In this example, we will characterize the performance of an interleaved junction microring modulator using CHARGE (electrical simulation) and FDTD (optical simulation).
Theory
In a microring structure, changes to the effective index of the waveguide will result in a shift in the resonant frequency, which can be used to modulate an optical signal [1]. This change in effective index is driven by modulation of the carrier density in an electrically active waveguide. To simulate the distribution of the charge carriers, a self-consistent simulation of the charge and electrostatic potential is performed using CHARGE. To calculate the effect of the change in the carrier density on the waveguide loss and effective index, an FDTD simulation will be run. An NP density grid attribute object in FDTD will take the carrier density information and calculate the corresponding changes in the real and imaginary parts of refractive index of the material according to a formulation in a work by Soref et al.. For a more detailed description of this structure group and the formula, please visit the section on Charge to index conversion.
Simulation Setup
The interleaved junction structure of the waveguide modulator can be set up in CHARGE. Open the interleaved_modulator.ldev project in CHARGE. The dimensions of the waveguide reflect the structures in the referenced paper by J. C. Rosenberg, et al. [1]. The component is fabricated in an SOI process, where a thin layer of silicon is formed epitaxially on a buried oxide. The silicon is then etched back to pattern the waveguide structures. The peak doping concentrations and dimensions of the doping profile in the referenced paper are used to define the interleaved PN junction profile in the waveguide. Analytic models are used to construct the doping profile, however doping profiles determined from a process simulation could also be used.
In the referenced device, interleaved PN junctions are formed by alternately doping p-type and n-type fingers along the length of the waveguide [1]. The resulting structure and doping profile has mirror symmetry along the lines in the middle of each doping finger. This is illustrated in the top-down view of the waveguide sketched in the figure below. Lines A-A' and B-B' are lines of mirror symmetry. Because the CHARGE simulation region conserves charge, these lines can be used to set the boundaries of the simulation region in the direction of optical propagation. The distance between the cross-sections is the half-pitch of the interleaved dopants (λ/2).
The layout for the optical simulation mirrors that of the electrical simulation. Open the interleaved_modulator.fsp file in FDTD. The electrical simulation is used to calculate the distribution of electrons and holes in response to the applied voltage. To represent a material whose index varies spatially in response the distribution of the electrical carrier density, NP density grid attribute objects are used. These objects import the spatial charge distribution data calculated with CHARGE and perform an interpolation to the rectilinear grid in FDTD. There are several properties associated with each of these materials that can be set through the properties of the structure group. Details of the properties can be found in the discussion of the Mach-Zehnder modulator example.
|
Note: the electrical simulation must first be run in CHARGE to generate the data required to construct the waveguide with a spatially varying index in FDTD. |
To calculate the phase and loss response of the waveguide as a function of the applied voltage, a parameter sweep is used. Users with multi-processor computers or access to concurrent computing resources (e.g. a compute cluster) can take advantage of these extra processors by configuring the resources in MODE to utilize all available cores during the parameter sweep. The results of the parameter sweep (phase and loss) can be analyzed and plotted using the ilmod_txresponse.lsf script file provided.
Results and Discussion
Carrier Density vs. Voltage
Load the project file interleaved_modulator.ldev, which is set-up for a DC bias sweep from 0-1V applied to the cathode, such that the diode is operated in reverse bias. The carrier density (electrons and holes) is required for the material index model in the optical simulation. The carrier densities will be extracted directly from a DC bias sweep by saving the values recorded by the charge monitor (monitor) into a ilmod_carriers.mat file. Note that mesh overrides have been added to the waveguide region to ensure a higher resolution of the charge density recorded for the optical simulation.
