Abstract

The Ge1−xSnx alloy is a promising material for optoelectronic applications. It offers a tunable wavelength in the infrared (IR) spectrum and high compatibility with complementary metal-oxide-semiconductor (CMOS) technology. However, difficulties in growing device quality Ge1−xSnx films has left the potentiality of this material unexplored. Recent advances in technological processes have renewed the interest toward this material paving the way to potential applications. In this work, we perform a numerical investigation on absorption coefficient, radiative recombination rate, and Auger recombination properties of intrinsic and doped Ge1−xSnx for application in the extended-short wavelength infrared and medium wavelength infrared spectrum ranges. We apply a Green’s function based model to the Ge1−xSnx full electronic band structure determined through an empirical pseudopotential method and determine the dominant recombination mechanism between radiative and Auger processes over a wide range of injection levels.

© 2016 Optical Society of America

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    [Crossref]
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    [Crossref]
  3. M. R. Bauer, J. Tolle, C. Bungay, A. V. Chizmeshya, D. J. Smith, J. Menéndez, and J. Kouvetakis, “Tunable band structure in diamond–cubic tin–germanium alloys grown on silicon substrates,” Solid State Commun. 127, 355–359 (2003).
    [Crossref]
  4. S. Wirths, R. Geiger, N. von den Driesch, G. Mussler, T. Stoica, S. Mantl, Z. Ikonic, M. Luysberg, S. Chiussi, J. Hartmann, H. Sigg, J. Faist, D. Buca, and D. Grützmacher, “Lasing in direct-bandgap GeSn alloy grown on Si,” Nature Photon. 9, 88–92 (2015).
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  18. N. Bouarissa and F. Annane, “Electronic properties and elastic constants of the ordered Ge1−x Snx alloys,” Mater. Sci. Eng. B 95, 100–106 (2002).
    [Crossref]
  19. J. D. Sau and M. L. Cohen, “Possibility of increased mobility in Ge–Sn alloy system,” Phys. Rev. B 75, 045208 (2007).
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    [Crossref]
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  23. T. Tsukamoto, N. Hirose, A. Kasamatsu, T. Mimura, T. Matsui, and Y. Suda, “Formation of GeSn layers on Si (001) substrates at high growth temperature and high deposition rate by sputter epitaxy method,” J. Mater. Sci. 50, 4366–4370 (2015).
    [Crossref]
  24. J. Mathews, R. Beeler, J. Tolle, C. Xu, R. Roucka, J. Kouvetakis, and J. Menéndez, “Direct-gap photoluminescence with tunable emission wavelength in Ge1−y Sny alloys on silicon,” Appl. Phys. Lett. 97, 221912 (2010).
    [Crossref]
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    [Crossref]
  26. H. Lin, R. Chen, W. Lu, Y. Huo, T. I. Kamins, and J. S. Harris, “Investigation of the direct band gaps in Ge1−x Snx alloys with strain control by photoreflectance spectroscopy,” Appl. Phys. Lett. 100, 102109 (2012).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  34. W. Bardyszewski and D. Yevick, “Compositional dependence of the Auger coefficient for InGaAsP lattice matched to InP,” J. Appl. Phys. 58, 2713–2723 (1985).
    [Crossref]
  35. F. Bertazzi, M. Goano, and E. Bellotti, “A numerical study of Auger recombination in bulk InGaN,” Appl. Phys. Lett. 97, 231118 (2010).
    [Crossref]
  36. F. Bertazzi, M. Goano, and E. Bellotti, “Numerical analysis of indirect Auger transitions in InGaN,” Appl. Phys. Lett. 101, 011111 (2012).
    [Crossref]
  37. F. Bertazzi, X. Zhou, M. Goano, G. Ghione, and E. Bellotti, “Auger recombination in InGaN/GaN quantum wells. A full-Brillouin-zone study,” Appl. Phys. Lett. 103, 081106 (2013).
    [Crossref]
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    [Crossref]
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    [Crossref]
  43. L. Huldt, “Phonon-assisted Auger recombination in germanium,” Phys. Status Solidi A 33, 607–614 (1976).
    [Crossref]
  44. A. Haug, “Phonon-assisted Auger recombination in degenerate semiconductors,” Solid State Commun. 22, 537–539 (1977).
    [Crossref]
  45. A. Haug, “Auger recombination of electron-hole drops,” Solid State Commun. 25, 477–479 (1978).
    [Crossref]
  46. C. Senaratne, P. Wallace, J. Gallagher, P. Sims, J. Kouvetakis, and J. Menendez, “Direct gap Ge1−y Sny alloys: Fabrication and design of mid-IR photodiodes,” J. Appl. Phys. 120, 025701 (2016).
    [Crossref]
  47. S. Dominici, H. Wen, F. Bertazzi, M. Goano, and E. Bellotti, “Numerical evaluation of Auger recombination coefficients in relaxed and strained germanium,” Appl. Phys. Lett. 108, 211103 (2016).
    [Crossref]

