Electro-absorption modulator

Model and simulate a Germanium-Silicon (GeSi) electro-absorption modulator (EAM) on Silicon-on-insulator (SOI). The eigenmode expansion (EME) and CHARGE solvers are used to simulate the modulator, and a compact model for the device is created in INTERCONNECT. Performance metrics of the EAM such as dark current, photocurrent, s-parameters and bandwidth are simulated. The operation of the modulator is also simulated in a simple test circuit.

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Overview

Understand the simulation workflow and key results

An electro-absorption modulator (EAM) modulates the amplitude of light thanks to the change in the absorption coefficient of semiconductor material with an applied external electric field. As an electro-optic device, characterizing an electro-absorption modulator requires both electrical and optical simulations. In this example, we model and simulate an electro-absorption modulator (EAM) based on Germanium-Silicon (GeSi) for Silicon-on-insulator (SOI) applications which is similar to the device presented in reference [2] and its structure is shown in the figure below:

Step 1: Electrical simulation

The first step is to calculate the electric field and charge distribution profile in GeSi as a function of applied voltage using the CHARGE solver. Optionally, the dark current and capacitance, and resistance of the device which are needed for bandwidth estimation can also be calculated in this step.

Step 2: Optical simulation

The carrier concentration and electric field profile obtained from the electrical simulation will be converted into a spatially dependent refractive index to determine how the optical properties (effective index) of the waveguide change as a function of applied voltage. The amount of light transmission through the device and the photogenerated carriers resulting from light absorption will then be extracted in the form of S-parameters and a generation rate profile, respectively. The taper attached to the device can also be simulated in a separate simulation. The optical simulation will be entirely based on the EME solver in MODE which offers superior performance compared to FDTD for simulating light propagation over long distances.

Step 3: 2nd electrical simulation (optional)

Another electrical simulation by CHARGE solver is needed to obtain the photocurrent generated by the device as a result of light absorption by using the generation rate data from the optical simulation. This step can be optional since photocurrent and responsivity are not the desired output of the EAM device (as opposed to output optical signal) and rather a byproduct of light absorption in the device. It is done only for the completeness of the compact model generated in the next step.

Step 4: Compact model creation and circuit simulation

The performance metrics calculated in the previous steps will be used to create a compact model for the EAM device in INTERCONNECT. A simple test circuit is set up in a time-domain simulation to evaluate those performance metrics.

Run and results

Instructions for running the model and discussion of key results

Step 1: Electrical simulation

The electric field profile at the cross-section of the EAM is shown in the figure below at a reverse bias voltage of 3V. It can be noticed that the electric field is mostly concentrated in the waveguide area, enabling efficient modulation of light propagation.

A resistance value of about 96 Ohm is returned in the Script Prompt window.

The junction capacitance and the RC-limited bandwidth of the EAM modulator as a function of applied reverse bias are plotted in the figures below. The bandwidth values are obtained using the slab resistance value from the previous step.

Step 2: Optical simulation

The script also plots the transmission (\(|S_{12}|^{2}\)) vs. bias voltage result from the sweep.  It is obvious that the transmission is reduced as the reverse bias voltage is increased due to the higher light absorption in GeSi, resulting from the increase in the absorption coefficient as the electric field increases at the wavelength in consideration (Franz-Keldysh effect).

The field profile from the “monitor_XZ_field” at bias voltage or -3 V (equivalent to the voltage index of 7) is shown below. The index profile near the junction of the Si and the GeSi/Si waveguide sections is also shown, together with the field profile.

Some points worth noting from the results include:

• The field is mostly confined in the high-indexed GeSi layer as the input mode from the Si waveguide propagates.
• The intensity of the field decays as it propagates due to the absorption in the GeSi layer.
• There is a mode-beating pattern along the propagation direction. This is because the injected mode of the Si waveguide couples into multiple modes that the GeSi/Si sections support.

The above points might need to be considered when trying to improve the current design for better modulator performance.

You can additionally consider the taper sections of the whole device for more realistic simulation results:

The figures below show the E-field intensity in the xy plane as well as at the input and output ports. The input mode at the 0.5 um wide straight waveguide gradually expands as it propagates towards the wider end (1 um) of the taper, lending itself for better coupling to the 1um wide straight modulator section.

