Abstract

We report the study of hydrogenated amorphous silicon-germanium (a-Si1-XGeX:H) films prepared by low-frequency plasma-enhanced chemical vapor deposition (LF-PECVD) varying the composition (0 ≤ X ≤ 1). Silicon and germanium content is determined by energy dispersion spectroscopy (EDS). Refractive index, absorption coefficient and optical gap are estimated by transmittance measurements as well as by the use of PUMA software. Absorption coefficients obtained by using this software and by the Beer-Lambert law show good agreement according to absorption region. Results indicate that refractive index exhibits a linear behavior on germanium content for atomic percent (at.%). Employing these films and by the use of the finite-element software COMSOL, a single-mode low-contrast rib optical waveguide operating at the wavelength of 1550 nm is simulated, and later fabricated by using photolithography and plasma etching techniques. Measured optical losses are 7.6 dB/cm.

© 2012 OSA

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2010

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-Based Plasmonics for On-Chip Photonics,” IEEE J. Sel. Top. Quantum Electron.16(1), 295–306 (2010).
[CrossRef]

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

M. Labrune, M. Moreno, and P. Roca i Cabarrocas, “Ultra-shallow junctions formed by quasi-epitaxial growth of boron and phosphorous-doped silicon films at 175°C by rf-PECVD,” Thin Solid Films518(9), 2528–2530 (2010).
[CrossRef]

2009

2008

R. Soref, R. E. Peale, and W. Buchwald, “Longwave plasmonics on doped silicon and silicides,” Opt. Express16(9), 6507–6514 (2008).
[CrossRef] [PubMed]

C. Iliescu and B. Chen, “Thick and low-stress PECVD amorphous silicon for MEMS applications,” J. Micromech. Microeng.18(1), 015024 (2008).
[CrossRef]

2007

A. Sabac, C. Gorecki, M. Jozwik, L. Nieradko, C. Meunier, and K. Gut, “Technology and performances of silicon oxynitride waveguides for optomechanical sensors fabricated by plasma-enhanced chemical vapor deposition,” J. Eur. Opt. Soc.2, 07026 (2007).
[CrossRef]

K. Ikeda, Y. Shen, and Y. Fainman, “Enhanced optical nonlinearity in amorphous silicon and its application to waveguide devices,” Opt. Express15(26), 17761–17771 (2007).
[CrossRef] [PubMed]

2006

R. Soref, “The Past, Present, and Future of Silicon Photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006).
[CrossRef]

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt.8(10), 840–848 (2006).
[CrossRef]

M. E. Wieser, “Atomic weights of the elements 2005 (IUPAC Technical Report),” Pure Appl. Chem.78(11), 2051–2066 (2006).
[CrossRef]

2005

A. M. Pérez, C. Zúñiga, F. J. Renero, and A. Torres, “Optical properties of amorphous silicon germanium obtained by low-frequency plasma-enhanced chemical vapor deposition from SiH4 + GeF4 and from SiH4 + GeH4,” Opt. Eng.44(4), 043801 (2005).
[CrossRef]

M. Lipson, “Switching light on a silicon chip,” Opt. Mater.27(5), 731–739 (2005).
[CrossRef]

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. Express13(8), 3129–3135 (2005).
[CrossRef] [PubMed]

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett.41(25), 1377–1378 (2005).
[CrossRef]

2003

A. Kosarev, A. Torres, C. Zúñiga, A. Abramov, P. Rosales, and A. Sibaja, “Effect of hydrogen dilution on electronic properties of a-SiHx films deposited by low-frequency plasma,” J. Mater. Res.18(08), 1918–1925 (2003).
[CrossRef]

D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” J. Phys. D Appl. Phys.36(15), 1850–1857 (2003).
[CrossRef]

2002

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

2000

A. Vonsovici, G. T. Reed, and A. G. R. Evans, “β-SiC-on insulator waveguide structures for modulators and sensors systems,” Mater. Sci. Semicond. Process.3(5-6), 367–374 (2000).
[CrossRef]

