We provide detailed analysis of a four terminal p + pnn + optical modulator integrated into a silicon-on-insulator (SOI) rib waveguide. The proposed depletion device has been designed to approach birefringence free operation. The modulation mechanism is the carrier depletion effect in a pn junction; carrier losses induced are minimised in our design and because we use a depletion device, the device is insensitive to carrier lifetime. The rise time and fall time of the proposed device have both been calculated to be 7 ps for a reverse bias of only 5 volts. A maximum excess loss of 2 dB is predicted for TE and TM due to the presence of p type and n type carriers in the waveguide.
©2005 Optical Society of America
The main reasons why silicon on insulator substrates have proven successful for integrated optics are that the material and the processing are relatively low cost with resulting waveguides having loss of around 0.15dB/cm; ; optical modulation devices can be fabricated in SOI; and that silicon micromachining allows hybrid circuits to include sources and detectors. Moreover, any technological development in either silicon or in the associated electronics industry can readily be transferred to silicon based integrated optics. The main methods to alter the refractive index in Si are the thermo-optic effect and plasma dispersion effect . The thermo-optic effect is rather slow and can be used only up to approximately 1-MHz modulation frequencies . For higher modulation frequencies, up to a few Gigahertz, devices based on the free carrier dispersion effect  are required. Table (1) shows a list of several silicon plasma dispersion modulators recently reported in the literature. The free carrier dispersion effect is used to change both the real refractive index and optical absorption coefficient.
In this paper, we analyse a depletion type device configuration for varying the refractive index of high-index-contrast SOI waveguides by using a four terminal pn junction diode. Section 2 describes the waveguide structure. Section 3 presents the electrical and optical models. In Section 4, the results from the simulations are presented and discussed.
2. Device structure
2.1. Modulator structure
Figure (1) below illustrates the optimised device structure to be considered in this paper. This design has been optically optimised in order to achieve birefringence free propagation, then optical and electrical simulations were used to determine the optimum positioning of n-type and p-type doping, as well as the doping concentrations involved in order to reach maximum bandwidth and minimum losses. It is a vertical optical phase modulator integrated into a low loss SOI rib waveguide. The device has an asymmetrical p-n structure where two slab regions are joined as a common cathode and two poly-silicon regions are joined as a common anode. It is referred to as p + pnn + device for obvious reasons. Both n + and p + regions were modelled as highly doped regions with peak doping concentrations of 1× 1019 ions/cm3. The structure is based around an overall silicon thickness of 0.45 μm etched rib waveguides 0.415 μm wide with a slab thickness of 0.10 μm. The silicon slab and the bottom part of the rib have an n-type background doping concentration of 4×1017 ions/cm3 and the top part of the rib has a p-type background doping concentration of 2×1017 ions/cm3. The oxide thickness was chosen to be 1μm which ensures sufficiently good optical confinement and a top silicon oxide cladding layer covers the whole structure. The n + doped regions are situated on both sides of the wave guiding region, in the slab, 1.5 microns away from the centre of the waveguide. Furthermore the poly-silicon p + doped regions are situated on both side of the top of the rib in order to reduce the losses resulting from the poly-silicon and aluminum contacts.
2.2. Mach Zehnder Interferometer Structure
The proposed device is inserted in both arms of a 1×1 Mach Zehnder Interferometer (MZI). On the MZI four phase shifters of the same geometry are present, two of them represent the depletion device proposed in Fig. (1), the next two are DC phase shifters and will bias the device by introducing a static π-phase shift in one arm. The length of a depletion phase shifter has been calculated to be 2.5 mm in order to achieve a π-phase shift when a reverse bias of 10 volts is applied. The MZI functions in a push-pull configuration. An ”Off state” means that both ac modulators are biased to a reverse bias of 5 volts and one of the DC ”injection type” phase shifters induces a π-phase shift in one arm enabling destructive interference at the output. The ”On state” corresponds to the ac modulators having a bias of 10 volts and 0 volts respectively, allowing constructive interference. Hence the MZI converts phase modulation into amplitude modulation. The DC ”injection type” phase shifter remains unchanged and could be used in a feedback loop in order to adjust the extinction ratio of the MZI. The second DC phase shifter located in the other arm of the MZI is present for redundancy reasons. Another possibility would be to increase the length of one waveguide by half a wavelength; this would create a native π-phase difference between the two arms of the MZI. The difficulty of this method is to obtain the necessary processing resolution and that is why a DC phase shifter has been chosen.
