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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 8 — Apr. 14, 2008
  • pp: 5218–5226
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Effect of carrier lifetime on forward-biased silicon Mach-Zehnder modulators

Gui-Rong Zhou, Michael W. Geis, Steven J. Spector, Fuwan Gan, Matthew E. Grein, Robert T. Schulein, Jason S. Orcutt, Jung U. Yoon, Donna M. Lennon, Theodore M. Lyszczarz, Erich P. Ippen, and Franz X. Kärtner  »View Author Affiliations


Optics Express, Vol. 16, Issue 8, pp. 5218-5226 (2008)
http://dx.doi.org/10.1364/OE.16.005218


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Abstract

We present a systematic study of Mach-Zehnder silicon optical modulators based on carrier-injection. Detailed comparisons between modeling and measurement results are made with good agreement obtained for both DC and AC characteristics. A figure of merit, static VπL, as low as 0.24Vmm is achieved. The effect of carrier lifetime variation with doping concentration is explored and found to be important for the modulator characteristics.

© 2008 Optical Society of America

1. Introduction

A silicon electro-optical modulator is an essential component for low cost optoelectronic circuits [1–2

1. L. Pavesi and D. J. Lockwood, Silicon Photonics, Topics in Applied Physics94, (Springer, New York, 2004). [CrossRef]

]. The electro-optic effect in silicon, induced by the free carrier plasma effect, has been studied experimentally by Soref [3

3. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. QE-23, 123–129 (1987). [CrossRef]

]. Over the past few years, several implementations of silicon modulators have been fabricated and show promising performance [4–12

4. C. K. Tang and G. T. Reed, “Highly efficient optical phase modulator in SOI waveguides,” Electron. Lett. 31, 451–452 (1995). [CrossRef]

].

Among the various types of silicon modulators, the carrier injection modulators have the advantages of being both compact and power efficient [8–12

8. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometer-scale silicon electro-optic modulator” Nature 435, 325–327 (2005). [CrossRef] [PubMed]

]. Here, we study the Mach-Zehnder (MZ) interferometer modulator [10–12

10. F. Gan and F. X. Kärtner, “High-speed silicon electrooptic modulator design,” IEEE Photon. Technol. Lett. 17, 1007–1009 (2005). [CrossRef]

]. Although it has a larger footprint than its ring resonator counterpart [8–9

8. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometer-scale silicon electro-optic modulator” Nature 435, 325–327 (2005). [CrossRef] [PubMed]

], a MZ modulator achieves the broadband optical response necessary for modulating sub-picosecond pulses. This is necessary for many applications such as high-speed photonic analog-to-digital converters using electronic-photonic integrated circuits (EPIC) [13

13. F. X. Kärtner, et al., “Silicon electronic photonic integrated circuits for high speed analog to digital conversion,” (Invited) in Technical Digiest of 3rd International Conference on Group IV Photonics, ThC3, (2006). [PubMed]

]. The availability of predictive models is of importance for subsequent optimization such as reducing RF-power consumption, increasing bandwidth, or improving other device parameters to meet the special requirements of various applications. Here, we present a detailed comparison of simulation and measurement results for both DC and AC characteristics of MZ silicon modulators that operate in carrier injection mode. Excellent agreement over a wide range of drive voltages and frequencies is obtained.

2. Device design and fabrication

The detailed design considerations and fabrication of MZ modulators with similar structures have been described previously [11–12

11. F. Gan and F. X. Kärtner, “Low insertion Loss, high-speed silicon electro-optic modulator design”, in Technical Digest of Integrated Photonics Research and Applications/Nanophotonics, ITuB2, (2006).