1999

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the Optical Constants and the Thickness of Thin Films Using Unconstrained Optimization,” J. Comput. Phys.151(2), 862–880 (1999), http://www.ime.usp.br/~egbirgin/puma/ .
[CrossRef]

1998

S. P. Pogossian, L. Vescan, and A. Vonsovici, “The Single-Mode Condition for Semiconductor Rib Waveguides with Large Cross Section,” J. Lightwave Technol.16(10), 1851–1853 (1998).
[CrossRef]

G. Cocorullo, F. G. Della Corte, R. De Rosa, I. Rendina, A. Rubino, and E. Terzini, “Amorphous Silicon Waveguides and Interferometers for Low-Cost Silicon Optoelectronics,” Proc. SPIE3278, 286–292 (1998).
[CrossRef]

Y. P. Chou and S. C. Lee, “Structural, optical, and electrical properties of hydrogenated amorphous silicon germanium alloys,” J. Appl. Phys.83(8), 4111–4123 (1998).
[CrossRef]

1996

1991

R. A. Soref, J. Schmidtchen, and K. Petermann, “Large Single-Mode Rib Waveguides in GeSi-Si and Si on SiO2,” IEEE J. Quantum Electron.27(8), 1971–1974 (1991).
[CrossRef]

1983

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E16(12), 1214–1222 (1983).
[CrossRef]

1981

K. Shimakawa, “On the compositional dependence of the optical gap in amorphous semiconducting alloys,” J. Non-Cryst. Solids43(2), 229–244 (1981).
[CrossRef]

1974

1921

L. Vegard, “Die Konstitution der Mischkristalle und die Raumfülung der Atome,” Z. Phys.5(1), 17–26 (1921).
[CrossRef]

Abramov, A.

A. Kosarev, A. Torres, C. Zúñiga, A. Abramov, P. Rosales, and A. Sibaja, “Effect of hydrogen dilution on electronic properties of a-SiHx films deposited by low-frequency plasma,” J. Mater. Res.18(08), 1918–1925 (2003).
[CrossRef]

Abramov, A. S.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Ahn, Q.

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

Alivisatos, A. P.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-Based Plasmonics for On-Chip Photonics,” IEEE J. Sel. Top. Quantum Electron.16(1), 295–306 (2010).
[CrossRef]

Ambrosio, R.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Asomoza, R.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-Based Plasmonics for On-Chip Photonics,” IEEE J. Sel. Top. Quantum Electron.16(1), 295–306 (2010).
[CrossRef]

Baets, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Birgin, E. G.

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the Optical Constants and the Thickness of Thin Films Using Unconstrained Optimization,” J. Comput. Phys.151(2), 862–880 (1999), http://www.ime.usp.br/~egbirgin/puma/ .
[CrossRef]

Buchwald, W.

Buchwald, W. R.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt.8(10), 840–848 (2006).
[CrossRef]

Chambouleyron, I.

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the Optical Constants and the Thickness of Thin Films Using Unconstrained Optimization,” J. Comput. Phys.151(2), 862–880 (1999), http://www.ime.usp.br/~egbirgin/puma/ .
[CrossRef]

Chen, B.

C. Iliescu and B. Chen, “Thick and low-stress PECVD amorphous silicon for MEMS applications,” J. Micromech. Microeng.18(1), 015024 (2008).
[CrossRef]

Cheng, J.

Chew, K.

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

Chou, Y. P.

Y. P. Chou and S. C. Lee, “Structural, optical, and electrical properties of hydrogenated amorphous silicon germanium alloys,” J. Appl. Phys.83(8), 4111–4123 (1998).
[CrossRef]

Cocorullo, G.

G. Cocorullo, F. G. Della Corte, R. De Rosa, I. Rendina, A. Rubino, and E. Terzini, “Amorphous Silicon Waveguides and Interferometers for Low-Cost Silicon Optoelectronics,” Proc. SPIE3278, 286–292 (1998).
[CrossRef]

G. Cocorullo, F. G. Corte, I. Rendina, C. Minarini, A. Rubino, and E. Terzini, “Amorphous silicon waveguides and light modulators for integrated photonics realized by low-temperature plasma-enhanced chemical-vapor deposition,” Opt. Lett.21(24), 2002–2004 (1996).
[CrossRef] [PubMed]

Corte, F. G.