Step 3: 2nd electrical simulation (optional)

The photocurrent of the EAM as a function of the reverse bias voltage is shown in the plot below. The higher photocurrent due to higher light absorption at higher bias voltages is clear from the plot.

Step 4: Compact model creation and circuit simulation

The circuit diagram below shows where in the circuit model each data file should be imported.

The results for the bias voltage of 2V and the impulse input signal for a 2V bias (visualized separately from OSC_3) are shown below. The time-domain output optical power of the EAM in response to an impulse reverse bias voltage shows the expected behavior (an impulse with a reduction in output power) and includes some delay in returning to the steady-state due to the limit in the speed of the device (RC limited bandwidth).  The steady-state value of output power also agrees with the S-parameter values obtained in the optical simulation for the corresponding bias voltage assuming a 1mW input power. The slight delay in the start of optical response with respect to the impulse input is a simulation artifact due to the use of digital filter in the optical s-parameter to fit the element transfer function in the time domain however it won’t affect the results which only depend on the decay of the pulse from its max value to steady-state as an assessment of the device’s speed.

The 3dB bandwidth (~300 GHz) result obtained from the time domain optical power impulse response matches very well with the cutoff-frequency result obtained in step 1.

The time-domain photocurrent of the EAM also shows the expected behavior (an impulse with an increase in photocurrent) and includes some delay which is longer than the delay in output optical power due to the additional limitations imposed by transit time-limited bandwidth (see “taking the model further” section for details). In addition, the steady-state value of photocurrent is in agreement with values obtained in step 3 for the corresponding bias voltage.

Important model settings

Description of important objects and settings used in this model

Geometry: As shown in the figure below, in this EAM device, GeSi is grown inside a recessed region created by anisotropic wet etch in a Si rib waveguide.  The length of GeSi along the propagation direction is a very important parameter to reduce the insertion loss in the ON state of the modulator (see appendix: length optimization). The device is meant to be connected to Si strip waveguides 500nm wide, so tapers are added at the input and output of the modulating section of the EAM as shown in the figure, but they can be simulated separately. The geometrical features of the device are parameterized in the optical simulation using setup scripts for “model” and other geometrical objects to ease the changes in the geometry of the device. The same approach can be done in electrical simulation however, it might not be as necessary since the simulation is 2D and only done at the cross-section of the device.

The tapers at both ends of the EAM modulator can be simulated along with the modulator or they can be simulated separately as in this example.  The figure below shows the geometry and structure of the taper simulated in this example. For an example of geometry optimization of a taper, see the waveguide taper example.

Electrical simulation

Charge monitor settings: There are two charge monitors used in the simulation. The “charge_total” is used to obtain the junction capacitance from the total charge measured across the junction by enabling the “integrated total charge option.” The “charge_Si” monitor over the Si waveguide region has the “save data” option enabled with a file name specified, allowing exporting the recorded charge profile in the Si waveguide to the ensuing optical simulation.

Electric field monitor settings: The “Efield” monitor records the electric field profile. You can visualize the result directly from the monitor, which retains the data on the finite element mesh grids (triangular or tetrahedral). This can typically result in ragged edges along with the bounds of the rectangular monitor region due to the lack of data points. Alternatively, you can interpolate the data onto rectilinear mesh grids for better visualization as was done in the script file EAM_Efield.lsf.

Bias voltage: In order to apply a reverse bias to the junction, a negative voltage can be applied to the “anode” contact of the device using an electrical boundary condition. To sweep over a range of bias, the sweep type can be chosen as “range” and the range and number bias points can be adjusted accordingly.

RC bandwidth calculation: The cutoff frequency of the modulator’s optical output is determined by its RC time constant since the Franz-Keldysh effect is almost instantaneous and the output switching speed is limited by how fast the fields can be switched by an applied voltage. To calculate the resistance (R) of the silicon slab, we place a metal contact (“ground” electrical boundary condition) in the center of the GeSi layer (covering the depletion region).  We ground the center and apply a voltage to each of the external contacts (anode and cathode) to measure the resistance of the p and n silicon slabs, separately. To obtain the capacitance, we sweep the bias (with small perturbations at each bias point, dV) and measure the total charge (positive and negative) in the semiconductor region.  We can calculate the capacitance of the modulator from the derivative of the charge with voltage (C=dQ/dV). The RC limited bandwidth then can be calculated using the formula (1/2πRC). All of these are done through the script used for junction capacitance calculation.