]. The devices studied here are illustrated in top-view in Fig. 1(a). Embedded on the two arms of MZ interferometer are silicon waveguide phase shifter sections, fabricated on SOI wafers. The cross-sectional structure, depicted in Fig. 1(b), consists of a ridge waveguide, 500 nm wide and 210 nm high, that is lightly n-type doped with a density of 1017cm-3. The two sidewalls (50 nm wide) are moderately doped with a density of 1018 cm-3 before connecting to the two heavily doped 1019 cm-3 thin slab regions (50nm thick). To ensure good Ohmic contact between the silicon and the Aluminum metal contacts, two thin slab regions ~1 µm away from both sides of the rib are heavily doped to 1021 cm-3. Figure 1(c) shows an SEM micrograph of the fabricated PnN junction phase shifter.

Fig. 1. (a). Top-view of Mach-Zehnder interferometer silicon modulator. (b) Schematic cross-sectional view and (c) SEM cross-sectional view of the fabricated SOI waveguide phase shifter with the inset shown as a magnified view of the silicon ridge waveguide.

3. Modeling and measurement: DC characteristics

For DC characterization, only the voltage on one arm of the MZ-modulator is varied while the voltage on the other arm is kept constant. The numerical simulations are performed with the software SENTAURUS™ (Synopsys) with which an induced carrier density can be calculated as a function of the applied DC voltage by solving the Poisson equation coupled to the electron and hole continuity equations that govern carrier transport. After obtaining the induced refractive index and absorption changes due to the free-carrier plasma effect, the resulting phase shift is then modeled by solving the optical mode profile (with its intensity distribution shown in the inset of Fig. 2(b)) and its propagation constant at 1550nm.

Fig. 2. Modeled and measured (a) current-voltage characteristics (b) normalized optical transmission-current characteristics for a MZ silicon modulator with 0.25-mm-long phase shifters embedded on both arms.

The measured and simulated current-voltage characteristics of a MZ silicon modulator with 0.25mm long phase shifters are shown in Fig. 2(a). Following [17

17. D. J. Roulston, N. D. Arora, and S. G. Chamberlain, “Modeling and measurement of minority-carrier lifetimes versus doping in diffused layers of n+-p silicon diodes,” IEEE Trans. Electron Devices 29, 961–964 (1982). [CrossRef]

] the carrier lifetime, τ, is dependent on the doping concentration with an empirical relation τ=(τ0/(1+N/N0)) to mainly incorporate the predominantly phonon-assisted recombination processes, where N is the doping level, and τ0 and N0 are parameters that can be varied. The defect level carrier recombination processes such as Shockley-Read-Hall (SRH), Auger as well as surface recombination are also considered in the simulation. Good agreement between the measurement and simulation is obtained by choosing τ0 as 2.4 ns and N0 as 5×1017 cm-3 (shown in red). The selection of the two parameters is found to be essential to accurately predicting device performance. As seen in Fig. 2(a), two additional simulated I–V characteristics are included with different choices for the lifetime parameters (τ0=2.4 ns, N0=1×1016 cm-3, shown in black) and (τ0=100 ps, N0=5×1017 cm-3, shown in green). These three lifetime doping relations are also plotted in Fig. 3(a) for comparison. The lifetime choice with smaller N0 (1×1016 cm-3) results in a much stronger doping dependence which essentially reduces the carrier lifetime for most of the carrier concentrations in the regime of interest. It’s worth mentioning that the default value of the reference doping level in Sentaurus is 1×1016 cm-3. Assuming this doping level and choosing a smaller parameter τ0 reduces the carrier lifetime but maintains the same doping dependence. In both cases, the induced current under the same applied voltage increases for the low bias voltage regime (below 0.8V) as shown in Fig. 2(a). To confirm the consistency of our choice (τ0=2.4 ns, N0=5×1017 cm-3) for the carrier lifetime doping dependence, we study the effective carrier lifetime τ,eff as a function of the effective carrier concentration experimentally and compare with the simulation results. The blue dots in Fig. 3(b) show the measured effective carrier lifetimes extracted from I–V measurements according to ref. [18