De Rosa, R.

G. Cocorullo, F. G. Della Corte, R. De Rosa, I. Rendina, A. Rubino, and E. Terzini, “Amorphous Silicon Waveguides and Interferometers for Low-Cost Silicon Optoelectronics,” Proc. SPIE3278, 286–292 (1998).
[CrossRef]

de Vries, T.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Della Corte, F. G.

G. Cocorullo, F. G. Della Corte, R. De Rosa, I. Rendina, A. Rubino, and E. Terzini, “Amorphous Silicon Waveguides and Interferometers for Low-Cost Silicon Optoelectronics,” Proc. SPIE3278, 286–292 (1998).
[CrossRef]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-Based Plasmonics for On-Chip Photonics,” IEEE J. Sel. Top. Quantum Electron.16(1), 295–306 (2010).
[CrossRef]

Emelett, S. J.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt.8(10), 840–848 (2006).
[CrossRef]

Evans, A. G. R.

A. Vonsovici, G. T. Reed, and A. G. R. Evans, “β-SiC-on insulator waveguide structures for modulators and sensors systems,” Mater. Sci. Semicond. Process.3(5-6), 367–374 (2000).
[CrossRef]

Fainman, Y.

Felter, T. E.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Franck, T.

Geluk, E.-J.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Gomez-Barojas, E.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Gorecki, C.

A. Sabac, C. Gorecki, M. Jozwik, L. Nieradko, C. Meunier, and K. Gut, “Technology and performances of silicon oxynitride waveguides for optomechanical sensors fabricated by plasma-enhanced chemical vapor deposition,” J. Eur. Opt. Soc.2, 07026 (2007).
[CrossRef]

Gut, K.

A. Sabac, C. Gorecki, M. Jozwik, L. Nieradko, C. Meunier, and K. Gut, “Technology and performances of silicon oxynitride waveguides for optomechanical sensors fabricated by plasma-enhanced chemical vapor deposition,” J. Eur. Opt. Soc.2, 07026 (2007).
[CrossRef]

Harke, A.

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett.41(25), 1377–1378 (2005).
[CrossRef]

Hernandez, Y.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Hodge, D.

Huybrechts, K.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Ikeda, K.

Iliescu, C.

C. Iliescu and B. Chen, “Thick and low-stress PECVD amorphous silicon for MEMS applications,” J. Micromech. Microeng.18(1), 015024 (2008).
[CrossRef]

Ilinski, A.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Jozwik, M.

A. Sabac, C. Gorecki, M. Jozwik, L. Nieradko, C. Meunier, and K. Gut, “Technology and performances of silicon oxynitride waveguides for optomechanical sensors fabricated by plasma-enhanced chemical vapor deposition,” J. Eur. Opt. Soc.2, 07026 (2007).
[CrossRef]

Keil, U. D.

Kimerling, L.

Kogelnik, H.

Kosarev, A.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

A. Kosarev, A. Torres, C. Zúñiga, A. Abramov, P. Rosales, and A. Sibaja, “Effect of hydrogen dilution on electronic properties of a-SiHx films deposited by low-frequency plasma,” J. Mater. Res.18(08), 1918–1925 (2003).
[CrossRef]

Krause, M.

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett.41(25), 1377–1378 (2005).
[CrossRef]

Kudriavtsev, Y.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Kumar, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Labrune, M.

M. Labrune, M. Moreno, and P. Roca i Cabarrocas, “Ultra-shallow junctions formed by quasi-epitaxial growth of boron and phosphorous-doped silicon films at 175°C by rf-PECVD,” Thin Solid Films518(9), 2528–2530 (2010).
[CrossRef]

Lee, S. C.

Y. P. Chou and S. C. Lee, “Structural, optical, and electrical properties of hydrogenated amorphous silicon germanium alloys,” J. Appl. Phys.83(8), 4111–4123 (1998).
[CrossRef]

Liao, L.

