Single Photon Avalanche Detectors (SPADs) are biased above breakdown causing a large avalanche current upon detecting even a single photon due to a very high multiplication gain. This avalanche is a source of secondary photons which can cause external (between SPAD arrays) and internal (between SPADs in an array) crosstalk. In this previous example we showed how to extract this secondary photon production spectrum by a combination of simulation and measurement. In this example, we show how to simulate crosstalk due to secondary photon production and we compare it to the measured crosstalk.
Overview
Understand the simulation workflow and key results
In the remainder of this article, we will refer to the optical crosstalk due to secondary emission as direct optical crosstalk. The mechanism responsible for the direct optical crosstalk is propagation of light away from the secondary photon production area and into other SPADs in the same array or out through the surface of the SPAD array. This crosstalk does not include other crosstalk mechanisms such as delayed diffusion of carriers generated in the substrate into the avalanche region, or delayed release of trapped charges.
Step 1: Obtain the photon production curve due to the SPAD secondary emission
In our previous example SPAD Secondary Emission and Absorption – Ansys Optics we showed how to extract the secondary photon production spectrum in the avalanche region. Here, we will use the extracted spectrum as the source for direct optical crosstalk simulation. Since the extracted spectrum is per charge carrier it must be multiplied with the multiplication gain at the given applied voltage to obtain the total photon production.
Step 2: Simulate optical crosstalk between individual SPADs in an array
Direct optical crosstalk is caused by secondary photons propagating away from the avalanche region. If they get absorbed by one of the neighboring SPADs they may trigger that SPAD leading to the internal optical crosstalk. If they leave though the surface of the SPAD array and get absorbed by any neighboring SPAD array they may trigger one of the SPADs in the neighboring array leading to the external crosstalk. This is illustrated in the image below:
In this example the focus is on the internal crosstalk. For the simulation of the angular distribution of the secondary emission in the far field, responsible for the external crosstalk, please take a look at our previous example SPAD Secondary Emission and Absorption – Ansys Optics .
To simulate the internal crosstalk, we perform a 2D FDTD simulation with three neighboring SPADs. 2D simulation is approximative since it effectively means that the source of light is cylindrical (extending uniformly in the third direction), instead of spherical, but it is much faster than the 3D simulation. The users are encouraged to try the 3D simulation after obtaining reasonable results with the 2D simulation. The three SPADs are numbered 0, 1, 2. SPAD 0 is the source of secondary emission. We simulate absorption in SPADs 1 and 2 and then post-process the results to extract the simulated crosstalk in SPADs 1 and 2 as a unitless number. SPAD 0 is a source of crosstalk in both SPADs 1 and 2, while SPAD 1 can be a source of crosstalk in SPAD 2. Due to the way the avalanche mechanism and the quenching circuit work, the higher-numbered SPADs, which are farther away from the source SPAD, cannot be a source of crosstalk in the lower-numbered SPADs.
Step 3: Compare to crosstalk measurements (verification step)
Crosstalk measurement is performed by illuminating one SPAD with a pulsed laser source and imaging the surface of the SPAD array with a camera to record photon emission from all SPADs in the array. This creates 2D emission maps which can be postprocessed to obtain the crosstalk. Since the imaging is done over a long period of time, the measured crosstalk includes both direct optical crosstalk as well as delayed crosstalk from carrier diffusion and de-trapping. So, this measurement represents the upper limit of crosstalk and we will use it as such to show that our simulated optical crosstalk is always smaller than the measurement.
Run and results
Instructions for running the model and discussion of key results
Step 1: Obtain the photon production curve due to the SPAD secondary emission
- Run step 2 of example SPAD Secondary Emission and Absorption – Ansys Optics . Remember to set the oxide thickness to 1.33 um.
- Change working folder to this example and open and run script [[save_photon_production_spectrum.lsf]]. This script will multiply the photon production spectrum per charge carrier from the previous step with the multiplication gain at the given applied voltage and save the result to a file to be used in the next step.
Step 2: Simulate optical crosstalk between individual SPADs in an array
- Open file [[fdtd_crosstalk_2d.fsp]].
- Open and run file [[run_crosstalk_sweep.lsf]].
- Open and run file [[analyze_crosstalk_sweep.lsf]].
Script in step 2.2 will do the following:
- It will set a sweep over 3 possible dipole orientations in SPAD 0. To speed up the simulation it is possible to do sweep points concurrently, by setting Capacity = 3 in Resource Configuration, and by setting the number of threads according to the remainder of available cores which will also speed up each sweep point by parallelizing the simulation domain. The product of threads and capacity should be less than the number of available cores.
- Then it will run sweep and record the continuous wave (CW) electron-hole pair generation rate in SPADs 1 and 2 to a file.
This generation rate is due to the absorption of the dipole source power from SPAD 0 that represents the secondary emission. The source power in this step (W/m for 2D simulation) is set to 1 photon/s at each frequency in the simulated bandwidth.
The number of direct optical crosstalk avalanches per one secondary photon created in SPAD 0 per wavelength are shown below.
Script in step 2.3 will do the following:
- It will multiply the electron-hole pair generation rate with the avalanche triggering probability and integrate over volume to obtain the number of crosstalk avalanches in SPADs 1 and 2 per photon from SPAD 0 per wavelength.
- Using the secondary photon production spectrum, it will then renormalize this result to the number of crosstalk avalanches in SPADs 1 and 2 per avalanche in SPAD 0 per wavelength.
- Once the previous result is integrated over source wavelengths it will give unitless optical crosstalk.
- The crosstalk contribution from the secondary avalanche in SPAD 1 will be taken into account in SPAD 2.
The number of direct optical crosstalk avalanches per one avalanche in SPAD 0 per wavelength are shown below.