3. Device modelling
3.1. Optical simulation
The device was optically modelled using Beamprop, the device simulation package from Rsoft . Beamprop incorporates computational techniques based on the beam propagation method (BPM), and utilises an implicit finite-difference scheme. The simulator has been used with full vectorial capability in order to characterise the birefringence-free, single mode waveguide. By fitting the change in refractive index provided by the simulation of the carrier distribution in the device into Beamprop and by using the effective index method, it has been possible to predict the effective index change of the waveguide for different voltages, hence calculating the optimum length in order to achieve a π-phase shift (Eq. (1) below). The same method has been used to characterise the phase-shift against transient time discussed later, see Fig. (8) and Fig. (9) below. The carrier concentration changes at any given bias, compared to a 5 volts reverse reference bias, are fitted into the Beamprop waveguide structure for different transient times, hence giving a dynamic change in effective index. Thus using Eq. (1) below, the phase shift induced against time can be evaluated. The loss variation induced by the change in carrier concentration is predicted using the Beam Propagation Method (BPM). The change in the absorption coefficient Δα is calculated using Eq. (3) below . The variation profile is then fitted to the waveguide in Beamprop in order to predict the losses for different reverse bias.
3.2. Electrical simulation
The device was modelled for both its static and dynamic behaviour using Atlas the device simulation package from Silvaco . Atlas device simulation framework predicts the electrical behavior of semiconductor devices. It provides a physics-based platform to analyse DC and time domain responses for semiconductor based technologies, by solving the equations which describe semiconductor physics such as Poisson’s equation and the charge continuity equations for holes and electrons. The simulator has been used to predict the free carrier concentrations in the wave guiding region of the device for both DC and transient biasing conditions. The change in free carriers profile is then converted to refractive index profile in the device using the expressions given by Soref and Bennett Eq. (2) and Eq. (3) of . To determine the voltage associated with a π-phase shift, the change in concentration of free carriers must be known. Assuming a non uniform change in refractive index, the active device length required to produce the refractive index change associated with a π-phase shift is obtained approximately from the change in effective index.
Where λ is the operating wavelength, Lπ is the active length of the modulator and Δneff is the change in effective index induced by the change in refractive index Δn in the waveguide. At 1.55 μm, which is also the operating wavelength for all simulations here, the refractive index change Δn and the change in absorption coefficient δα are given by
where Δne is the change in refractive index resulting from the change in free electron concentration, Δnh is the change in refractive index resulting from the change in free hole concentration, Δα e is the change in absorption resulting from change in free electron concentration and Δα h is the change in absorption resulting from change in free hole concentration. The following parameters have been used in order to perform Silvaco simulations.
The carrier lifetime time has been chosen to be large in order to show that it doesn’t play a dominant role on the speed of depletion devices. A range of simulations have been processed with carrier lifetime varying between the value in table 3 and 10 ps and the changes on the switching current or switching time were barely noticeable.
4. Results an discussions
4.1. Optical simulation
The device has been designed in order to achieve birefringence free behaviour where the effective index of the waveguide is equal for TE and TM polarisation. This result is obtained by varying the slab thickness and the rib width of the rib waveguide chosen to have a height of 450 nm. The birefringence free is achieved by taking a rib width of 410 nm and a slab height of 100 nm. These dimensions have been chosen as a trade off between the capacitance and the sidewall roughness. The rib width has a direct influence on the capacitance of the device (Eq. (4) below) as the p-n junction is situated in the rib of the waveguide, so diminishing the rib width reduces the capacitance. On the other hand reducing the rib width increases further the losses induced by the sidewall roughness . In order to ensure that the device is to operate with only a single mode, the optical mode spectrum was also modelled. The results of this method demonstrated that the waveguide designed supports only one mode for TE and TM polarisation.In order to simulate the change in loss in the waveguide, the carrier concentration profiles obtained for a specific bias voltage from Atlas is converted into a change in absorption coefficient profiles using Eq. (3). The different absorption profiles plotted in Fig. (2) below are then included into Beamprop simulations. The carrier concentration is considered as uniform along the horizontal axis and the profiles displayed in Fig. (2) below start from the base of the waveguide (height = 0) to the top of the rib (height = 450 nm). The absorption coefficient profile in the rib Fig. (2) demonstrate the difference in loss induced by hole and electrons. Loss in the n-type region situated between 0 and 100 nm is about three times greater than in the p-type region situated between 100 nm and 450 nm. This is one of the reasons why the n-type doping region has been minimised in the waveguide. The results of the loss simulation in Beamprop are illustrated in Fig. (3) below. On this graph, we can see the evolution of the loss created by carriers for both polarisations and different reverse bias. The carriers present in the wave-guiding region induce a loss of about 2 dB for TE and TM when no potential is applied. The losses decrease to around 1 dB for a reverse bias of 10 volts for TE and TM. The modulator is being operated as a push pull device. During an ”On state”, the loss of one modulator will increase to 2dB while the other is decreased to 1dB. This imbalance in power between the two arms for an ”On state” slightly reduces the maximum modulation depth, that would be achieved if no absorption loss was present.
