Simulate an Avalanche photodetector (APD) based on a Ge-on-Si heterostructure with the avalanche multiplication taking place in the Si layer. The optical simulation will be performed using FDTD and the electrical simulation using CHARGE solver. The performance figures of merit of the detector obtained from these simulations, including dark current, responsivity and gain, will be incorporated in an APD compact model which will then be used in a simple circuit simulation in INTERCONNECT.

## Overview

Understand the simulation workflow and key results

This example covers the simulation of a Ge-on-Si Avalanche photodetector with Si multiplication layer reported in reference [1]. The FDTD solver is used to obtain the optical generation rate profile inside the detector due to light absorption. This optical generation data is then fed into the CHARGE solver to characterize various figures of merit (FOM) for Avalanche photodetectors including dark and photocurrent, responsivity and gain. A compact model based on these parameters is created in INTERCONNECT and evaluated by characterizing the response of the device to a modulated signal.

### Step 1: Optical simulation

Light absorption in the Ge layer of the detector is simulated in this step (assuming a single mode TE light arriving within the waveguide for detection) and is converted into a generation rate profile which is needed for the electrical simulation in the next step. This generation rate is averaged over the length of the device in order to simplify the electrical simulation by performing the simulation at the cross section of the device in a 2D plane.

### Step 2: Electrical simulation

The optical generation rate data (representing the concentration of photogenerated carriers inside the device) is imported into the electrical simulation. Then, two separate simulations are carried out to obtain the dark current and the photo-current by disabling and enabling the optical generation object, respectively. The responsivity and multiplication gain of the detector can be calculated from the photocurrent result.

### Step 3: Circuit simulation

The performance figures of merit obtained from the previous step are used to define a compact model for the Avalanche photodetector (APD) in INTERCONNECT to simulate the operation of the device for detecting a low power modulated signal. An eye diagram is employed to evaluate the quality of the detected signal and its variation as a function of the detector gain (multiplication factor).

## Run and results

Instructions for running the model and discussion of key results

### Step 1: Optical simulation

- Open and run the APD simulation file (avalanche_photodetector_optical.fsp) using FDTD
- Right click on "generation rate" analysis group object and select "Run analysis". This will calculate the optical generation data and export the data to a file (apd_10um_G_Ge.mat) that will be imported into CHARGE.
- To visualize the exported generation data, right click on “generation rate” analysis group and select “Visualize” then “G_export”. To have a plot of the generation data at the cross section of the device, select the parameter “z” from the list of parameters in the visualizer window and set the “action” field to “plot y axis”.

### Generation rate profile

The average generation rate profile at a cross-section of the device calculated by the optical simulation is illustrated in the figure below.

### Step 2: Electrical simulation

- Open and run the electrical simulation file (avalanche_photodetector_electrical.ldev). Note that the simulation will eventually diverge at a voltage close to the APD's breakdown, at which point the job in the job manager window will get red in color and job status changes to Job error (as shown below). This is indicative of reaching the breakdown and not a problem in the simulation. Then you need to click "Quit & Save" to preserve the results for the next step.

- Run the script file avalanche_photodetector_IV.lsf to plot and save the dark current results and compare with the reference publication
- Switch back to Layout mode and enable the optical generation rate object. Optionally import the generation rate data into the object (the object is preloaded with data)
- Run the simulation. The simulation will diverge again close to breakdown. Click "Quit & Save" to preserve the results for the next step.
- Run the script file avalanche_photodetector_IV.lsf to plot and save the photo current results and compare with the reference publication
- To obtain the figures of merit of the detector, including responsivity and multiplication gain, run the script file avalanche_photodetector_FOMs.lsf

### Dark and photo current characteristics

The dark and photo current plots as a function of bias voltage obtained from the electrical simulation are compared with the same results from the reference publication [1] (the device with 10um length) in the figures below. It can be seen that the results are in close agreement with the experimental values. It is obvious from the plots that the breakdown voltage of the device is around 10V in reverse bias where there is a sudden jump in the amount of current due to Avalanche effects.

### Responsivity and multiplication gain

Responsivity of the APD (which is defined as the ratio of the photocurrent to the optical input power) as a function of bias voltage is depicted in the figure below (left). It is clear that the responsivity follows the same trend as the photocurrent and will be increased as the bias gets closer to the breakdown voltage. In addition, multiplication gain of the APD (see the Important model settings section for details about how the multiplication gain is defined) as a function of bias voltage can be obtained from the electrical simulation results and is shown in the figure below (right). It can be depicted from the results that the unity gain voltage of the APD (the voltage at which the gain is equal to 1) is about 2V which is consistent with the result in the reference. It is also clear that gains in the order of hundreds can be achieved with this device when operating close to breakdown.