Ligatchev, E. J.

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

Lipson, M.

M. Lipson, “Switching light on a silicon chip,” Opt. Mater.27(5), 731–739 (2005).
[CrossRef]

Liu, A.

Liu, L.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Martínez, J. M.

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the Optical Constants and the Thickness of Thin Films Using Unconstrained Optimization,” J. Comput. Phys.151(2), 862–880 (1999), http://www.ime.usp.br/~egbirgin/puma/ .
[CrossRef]

Meunier, C.

A. Sabac, C. Gorecki, M. Jozwik, L. Nieradko, C. Meunier, and K. Gut, “Technology and performances of silicon oxynitride waveguides for optomechanical sensors fabricated by plasma-enhanced chemical vapor deposition,” J. Eur. Opt. Soc.2, 07026 (2007).
[CrossRef]

Michel, J.

Minarini, C.

Moreno, M.

M. Labrune, M. Moreno, and P. Roca i Cabarrocas, “Ultra-shallow junctions formed by quasi-epitaxial growth of boron and phosphorous-doped silicon films at 175°C by rf-PECVD,” Thin Solid Films518(9), 2528–2530 (2010).
[CrossRef]

Morse, M.

Morthier, G.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Mueller, J.

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett.41(25), 1377–1378 (2005).
[CrossRef]

Nieradko, L.

A. Sabac, C. Gorecki, M. Jozwik, L. Nieradko, C. Meunier, and K. Gut, “Technology and performances of silicon oxynitride waveguides for optomechanical sensors fabricated by plasma-enhanced chemical vapor deposition,” J. Eur. Opt. Soc.2, 07026 (2007).
[CrossRef]

Osipowicz, T.

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

Peale, R. E.

Pérez, A. M.

A. M. Pérez, C. Zúñiga, F. J. Renero, and A. Torres, “Optical properties of amorphous silicon germanium obtained by low-frequency plasma-enhanced chemical vapor deposition from SiH4 + GeF4 and from SiH4 + GeH4,” Opt. Eng.44(4), 043801 (2005).
[CrossRef]

Petermann, K.

R. A. Soref, J. Schmidtchen, and K. Petermann, “Large Single-Mode Rib Waveguides in GeSi-Si and Si on SiO2,” IEEE J. Quantum Electron.27(8), 1971–1974 (1991).
[CrossRef]

Poelman, D.

D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” J. Phys. D Appl. Phys.36(15), 1850–1857 (2003).
[CrossRef]

Pogossian, S. P.

Ramaswamy, V.

Reed, G. T.

A. Vonsovici, G. T. Reed, and A. G. R. Evans, “β-SiC-on insulator waveguide structures for modulators and sensors systems,” Mater. Sci. Semicond. Process.3(5-6), 367–374 (2000).
[CrossRef]

Regreny, P.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Rendina, I.

G. Cocorullo, F. G. Della Corte, R. De Rosa, I. Rendina, A. Rubino, and E. Terzini, “Amorphous Silicon Waveguides and Interferometers for Low-Cost Silicon Optoelectronics,” Proc. SPIE3278, 286–292 (1998).
[CrossRef]

G. Cocorullo, F. G. Corte, I. Rendina, C. Minarini, A. Rubino, and E. Terzini, “Amorphous silicon waveguides and light modulators for integrated photonics realized by low-temperature plasma-enhanced chemical-vapor deposition,” Opt. Lett.21(24), 2002–2004 (1996).
[CrossRef] [PubMed]

Renero, F. J.

A. M. Pérez, C. Zúñiga, F. J. Renero, and A. Torres, “Optical properties of amorphous silicon germanium obtained by low-frequency plasma-enhanced chemical vapor deposition from SiH4 + GeF4 and from SiH4 + GeH4,” Opt. Eng.44(4), 043801 (2005).
[CrossRef]

Roca i Cabarrocas, P.