4.2. Electrical simulations
4.2.1. Static characteristics
In order to define the characteristics of the device, the carrier distribution in the waveguide has been modelled with Atlas. Figure (4) below illustrates the refractive index change induced by the depletion of carriers in the ridge region. In common with Fig. (2) the carrier profile is taken at the centre of the waveguide and the carrier distribution change for different bias is considered constant in the horizontal direction. The refractive index change profile is obtained from the carrier profile using Eq. (2). The two doping regions have been designed to induce a maximum change in effective index, resulting in a p-type region overlapping the centre of the mode. Using data from Fig. (4) below, the effective index of the waveguide for different voltages and for TE and TM polarisations has been calculated. The main characteristic to be derived from these results is the phase shift obtained for a specific length by using Eq. (1). For the present device the optimum length has been calculated to be around 2.5 mm in order to reach a π-phase shift for a reverse bias voltage of 10 volts. The phase shift obtained for TE and TM polarisation and for a length of 2.5 mm is illustrated in Fig. (5) below.
4.2.2. Depletion Capacitance
The small signal transient response determines the feasibility of the device to be used for high speed data modulation. In the studied configuration, the small signal response will be defined by the depletion or junction capacitance CJ. The capacitance associated with depletion region CJ is given by Eq. (4) below.
Where A is the area of the junction, q is the electronic charge, εS is the permittivity of silicon, NA is the concentration of acceptors, ND is the concentration of donors, ϕB is the potential difference at the junction and VD is the voltage applied to the junction. The Capacitance of the device has been simulated using Atlas and is equal to 1.14 pf for a reverse bias of 5 volts. The capacitance of our modulator is significantly smaller that that reported in  which suggests an increase in performance in terms of RC cutoff frequency by at least an order of magnitude.
4.2.3. Transient Current
The following simulation demonstrates a worst case scenario. The simulated square wave voltage function in Silvaco was chosen to have a 0–100 percent rise time of 5ps , thus creating the current pulse illustrated in Fig. (6). If the voltage rise time is increased, the switching current involved is largely reduced as illustrated in Fig. (7). During static reverse bias operation the device current is considered as negligible, and a maximum leakage current of 12 nA goes through the 2.5 mm long device for a reverse bias of 10 volts. During switching operation, the waveg-uiding region where the pn junction is located is depleted of carriers. The currents generated by the switching process are illustrated Fig. (6), where the current against transient time has been calculated during transient simulations, Fig. (8) and Fig. (9) below. Considering the MZI of Fig. (1), both modulators are biased to 5 volts as an ”Off state”, when switching to an ”On state” one of the modulators is switched to a zero reverse bias while the other one is switched to 10 volts. This creates a pulse of current needed to drive both modulators. When switching from a bias of 5 volts to a bias of 10 volts the pick current is 1.2 amps while when switching from a bias of 5 volts to a bias of 0 volts the peak current is 1.4 amps. A maximum switching current of 2.6 Amps with a bias of 5 volts is then needed in order to drive the modulator, where the transient time for a pulse of current is in the order of 7 picoseconds for both arms. In practical application switching such a high current in a short time will be impossible, although the intrinsic bandwidth of about 60 GHz means that multi GHz modulation remains a possibility. Another important factor to take into account is the heat generated by raising the average current hence inducing a thermo-optic effect which would lead to changing the characteristics of the device.
