an example, the light emitted by one SPAD and reflecting off the bottom of the chip is shown in yellow).The black absorbing rings keep the effect of the isolation into account, whereas the bulk is modeled as a piece of uniform material (n-type silicon).

Fig. 13 .Fig. 14 .
Fig. 13.Simulated 2D intensity profile at the top surface of the chip with one emitter (at the top right of the image) turned on.The black dots represents the SPADs position on the array.