2016 (7)

H. Tran, W. Du, S. A. Ghetmiri, A. Mosleh, G. Sun, R. A. Soref, J. Margetis, J. Tolle, B. Li, H. A. Naseem, and S. Q. Yu, “Systematic study of Ge1−x Snx absorption coefficient and refractive index for the device applications of Si-based optoelectronics,” J. Appl. Phys. 119, 103106 (2016).
[Crossref]

C. Senaratne, P. Wallace, J. Gallagher, P. Sims, J. Kouvetakis, and J. Menendez, “Direct gap Ge1−y Sny alloys: Fabrication and design of mid–IR photodiodes,” J. Appl. Phys. 120, 025701 (2016).
[Crossref]

Y. Zhou, W. Dou, W. Du, T. Pham, S. A. Ghetmiri, S. Al-Kabi, A. Mosleh, M. Alher, J. Margetis, J. Tolle, G. Sun, R. Soref, B. Li, M. Mortazavi, H. Naseem, and S. Q. Yu, “Systematic study of GeSn heterostructure-based light-emitting diodes towards mid–infrared applications,” J. Appl. Phys. 120, 023102 (2016).
[Crossref]

S. Wirths, D. Buca, and S. Mantl, “Si–Ge–Sn alloys: From growth to applications,” Progress in Crystal Growth and Characterization of Materials 62, 1–39 (2016).
[Crossref]

F. Freitas, J. Furthmüller, F. Bechstedt, M. Marques, and L. Teles, “Influence of the composition fluctuations and decomposition on the tunable direct gap and oscillator strength of Ge1−x Snx alloys,” Appl. Phys. Lett. 108, 092101 (2016).
[Crossref]

C. Senaratne, P. Wallace, J. Gallagher, P. Sims, J. Kouvetakis, and J. Menendez, “Direct gap Ge1−y Sny alloys: Fabrication and design of mid-IR photodiodes,” J. Appl. Phys. 120, 025701 (2016).
[Crossref]

S. Dominici, H. Wen, F. Bertazzi, M. Goano, and E. Bellotti, “Numerical evaluation of Auger recombination coefficients in relaxed and strained germanium,” Appl. Phys. Lett. 108, 211103 (2016).
[Crossref]

2015 (6)

H. Wen and E. Bellotti, “Optical absorption and intrinsic recombination in relaxed and strained InAs1−x Sbx alloys for mid-wavelength infrared application,” Applied Physics Letters 107, 222103 (2015).
[Crossref]

T. Tsukamoto, N. Hirose, A. Kasamatsu, T. Mimura, T. Matsui, and Y. Suda, “Formation of GeSn layers on Si (001) substrates at high growth temperature and high deposition rate by sputter epitaxy method,” J. Mater. Sci. 50, 4366–4370 (2015).
[Crossref]

H. Wen and E. Bellotti, “Rigorous theory of the radiative and gain characteristics of silicon and germanium lasing media,” Phys. Rev. B 91, 035307 (2015).
[Crossref]

H. Wen, B. Pinkie, and E. Bellotti, “Direct and phonon-assisted indirect Auger and radiative recombination lifetime in HgCdTe, InAsSb, and InGaAs computed using Green’s function formalism,” J. Appl. Phys. 118, 015702 (2015).
[Crossref]

D. Stange, S. Wirths, N. Von DenDriesch, G. Mussler, T. Stoica, Z. Ikonic, J. M. Hartmann, S. Mantl, D. Grützmacher, and D. Buca, “Optical transitions in direct-bandgap Ge1−x Snx alloys,” ACS Photon. 2, 1539–1545 (2015).
[Crossref]

