Abstract

We investigated influence of carrier lifetime on performance of silicon (Si) p-i-n variable optical attenuators (VOAs) on submicrometer Si rib waveguides. VOAs were fabricated with and without intentional implantation of lattice defects into their intrinsic region. Carrier lifetime was measured by pulse responses for normal incidence of picosecond laser pulse of 775 nm to the VOA, as ~1 ns and ~7 ns for the VOAs with and without defects, respectively. Carrier lifetime is determined by the sum of surface recombination and Auger recombination for VOAs without defects, while Schockley-Read-Hall recombination is dominant for the VOA with defects. As a result, attenuation efficiency (dB/mA) is 0.2 - 0.7 and 0.04 - 0.1, while 3-dB bandwidth is 40 - 100 MHz and over 200 MHz for the VOAs with and without defects, respectively. There is a trade-off relation between attenuation and response speed of the VOAs with respect to carrier lifetime i.e., attenuation efficiency is linearly proportional to the carrier lifetime, whereas response speed is inversely proportional to it.

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2008 (3)

D. W. Zheng, B. T. Smith, and M. Asghari, “Improved efficiency Si-photonic attenuator,” Opt. Express 16(21), 16754–16765 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16754 .
[CrossRef] [PubMed]

D. W. Zheng, B. T. Smith, J. Dong, and M. Asghari, “On the effective carrier lifetime of a silicon p-i-n diode optical modulator,” Semicond. Sci. Technol. 23(6), 064006 (2008).
[CrossRef]

S. Nishihara, M. Nakamura, K. Nishimura, K. Kishine, S. Kimura, and K. Kato, “10.3 Gbit/s burst-mode PIN-TIA module with high sensitivity, wide dynamic range and quick response,” Electron. Lett. 44(3), 222 (2008).
[CrossRef]

2005 (4)

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86(7), 071115 (2005).
[CrossRef]

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

H. Cai, X. M. Zhang, C. Lu, A. Q. Liu, and E. H. Khoo, “Linear MEMS variable optical attenuator using reflective elliptical mirror,” IEEE Photon. Technol. Lett. 17(2), 402–404 (2005).
[CrossRef]

2004 (1)

2003 (1)

T. Kuwayama, M. Ichimura, and E. Arai, “Interface recombination velocity of silicon-on-insulator wafers measured by microwave reflectance photoconductivity decay method with electric field,” Appl. Phys. Lett. 83(5), 928 (2003).
[CrossRef]

1998 (2)

T. Kawai, M. Koga, M. Okuno, and T. Kitoh, “PLC type compact variable optical attenuator for photonic transport network,” Electron. Lett. 34(3), 264 (1998).
[CrossRef]

J. Linnros, “Carrier lifetime measurements using free carrier absorption transients. I. Principle and injection dependence,” J. Appl. Phys. 84(1), 275 (1998).
[CrossRef]

1997 (1)

D. K. Schroder, “Carrier lifetime in silicon,” IEEE Trans. Electron. Dev. 44(1), 160–170 (1997).
[CrossRef]

1994 (1)

C. K. Tang, G. T. Reed, A. J. Walton, and A. G. Rickman, “Low-loss, single-model optical phase modulator in SIMOX material,” J. Lightwave Technol. 12(8), 1394–1400 (1994).
[CrossRef]

1987 (1)

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

1977 (1)

J. Dziewior and W. Schmid, “Auger coefficients for highly doped and highly excited silicon,” Appl. Phys. Lett. 31(5), 346 (1977).
[CrossRef]

Arai, E.

T. Kuwayama, M. Ichimura, and E. Arai, “Interface recombination velocity of silicon-on-insulator wafers measured by microwave reflectance photoconductivity decay method with electric field,” Appl. Phys. Lett. 83(5), 928 (2003).
[CrossRef]

Asghari, M.

D. W. Zheng, B. T. Smith, J. Dong, and M. Asghari, “On the effective carrier lifetime of a silicon p-i-n diode optical modulator,” Semicond. Sci. Technol. 23(6), 064006 (2008).
[CrossRef]

D. W. Zheng, B. T. Smith, and M. Asghari, “Improved efficiency Si-photonic attenuator,” Opt. Express 16(21), 16754–16765 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16754 .
[CrossRef] [PubMed]

Bennett, B.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Cai, H.

