In this example, we will study the performance of an avalanche photodetector (APD) with a low breakdown voltage of around 10 V using full electro-optical simulations. This APD is 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. The photodetector under investigation is taken from the work of Z. Huang, et.al. published in Ref. .
The APD structure is set up as a 3D simulation in FDTD with the lengths of 10 and 50 microns, which can be seen in the apd_optical_10um.fsp and apd_optical_50um.fsp files. Ref.  reports dark and photo I-V curves for a 10 um long APD and responsivity for a 50 um long APD, which we can verify using both simulations. We have taken advantage of the symmetry of the cross section in the x direction. An anti-symmetric boundary is used in the x min direction to preserve the TE mode.
In the FDTD simulation, we have used a mode source with the telecommunications wavelength of 1.55 um to model the optical input. The injected mode then propagates down the device and the absorbed light in the Germanium is measured using the analysis group, "generation rate." It contains a script that calculates the number of absorbed photons. The source intensity as well as the name of the output file can be specified. 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. A .mat file is generated once the script of the analysis group is run. This .mat file is then imported and superimposed onto the structure in CHARGE as an optical generation object.
The CHARGE project files apd_electrical_10um_dark.ldev, apd_electrical_10um_photo.ldev, apd_electrical_50um_dark.ldev, and apd_electrical_50um_photo.ldev contain the setup for the electrical simulation which is identical to the optical setup. The only difference is that the electrical simulation files include the metal contacts so that the APD can be biased. The simulation setup in the four project files are identical except for the optical generation rate where the .mat file from simulating the 10 um APD in FDTD (apd_10um_G_Ge.mat) is loaded into the first two files and the one from the 50 um APD simulation (apd_50um_G_Ge.mat) is loaded into the latter two. The "norm length" of the CHARGE solver is also set to the appropriate value for each file (10 um for the first two and 50 um for the latter two). For the files simulating the dark currents the optical generation rate is kept disabled. The uniformity of the APD along the propagation direction of light is utilized by performing a 2D simulation instead of 3D. The length of the APD is set to 10 (or 50) micron through the "norm length" property of the solver region. An import generation object is used to import the generation data saved by FDTD. The "scale factor" parameter of the import generation object can be used to set the input optical power since the input data was created for unity power in the optical simulation.
Multiple doping objects are used to model the doping profile of the APD. The p+ doping in Ge layer is assumed to have a Gaussian profile in the CHARGE simulation with the doping concentration at 1e18 cm-3 at the surface and 1e17 cm-3 at the bottom. This agrees well with Ref.  where they report a doping density of 1e18 cm-3 in Ge using ion implantation (which has approximately Gaussian profile). The p- doping in Si is assumed to be constant in the CHARGE simulation and equal to 2e17 cm-3.
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 "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 is turned on.
NOTE: As the electrical simulation gets closer to the breakdown voltage, convergence becomes harder and a smaller voltage step becomes necessary to avoid divergence. The electrical 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. With this option enabled, the solver will indicate a divergence at the end of the simulation (depicting that the breakdown has been reached). Users should click "Quit and Save" at this point to ensure the simulation data up to the breakdown point is saved in the file.
Results and Discussion
Open the apd_optical_10um.fsp file and run it. Once the simulation is done, run the script in the analysis group, "generation rate." The script can be run by either entering the command runanalysis; at the script prompt, or open the edit window for the generation rate object and click on the run script button. The script in the analysis group should make a plot of the generation rate versus position for the planar structure. Several plots will be generated, including the plot of G that is exported to a .mat file for CHARGE.
In the script prompt of the generation rate object, the generated current and the maximum generation value amongst others will be printed. The script then saves the results of the generation rate versus position to apd_10um_G_Ge.mat. The simulation file for the 50 um long APD simulation is also included (apd_optical_50um.fsp).
Open the apd_electrical.lsf script file in CHARGE. The script is set up to load the apd_electrical_10um_dark.ldev and the apd_electrical_10um_photo.ldev files and plot the dark and photocurrent of the 10 um long APD. Prior to using this script ldev project files should be run. Once the simulations end the script will plot the dark and photocurrent from the simulation along with the dark and photocurrent values reported in Ref.  for a 10 um long device (saved in the apd_10um_IV_data_publication.mat file). As shown in the figure below the simulation results are in very good agreement with the reported experimental values . The breakdown voltage of the APD is found to be about -10 V.
NOTE: 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.
The script will also calculate the responsivity of the APD 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) and plot it as a function of bias voltage (Fig. below-left). Finally, the script will calculate the multiplication gain of the APD 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 ), and then by dividing the photocurrent versus voltage curve with the photocurrent at the bias point with multiplication gain = 1. The plot below (right) shows the multiplication gain of the 10 um APD as a function of bias voltage. The unit gain voltage from our simulation is -3 V, which compares well with the reported experimental unit gain voltage of -2 V from ref.  considering that they are both much smaller than the breakdown voltage of approximately -10 V.
The script file apd_electrical.lsf can also be used in a similar manner to plot the dark and photocurrent of the 50 um long APD along with its responsivity and multiplication gain curves. To do this simply change the length parameter (L) to 50 (um) and the script will use the corresponding files. The plots below show the I-V characteristics, responsivity, and multiplication gain of the 50 um long APD. The responsivity at unit gain voltage for the 50 um long APD is around 1 A/W in our simulation and it is in a very good agreement with the experimentally measured value of 1.05 A/W reported in Ref. .
- 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.