M. Labrune, M. Moreno, and P. Roca i Cabarrocas, “Ultra-shallow junctions formed by quasi-epitaxial growth of boron and phosphorous-doped silicon films at 175°C by rf-PECVD,” Thin Solid Films518(9), 2528–2530 (2010).
[CrossRef]

Roelkens, G.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Rosales, P.

A. Kosarev, A. Torres, C. Zúñiga, A. Abramov, P. Rosales, and A. Sibaja, “Effect of hydrogen dilution on electronic properties of a-SiHx films deposited by low-frequency plasma,” J. Mater. Res.18(08), 1918–1925 (2003).
[CrossRef]

Rubin, D.

Rubino, A.

G. Cocorullo, F. G. Della Corte, R. De Rosa, I. Rendina, A. Rubino, and E. Terzini, “Amorphous Silicon Waveguides and Interferometers for Low-Cost Silicon Optoelectronics,” Proc. SPIE3278, 286–292 (1998).
[CrossRef]

G. Cocorullo, F. G. Corte, I. Rendina, C. Minarini, A. Rubino, and E. Terzini, “Amorphous silicon waveguides and light modulators for integrated photonics realized by low-temperature plasma-enhanced chemical-vapor deposition,” Opt. Lett.21(24), 2002–2004 (1996).
[CrossRef] [PubMed]

Rusli, S. F.

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

Sabac, A.

A. Sabac, C. Gorecki, M. Jozwik, L. Nieradko, C. Meunier, and K. Gut, “Technology and performances of silicon oxynitride waveguides for optomechanical sensors fabricated by plasma-enhanced chemical vapor deposition,” J. Eur. Opt. Soc.2, 07026 (2007).
[CrossRef]

Samara-Rubio, D.

Schmidtchen, J.

R. A. Soref, J. Schmidtchen, and K. Petermann, “Large Single-Mode Rib Waveguides in GeSi-Si and Si on SiO2,” IEEE J. Quantum Electron.27(8), 1971–1974 (1991).
[CrossRef]

Sheldon, M. T.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-Based Plasmonics for On-Chip Photonics,” IEEE J. Sel. Top. Quantum Electron.16(1), 295–306 (2010).
[CrossRef]

Shen, Y.

Shimakawa, K.

K. Shimakawa, “On the compositional dependence of the optical gap in amorphous semiconducting alloys,” J. Non-Cryst. Solids43(2), 229–244 (1981).
[CrossRef]

Sibaja, A.

A. Kosarev, A. Torres, C. Zúñiga, A. Abramov, P. Rosales, and A. Sibaja, “Effect of hydrogen dilution on electronic properties of a-SiHx films deposited by low-frequency plasma,” J. Mater. Res.18(08), 1918–1925 (2003).
[CrossRef]

Silva-Gonzalez, R.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Smet, P. F.

D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” J. Phys. D Appl. Phys.36(15), 1850–1857 (2003).
[CrossRef]

Soref, R.

R. Soref, R. E. Peale, and W. Buchwald, “Longwave plasmonics on doped silicon and silicides,” Opt. Express16(9), 6507–6514 (2008).
[CrossRef] [PubMed]

R. Soref, “The Past, Present, and Future of Silicon Photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006).
[CrossRef]

Soref, R. A.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt.8(10), 840–848 (2006).
[CrossRef]

R. A. Soref, J. Schmidtchen, and K. Petermann, “Large Single-Mode Rib Waveguides in GeSi-Si and Si on SiO2,” IEEE J. Quantum Electron.27(8), 1971–1974 (1991).
[CrossRef]

Spuesens, T.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Sun, R.

Swanepoel, R.

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E16(12), 1214–1222 (1983).
[CrossRef]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-Based Plasmonics for On-Chip Photonics,” IEEE J. Sel. Top. Quantum Electron.16(1), 295–306 (2010).
[CrossRef]

Teo, T.

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

Terzini, E.