S. Wirths, R. Geiger, N. von den Driesch, G. Mussler, T. Stoica, S. Mantl, Z. Ikonic, M. Luysberg, S. Chiussi, J. Hartmann, H. Sigg, J. Faist, D. Buca, and D. Grützmacher, “Lasing in direct-bandgap GeSn alloy grown on Si,” Nature Photon. 9, 88–92 (2015).
[Crossref]

2014 (3)

W. Du, Y. Zhou, S. A. Ghetmiri, A. Mosleh, B. R. Conley, A. Nazzal, R. A. Soref, G. Sun, J. Tolle, J. Margetis, H. A. Naseem, and S. Q. Yu, “Room-temperature electroluminescence from Ge/Ge1−x Snx/Ge diodes on Si substrates,” Appl. Phys. Lett. 104, 241110 (2014).
[Crossref]

W. Du, Y. Zhou, S. A. Ghetmiri, A. Mosleh, B. R. Conley, A. Nazzal, R. A. Soref, G. Sun, J. Tolle, J. Margetis, H. A. Naseem, and S. Q. Yu, “Room–temperature electroluminescence from Ge/Ge1−x Snx/Ge diodes on Si substrates,” Appl. Phys. Lett. 104, 241110 (2014).
[Crossref]

S. A. Ghetmiri, W. Du, J. Margetis, A. Mosleh, L. Cousar, B. R. Conley, L. Domulevicz, A. Nazzal, G. Sun, R. A. Soref, J. Tolle, B. Li, H. A. Naseem, and S. Q. Yu, “Direct-bandgap GeSn grown on silicon with 2230 nm photoluminescence,” Appl. Phys. Lett. 105, 151109 (2014).
[Crossref]

2013 (3)

F. Bertazzi, X. Zhou, M. Goano, G. Ghione, and E. Bellotti, “Auger recombination in InGaN/GaN quantum wells. A full-Brillouin-zone study,” Appl. Phys. Lett. 103, 081106 (2013).
[Crossref]

J. P. Gupta, N. Bhargava, S. Kim, T. Adam, and J. Kolodzey, “Infrared electroluminescence from GeSn heterojunction diodes grown by molecular beam epitaxy,” Appl. Phys. Lett. 102, 251117 (2013).
[Crossref]

A. A. Tonkikh, C. Eisenschmidt, V. G. Talalaev, N. D. Zakharov, J. Schilling, G. Schmidt, and P. Werner, “Pseudomorphic GeSn/Ge (001) quantum wells: Examining indirect band gap bowing,” Appl. Phys. Lett. 103, 032106 (2013).
[Crossref]

2012 (2)

F. Bertazzi, M. Goano, and E. Bellotti, “Numerical analysis of indirect Auger transitions in InGaN,” Appl. Phys. Lett. 101, 011111 (2012).
[Crossref]

H. Lin, R. Chen, W. Lu, Y. Huo, T. I. Kamins, and J. S. Harris, “Investigation of the direct band gaps in Ge1−x Snx alloys with strain control by photoreflectance spectroscopy,” Appl. Phys. Lett. 100, 102109 (2012).
[Crossref]

2010 (2)

J. Mathews, R. Beeler, J. Tolle, C. Xu, R. Roucka, J. Kouvetakis, and J. Menéndez, “Direct-gap photoluminescence with tunable emission wavelength in Ge1−y Sny alloys on silicon,” Appl. Phys. Lett. 97, 221912 (2010).
[Crossref]

F. Bertazzi, M. Goano, and E. Bellotti, “A numerical study of Auger recombination in bulk InGaN,” Appl. Phys. Lett. 97, 231118 (2010).
[Crossref]

2009 (1)

J. Xie, J. Tolle, V. D’Costa, C. Weng, A. Chizmeshya, J. Menendez, and J. Kouvetakis, “Molecular approaches to p-and n-nanoscale doping of Ge1−y Sny semiconductors: Structural, electrical and transport properties,” Semiconductor Sci. Tech. 53, 816–823 (2009).