### Step 3: Compact model creation and circuit simulation

- Open the APD simulation file (avalanche_photodetector_circuit.icp) using INTERCONNECT.
- Select the APD component in the circuit, set the parameters "multiplication factor" and "dark current" in Property View window according to the desired bias point. The responsivity parameter is the unity gain responsivity (responsivity for a gain of 1) and should always remain the same regardless of the bias point
- Run the simulation
- Visualize the eye diagram
- Optionally repeat steps 2-4 for a different APD gain after switching to the Design mode

### Eye diagram as a function of detector gain

The variation of eye diagram as a function of the detector gain (multiplication factor ,M) is illustrated in the figures below. It can be seen that the signal detection quality increases as the gain increases and at a gain of 1 (no gain) it is almost impossible the distinguish the signal from noise. This demonstrates the advantage of an APD device compared to a normal detector for detection of low power signals.

## Important model settings

Description of important objects and settings used in this model

### Step 1: Optical simulation

** Boundary conditions **

We have taken advantage of the symmetry of the cross section of the device in the x direction by simulating only half of the device. This has been accomplished by using an anti-symmetric boundary condition in the x min direction which will allow us to preserve the TE modes propagating in the waveguide.

** Source **

A mode source is used with the telecommunications wavelength of 1.55 um to model the optical input. To represent a single mode TE light arriving within the waveguide for detection, the source is set to inject only in fundamental mode of the waveguide which will be a TE mode due to the anti-symmetric boundary condition used in the simulation.

** Generation rate **

The absorbed light in the Germanium is measured using the analysis group "generation rate". This analysis group contains a script that calculates the number of absorbed photons (see appendix) needed for generation rate calculation. The source intensity as well as the name of the output file for generation data can be specified as inputs for this object. The source intensity is set in such a way so that the generation rate is calculated for unity input power (1 W), ie, intensity = input_power / (x_span*z_span). The generation rate profile is averaged in the y direction to export the results in the XZ cross section in CHARGE, which enables us to perform a 2D electrical simulation instead of 3D to increase the simulation speed. Users have the option to consider a full 3D generation data (and thus 3D electrical simulation) without averaging in propagation direction to increase the accuracy but the averaging is proven to be a good approximation based on experience.

### Step 2: Electrical simulation

** Metal contacts **

The device structure in electrical simulation is identical to the optical setup except that the electrical simulation includes the side contacts so that the APD can be biased properly. Since the arms (which include the side contacts) are not involved in light propagation, there is no need to include the side contacts in the optical simulation.

** Norm length **

The "norm length" setting of the CHARGE solver is set to a value same as the length of the APD so that the results are calculated correctly. This is essential since the electrical simulation is performed in 2D and the length of the device in the third dimension should be specified for the solver.

** Dark current simulation **

Since dark current represents the device's output under lack of illumination, for simulating the dark current, the optical generation rate object should be kept disabled.

** Generation rate scale factor **

The "scale factor" parameter of the import generation object can be used to set the actual input optical power since the generation data obtained from optical simulation was created for unity power.

** Recombination material models **

The recombination models for "impact ionization" and "band-to-band tunneling" are enabled in both Ge and Si to model the avalanche breakdown in the APD. The bulk carrier lifetime is reduced to 2e-10 s in both Si and Ge to match the dark current. Since both the Si and Ge layers are less than a micron thin their carrier lifetimes are expected to be much smaller than the bulk values. The surface recombination velocities are set to typical values for the corresponding material combinations.

** Solver settings for better convergence **

The "gradient mixing" option in the Advanced tab of the CHARGE solver has been enabled to help with the convergence of Poisson's and drift-diffusion equations when impact ionization model is turned on.

** Range backtracking for bias voltage **

As the electrical simulation gets closer to the breakdown voltage, convergence becomes harder and a smaller voltage step becomes necessary to avoid divergence. The CHARGE solver therefore needs to dynamically reduce the voltage step size as the simulation get closer and closer to breakdown. This was achieved by enabling the "range backtracking" option in the (collector) electrical contact. The "min interval (V)" parameter was reduced to 1 mV from its default value since the solver requires extremely small voltage steps for convergence at breakdown.

** Responsivity calculation **

The responsivity of the APD was obtained by taking the ratio of the photocurrent with respect to the optical input power (set as the scale factor of the optical generation rate object).

** Multiplication gain calculation **

The multiplication gain of the APD was calculated by first locating the bias point where multiplication gain equals 1 (defined as the bias point where the second derivative of the photocurrent with respect to the voltage is zero [2]), and then by dividing the photocurrent versus voltage curve with the photocurrent at the bias point with unity multiplication gain.

### Step 3: Circuit simulation

** Bit rate **

Random sequence of bits drives the amplitude modulation of the CW laser source. This quantity should be set at the root element, so that it can be accessed by all the elements requiring the bit sequence information such as PRBS generator and the eye diagram analyzer. To set the bit rate at these lower level elements set their bitrate expression field to %bitrate%.