G. Cocorullo, F. G. Della Corte, R. De Rosa, I. Rendina, A. Rubino, and E. Terzini, “Amorphous Silicon Waveguides and Interferometers for Low-Cost Silicon Optoelectronics,” Proc. SPIE3278, 286–292 (1998).
[CrossRef]

G. Cocorullo, F. G. Corte, I. Rendina, C. Minarini, A. Rubino, and E. Terzini, “Amorphous silicon waveguides and light modulators for integrated photonics realized by low-temperature plasma-enhanced chemical-vapor deposition,” Opt. Lett.21(24), 2002–2004 (1996).
[CrossRef] [PubMed]

Torres, A.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

A. M. Pérez, C. Zúñiga, F. J. Renero, and A. Torres, “Optical properties of amorphous silicon germanium obtained by low-frequency plasma-enhanced chemical vapor deposition from SiH4 + GeF4 and from SiH4 + GeH4,” Opt. Eng.44(4), 043801 (2005).
[CrossRef]

A. Kosarev, A. Torres, C. Zúñiga, A. Abramov, P. Rosales, and A. Sibaja, “Effect of hydrogen dilution on electronic properties of a-SiHx films deposited by low-frequency plasma,” J. Mater. Res.18(08), 1918–1925 (2003).
[CrossRef]

Van Thourhout, D.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Vegard, L.

L. Vegard, “Die Konstitution der Mischkristalle und die Raumfülung der Atome,” Z. Phys.5(1), 17–26 (1921).
[CrossRef]

Vescan, L.

Vonsovici, A.

A. Vonsovici, G. T. Reed, and A. G. R. Evans, “β-SiC-on insulator waveguide structures for modulators and sensors systems,” Mater. Sci. Semicond. Process.3(5-6), 367–374 (2000).
[CrossRef]

S. P. Pogossian, L. Vescan, and A. Vonsovici, “The Single-Mode Condition for Semiconductor Rib Waveguides with Large Cross Section,” J. Lightwave Technol.16(10), 1851–1853 (1998).
[CrossRef]

Watt, F.

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

Wieser, M. E.

M. E. Wieser, “Atomic weights of the elements 2005 (IUPAC Technical Report),” Pure Appl. Chem.78(11), 2051–2066 (2006).
[CrossRef]

Yoon, J.

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

Zhang, V.

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

Zuniga, C.

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

Zúñiga, C.

A. M. Pérez, C. Zúñiga, F. J. Renero, and A. Torres, “Optical properties of amorphous silicon germanium obtained by low-frequency plasma-enhanced chemical vapor deposition from SiH4 + GeF4 and from SiH4 + GeH4,” Opt. Eng.44(4), 043801 (2005).
[CrossRef]

A. Kosarev, A. Torres, C. Zúñiga, A. Abramov, P. Rosales, and A. Sibaja, “Effect of hydrogen dilution on electronic properties of a-SiHx films deposited by low-frequency plasma,” J. Mater. Res.18(08), 1918–1925 (2003).
[CrossRef]

Appl. Opt.

Electron. Lett.

A. Harke, M. Krause, and J. Mueller, “Low-loss singlemode amorphous silicon waveguides,” Electron. Lett.41(25), 1377–1378 (2005).
[CrossRef]

IEEE J. Quantum Electron.

R. A. Soref, J. Schmidtchen, and K. Petermann, “Large Single-Mode Rib Waveguides in GeSi-Si and Si on SiO2,” IEEE J. Quantum Electron.27(8), 1971–1974 (1991).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

R. Soref, “The Past, Present, and Future of Silicon Photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-Based Plasmonics for On-Chip Photonics,” IEEE J. Sel. Top. Quantum Electron.16(1), 295–306 (2010).
[CrossRef]

J. Appl. Phys.

Y. P. Chou and S. C. Lee, “Structural, optical, and electrical properties of hydrogenated amorphous silicon germanium alloys,” J. Appl. Phys.83(8), 4111–4123 (1998).
[CrossRef]

K. Chew, S. F. Rusli, J. Yoon, Q. Ahn, V. Zhang, E. J. Ligatchev, T. Teo, T. Osipowicz, and F. Watt, “Gap state distribution in amorphous hydrogenated silicon carbide films deduced from photothermal deflection spectroscopy,” J. Appl. Phys.91(7), 4319–4325 (2002).
[CrossRef]

