## Abstract

An original method to simulate depletion-based silicon modulators based on an analytical description of the active region is presented. This method is fast and efficient in particular for performance optimization. It is applied for a lateral diode integrated in a rib waveguide, and a comparison is performed with classical 2D numerical simulation. A very good agreement is obtained, showing the accuracy and efficiency of this analytical method.

©2011 Optical Society of America

## 1. Introduction

Silicon photonics could revolutionize the future of optical telecommunications and computing. Its development requires the demonstration of high-speed optical link using optical sources, modulators and photodetectors integrated in the silicon platform [1,2]. For silicon modulators, more investigations are required to obtain devices showing at the same time, large bandwidth, low insertion loss and compactness (i.e. large efficiency). Up to now, most of the silicon modulators use free carrier concentration variation to obtain refractive index variation [3] exploiting carrier accumulation in MOS capacitor [4], carrier injection in pin diode [5] or depletion inside PN, or PIPIN diodes [6–8]. All these structures need to be optimized as a function of the application in terms of structure geometry and doping level. The number of parameters is considerable which requires efficient and fast simulating tools.

This paper reports a new, efficient and rapid method to simulate silicon modulators in order to calculate key parameters (V_{π}L_{π} and insertion loss). The first step in the simulation process is to define appropriate test structures. As a carrier depletion-based lateral diode is used for illustration, the choice of test structures is based on an optimization of the modulation efficiency previously reported in [9]. The main points of the modulation efficiency optimization and the main results are recalled in section 2. Section 3 provides complete details of the proposed analytical method to simulate the modulator performance, which is based on an analytical description of the active region and of the optical mode. Modulation performance using this method is evaluated in section 4, in comparison with results obtained from standard 2D simulation numerical tools.

## 2. Optimization of the modulation efficiency in a carrier depletion modulator

Different active regions have been used and compared to achieve a depletion based silicon modulator [10]. In the following, we focus on a lateral PIN diode inserted into a rib waveguide in which SiO_{2} cladding is used, as presented in Fig. 1a
. Such a structure covers all different active regions from PN to PIPIN diodes. Indeed, PN diode is a particular case of PIN where the width of the intrinsic region is reduced to zero. PIPIN diode [7] can be seen as a PIN diode where an intrinsic region is added in the P part. This additional intrinsic region directly influences the bandwidth and the optical losses, which are reduced in comparison with PIN diode, but it does not influence the modulator efficiency, which depends only on the junction itself.

Three parameters are used to describe the PIN junction: the width of the intrinsic region w_{i} and the concentrations of the p- and n-doped regions N_{A} and N_{D}, respectively. The rib waveguide is also defined by three parameters: the width of the rib W, the height of the silicon under the rib H, and the height of the slab h (Fig. 1a). The aspect ratio r is defined as the ratio between h and H (r = h/H), and is used to characterize the different waveguide structures.

The modulation efficiency optimization method is separated into three steps. First, the waveguide geometry is optimized to choose waveguide geometries leading to single mode propagation with large optical confinement in the region where the carrier depletion occurs. In a second step the active region is optimized to adjust the doping level in accordance to the modulator parameters. Finally, both optimizations are used together to define the optimal phase shifters for modulation efficiency.

To obtain a large effective index variation, the optical mode confinement should be as large as possible. To calculate the optical confinement as a function of the waveguide geometry, it is possible to calculate Filling Factor (FF), which gives the fraction of the optical power located in a defined space region. The FF calculation is done hereafter for silicon on insulator waveguides having a rib height H of 400 nm and for different values of width (W) and aspect ratio (r). The region Γ used to calculate the filling factor is located in the middle of the rib and it is 100 nm wide as represented in Fig. 1b. For each value of r, the W that provides the maximum FF is called W_{FF} and is reported in Fig. 2a
. In addition, the waveguide geometry has to ensure single mode condition. For each value of r, single mode waveguides are obtained when the width of the rib is lower than W_{single mode}, which is also reported in Fig. 2a. For r above 0.5 W_{single:mode} comes from Ref 11, whereas mode solver calculations have been used to calculate W_{single mode} for r below 0.5.