When the simulation is complete, execute the following script commands to export the data to a file which will be read by FDTD for the optical simulation.
carrier_density = getresult('CHARGE::monitor','charge');
V = pinch(carrier_density.V_cathode);
matlabsave('ilmod_carriers',carrier_density,V);
DC Capacitance vs. Voltage
The PN junction capacitance is simulated at DC by calculating the numerical derivative: C = dQ/dV. The total charge can be easily calculated using a charge monitor, and enabling the total charge calculation. The monitor will integrate the carrier density (electrons and holes) over the volume of simulation. By running two simulations at bias voltage V and V+ΔV, the capacitance can be estimated as
$$C_{n, p} \approx \frac{Q_{n, p}(V+\Delta V)-Q_{n, p}(V)}{\Delta V}$$
The script file ilmod_CV_analysis.lsf can be used to set up a DC sweep over the range from 0-1V with a perturbation of ΔV=25mV at each step. The same script file will also perform the capacitance calculation and plot the results as shown below. In the referenced paper, a value of 0.65fF/um was reported at a 1V bias [1].
|
Note: the total charge integration is sensitive to the mesh density, particularly in the region containing the PN junctions. It is recommended that the simulation first be run with a coarse mesh to ensure proper configuration and confirm the trend in the results. A coarse mesh may result in a discrepancy between the capacitance calculated from the electron density and that calculated from the hole density. Once the simulation configuration has been verified, a mesh constraint may be applied in the waveguide region, and the mesh refined further until the result converges (namely Cn = Cp). |
|
Note: the calculated PN junction capacitance will depend strongly on the doping profile. In this example, peak doping concentrations were chosen to match the values reported in the referenced paper, but analytic doping profiles were used to mimic the implants in the fabricated devices. This will affect the simulated capacitance. |
Loss and Vπ-L
Open the project interleaved_modulator.fsp in FDTD. Each of the semiconductor geometries from the CHARGE simulation are constructed as an NP density grid attribute object which account for the spatial variation in the index as a function of the applied voltage. After opening the simulation file, run the following lines of script which will load the charge density data from CHARGE simulation into the NP density grid attribute objects in proper direction along the waveguide length. Before running the script, make sure the ilmod_carriers.mat file generated by CHARGE in the previous simulation is located is the same folder as the FDTD simulation file:
matlabload("ilmod_carriers");
elements = carrier_density.elements;
n = pinch(carrier_density.n);
p = pinch(carrier_density.p);
x = carrier_density.x;
y = carrier_density.y;
z = carrier_density.z;
V = carrier_density.V_cathode;
y = -y;
carrier_density_mirrored = unstructureddataset("carrier_density",x,y,z,elements);
carrier_density_mirrored.addparameter("V_cathode",V);
carrier_density_mirrored.addattribute("n",n);
carrier_density_mirrored.addattribute("p",p);
select("wg1");
importdataset(carrier_density_mirrored);
select("wg2");
importdataset(carrier_density);
select("wg3");
importdataset(carrier_density_mirrored);
select("wg4");
importdataset(carrier_density);
To sweep the range of applied bias voltages, a parameter sweep is used. Choose the parameter sweep "voltage" and click Run. The sweep will run over the bias voltages from the electrical simulation with CHARGE, and will update the material index in the waveguide at each step.
After the simulation is complete, the results of the parameter sweep can be plotted using the ilmod_txresponse.lsf script file provided. The script will analyze the electric field profile and transmission after one period in the waveguide. The fundamental mode in the wave guide is a TE mode. Therefore, the phase from the x-component of the electric field (Ex) at the midpoint of the waveguide is averaged to determine the change in phase over one period of the interleaved dopants. The fraction a of a full π phase shift can then be determined, yielding the full length required for a phase shift of π radians (Lπ):
$$\Delta \varphi(V)=\frac{2 \pi}{\lambda_{0}} \Delta n_{e f f}(V) L_{\lambda}=a \pi \rightarrow L_{\pi}=\frac{L_{\lambda}}{a}$$
The figure of merit VπL is the product of Lπ and the corresponding voltage V. The loss can be calculated directly from the transmission, and normalized to the doping period (560nm) to obtain the values in dB/cm. The figure of merit and loss are plotted below. These results are consistent with the values reported in the referenced paper: VπL 0.76V-cm at 1V and a loss of 35dB/cm [1]. The difference between simulated and measured results can be attributed to the lack detailed information about the doping profile as well as additional loss in experiment due to surface roughness in waveguide.
(a)
(b)
Reference
J. C. Rosenberg, et al., "A 25 Gbps silicon microring modulator based on
an interleaved junction," Optics Express, 20, 25411 (2012)