18. D. W. Zheng, B. T. Smith, and M. Asghari, “Assessment of the effective carrier lifetime in a SOI p-i-n diode silicon modulator using the reverse recovery method,” Proc. of SPIE 6477, 647711 (2007). [CrossRef]

] for a series of device samples with different device lengths. When fitting the empirical model to the measured values for the effective carrier lifetime, we obtain for the optimum model parameters τ0=2.4ns and N0=4.9×1017cm-3, which confirms our parameter choice for carrier lifetime doping dependence used for the Sentaurus simulation. It is also worth mentioning that our I–V characteristics measured for a series of device length ranging from 0.25mm to 4mm also reveal a low contact series resistance of ~5Ω.

Figure 2(b) shows the normalized optical transmission as a function of current level for the same device. After evaluating both phase Δϕ and absorption changes Δα with the well known free carrier plasma effect formulation [2

2. G. T. Reed and A. P. Knights, Silicon Photonics: an introduction (John Wiley, Chichester, 2004). [CrossRef]

] [3

3. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. QE-23, 123–129 (1987). [CrossRef]

] [5

5. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004). [CrossRef] [PubMed]

], the normalized transmission is then calculated by 0.5e-ΔαL(cos(Δϕ)+1) [14

14. S. L. Chuang, Physics of Optoelectronic Devices. (John Wiley, New York, 1995).

]. A current level of ~3mA or voltage level of ~0.94V is required to achieve a π phase shift. The static Vπ L-product obtained from the simulations is 0.024 Vcm and the low frequency RF Vπ L-product is as small as 0.0025Vcm with the DC operating point chosen as ~0.9V. The excellent agreement between measurement and simulation for the DC-characteristics of the modulator shown in Fig. 2(b) confirms that the choice for the carrier lifetime doping dependence is correct and consistent.

Fig. 3. (a). Variation of carrier lifetime as a function of effective carrier concentration and (b) Measured and fitted effective carrier lifetime vary as a function of effective carrier concentration. Measurements, (blue dots); best-fit curve to the empirical model (red) for device #1; model prediction for two additional parameter choices (green and black)..

4. Modeling and measurement: AC characteristics

The experimental setup for high frequency AC measurement is illustrated in Fig. 4. The MZ silicon modulator used for the measurement contains two 0.5 mm long phase shifters that have been driven in push-pull configuration. A 1550 nm laser beam generated by a laser diode with maximum optical power ~15mW is amplified in a fiber amplifier and coupled into the modulator via a lensed fiber. A fiber polarization controller is used to match the polarization of the input laser to the silicon waveguide. The RF signal is generated by a network analyzer, amplified by a RF power amplifier with bandwidth ~20GHz and a gain of ~18dBm, and connected to the modulator chip through a bias Tee. The output beam from the modulator chip is collected by an aspheric lens, goes through a fiber preamplifier, a 2 nm bandpass filter, another fiber amplifier and then finally detected with a ~50 GHz InGaAs photodetector.

Fig. 4. Schematic diagram of the experimental setup for MZI silicon modulator AC frequency response measurement.

Under the assumption that the parasitic effects due to packaging and the external RF drive circuits are negligible, modeling of the small signal AC frequency response is essentially evaluation of the amount of free carriers stored due to the applied AC signal at angular frequency ω. We have developed an analytical and a numerical model to study this AC characteristic of the modulator. The analytical model is based on the small signal equivalent circuit of a semiconductor pn-junction [15–16

15. R. F. Pierret, Semiconductor Device Fundamentals (Addison Wesley, 1996), Part IIA.

] as shown in Fig. 5 (a conductor GD in parallel with a capacitor CD and in series with Rs). We evaluate the normalized power response |S21|2 by calculating the stored charge Q on CD under the applied voltage u=(Vdc+νsinωt) as shown in Eq. (1). The phase shift due to the refractive index change is proportional to the charge Q according to [3

3. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. QE-23, 123–129 (1987). [CrossRef]

]. For small signal modulation, the output amplitude of the MZ interferometer will depend linearly on the phase shift. Note that both GD and CD are functions of ω as described in Eq. (2) according to [15

15. R. F. Pierret, Semiconductor Device Fundamentals (Addison Wesley, 1996), Part IIA.

]. G0 is the low frequency conductance evaluated from the DC I–V characteristics. A constant effective lifetime τ of 0.82 ns is assumed in the analytical model and Rs is chosen as 55Ω (5Ω of extracted series resistance from DC characteristics and 50Ω from the connecting cable).