** Photodetector dark current and responsivity **

The photodetector circuit element accepts responsivity parameter as a constant or as a frequency dependent table. Here, In each simulation, we set a constant responsivity that was calculated in step 3 for a single frequency (1.55 um). The responsivity parameter is the unity gain responsivity (responsivity for a gain of 1) and should always remain the same regardless of the bias point fro the detector in the simulation. Dark current is also set as a constant at the desired bias.

** Photodetector bandwidth **

The easiest way to include a finite bandwidth of the photodetector in INTERCONNECT is by adding a low pass RC filter element, connect it to the output of the photodetector element, and set its cutoff frequency to the value of the 3dB bandwidth of the detector. Here, the bandwidth is taken from the value reported in ref [1].

** CW laser frequency and power **

This frequency should correspond to the optical source frequency in step 1 (1.55 um). The power should also represent the signal power to be detected by the detector.

** Photodetector frequency **

This represents the frequency to which other parameters of the detector element correspond to and should be the same as the CW laser frequency.

** Photodetector multiplication factor **

The gain at which the detector currently operates is determined by this parameter and the dark current should also change according to the chosen gain value for most accurate results as the dark current depends on the bias voltage which is based on the desired gain.

## Updating the model with your parameters

Instructions for updating the model based on your device parameters

When updating the model to match your design, it is important to remember that multiple solvers and simulation files are involved in the workflow. Changes must be made consistently across the workflow. Some key parameters are listed below:

Photodetector geometry: Use a different photodetector geometry (typically length or width). This might require changes to the dimensions of the simulation region, generation rate analysis group, mode source and doping profiles as applicable to the simulation type (optical or electrical)

Optical source frequency: Change the optical source frequency to accommodate your own operating frequency

** Material: ** Change photodetector material based on your own design including the absorbing and multiplication layers and also the electrical contacts. You will need to change the materials in both optical and electrical simulations and define appropriate optical and electrical properties if not using the default materials from the database.

** Doping profile: ** Change the doping profile in electrical simulations to represent your own doping profile design. Profiles established using ion implantation can be modeled using the "Implant" doping objects or the doping profile from a process simulation can be imported using the "Import" doping object.

## Taking the model further

Information and tips for users that want to further customize the model

** Quantum efficiency calculation: ** To obtain the quantum efficiency of the detector as a function of bias voltage (or detector length), another post processing step can be performed using the obtained responsivity data. The quantum efficiency (defined as the ratio of collected electron-hole pairs to the number of incident photons) can be calculated by using the following equation: \(QE=Responsivity\times\frac{hc}{e\lambda}\) where \(h\) is the Planck's constant, \(c\) denotes the light speed in vacuum, \(e\) represents the electron charge and \(\lambda\) is the light wavelength.

** 3D electrical simulation: ** In the cases where the device’s cross-section is varying along the light propagation direction (for example, changes in doping profile or device’s dimensions), the electrical simulation should be in 3D. In these cases, the generation data from optical simulation should not be averaged and directly imported into electrical simulation.

## Additional resources

Additional documentation, examples and training material

### Related publications

- Z. Huang, et. al. "25 Gbps low-voltage waveguide Si-Ge avalanche photodiode," Optica, vol. 3, no. 8, pp. 793-798, Aug. 2016.
- H. T. J. Meier, "Design, characterization and simulation of avalanche photodiodes," Ph.D. dissertation, ETH Zurich, no. 19519, 2011.

### See also

### Related Lumerical University courses

## Appendix: Calculating power absorption

Additional background information and theory

As light is incident on the Germanium layer, it gets absorbed. The absorption per unit volume can be calculated from the divergence of the Poynting vector,

$$P_{abs}=-0.5\textrm{real}\left(\vec{\nabla}\cdot\vec{P} \right)$$

It is possible to calculate the absorption directly from this formula, but divergence calculations tend to be very sensitive to numerical problems. Fortunately, there is a more numerically stable form. It can be shown that the above formula is equivalent to

$$P_{abs}=0.5 \textrm{real}\left(i\omega\vec{E}\cdot\vec{D}^* \right)$$

With a little more work, we get the desired result

$$P_{abs}=-0.5\omega\left|{E} \right|^2\textrm{imag}\left(\varepsilon \right)$$

To calculate the absorption as a function of space and frequency, we only need to know the electric field intensity and the imaginary part of the permittivity. Both quantities are easy to measure in an FDTD simulation. The number of absorbed photons per unit volume can then be calculated by dividing this value by the energy per photon:

$$g=\frac{P_{abs}}{\hbar\omega}=\frac{-0.5\left|E \right|^2\textrm{imag}\left(\varepsilon \right)}{\hbar}$$

The absorbed photons will generate electron hole pairs which will be separated out of the depletion region by the electric field and produce a flow of current.