Import generation scale factor: The “scale factor” parameter of the optical generation rate import object needs to be set appropriately to account for the actual input optical power to the modulator. As the optical simulation by default assumes an input power of 1 W, the current “scale factor” value of 0.001 corresponds to 1 mW input optical power for the device.

Responsivity calculation: The responsivity of the modulator is calculated by dividing the photocurrent by the input optical power. The script finds the responsivity for 0 V and saves it in a text file (EAM_responsivity_0V.txt). This is done based on the fact that the photodiode element used in the compact model of EAM can only take one value for responsivity and can’t be voltage-dependent. Next, the script calculates the effect of bias on the responsivity by taking the ratio of the responsivity at different biases to the responsivity at 0V and saves this scaling factor in a text file (EAM_scale_resp.txt) to be used in a lookup table in a compact model to account for the variation of responsivity as a function bias voltage. In addition, since the generation rate is voltage-dependent, it should be updated for each bias point and this is done by the script as well.

GeSi material: The direct bandgap of silicon (0.365um) is too far from the desired operational wavelength (1550nm).  Therefore, we use germanium (direct bandgap at 1580nm in a pure state) and add a small percentage of silicon (Ge1-xSix with x=0.75%) to shift the direct bandgap to a wavelength smaller than 1550nm, so that there will be no direct-bandgap absorption in GeSi in the absence of an applied field at this wavelength [1,2]. GeSi is defined as a binary alloy formed of Ge and Si as base materials and the fraction of Ge (1-x) is determined under the material settings of the GeSi geometry object.

Norm length: The “norm length” parameter of the CHARGE solver object determines the dimension of the device along propagation direction (since the simulation is 2D) and is set to the optimized length obtained for the device (see appendix: length optimization)

Optical simulation

Operation wavelength: The operating wavelength for this example was set to 1550 nm based on the reference [2]. It can be set in the “General” tab of the EME solver and is also a required parameter for the nk import object “GeSi” for the correct calculation of the (n,k) values.

Material refractive indices: An n,k import material is created to model the field-dependent GeSi material.  The real part of the index is set to the index of bulk GeSi (n=4.1-0.64x [2]) as n0 parameter in GeSi group object and the imaginary part of the index is calculated from the imported electric field profile based on the Franz-Keldysh model [1] using a lookup table of absorption coefficient as a function of electric field obtained from the model. The charge profile is also imported using an np density grid attribute.  An index perturbation type material is created for silicon that calculates the index using the imported charge density data and the Soref-Bennett model [3]. The parameter sweep changes “V_index” parameter of the electric field and charge density import objects for different bias voltages to account for index perturbations.

S-parameter calculation: The EME ports are used to calculate the optical s-parameters of the modulator.  The ports use a mode source to inject light into the modulator and mode expansion monitors to calculate the s-parameters.

Optical generation rate calculation: The “Generation_rate_EME” analysis group calculates the optical generation rate (photogenerated electron-hole pair density) in the GeSi layer, which is proportional to the optical field intensity. The object always assumes the input optical power of 1W when calculating the generation rate. The dimensions of the object should be such that it covers the device entirely. Since the electrical simulation is 2D, the generation rate will be averaged along the propagation direction for a fair representation of the optical generation in 3D. 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.

EME setup: Due to the geometry of the EAM (see the schematic of the device in this section) and its varying cross-section along the propagation direction, 5 group spans are needed for the EME solver. Two for areas between the tapers and GeSi layer, two for the areas with grooves at the two ends of GeSi layer and one for the flat area in GeSi. The group spans covering the grooves need more than one cell since the cross-section is constantly changing as light is propagating in those areas. Also due to this varying cross-section, the CVCS subcell method would be a better choice for these areas.

Circuit simulation

Compact model: The EAM compact model is created using a compound element in INTERCONNECT. There are two optical ports (opt_1 and opt_2) and two electrical ports (elec_anode and elec_cathode). The main element of the optical part of the circuit is a voltage-dependent s-parameter element (EAM) that loads the s-parameters obtained from optical simulation for all biases. We also add two s-parameter elements (taper_in, taper_out) for the tapers, with data from taper simulation. The electrical part of the circuit has three main portions:

• A voltage-dependent low-pass filter element (LPF_1) to model the RC limited bandwidth of the modulator.
• A look-up table (TABLE_1) to model the dark current as a function of applied voltage
• A photodiode (PIN_1) in conjunction with a look-up table (TABLE_2) to model the photocurrent.  The photodiode models the photocurrent at 0 V. The current from the photodiode is then multiplied with the scale factor loaded in the look-up table to get the photocurrent at different applied voltages.  The photocurrent and dark current are then added and connected to the electrical ports.