2007 (2)

P. Moontragoon, Z. Ikonić, and P. Harrison, “Band structure calculations of Si–Ge–Sn alloys: achieving direct band gap materials,” Semiconductor Sci. Tech. 22, 742 (2007).
[Crossref]

J. D. Sau and M. L. Cohen, “Possibility of increased mobility in Ge–Sn alloy system,” Phys. Rev. B 75, 045208 (2007).
[Crossref]

2006 (2)

V. R. D’Costa, C. S. Cook, A. Birdwell, C. L. Littler, M. Canonico, S. Zollner, J. Kouvetakis, and J. Menéndez, “Optical critical points of thin-film Ge1−y Sny alloys: A comparative Ge1−y Sny/Ge1−x Snx study,” Phys. Rev. B 73, 125207 (2006).
[Crossref]

A. Chizmeshya, C. Ritter, J. Tolle, C. Cook, J. Menendez, and J. Kouvetakis, “Fundamental studies of P (GeH3)3, As (GeH3)3, and Sb (GeH3) 3: Practical n–dopants for new group IV semiconductors,” Chem. Mater. 18, 6266–6277 (2006).
[Crossref]

2003 (1)

M. R. Bauer, J. Tolle, C. Bungay, A. V. Chizmeshya, D. J. Smith, J. Menéndez, and J. Kouvetakis, “Tunable band structure in diamond–cubic tin–germanium alloys grown on silicon substrates,” Solid State Commun. 127, 355–359 (2003).
[Crossref]

2002 (2)

N. Bouarissa and F. Annane, “Electronic properties and elastic constants of the ordered Ge1−x Snx alloys,” Mater. Sci. Eng. B 95, 100–106 (2002).
[Crossref]

M. Bauer, J. Taraci, J. Tolle, A. Chizmeshya, S. Zollner, D. J. Smith, J. Menendez, C. Hu, and J. Kouvetakis, “Ge–Sn semiconductors for band-gap and lattice engineering,” Appl. Phys. Lett. 81, 2992–2994 (2002).
[Crossref]

1997 (1)

G. He and H. A. Atwater, “Interband transitions in Snx Ge1−x alloys,” Phys. Rev. Lett. 79, 1937 (1997).
[Crossref]

1996 (1)

M. V. Fischetti and S. E. Laux, “Band structure, deformation potentials, and carrier mobility in strained Si, Ge, and SiGe alloys,” J. Appl. Phys. 80, 2234–2252 (1996).
[Crossref]

1989 (1)

M. L. Cohen, J. R. Chelikowsky, and J. R. Meyer-Arendt, “Electronic structure and optical properties of semiconductors,” Applied Optics 28, 2388 (1989).

1985 (1)

W. Bardyszewski and D. Yevick, “Compositional dependence of the Auger coefficient for InGaAsP lattice matched to InP,” J. Appl. Phys. 58, 2713–2723 (1985).
[Crossref]

1983 (1)

C. Goodman, “Device applications of direct gap group IV semiconductors,” Japan. J. Appl. Phys. 22, 583 (1983).
[Crossref]

1982 (2)

M. Takeshima, “Unified theory of the impurity and phonon scattering effects on Auger recombination in semiconductors,” Phys. Rev. B 25, 5390 (1982).
[Crossref]

M. Takeshima, “Green’s-function formalism of band-to-band Auger recombination in semiconductors. correlation effect,” Phys. Rev. B 26, 917 (1982).
[Crossref]

1978 (1)

A. Haug, “Auger recombination of electron-hole drops,” Solid State Commun. 25, 477–479 (1978).
[Crossref]

1977 (1)

A. Haug, “Phonon-assisted Auger recombination in degenerate semiconductors,” Solid State Commun. 22, 537–539 (1977).
[Crossref]

1976 (1)

L. Huldt, “Phonon-assisted Auger recombination in germanium,” Phys. Status Solidi A 33, 607–614 (1976).
[Crossref]

1974 (1)

L. Huldt, “Auger recombination in germanium,” Phys. Status Solidi A 24, 221–229 (1974).
[Crossref]

1972 (1)

J. P. Walter and M. L. Cohen, “Frequency and wave-vector dependent dielectric function for silicon,” Phys. Rev. B 5, 3101 (1972).
[Crossref]

1959 (1)

P. Landsberg and A. Beattie, “Auger effect in semiconductors,” J. Phys. Chem. Solids 8, 73–75 (1959).
[Crossref]

Adam, T.