J. Comput. Phys.

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, “Estimation of the Optical Constants and the Thickness of Thin Films Using Unconstrained Optimization,” J. Comput. Phys.151(2), 862–880 (1999), http://www.ime.usp.br/~egbirgin/puma/ .
[CrossRef]

J. Eur. Opt. Soc.

A. Sabac, C. Gorecki, M. Jozwik, L. Nieradko, C. Meunier, and K. Gut, “Technology and performances of silicon oxynitride waveguides for optomechanical sensors fabricated by plasma-enhanced chemical vapor deposition,” J. Eur. Opt. Soc.2, 07026 (2007).
[CrossRef]

J. Lightwave Technol.

J. Mater. Res.

A. Kosarev, A. Torres, C. Zúñiga, A. Abramov, P. Rosales, and A. Sibaja, “Effect of hydrogen dilution on electronic properties of a-SiHx films deposited by low-frequency plasma,” J. Mater. Res.18(08), 1918–1925 (2003).
[CrossRef]

A. Kosarev, A. Torres, Y. Hernandez, R. Ambrosio, C. Zuniga, T. E. Felter, R. Asomoza, Y. Kudriavtsev, R. Silva-Gonzalez, E. Gomez-Barojas, A. Ilinski, and A. S. Abramov, “Silicon-germanium films deposited by low-frequency plasma-enhanced chemical vapor deposition: Effect of H2 and Ar dilution,” J. Mater. Res.21(01), 88–104 (2006).
[CrossRef]

J. Micromech. Microeng.

C. Iliescu and B. Chen, “Thick and low-stress PECVD amorphous silicon for MEMS applications,” J. Micromech. Microeng.18(1), 015024 (2008).
[CrossRef]

J. Non-Cryst. Solids

K. Shimakawa, “On the compositional dependence of the optical gap in amorphous semiconducting alloys,” J. Non-Cryst. Solids43(2), 229–244 (1981).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt.8(10), 840–848 (2006).
[CrossRef]

J. Phys. D Appl. Phys.

D. Poelman and P. F. Smet, “Methods for the determination of the optical constants of thin films from single transmission measurements: a critical review,” J. Phys. D Appl. Phys.36(15), 1850–1857 (2003).
[CrossRef]

J. Phys. E

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E16(12), 1214–1222 (1983).
[CrossRef]

Mater. Sci. Semicond. Process.

A. Vonsovici, G. T. Reed, and A. G. R. Evans, “β-SiC-on insulator waveguide structures for modulators and sensors systems,” Mater. Sci. Semicond. Process.3(5-6), 367–374 (2000).
[CrossRef]

Nat. Photonics

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E.-J. Geluk, T. de Vries, P. Regreny, D. Van Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010).
[CrossRef]

Opt. Eng.

A. M. Pérez, C. Zúñiga, F. J. Renero, and A. Torres, “Optical properties of amorphous silicon germanium obtained by low-frequency plasma-enhanced chemical vapor deposition from SiH4 + GeF4 and from SiH4 + GeH4,” Opt. Eng.44(4), 043801 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Mater.

M. Lipson, “Switching light on a silicon chip,” Opt. Mater.27(5), 731–739 (2005).
[CrossRef]

Proc. SPIE

G. Cocorullo, F. G. Della Corte, R. De Rosa, I. Rendina, A. Rubino, and E. Terzini, “Amorphous Silicon Waveguides and Interferometers for Low-Cost Silicon Optoelectronics,” Proc. SPIE3278, 286–292 (1998).
[CrossRef]

Pure Appl. Chem.