Test circuit: The test circuit is comprised of a 1mW CW laser at 1.55um wavelength as the optical source and an electrical impulse with small amplitude to simulate the impulse response of the EAM in the time domain and obtain its 3dB bandwidth by taking a Fourier transform.

Impulse input signal bias: The “bias” value of the electrical impulse input signal will determine the bias voltage of the EAM and its steady-state response. In the parameter sweep, however, the last bias point is set to 2.9V (plus an amplitude of 0.1V) rather than 3V since the compact model does not contain data for any voltage above 3V and might behave unexpectedly when the input voltage at any point in time is above 3V. A 2.9 V bias should have a minimal difference from a 3V bias in terms of the response. An alternative would be a 3V bias and a negative amplitude to avoid going above 3V.

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:

Device geometry: Use a different geometry (typically length or width) based on your design. This might require changes to the dimensions of the simulation region, generation rate analysis group, field and charge monitors, and doping profiles as applicable to the simulation type (optical or electrical)

Doping: Update the model with your own doping profile. In this example, the rib waveguide is doped to create a p-i-n junction such that a strong electric field can be applied inside GeSi. The recess region in Si helps to optimize the overlap of the optical mode in GeSi with the applied electric field while improving the coupling of light in and out of GeSi.  Note that there is small additional doping in GeSi to achieve a more homogenous field profile inside GeSi; it is important to keep this doping small to reduce free-carrier absorption in GeSi.

Surface recombination: Provide surface recombination values for your own design. Surface recombination at various interfaces (especially between Si and GeSi) can be very dependent on the fabrication process, design and material properties so it is important to make sure the values used in the simulation are close to your actual design.

Operation wavelength: To design for a different wavelength, change the frequency/wavelength for the EME solver and specify the wavelength in nk import object. Note that the built-in Soref and Bennett model used for charge-dependent index perturbation only supports two communication wavelengths (1550 and 1310 nm). The model coefficients for other wavelengths need to be defined by the user. Alternatively, the less accurate Drude model can be used which supports a wide range of wavelengths.

Material: Change the materials based on your own design including the absorbing and waveguide layers and 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. An important point to note here is that the absorption coefficient vs. electric field data provided in this example is only valid GeSi with 0.75% Si content. For any different contents of GeSi or a different material, a separate set of absorption coefficient vs. electric field data should be used to accurately model the Franz-Keldysh effect in that material. See reference [1] for more information.  Also, for the waveguide, a different charge to index conversion model might be needed depending on its material.

Taking the model further

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

Charge-dependent index perturbation in GeSi: The effect of doping (charge density) on the index of the GeSi layer is ignored in the simulation since the GeSi layer is very lightly doped. In a different design, however, it might be necessary to include this effect using a similar approach used for the Si layer.

Transit time-limited bandwidth: The bandwidth of the output photocurrent for this device will be limited by the transit time of the photo-generated carriers rather than the RC time constant.  This can be modeled by placing an additional low-pass filter in the compact model (already available in the file as LPF_2 element). The transit time-limited bandwidth can be calculated by performing a transient electrical simulation, where the optical generation object is turned on using an optical shutter and the response of the photocurrent at the contacts is calculated. See the vertical photodetector example for an example of bandwidth estimation using this approach. In the example file, the transit time-limited bandwidth for the modulator is assumed to be 100 GHz and is expected to be very weakly dependent on bias voltage. Since the main output of the EAM device is the optical signal, accurate modeling of the bandwidth limit on the photocurrent may not be necessary.

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 nk import object should be modified such that it can take 3D field data rather than 2D and the generation rate data from optical simulation should not be averaged and directly imported into the electrical simulation.