The following figure is the total normalized direct optical crosstalk in SPADs 1 and 2 due to 1 avalanche in SPAD 0.
Step 3: Compare to crosstalk measurements (verification step)
- Compare the simulated crosstalk to the measured crosstalk.
The measured crosstalk is based on recording emission maps from the surface of the SPAD array after the center SPAD is illuminated with a pulsed laser source stimulating avalanches that cause optical crosstalk in neighboring SPADs.
The left figure above is the emission map of a SPAD array after 600 s exposure of the center SPAD to a 405 nm pulsed laser light (500 KHz). A long pass filter filters out any reflected 405nm laser light and shows only the emission above 550nm. The distinct pattern of emission is caused by optical crosstalk due to secondary emission in the center SPAD. However, the image also includes secondary crosstalk from secondary fired SPADs (crosstalk of crosstalk), delayed crosstalk from electron-hole pairs created in the Si bulk, trapped and released CHARGE in each SPAD, and potential stray photons from the laser. To characterize the probability of each SPAD firing due to crosstalk, in the right figure above we sum up all the photons present from each SPAD. This figure shows the grid used.
After image processing, we can extract the average optical power from each SPAD (squares in the image below).
The left image above is after summation of all pixels in each SPAD region, representing lumped photon emission from each SPAD. The image box is cropped since the total events outside this boundary are small and are just noise. The right image is the emission probability in each SPAD after normalizing the center SPAD to the number of avalanches in it, representing the crosstalk probability.
Finally, we can take a 1D cut along horizontal and vertical directions and normalize the results to the center SPAD to plot the crosstalk.
The following plot is the horizontal and vertical cuts of the emission probability from the cropped 2D map.
This crosstalk is a result of integration over a long period of time and therefore includes all other sources of crosstalk in addition to direct optical crosstalk, such as delayed crosstalk from carrier diffusion in the substrate and delayed carrier de-trapping in the multiplication region. So, the measured crosstalk represents the absolute upper limit of the direct optical crosstalk. By comparing the simulated direct optical crosstalk to the measured crosstalk, we can see that the simulated crosstalk is always smaller, as it should be.
Important model settings
Description of important objects and settings used in this model
Multiplication gain
Multiplication gain depends on applied overvoltage. It is required for the calculation of the total secondary photon production, by multiplying the number of secondary photons produced per charge carrier with the multiplication gain to obtain the total number of photons resulting from the total number of charge carriers. This is done in step 1.2.
Avalanche triggering probability
The avalanche triggering probability (ATP) is a probability of an electron-hole pair generated at some location to cause a self-sustaining avalanche. This is needed to convert the electron-hole generation rate due to secondary photons absorbed in neighboring SPADs 1 and 2, as calculated by FDTD, to a probability of triggering an avalanche. This is done in step 2.3.
ATP can be obtained by solving the ATP equation for the given electric field distribution and impact ionization model. The required electric field distribution can be obtained from a CHARGE simulation with the impact ionization model turned off. If you would like to be able to solve the ATP equation please contact Lumerical support for further information. For an example of this calculation see references 1 and 2.
CW generation rate analysis object
The analysis script in this object has been modified after adding the object to the simulation from the default library. The default analysis script is designed for a continuous wave source with single frequency. This requires doing a sweep over frequencies to cover the secondary emission bandwidth of interest. To speed up the simulation we changed the analysis script to include multiple frequencies at once, where the power at each frequency is normalized to 1 photon/s. If a new project file is created the analysis script from the project file in this example will have to be copied, since it is not included in the default object library.
Simulation time
Simulation time should be set sufficiently long to include the longest possible distance light can travel taking into account the simulation domain size. However, since we use a dipole source of finite duration there is auto shutoff based on the total power in the simulation domain dropping below certain level. So, the actual simulation time may be just a certain fraction of the estimated time in the job manager.
Taking the model further
Information and tips for users that want to further customize the model
Reflections from the substrate bottom interface
For longer wavelengths close to the Si band gap the waves will be able to travel sufficiently long before being absorbed to backscatter from the bottom interface of the thick substrate and cause crosstalk when they return close to the surface where the SPAD active regions are located. In this example, this effect is ignored, which may be a good approximation.
To include this effect the simulation domain will have to be extended to include the substrate bottom interface. This will make the simulation much longer. To speed up the simulation, we can isolate the bottom reflections as a separate simulation and set the dipole source bandwidth to include only those wavelengths for which the absorption length is long enough so they can reach the bottom interface and reflect. By including only the required longer wavelengths the mesh will be coarser, which will speed up the simulation.
Since the source bandwidth will also be used for material index fitting, to ensure that the material index fitting is accurate enough when using a narrower source bandwidth with longer frequencies, it is possible to override the fit range in the advanced options of Material Explorer for the material in question (here Si). This will require to make a copy of the material (automatic) and continue using the copy in the project file. The new fit range should include the source bandwidth and be large enough to ensure accurate fitting in the source bandwidth range.
Additional resources
Additional documentation, examples and training material
- SPAD Secondary Emission and Absorption – Ansys Optics
- Far field projections in FDTD overview – Ansys Optics
- Integrating power in far field projections – Ansys Optics
- Incoherent unpolarized dipole – Ansys Optics
- Avalanche photodetector – Ansys Optics
- Vertical photodetector – Ansys Optics
Related publications
Reference and related publication format:
- Electro-Optical Simulation and Characterization of DCR and secondary emission in SPADs | youtube video
- B. Novakovic, K. Raymond, G. Gallina, L. Xie, F. Retiere and D. McGuire, "Electro-Optical Simulation and Characterization of DCR and secondary emission in SPADs," 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) , 2022, pp. 3-4, doi: 10.1109/NUSOD54938.2022.9894805.