H. Cai, X. M. Zhang, C. Lu, A. Q. Liu, and E. H. Khoo, “Linear MEMS variable optical attenuator using reflective elliptical mirror,” IEEE Photon. Technol. Lett. 17(2), 402–404 (2005).
[CrossRef]

Claps, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86(7), 071115 (2005).
[CrossRef]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in Silicon waveguides,” Opt. Express 12(12), 2774–2780 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-12-2774 .
[CrossRef] [PubMed]

Cohen, O.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Dimitropoulos, D.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86(7), 071115 (2005).
[CrossRef]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in Silicon waveguides,” Opt. Express 12(12), 2774–2780 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-12-2774 .
[CrossRef] [PubMed]

Dong, J.

D. W. Zheng, B. T. Smith, J. Dong, and M. Asghari, “On the effective carrier lifetime of a silicon p-i-n diode optical modulator,” Semicond. Sci. Technol. 23(6), 064006 (2008).
[CrossRef]

Dziewior, J.

J. Dziewior and W. Schmid, “Auger coefficients for highly doped and highly excited silicon,” Appl. Phys. Lett. 31(5), 346 (1977).
[CrossRef]

Fang, A.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Hak, D.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Ichimura, M.

T. Kuwayama, M. Ichimura, and E. Arai, “Interface recombination velocity of silicon-on-insulator wafers measured by microwave reflectance photoconductivity decay method with electric field,” Appl. Phys. Lett. 83(5), 928 (2003).
[CrossRef]

Jalali, B.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86(7), 071115 (2005).
[CrossRef]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in Silicon waveguides,” Opt. Express 12(12), 2774–2780 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-12-2774 .
[CrossRef] [PubMed]

Jhaveri, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86(7), 071115 (2005).
[CrossRef]

Jones, R.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Kato, K.

S. Nishihara, M. Nakamura, K. Nishimura, K. Kishine, S. Kimura, and K. Kato, “10.3 Gbit/s burst-mode PIN-TIA module with high sensitivity, wide dynamic range and quick response,” Electron. Lett. 44(3), 222 (2008).
[CrossRef]

Kawai, T.

T. Kawai, M. Koga, M. Okuno, and T. Kitoh, “PLC type compact variable optical attenuator for photonic transport network,” Electron. Lett. 34(3), 264 (1998).
[CrossRef]

Khoo, E. H.

H. Cai, X. M. Zhang, C. Lu, A. Q. Liu, and E. H. Khoo, “Linear MEMS variable optical attenuator using reflective elliptical mirror,” IEEE Photon. Technol. Lett. 17(2), 402–404 (2005).
[CrossRef]

Kimura, S.

S. Nishihara, M. Nakamura, K. Nishimura, K. Kishine, S. Kimura, and K. Kato, “10.3 Gbit/s burst-mode PIN-TIA module with high sensitivity, wide dynamic range and quick response,” Electron. Lett. 44(3), 222 (2008).
[CrossRef]

Kishine, K.

S. Nishihara, M. Nakamura, K. Nishimura, K. Kishine, S. Kimura, and K. Kato, “10.3 Gbit/s burst-mode PIN-TIA module with high sensitivity, wide dynamic range and quick response,” Electron. Lett. 44(3), 222 (2008).
[CrossRef]

Kitoh, T.

T. Kawai, M. Koga, M. Okuno, and T. Kitoh, “PLC type compact variable optical attenuator for photonic transport network,” Electron. Lett. 34(3), 264 (1998).
[CrossRef]

Koga, M.

T. Kawai, M. Koga, M. Okuno, and T. Kitoh, “PLC type compact variable optical attenuator for photonic transport network,” Electron. Lett. 34(3), 264 (1998).
[CrossRef]

Kuwayama, T.

T. Kuwayama, M. Ichimura, and E. Arai, “Interface recombination velocity of silicon-on-insulator wafers measured by microwave reflectance photoconductivity decay method with electric field,” Appl. Phys. Lett. 83(5), 928 (2003).
[CrossRef]

Linnros, J.

J. Linnros, “Carrier lifetime measurements using free carrier absorption transients. I. Principle and injection dependence,” J. Appl. Phys. 84(1), 275 (1998).
[CrossRef]

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Liu, A.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Liu, A. Q.

H. Cai, X. M. Zhang, C. Lu, A. Q. Liu, and E. H. Khoo, “Linear MEMS variable optical attenuator using reflective elliptical mirror,” IEEE Photon. Technol. Lett. 17(2), 402–404 (2005).
[CrossRef]

Lu, C.

H. Cai, X. M. Zhang, C. Lu, A. Q. Liu, and E. H. Khoo, “Linear MEMS variable optical attenuator using reflective elliptical mirror,” IEEE Photon. Technol. Lett. 17(2), 402–404 (2005).
[CrossRef]

Nakamura, M.