J. P. Gupta, N. Bhargava, S. Kim, T. Adam, and J. Kolodzey, “Infrared electroluminescence from GeSn heterojunction diodes grown by molecular beam epitaxy,” Appl. Phys. Lett. 102, 251117 (2013).
[Crossref]

Alher, M.

Y. Zhou, W. Dou, W. Du, T. Pham, S. A. Ghetmiri, S. Al-Kabi, A. Mosleh, M. Alher, J. Margetis, J. Tolle, G. Sun, R. Soref, B. Li, M. Mortazavi, H. Naseem, and S. Q. Yu, “Systematic study of GeSn heterostructure-based light-emitting diodes towards mid–infrared applications,” J. Appl. Phys. 120, 023102 (2016).
[Crossref]

Al-Kabi, S.

Y. Zhou, W. Dou, W. Du, T. Pham, S. A. Ghetmiri, S. Al-Kabi, A. Mosleh, M. Alher, J. Margetis, J. Tolle, G. Sun, R. Soref, B. Li, M. Mortazavi, H. Naseem, and S. Q. Yu, “Systematic study of GeSn heterostructure-based light-emitting diodes towards mid–infrared applications,” J. Appl. Phys. 120, 023102 (2016).
[Crossref]

Annane, F.

N. Bouarissa and F. Annane, “Electronic properties and elastic constants of the ordered Ge1−x Snx alloys,” Mater. Sci. Eng. B 95, 100–106 (2002).
[Crossref]

Atwater, H. A.

G. He and H. A. Atwater, “Interband transitions in Snx Ge1−x alloys,” Phys. Rev. Lett. 79, 1937 (1997).
[Crossref]

Bardyszewski, W.

W. Bardyszewski and D. Yevick, “Compositional dependence of the Auger coefficient for InGaAsP lattice matched to InP,” J. Appl. Phys. 58, 2713–2723 (1985).
[Crossref]

Bauer, M.

M. Bauer, J. Taraci, J. Tolle, A. Chizmeshya, S. Zollner, D. J. Smith, J. Menendez, C. Hu, and J. Kouvetakis, “Ge–Sn semiconductors for band-gap and lattice engineering,” Appl. Phys. Lett. 81, 2992–2994 (2002).
[Crossref]

Bauer, M. R.

M. R. Bauer, J. Tolle, C. Bungay, A. V. Chizmeshya, D. J. Smith, J. Menéndez, and J. Kouvetakis, “Tunable band structure in diamond–cubic tin–germanium alloys grown on silicon substrates,” Solid State Commun. 127, 355–359 (2003).
[Crossref]

Beattie, A.

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[Crossref]

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[Crossref]

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D. Stange, S. Wirths, N. Von DenDriesch, G. Mussler, T. Stoica, Z. Ikonic, J. M. Hartmann, S. Mantl, D. Grützmacher, and D. Buca, “Optical transitions in direct-bandgap Ge1−x Snx alloys,” ACS Photon. 2, 1539–1545 (2015).
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J. Mathews, R. Beeler, J. Tolle, C. Xu, R. Roucka, J. Kouvetakis, and J. Menéndez, “Direct-gap photoluminescence with tunable emission wavelength in Ge1−y Sny alloys on silicon,” Appl. Phys. Lett. 97, 221912 (2010).
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D. Stange, S. Wirths, N. Von DenDriesch, G. Mussler, T. Stoica, Z. Ikonic, J. M. Hartmann, S. Mantl, D. Grützmacher, and D. Buca, “Optical transitions in direct-bandgap Ge1−x Snx alloys,” ACS Photon. 2, 1539–1545 (2015).
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Figures (17)