Fig. 5. Small signal equivalent circuit of the forward biased silicon PiN diode
NormalizedS212Qu2=CD(ω)1+RSGD(ω)+jωRSCD(ω)
(1)
CD=G02ω(1+ω2τ21)12andGD=G02(1+ω2τ2+1)12
(2)

The numerical modeling of the AC characteristic is based on calculating the induced AC carrier densities for an applied small signal AC input (νsin(ωt)) signal, similar to the modeling of the DC characteristics. The normalized power response |S21|2 is then obtained by evaluating the transmitted light intensity using the calculated AC free carrier densities. The operating conditions for both simulations and experiments are 0.9 V DC bias voltage and 3mV AC amplitude.

Figure 6 shows the measured and modeled small signal response of a 0.5 mm long modulator under forward biased operation for two types of devices. The baseline device (#1) has been described previously and device #2 has been implanted with silicon (with an area dose of 1014cm2) to reduce the carrier lifetime. Note, that the curves have been normalized to the response level at low frequency for device #1. For the modeling results, the carrier lifetime choice for device #1 is taken from curve #1 in Fig. 3 whereas for device #2, the curve #2 in Fig. 3 is used accordingly. For device #1, both analytical and numerical models predict an accurate 3-dB bandwidth of ~0.3GHz but the analytical model deviates for frequencies above 1GHz. On the other hand, the simulation results are in good agreement with measurements up to 20GHz (beyond which the bandwidth was limited due to RF parasitics in the measurement apparatus). For device #2, both modeling and measurement results show that after carrier lifetime reduction the device bandwidth can be increased to ~2GHz but with a reduced response level at lower frequencies. An alternative technique using a pre-compensation filter has been presented previously to extend the bandwidth to ~5GHz [3

3. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. QE-23, 123–129 (1987). [CrossRef]

], a bandwidth necessary to achieve 10Gbps of data transmission rate.

Fig. 6. Modeled and measured small signal AC characteristics of the Mach-Zehnder silicon modulator with 0.5mm phase shifters.

To further understand the simulated AC characteristic shown in Fig. 6, the magnitude of the total AC carrier concentrations along the center region are plotted in Fig. 7. The y-slicing plane is chosen in the middle of the center region where the optical mode exhibits maximum intensity as seen in the insets of Fig. 2(b). For the same magnitude of the AC voltage (3mV) and frequencies less than 5GHz, more carriers are induced. The carrier modulation rolls off faster with frequency for the structure with longer lifetime (2.4ns). This corresponds to a larger response level but faster response roll-off versus frequency, which is consistent with Fig. 6. Additionally, for frequencies beyond 5GHz, the AC carriers have similar distribution for the two structures and both decrease with frequency at a faster speed. The 2-D AC carrier concentration distribution graphs at frequencies (0.01GHz and 10GHz) are provided in the insets of Fig. 7.

Fig. 7. Simulated magnitude of total AC carrier density at different frequencies for two carrier lifetime choices in Fig. 3 (the left is for lifetime #1 and the right is for lifetime #2). The 2-D carrier distribution graphs at two end frequencies are shown in insets.

As seen from Fig. 6, the analytical model can not capture the complex carrier diffusion and recombination processes beyond 1GHz, correctly described by the numerical model. While development and refinement of the analytical model will continue to aid in understanding and developing design intuition, accurate numerical simulation will play a key role to achieve quantitative results.

