This example has been updated. Find the latest version at Travelling Wave Mach-Zehnder Modulator.
In a Mach-Zehnder structure, changes to the effective index of the waveguide will result in a differential change in phase of the light propagating in that arm, which can be used to modulate an optical signal . 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 the CHARGE solver. To calculate the effect of the change in the carrier density on the waveguide loss and effective index, a finite-element eigenmode simulation will be run using the FEEM solver. An (n,k) Material grid attribute in FEEM will be used to model the changes in the real and imaginary parts of refractive index of the material (silicon) according to a formulation in a work by Soref et al.. For a more detailed description of this model, please visit the section on Charge to index conversion.
The cross-section of the waveguide with lateral PN junction can be set up in CHARGE. Open the tw_modulator.ldev project in CHARGE. The dimensions of the waveguide reflect the structures in the referenced paper by T. Baehr-Jones, et al. . 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 PN junction profile in the waveguide. Analytic models for ion implantation doping are used to construct the doping profile, however doping profiles determined from a process simulation could also be used.
To establish an accurate correspondence between the simulations and measured results, the measured capacitance vs. voltage characteristic was fit by setting the PN junction profile in the waveguide. The size and position of the analytic models were adjusted until a match between simulation and experiment was obtained. Please see the section on C-V calculations (below) for details on the accurate simulation of PN junction capacitance.
Load the project file tw_modulator.ldev, which is set-up for a DC bias sweep from -0.5 to 4 V applied to the cathode, such that the diode is swept from forward to reverse bias. The carrier densities (electrons and holes) are required for the material index model in the optical simulation. The carrier densities can be extracted directly from a DC bias sweep by saving the values recorded by the charge monitor on the waveguide (monitor_charge). The monitor specifies a data file where the results will be recorded. Running the simulation will automatically save the data to the appropriate file.
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
The script file twmod_junction_capacitance.lsf can be used to set up a DC sweep over the range from -0.4 to 4 V with a perturbation of ΔV = 25 mV at each step. The mesh-override in the waveguide region will be enabled, and the option to save the recorded carrier density in the charge monitors will be disabled to prevent overwriting the previous simulation results. The same script file will also perform the capacitance calculation and plot the results. In the plot below, the result is compared to the value obtained in the reference .
The silicon slab introduces a series resistance in series with the junction capacitance of the Mach-Zenhder structure. To calculate this resistance, we can run a steady state simulation on each half of the slab and combine the resulting resistances. The script file tw_modulator_Rslab.lsf places a metal contact "ground" in the place of the pn junction (rib waveguide) and runs two simulation to calculate the slab resistances on the p and n side of the device. It then calculates the total resistance of the silicon slab. The value of Rslab from the simulation is found to be approximately 1.9 ohm.
The small-signal equivalent circuit of the Mach-Zehnder structure is similar to that of a pn junction diode in reverse bias. The pn-junction can be modeled by capacitor and in parallel with a resistor. The capacitor models the junction capacitance of the pn junction and the resistor models the dynamic resistance. The capacitor-resistor combination is in series with another resistor that models the resistance of the silicon slab. In this section of the example, we will use the small-signal ac solver (ssac) in CHARGE to generate the small-signal equivalent circuit of the Mach-Zehnder structure.
Download the tw_modulator_ssac.ldev and twmod_ac_capacitance.lsf files in the same folder. Run the script file. It will sweep the DC bias voltage from -0.4 to 4 V in 12 steps and perform small signal analysis at each point. The frequency of the ac signal will be swept from 1 MHz to 10 GHz. The script will then calculate the impedance Ztotal for the device. Using the equivalent circuit of the structure, we can see that by subtracting the slab resistance (Rslab) from Ztotal, we can get the impedance of the pn junction (Cj || Rd). From that the script calculates the junction capacitance of the pn junction at different frequencies and plots it along with reference and steady state simulation data.
The plot above shows that the junction capacitance remains practically unchanged up to very high frequencies of operation. This is expected since the pn junction is operating in reverse bias. The small-signal and steady state simulation results are all in good agreement with the measured data from Ref. . The script also plots the impedance of the modulator on a smith chart for a reverse DC bias of 4 V. The plot below was generated for a much higher resolution in the frequency spectrum (10 points per decade).
Simulation tip: 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.