4.2.4. Transient phase shift
In order to study the transient response of the MZI, the rise time and fall time of a modulator has been studied. To calculate the rise time of the MZI, the rise time for a modulator to switch from 5volts to 0 volts and from 5 volts to 10 volts must be known. Following the same process, the fall time of the MZI ”Off state” is calculated by overlapping the fall time of modulators biased to 10 volts and 0 volt switching to a 5 volt bias. The transient time is first characterised by Atlas in terms of carrier concentration against time. The carrier concentration profile for a reverse bias of 5 volts is taken for reference. The change in carrier concentration compared to a reverse bias of 5 volts against time is converted to a change in refractive index profile against time using Eq. (2). This profile is then used in Beamprop waveguide structure and by using the effective index method from Beamprop, the change in effective index is converted into a change in phase shift with time using Eq. (1). The difference of phase between the two modulators at any given time is equal to the π-phase shift induced by the DC phase shifter and the difference of phase induced by the two depletion phase shifters. Fig. (8) and Fig. (9) illustrate respectively the ”turn On” and ”turn Off” transient time of the MZI for TE an TM polarisations. The ”turn off” or ”turn On” time is calculated by measuring the time taken to achieve 10 percent to 90 percent of the maximum phase shift created between the arms of the MZI. The total switching time (”turn On” + ”turn Off” time) of the MZI has been calculated to be around 14 picoseconds, where both have an equal transient time of 7 picoseconds.
We have studied a 2.5 mm long optical modulator based on the depletion of a pn junction integrated into a SOI high index contrast rib waveguide for 1.55 μm operation wavelength. The real refractive index and the absorption coefficient of the core waveguide are changed using the free carrier dispersion effect via depletion of a pn junction. A rib waveguide with a height of 450 nm, a width of 415 nm and a slab of 100 nm has been shown as a good trade off in order to achieve a birefringence-free and single-mode operation for TE and TM polarizations. Other modulators such as the one proposed by  supports only one polarization, or show very high polarization dependence. The losses, which are of major importance in sub-micrometer devices where insertion loss can be a problem, have been optimised in order to achieve the best extinction ratio. The loss and the phase shift induced by the phase shifter are approximately equal for both polarisations which makes it easier to drive the modulator in case of random polarisation input (e.g. from fibre). The main improvement is of course the rise and fall times, predicted to be in the order of 7 ps for a switch bias of only 5 volts, which represents one the fastest Si photonic modulators yet reported (table (1)). The studied carrier dispersion modulator is therefore a serious candidate for integrated circuits in Si nanophotonics. Future work will include further optimisation in order to minimise the losses. Simulation work on the driver circuit and on-chip interconnects will be performed in order to maximise data rate transmission.
References and links
1 . C. Cocorullo , M. Iodice , I. Rendina , and P.M. Sarro , “ Silicon thermo-optical micro-modulator with 700 khz 3db bandwidth ,” IEEE Photonics Technol. Lett. 7 , 363 – 365 ( 1995 ). [CrossRef]
2 . R. A. Soref and B.R Bennett , “ Electrooptical effects in silicon ,” IEEE J. Quantum Electron. 23 , 123 – 129 ( 198 ). [CrossRef]
3 . C. E. Png , G. T. Reed , R. M. Atta , G. J. Ensell , and A. G. R. Evans , “ Development of small silicon modulators in silicon-on-insulator (SOI) ,” Proc. SPIE 4997 , 190 – 197 ( 2003 ). [CrossRef]
4 . A. Liu , R. Jones , L. Liao , D. S. Rubio , D. Rubin , O. Cohen , R. Nicolaescu , and M. Paniccia , “ High-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor ,” Nature (London) 427 , 615 – 618 ( 2004 ). [CrossRef]
5 . L. Liao , D. Samara-Rubio , M. Morse , A. Liu , D. Hodge , D. Rubin , U. D. Keil , and T. Franck , “ High speed silicon mach-zehnder modulator ,” Opt. Express 13 , 3129 – 3135 ( 2005 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-3129 . [CrossRef] [PubMed]
6 . Q. Xu , B. Shmidt , S. Pradhan , and M. Lipson , “ Micrometre-scale silicon electro-optic modulator ,” Nature (London) 435 , 325 – 327 ( 2005 ). [CrossRef]
7 . F. Gan and F. X. Kartner , “ High-speed silicon electrooptic Modulator design ,” IEEE Photonics Technol. Lett. 17 , 1007 – 1009 ( 2005 ). [CrossRef]
8 . http://www.rsoftinc.com .
9 . R. A. Soref and B. R. Bennett , “ Kramers-Kronig analysis of E-O switching in silicon ,” SPIE Integrated Opt. Circuit Eng 704 , 32 – 37 ( 1986 ).
10 . Silvaco Internationnal, 4701 Patrick Henry drive,Bldg 1, Santa Clara, CA 94054, http://www.silvaco.com .
11 . K. K. Lee , D. R. Lim , H. C. Luan , A. Agarwal , J. Foresi , and L. C. Kimerling , “ Effect of size and roughness on light transmission in a Si/SiO 2 waveguide:experiments and model ,” Appl. Phys. Lett. 77 , 1617 – 1619 ( 2000 ). [CrossRef]