Finally, optimal waveguides are single mode waveguides with large optical confinement. If W_{FF} is lower than W _{single-mode} then W_{FF} is the width of a single mode waveguide with maximum confinement. If W_{FF} is larger than W _{single-mode,} then the waveguide with maximum confinement cannot be used as it is multimode, and W_{single-mode} is the width of single mode waveguide with the largest optical confinement. For each value of r, the optimal width is then the lowest value between W_{single mode} and W_{FF}.

With no other consideration, it could be possible to compare the filling factor for the different optimal waveguides, and to choose r and W to achieve the maximum optical confinement. However, due to the complexity of the modulator active region it is preferable to keep all possible values for the aspect ratio (r), associated with the optical width as defined just above. These different geometries will be compared next.

Effective index variation of the guided mode is obtained thanks to the removal of carriers originally present at equilibrium in the structure. It has been shown in Ref. 9 that the modulation efficiency optimization of the active region relies on the optimization of the product Δn × δ where Δn is the local refractive index variation, and δ the width of the region where the refractive index variation occurs. For carrier concentration variations below 10^{19}cm^{−3}, holes are the most efficient carriers for refractive index variation [3] so δ is the depleted region in the p side of the diode when bias goes from 0 to V_{π} (hereafter called δ_{p}).

We can notice that if the space charge regions stay inside the rib of the waveguide when the reverse voltage increases, the carrier density in the rib is almost uniform in silicon along the y direction. A 1D model can then be used to calculate δ_{p} by solving the Poisson equation, and in the simpler case of PN diode, δ_{p} is given by:

_{b}is the built-in voltage, and V the applied voltage. For a PIN diode, such an analytical expression cannot be deduced, but the Poisson equation can be solved numerically to obtain δ

_{p}as a function of N

_{A}, N

_{D}and w

_{i}, the width of the intrinsic region between P and N parts of the diode. Δn is the refractive index variation due to depletion in this δ

_{p}region, and is calculated using equations from Ref. 3 assuming that the hole concentration variation is uniform (equals N

_{A}) inside the depleted region (δ

_{p}). Product Δn × δ

_{p}for V

_{π}= 5 V has then been determined as a function of N

_{A}, N

_{D}and for different values of parameter w

_{i}. Couples (N

_{A}, N

_{D}) giving the maximal value of Δn × δ

_{p}are the optimal doping levels for any optical modulator based on carrier depletion in a PIN junction with bias voltage of 5 V. Optimal values of N

_{A}which achieve the maximum modulation efficiency for each w

_{i}and N

_{D}are plotted in Fig. 2b.

The last part of the modulation efficiency optimization consists in using optimized waveguide geometries (W and r as reported in Fig. 2a) and optimized active regions (N_{A}, N_{D} and wi as reported in Fig. 2b) to put optimal active regions inside optimal waveguides. The maximal value of Δn_{eff} is obtained when the active region is centered under the rib. This condition imposes placing the p-doped depleted region in the middle of the rib. The junction position depends then on the width of the region that can be depleted by the maximum voltage. Indeed, to obtain good depletion efficiency, the n-doped depleted region should be kept inside the rib when the bias increases to ensure a complete depletion of the p-doped region from the bottom to the top of the waveguide. By fulfilling the above conditions, each optimized active region can be placed inside an optimized waveguide. Figure 3
presents optimized N_{A} and N_{D} doping concentrations as a function of r obtained for several w_{i}. It can be noted that all waveguide width possibilities described in Fig. 2a are not accessible for each different values of w_{i}. For example, with w_{i} = 200 nm, maximum N_{A} and N_{D} doping levels are 1.5x10^{17} cm^{−3} and 1x10^{19} cm^{−3}, respectively. The waveguide width has to then be higher than 500 nm.

This method finally gives optimal doping concentrations as a function of the waveguide geometry to maximize modulation efficiency. Performance evaluation of these structures is carried out using an analytical model that is described in next section.

## 3. Analytic model for performance evaluation

The usually used method to evaluate modulator performance in term of modulation efficiency and optical loss is to use commercial softwares like Silvaco or ISE. They calculate carrier density in the diode as a function of the bias voltage. To convert the carrier density into effective index and absorption coefficient variations, an optical mode solver is used [9]. This method requires time and computing resources. For example, to change only one doping concentration of the active region, we need to calculate the carrier density in the structure using the electrical software, and to run the mode solver for each bias voltage. In this section we propose a simplified and accurate model to evaluate the performance of different phase shifter structures in a reduced time. In silicon depletion-based modulators the effective index variation Δn_{eff} is low and can be approximated by [12]:

^{2}is the normalized optical intensity of the fundamental guided mode and Δn(x,y) the local variation of the refractive index in silicon.