M. E. Wieser, “Atomic weights of the elements 2005 (IUPAC Technical Report),” Pure Appl. Chem.78(11), 2051–2066 (2006).
[CrossRef]

Thin Solid Films

M. Labrune, M. Moreno, and P. Roca i Cabarrocas, “Ultra-shallow junctions formed by quasi-epitaxial growth of boron and phosphorous-doped silicon films at 175°C by rf-PECVD,” Thin Solid Films518(9), 2528–2530 (2010).
[CrossRef]

Z. Phys.

L. Vegard, “Die Konstitution der Mischkristalle und die Raumfülung der Atome,” Z. Phys.5(1), 17–26 (1921).
[CrossRef]

Other

G. Y. Sung, N. M. Park, T. Y. Kim, K. H. Kim, K. S. Cho, and J. H. Shin, “High Efficiency Silicon Visible Light Emitter using Silicon Nanocrystals in Silicon Nitride Matrix and Transparent Doping Layer,” 2nd International Conference on Group IV Photonics, Belgium, September 2005.

http://en.wikipedia.org/wiki/Beer-Lambert_law .

G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003), Chap. 2.

J. Singh, Optical Properties of Condensed Matter and Applications (Wiley, 2006), Chap. 3.

L. Pavesi and D. J. Lockwood, eds., Silicon Photonics (Springer-Verlag, 2004).

G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (Wiley, 2004), Chap. 4.

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2010).

A. Scandurra, M. Lenzi, R. Guerra, F. G. Della Corte, and M. A. Nigro, “Optical Interconnects for Network on Chip,” 1st International Conference on Nano-Networks and Workshops, Lausanne, September 2006.

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Figures (11)

Fig. 1
Fig. 1

EDS measurements of the samples 847-850, (a-d) respectively. Horizontal axis is in the range 0-5 keV; full-scale of vertical axis is 20856 cts.

Fig. 2
Fig. 2

(a) Deposition rate, Vd, as a function of germanium content in solid phase, Xw. (b) Germanium content on solid phase, X, as a function of germanium content on gas phase, XGe, of the a-Si1-XGeX:H films.

Fig. 3
Fig. 3

(a) Conventional transmittance in the UV-Vis region, (b) FTIR transmittance in the NIR region of the a-Si1-XGeX:H films.

Fig. 4
Fig. 4

Absorption coefficient from transmittance in the UV-Vis region, using the results of PUMA software and the Beer-Lambert law for (a) common and (b) natural log functions.

Fig. 5
Fig. 5

Absorption coefficient from FTIR transmittance in the NIR region, using the results of PUMA and the Beer-Lambert law for (a) common log and (b) natural log functions.

Fig. 6
Fig. 6

Estimation of germanium content in solid phase from optical gap by the use of Shimakawa relation.

Fig. 7
Fig. 7

Refractive index estimated by means of the PUMA software from (a) UV-Vis conventional transmittance, and (b) FTIR transmittance. Vertical dashed lines are located at 830, 1310 and 1550 nm of wavelength.

Fig. 8
Fig. 8

Refractive index as a function of germanium content in solid phase of the a-Si1-XGeX:H films.

Fig. 9
Fig. 9

Simulations of the single-mode low-contrast rib optical waveguide for (a) quasi-TE and (b) quasi-TM mode.

Fig. 10
Fig. 10

(a) Etch rate as a function of germanium content in solid phase, Xw, for SF6/O2 and CF4/O2 gases mixtures. (b) Structure of 3 µm of width from SF6/O2. (c) Structure of 1.5 μm of width from CF4/O2.

Fig. 11
Fig. 11

(a) Propagation losses obtained from the low-contrast single-mode rib optical waveguide by cut-back technique. (b) Optical waveguides of 1.5 µm of width. (c) Optical waveguide of 15 µm of width.

Tables (3)

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Table 1 Gas Flows for Deposition and Thickness of the a-Si1-XGeX:H Films

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Table 2 Germanium and Silicon Content of the a-Si1-XGeX:H Films from EDS Measurements

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Table 3 Optical Gap, Germanium Content and B Parameter of the a-Si1-XGeX:H Films

Equations (6)

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a AB (X)=X a A +(1X) a B
E AB (Y)=Y E A +(1Y) E B
n SiGe (X)=X n Ge +(1X) n Si
X at = X w A r (Si) X w A r (Si)+(1 X w ) A r (Ge)
X w = X at A r (Ge) X at A r (Ge)+(1 X at ) A r (Si)
Γ= 1 L 10log( 10 αL )=10α

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