In this example, we will study the performance of an avalanche photodetector (APD) with a low breakdown voltage of around 5 V using full electro-optical simulations. This APD is based on a Ge-on-Si heterostructure with the peak electric field being near the Ge-Si interface, so that both materials contribute to the avalanche multiplication. The optical simulation will be performed using FDTD and the electrical simulation using CHARGE. The photodetector under investigation is taken from the work of H. T. Chen, et.al. published in ref. .
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The APD structure is set up as a 3D simulation in FDTD with the length of 10 microns, which can be seen in the apd5V_optical_10um.fsp file. 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 1.31 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. File apd5V_10um_G_Ge.mat 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 apd5V_electrical_10um_dark.ldev and apd5V_electrical_10um_photo.ldev contain the setup for the electrical simulation which is identical to the optical setup, except for the following two differences. The first difference is that the electrical simulation files include the metal contacts so that the APD can be biased. The second difference is that the Ge layer is on top of the Si layer, instead of being slightly buried into the Si layer. This was done to allow an unambiguous import of the 1D doping profile given in ref.  along the growth direction. In turn, Si layer was made thinner, because there is no overlap between Si and Ge anymore. This does not affect the electrical characteristics much.
The simulation setup in the two project files are identical except for the optical generation rate where the .mat file from simulating the 10 um APD in FDTD (apd5V_10um_G_Ge.mat) is enabled only in the second file. The "norm length" of the CHARGE solver is also set to 10 um for each file. For the files simulating the dark currents the optical generation rate is kept disabled. The uniformity of the VPD along the propagation direction of light is utilized by performing a 2D simulation instead of 3D. The length of the VPD is set to 10 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 the Ge layer and n+ doping in the upper part of the Si layer are imported from ref.  as 1D doping profiles along the growth direction. The n+ doping in the lower part of the Si layer is a continuation of the imported doping profile and is assumed to be Gaussian with 1e19 cm-3 at the lower surface. The n++ doping region around the Si-Al contact is assumed constant and equal to 1e20 cm-3.
The recombination models for "impact ionization" and "band-to-band tunneling" are enabled in Ge and Si material regions to model the avalanche breakdown in the APD. Impact ionization parameters for Ge are taken from Decker and Dunn (ref. ). The measured junction temperature in ref.  is 75 K, while we assumed the junction temperature in the simulated device is room temperature (the authors in ref.  state that they used external current limiting circuit). Since the breakdown voltage for impact ionization has a positive temperature gradient, we increased the impact ionization coefficient from ref.  by around 20% to take into account the effect of reduced temperature in the simulated device. Band-to-band-tunneling (BTBT) parameters in the Ge layer are taken from Kao et al. (ref. ). For BTBT parameters, instead of pure Ge we assume Si0.01Ge0.99 and use linear interpolation in alloy content. This gives a better match for the slope of the IV curve and can be justified by the fact that the Ge layer is buried into the Si layer, so there may be some alloying at the interface.
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 (anode) 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 apd5V_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 apd5V_10um_G_Ge.mat.
Open the apd5V_electrical.lsf script file in CHARGE. The script is set up to load the apd5V_electrical_10um_dark.ldev and the apd5V_electrical_10um_photo.ldev files and plot the dark and photocurrent of the 10 um long APD. Prior to using this script the 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 apd5V_10um_IV_publication.mat file). As shown in the figure below the simulation results for the dark current are in a very good agreement with the reported experimental value in ref. , while the photocurrent (at -19.8 dBm) has a larger discrepancy, the reasons for which will be given below. The breakdown voltage of the APD is found to be about -5.5 V.
NOTE: The bulk carrier lifetime is reduced to 4e-10 s in Ge to match the dark current. Since the Ge layer is 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 (ref. )), 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 gain vs. bias curve from our simulation compares well with the reported experimental gain.
NOTE: The experimental responsivity is reported as 0.3 A/W at -1 V bias in ref. . Our simulated responsivity is several times higher at around 1.1 A/W as can be seen in the plot above left. The discrepancy comes from the fact that our FDTD simulation at 1.31 um shows almost complete absorption in 10 um long Ge waveguide, so that the resulting responsivity is very high (our photocurrent is much larger for the given input optical power). Better optical parameters (refractive index) may be needed to more accurately model the Ge layer from ref. . In addition, the authors in ref.  commented on the fact that their responsivity is suboptimal and discussed several possible loss mechanisms responsible for low responsivity: light absorption in the contacts, free-carrier absorption, valence band-filling induced Ge absorption reduction, and a low photo-carrier collection efficiency. We did not attempt to model these effects.
 H. T. Chen, et. al. "25-Gb/s 1310-nm Optical Receiver Based on a Sub-5-V Waveguide-Coupled Germanium Avalanche Photodiode," IEEE Photonics Journal, vol. 7, no. 4, p. 7902909, Aug. 2015.
 David R. Decker and Charles N. Dunn, "Determination of Germanium Ionization Coefficients from Small-Signal IMPATT Diode Characteristics," IEEE Trans. Electron Devices, vol. ed-17, no. 4, Apr. 1970.
 K.-H. Kao, et al., "Direct and Indirect Band-to-Band Tunneling in Germanium-Based TFETs," IEEE Trans. Electron Devices, vol. 59, no. 2, Feb. 2012.
 H. T. J. Meier, "Design, characterization and simulation of avalanche photodiodes," Ph.D. dissertation, ETH Zurich, no. 19519, 2011.