This equation can be simplified by three different assumptions that are described hereafter. The first one is to consider the problem uniform along one specific direction. Due to the diode orientation (Fig. 1), the structure is supposed uniform along the y direction which is the direction parallel to the active junction. Effective index variation is then given by:

_{eff}which is given in the following equation as a function of α(x), the local value of the absorption coefficient in the waveguide, assumed to be uniform along the y direction.

The second assumption is to approximate hole and electron distribution variations with uniform regions, whose widths are δ_{p} and δ_{n} (widths of the space charge extension variation in the P and N regions which are calculated using the Poisson equation in 1D problem). Hole and electron concentration variations are assumed to be N_{A} and N_{D} in these depleted regions. Refractive index variation and absorption coefficient are deduced using Ref. 3.

The third approximation is to analytically express the optical mode distribution by fitting the mode profile in the x direction. A Gaussian distribution is a well-known distribution but it is only appropriate for weakly confined guided modes, because it is not able to fit correctly evanescent components of the optical mode. When the waveguide dimensions are chosen to achieve strong light confinement under the rib, a Pearson VII distribution is more suitable. We can notice that a Gaussian distribution is a particular case of the Pearson VII distribution, which is given by the following equation:

Using these analytic descriptions of the optical mode and carrier concentration in the junction, effective index variations and absorption coefficient are immediately obtained using Eq. (3) and (4). The same relation is used to determine the absorption coefficient. It is important to note that the mode solver is used only one time to determine the Pearson VII distribution coefficients for the waveguide considered. It is then easy to obtain performance of different active regions, for example, by changing the doping concentration of the active region in a given waveguide geometry. This requires only to calculate the carrier concentration in a 1D PIN diode and to solve a 1D integral.

## 4. Evaluation of modulator performance

The analytic model described in section 3 has been applied on optimized structures described in section 2. The goal is to evaluate final performance of such components in terms of modulation efficiency and optical loss. Modulation efficiency is characterized by the factor V_{π}L_{π}, which is the product between the voltage and the length needed to obtain a π phase shift, is given by:

_{π}and it is calculated using:where α

_{eff}(0V) is the effective absorption coefficient under 0V, due to free carrier absorption in the waveguide. V

_{π}L

_{π}and IL have been calculated for both PN and PIN diodes using the designs from Fig. 3. The results are plotted in Fig. 4 for PN diode (w

_{i}= 0nm) and PIN diode (w

_{i}= 150 nm), as a function of the waveguide aspect ratio r.

To evaluate the relevance of this analytic model, the structures have also been simulated using a physical device simulation package ISE-Dessis. This software provides hole and electron distributions for a given 2D system. Optical simulations are done using a complex Finite Difference Frequency Domain (FDFD) mode solver based on the method described in Ref 13. 2D permittivity maps obtained from the carrier distributions are used to calculate the effective index variation as a function of the voltage. The 2D simulations results are presented in Fig. 4 for the same designs. A very good agreement of V_{π}L_{π} and IL is obtained between the results obtained with the analytic model and the standard 2D simulation tools, for both values of w_{i}. For the modulation efficiency, the mean value of the difference between analytical and 2D models is as low as 0.2 V.cm for r from 0.1 to 0.8. Regarding optical loss, the error mean value is only 0.5 dB for r from 0.1 to 0.8. The presented analytical model allows good evaluation of silicon modulator performance. Considering the time and resources saved in comparison with a full 2D simulation method, the analytical model is a valuable tool to select high performance designs.

## 5. Conclusion

Efficient and fast simulation tools are a key to develop high performance optoelectronic devices. This paper presents an original and accurate method to optimize and study silicon optical modulators. This method is based on an analytical description of both the optical mode and the carrier concentration in the diode. This method is extremely fast in comparison with classical 2D numerical simulation. It has been applied on a specific design of carrier depletion based modulator, and a very good agreement is obtained in comparison with the results achieved by full 2D simulation tools. Indeed, a mean error of only 0.5 dB on insertion loss and 0.2 V × cm on V_{π}L_{π} is obtained

## Acknowledgements

The research leading to these results has received funding from the European Community under grant agreement n° 224312 HELIOS and from the French ANR under project SILVER.

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