5. Conclusion

In conclusion, we have found that the dependence of carrier lifetime on doping concentration is critical to accurately predict the DC and AC performance of the silicon modulators. This is verified by comparing measured and simulated DC and AC characteristics of a forward biased silicon electro-optic Mach-Zehnder modulator.

Acknowledgments

References and links

1.

L. Pavesi and D. J. Lockwood, Silicon Photonics, Topics in Applied Physics94, (Springer, New York, 2004). [CrossRef]

2.

G. T. Reed and A. P. Knights, Silicon Photonics: an introduction (John Wiley, Chichester, 2004). [CrossRef]

3.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. QE-23, 123–129 (1987). [CrossRef]

4.

C. K. Tang and G. T. Reed, “Highly efficient optical phase modulator in SOI waveguides,” Electron. Lett. 31, 451–452 (1995). [CrossRef]

5.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004). [CrossRef] [PubMed]

6.

L. Liao, D. Samara-Rubio, M. Morse, A. Liu, H. Hodge, D. Rubin, U. D. Keil, and T. Franck, “High-speed silicon Mach-Zehnder modulator,” Opt. Express 13, 3129–3135 (2005). [CrossRef] [PubMed]

7.

A. Liu, L. Liao, D. Rubin, H. Nguyen, G. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide” Opt. Express 15, 660–668 (2007). [CrossRef] [PubMed]

8.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometer-scale silicon electro-optic modulator” Nature 435, 325–327 (2005). [CrossRef] [PubMed]

9.

Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon microring silicon modulators” Opt. Express 15, 430–436 (2007). [CrossRef] [PubMed]

10.

F. Gan and F. X. Kärtner, “High-speed silicon electrooptic modulator design,” IEEE Photon. Technol. Lett. 17, 1007–1009 (2005). [CrossRef]

11.

F. Gan and F. X. Kärtner, “Low insertion Loss, high-speed silicon electro-optic modulator design”, in Technical Digest of Integrated Photonics Research and Applications/Nanophotonics, ITuB2, (2006).

12.

S. J. Spector, M. E. Grein, R. T. Schulein, M. W. Geis, J. U. Yoon, D. E. Lennon, F. Gan, F. X. Kartner, and T. M. Lyszczarz, “Compact carrier injection based Mach-Zehnder modulator in silicon,” in Technical Digest of 2007 Integrated Photonics Research, ITuE5 (2007).

13.

F. X. Kärtner, et al., “Silicon electronic photonic integrated circuits for high speed analog to digital conversion,” (Invited) in Technical Digiest of 3rd International Conference on Group IV Photonics, ThC3, (2006). [PubMed]

14.

S. L. Chuang, Physics of Optoelectronic Devices. (John Wiley, New York, 1995).

15.

R. F. Pierret, Semiconductor Device Fundamentals (Addison Wesley, 1996), Part IIA.

16.

S. M. Sze, Physics of Semiconductor Devices, (John Wiley, New York, 1981), Chap. 2.

17.

D. J. Roulston, N. D. Arora, and S. G. Chamberlain, “Modeling and measurement of minority-carrier lifetimes versus doping in diffused layers of n+-p silicon diodes,” IEEE Trans. Electron Devices 29, 961–964 (1982). [CrossRef]

18.

D. W. Zheng, B. T. Smith, and M. Asghari, “Assessment of the effective carrier lifetime in a SOI p-i-n diode silicon modulator using the reverse recovery method,” Proc. of SPIE 6477, 647711 (2007). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.2090) Optical devices : Electro-optical devices
(250.5300) Optoelectronics : Photonic integrated circuits
(250.7360) Optoelectronics : Waveguide modulators

ToC Category:
Optoelectronics

History
Original Manuscript: January 17, 2008
Revised Manuscript: March 22, 2008
Manuscript Accepted: March 25, 2008
Published: April 1, 2008