S. Nishihara, M. Nakamura, K. Nishimura, K. Kishine, S. Kimura, and K. Kato, “10.3 Gbit/s burst-mode PIN-TIA module with high sensitivity, wide dynamic range and quick response,” Electron. Lett. 44(3), 222 (2008).
[CrossRef]

Nicolaescu, R.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Nishihara, S.

S. Nishihara, M. Nakamura, K. Nishimura, K. Kishine, S. Kimura, and K. Kato, “10.3 Gbit/s burst-mode PIN-TIA module with high sensitivity, wide dynamic range and quick response,” Electron. Lett. 44(3), 222 (2008).
[CrossRef]

Nishimura, K.

S. Nishihara, M. Nakamura, K. Nishimura, K. Kishine, S. Kimura, and K. Kato, “10.3 Gbit/s burst-mode PIN-TIA module with high sensitivity, wide dynamic range and quick response,” Electron. Lett. 44(3), 222 (2008).
[CrossRef]

Okuno, M.

T. Kawai, M. Koga, M. Okuno, and T. Kitoh, “PLC type compact variable optical attenuator for photonic transport network,” Electron. Lett. 34(3), 264 (1998).
[CrossRef]

Paniccia, M.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Raghunathan, V.

Reed, G. T.

C. K. Tang, G. T. Reed, A. J. Walton, and A. G. Rickman, “Low-loss, single-model optical phase modulator in SIMOX material,” J. Lightwave Technol. 12(8), 1394–1400 (1994).
[CrossRef]

Rickman, A. G.

C. K. Tang, G. T. Reed, A. J. Walton, and A. G. Rickman, “Low-loss, single-model optical phase modulator in SIMOX material,” J. Lightwave Technol. 12(8), 1394–1400 (1994).
[CrossRef]

Rong, H.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Schmid, W.

J. Dziewior and W. Schmid, “Auger coefficients for highly doped and highly excited silicon,” Appl. Phys. Lett. 31(5), 346 (1977).
[CrossRef]

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Schroder, D. K.

D. K. Schroder, “Carrier lifetime in silicon,” IEEE Trans. Electron. Dev. 44(1), 160–170 (1997).
[CrossRef]

Smith, B. T.

D. W. Zheng, B. T. Smith, and M. Asghari, “Improved efficiency Si-photonic attenuator,” Opt. Express 16(21), 16754–16765 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16754 .
[CrossRef] [PubMed]

D. W. Zheng, B. T. Smith, J. Dong, and M. Asghari, “On the effective carrier lifetime of a silicon p-i-n diode optical modulator,” Semicond. Sci. Technol. 23(6), 064006 (2008).
[CrossRef]

Soref, R.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Tang, C. K.

C. K. Tang, G. T. Reed, A. J. Walton, and A. G. Rickman, “Low-loss, single-model optical phase modulator in SIMOX material,” J. Lightwave Technol. 12(8), 1394–1400 (1994).
[CrossRef]

Walton, A. J.

C. K. Tang, G. T. Reed, A. J. Walton, and A. G. Rickman, “Low-loss, single-model optical phase modulator in SIMOX material,” J. Lightwave Technol. 12(8), 1394–1400 (1994).
[CrossRef]

Woo, J. C. S.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86(7), 071115 (2005).
[CrossRef]

Xu, Q.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Zhang, X. M.

H. Cai, X. M. Zhang, C. Lu, A. Q. Liu, and E. H. Khoo, “Linear MEMS variable optical attenuator using reflective elliptical mirror,” IEEE Photon. Technol. Lett. 17(2), 402–404 (2005).
[CrossRef]

Zheng, D. W.

D. W. Zheng, B. T. Smith, and M. Asghari, “Improved efficiency Si-photonic attenuator,” Opt. Express 16(21), 16754–16765 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16754 .
[CrossRef] [PubMed]

D. W. Zheng, B. T. Smith, J. Dong, and M. Asghari, “On the effective carrier lifetime of a silicon p-i-n diode optical modulator,” Semicond. Sci. Technol. 23(6), 064006 (2008).
[CrossRef]

Appl. Phys. Lett. (3)

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86(7), 071115 (2005).
[CrossRef]

J. Dziewior and W. Schmid, “Auger coefficients for highly doped and highly excited silicon,” Appl. Phys. Lett. 31(5), 346 (1977).
[CrossRef]

T. Kuwayama, M. Ichimura, and E. Arai, “Interface recombination velocity of silicon-on-insulator wafers measured by microwave reflectance photoconductivity decay method with electric field,” Appl. Phys. Lett. 83(5), 928 (2003).
[CrossRef]

Electron. Lett. (2)