Convergence

Tips for ensuring that your model is giving accurate results

The default settings of the example provide a reasonable balance between accuracy and simulation time.  The following changes may provide higher accuracy, at the expense of longer simulation time and more required memory:

Electrical simulation

Since the electric field and charge distribution are the ultimate output of the electrical simulation, an adequately refined mesh is essential to ensure the accurate representation of data. This can be verified by looking at the electron and hole capacitance values (Cn and Cp) returned by the junction capacitance script. A reasonably refined mesh will make sure these values are as close as possible. Local mesh refinement around the waveguide and GeSi layer is recommended as this is the area where field and charge distributions overlap with confined optical modes.

Optical simulation

A converged electrical simulation is a requirement for convergence in optical simulation. This means that convergence for the electrical simulation should be reached first and then the electric and charge distribution from the converged electrical simulation should be imported into the optical simulation for further convergence testing.

EME number of eigenmodes: The EME method relies on the modal decomposition of fields into a basis set of eigenmodes; we can specify the maximum number of these basis modes. In the limit where the number of eigenmodes is infinite, the amount of error associated with the EME calculation goes to zero. In reality, only a limited number of modes can be used in an EME calculation, since more modes mean longer simulation time and more memory. It is always a good idea to start with a small number of modes for the initial calculations and increase it as necessary until the result converges. With a large enough number of eigenmodes, even free space propagation can be simulated. The convergence test for the number of eigenmodes can be done very efficiently with the mode convergence sweep tool .

EME Number of cells: If the cross-section of the structure or the material properties is continuously varying along the propagation direction, more cells in the corresponding group span will allow for a more accurate representation of the geometry in the longitudinal direction at the expense of simulation performance.

Mesh refinement in transverse direction: Since the index perturbation will be calculated on a rectilinear mesh and the imported data from the electrical simulation is located on a finite-element mesh, interpolation is necessary which demands an adequately refined optical mesh to ensure accurate interpolation. Generally, a mesh grid size of equal or less than that of the electrical simulation is recommended for the optical simulation to reach convergence. Local refinement of the mesh around the waveguide and GeSi layer through mesh override objects can ensure adequate refinement only in areas necessary.

Additional resources

Additional documentation, examples and training material

Related publications

1. J. Liu, “GeSi Photodetectors and Electro-absorption Modulators for Si Electronic-photonic Integrated Circuits,” Ph.D. dissertation, Dept. of Mat. Sci. and Engg., Massachusetts Ins. of Tech., MA, 2007.
2. S. A. Srinivasan, et al., “50Gb/s C-band GeSi Waveguide Electro-Absorption Modulator,” Optical Fiber Communication Conference, Tu3D.7, 2016.
3. R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quant. Electron., vol. QE-23, no. 1, 1987.

See also

• Charge to index conversion
• Travelling Wave Modulator
• Ring Modulator

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Appendix

Additional background information and theory

Working principle

An electro-absorption modulator (EAM) works based on the change in the absorption coefficient of semiconductor material with an applied external electric field.  The electric field tilts the bands of the semiconductor allowing for photon-assisted tunneling from the valence to the conduction band.  The consequence of this process is an exponential tail below the zero-field bandgap, as shown in the figure below for a typical direct bandgap semiconductor. Note the exponential tail below the zero-field bandgap (EG) and the oscillations above it.  Only the behavior near the bandgap is important for the operation of the EAM. This sub-picosecond effect, known as the Franz-Keldysh effect, can be used to switch the state of a semiconductor from transparent (ON state) to absorbing (OFF state) extremely fast, resulting in high modulation speed, usually only limited by how fast the electric field can be applied [1,2].

Length optimization

Due to the recurring coupling of light between Si and GeSi layers in the EAM device, the length of the GeSi layer is important to ensure efficient coupling of light from GeSi to Si waveguide at the far end of GeSi layer and minimize reflections. The length optimization is independent of the absorption coefficient (imaginary part of refractive index) and thus can be performed prior to step 1. The electrical simulation needs the device’s length as “norm length” parameter. The propagation sweep capability of EME solver can be used to optimize the length of GeSi layer without the need to run a separate simulation for each length value. Follow the steps below to obtain the sweep results:

The sweep results are shown in the figure below. From the results, it is obvious that at certain length values, there are some maxima in transmitted and some minima at which no light is coupled to Si at the end of GeSi layer. So any length value corresponding to a peak in transmitted power would be suitable for the design however, longer lengths would negatively affect the bandwidth of the device due to larger capacitance while offering better modulation efficiency because of higher light absorption.