Citation
Gui-rong Zhou, Michael W. Geis, Steven J. Spector, Fuwan Gan, Matthew E. Grein, Robert T. Schulein, Jason S. Orcutt, Jung U. Yoon, Donna M. Lennon, Theodore M. Lyszczarz, Erich P. Ippen, and Franz X. Käertner, "Effect of carrier lifetime on forward-biased silicon Mach-Zehnder modulators," Opt. Express 16, 5218-5226 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5218


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References

  1. L. Pavesi, D. J. Lockwood, Silicon Photonics, Topics in Applied Physics 94, (Springer, New York, 2004). [CrossRef]
  2. G. T. Reed, A. P. Knights, Silicon Photonics: an introduction (John Wiley, Chichester, 2004). [CrossRef]
  3. R. A. Soref, B. R. Bennett, "Electrooptical effects in silicon," IEEE J. Quantum Electron. QE-23, 123-129 (1987). [CrossRef]
  4. C. K. Tang, G. T. Reed, "Highly efficient optical phase modulator in SOI waveguides," Electron. Lett. 31, 451-452 (1995). [CrossRef]
  5. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, M. Paniccia, "A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor," Nature 427, 615-618 (2004). [CrossRef] [PubMed]
  6. L. Liao, D. Samara-Rubio, M. Morse, A. Liu, H. Hodge, D. Rubin, U. D. Keil, T. Franck, "High-speed silicon Mach-Zehnder modulator," Opt. Express 13, 3129-3135 (2005). [CrossRef] [PubMed]
  7. A. Liu, L. Liao, D. Rubin, H. Nguyen, G. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, "High-speed optical modulation based on carrier depletion in a silicon waveguide," Opt. Express 15, 660-668 (2007). [CrossRef] [PubMed]
  8. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometer-scale silicon electro-optic modulator," Nature 435, 325-327 (2005). [CrossRef] [PubMed]
  9. Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya and M. Lipson, "12.5 Gbit/s carrier-injection-based silicon microring silicon modulators," Opt. Express 15, 430-436 (2007). [CrossRef] [PubMed]
  10. F. Gan, F. X. Kärtner, "High-speed silicon electrooptic modulator design," IEEE Photon. Technol. Lett. 17, 1007-1009 (2005). [CrossRef]
  11. F. Gan and F. X. Kärtner, "Low insertion Loss, high-speed silicon electro-optic modulator design," in Technical Digest of Integrated Photonics Research and Applications/ Nanophotonics, ITuB2, (2006).
  12. S. J. Spector, M. E. Grein, R. T. Schulein, M. W. Geis, J. U. Yoon, D. E. Lennon, F. Gan, F. X. Kartner, and T. M. Lyszczarz, "Compact carrier injection based Mach-Zehnder modulator in silicon," in Technical Digest of 2007 Integrated Photonics Research, ITuE5 (2007).
  13. F. X. Kärtner,  et al., "Silicon electronic photonic integrated circuits for high speed analog to digital conversion," (Invited) in Technical Digiest of 3rd International Conference on Group IV Photonics, ThC3, (2006). [PubMed]
  14. S. L. Chuang, Physics of Optoelectronic Devices. (John Wiley, New York, 1995).
  15. R. F. Pierret, Semiconductor Device Fundamentals (Addison Wesley, 1996), Part IIA.
  16. S. M. Sze, Physics of Semiconductor Devices, (John Wiley, New York, 1981), Chap. 2.
  17. D. J. Roulston, N. D. Arora, and S. G. Chamberlain, "Modeling and measurement of minority-carrier lifetimes versus doping in diffused layers of n+-p silicon diodes," IEEE Trans. Electron Devices 29, 961-964 (1982). [CrossRef]
  18. D. W. Zheng, B. T. Smith, M. Asghari, "Assessment of the effective carrier lifetime in a SOI p-i-n diode silicon modulator using the reverse recovery method," Proc. SPIE 6477, 647711 (2007). [CrossRef]

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