T. Kawai, M. Koga, M. Okuno, and T. Kitoh, “PLC type compact variable optical attenuator for photonic transport network,” Electron. Lett. 34(3), 264 (1998).
[CrossRef]

S. Nishihara, M. Nakamura, K. Nishimura, K. Kishine, S. Kimura, and K. Kato, “10.3 Gbit/s burst-mode PIN-TIA module with high sensitivity, wide dynamic range and quick response,” Electron. Lett. 44(3), 222 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H. Cai, X. M. Zhang, C. Lu, A. Q. Liu, and E. H. Khoo, “Linear MEMS variable optical attenuator using reflective elliptical mirror,” IEEE Photon. Technol. Lett. 17(2), 402–404 (2005).
[CrossRef]

IEEE Trans. Electron. Dev. (1)

D. K. Schroder, “Carrier lifetime in silicon,” IEEE Trans. Electron. Dev. 44(1), 160–170 (1997).
[CrossRef]

J. Appl. Phys. (1)

J. Linnros, “Carrier lifetime measurements using free carrier absorption transients. I. Principle and injection dependence,” J. Appl. Phys. 84(1), 275 (1998).
[CrossRef]

J. Lightwave Technol. (1)

C. K. Tang, G. T. Reed, A. J. Walton, and A. G. Rickman, “Low-loss, single-model optical phase modulator in SIMOX material,” J. Lightwave Technol. 12(8), 1394–1400 (1994).
[CrossRef]

Nature (2)

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature 433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Opt. Express (2)

Semicond. Sci. Technol. (1)

D. W. Zheng, B. T. Smith, J. Dong, and M. Asghari, “On the effective carrier lifetime of a silicon p-i-n diode optical modulator,” Semicond. Sci. Technol. 23(6), 064006 (2008).
[CrossRef]

Other (7)

E. D. Palik, Handbook of Optical Constants of Solid (Academic Press, 1985), p. 565.

S. M. Sze, Physics of Semiconductor Devices (John Wiley & Sons, 1981), Chap 7.

I. E. Day, I. Evans, A. Knights, F. Hopper, S. Roberts, J. Johnston, S. Day, J. Luff, H. K. Tsang, and M. Asghari, “Tapered silicon waveguides for low insertion loss highly-efficient high-speed electronic variable optical attenuator,” in Proceedings of Optical Fiber Communication Conference, (Institute of Electrical and Electronics Engineers, Atlanta, USA, 2003), pp. 249–251.

T. Tsuchizawa, K. Yamada, T. Watanabe, H. Shinojima, H. Nishi, S. Itabashi, S. Park, Y. Ishikawa, and K. Wada, “Monolithic Integration of Germanium Photodetectors and Silicon Wire Waveguides with Carrier Injection Structures,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CTuV2.

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic cross-section of fabricated Si p-i-n VOAs. (b) Optical microscope image of fabricated VOAs with different VOA lengths.

Fig. 2
Fig. 2

(a) I-V curves of the “defect” VOA with varying temperature. (b) Arrhenius plot of temperature-dependent reverse current of the “defect” VOA.

Fig. 3
Fig. 3

Plots of (a) attenuation as a function of injected current to the VOA, and (b) attenuation divided by VOA length with respect to injection current divided by VOA length. On the left y axis, the corresponding carrier density is shown.

Fig. 4
Fig. 4

(a) Measurement setup for 3-dB cut-off frequency with a network analyzer (b) Frequency responses of four different VOAs at injection current of 50 mA.

Fig. 5
Fig. 5

Measured 3-dB cut-off frequencies as a function of carrier density.

Fig. 6
Fig. 6

(a) Measurement setup for pulse response by normal incidence of 775 nm in wavelength from a second-harmonic generator (SHG). Initial wavelength from mode-lock laser (MML) is 1550 nm. (b) Impulse responses of 1-mm-long VOAs with (solid) and without (dotted) defect implantation at 5 mA injection.

Fig. 7
Fig. 7

Measured carrier lifetimes as a function of carrier density. Theoretical fitting lines including a sum of Auger recombination and surface recombination (Supper ) for the “no defects” VOAs and SRH recombination for the “defect” VOA are also shown.

Fig. 8
Fig. 8

Plot of attenuation efficiency as a function of measured carrier lifetime.

Equations (3)

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Atten .  [dB] = 10 ln 10 ( 6.0 × 10 18 p + 8.5 × 10 18 n ) L .
1 τ r e c = 1 τ S R H + 1 τ r a d + 1 τ A u g e r + S l o w e r H + S u p p e r H .
τ r e c = Q I = q A n L I .

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