Fig. 1
Fig. 1 Experimental and calculated energy gaps at Γ and L valleys as a function of the Sn molar fraction. Shaded area and dashed lines represent standard deviation and average value of results from Freitas and coworkers. [21] The solid lines represent the values computed in this work. A number of values presented in literature for the energy gap have been reported. [4,8,12,22–26]
Fig. 2
Fig. 2 Sn molar fraction required to obtain the proper cutoff wavelength, function of temperature, and the corresponding intrinsic carrier concentration.
Fig. 3
Fig. 3 Effective masses at critical points for Sn molar fractions comprised between 0% and 20%.
Fig. 4
Fig. 4 Absorption coefficient for intrinsic Ge1−xSnx in E-SWIR and MWIR configurations.
Fig. 5
Fig. 5 Radiative recombination rate for intrinsic Ge1−xSnx in the E-SWIR configuration at 240K under different injection conditions.
Fig. 6
Fig. 6 Radiative recombination rate for intrinsic Ge1−xSnx in the MWIR configuration at 140K under different injection conditions.
Fig. 7
Fig. 7 Auger and radiative lifetimes function of excess carrier concentrations for doped Ge0.91Sn0.09 (E-SWIR configuration). The doping level is 1015 cm−3 for both n-type and p-type.
Fig. 8
Fig. 8 Auger (eeh and hhe) and radiative lifetimes over a wide range of excess carrier concentrations in case of doped Ge0.91Sn0.09 (E-SWIR configuration). The doping level is 1017 cm−3 for both n-type and p-type.
Fig. 9
Fig. 9 Auger (eeh and hhe) and radiative lifetimes over a wide range of excess carrier concentrations in case of doped Ge0.82Sn0.18 (MWIR configuration). The doping level is 1015 cm−3 for both n-type and p-type.
Fig. 10
Fig. 10 Auger (eeh and hhe) and radiative lifetimes over a wide range of excess carrier concentrations in case of doped Ge0.82Sn0.18 (MWIR configuration). The doping level is 1017 cm−3 for both n-type and p-type.
Fig. 11
Fig. 11 Auger recombination coefficient as a function of excess carrier concentration for intrinsic Ge1−xSnx at 300 K and three molar fractions: x = 0.04, x = 0.06, and x = 0.08. The Auger coefficients decrease with increasing Sn molar fractions and the contribution from CHSH processes in case of Ge0.91Sn0.09 and Ge0.82Sn0.18 is negligible.
Fig. 12
Fig. 12 Auger and radiative recombination rate function of excess carrier concentrations for doped Ge0.91Sn0.09 (E-SWIR configuration). The doping level is 1017 cm−3 for both n-type and p-type. The first label in each legend entry refers to indirect (i) and direct (d) processes, respectively. The second label refers to radiative (r) or Auger class (eeh or hhe). The last label, reported for radiative processes only, refers to the type of doping (n or p respectively).
Fig. 13
Fig. 13 Auger and radiative recombination rate function of excess carrier concentrations for doped Ge0.82Sn0.18 (MWIR configuration). The doping level is 1017 cm−3 for both n-type and p-type. The first label in each legend entry refers to indirect (i) and direct (d) processes, respectively. The second label refers to radiative (r) or Auger class (eeh or hhe). The last label, reported for radiative processes only, refers to the type of doping (n or p respectively).
Fig. 14
Fig. 14 Radiative recombination coefficient function of injection level for doped Ge1−xSnx in E-SWIR and MWIR configurations. The doping level is 1015 cm−3 for both n-type and p-type.
Fig. 15
Fig. 15 Radiative recombination coefficient function of injection level for doped Ge1−xSnx in E-SWIR and MWIR configurations. The doping level is 1017 cm−3 for both n-type and p-type.
Fig. 16
Fig. 16 Auger recombination coefficient function of injection level for doped Ge1−xSnx in E-SWIR and MWIR configurations. The doping level is 1015 cm−3 for both n-type and p-type.
Fig. 17
Fig. 17 Auger recombination coefficient function of injection level for doped Ge1−xSnx in E-SWIR and MWIR configurations. The doping level is 1017 cm−3 for both n-type and p-type.

Tables (3)

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Table 1 Alloy parameters for E-SWIR (9%) and MWIR (18%) Ge1−xSnx. [30, 41] The values for the alloy has been obtained using a linear interpolation in the molar fraction.

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Table 2 Zeroth order interpolation coefficients for direction dependent effective masses. Curves were fitted to Sn molar fraction comprised between 0% and 20%.

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Table 3 First order interpolation coefficients for direction dependent effective masses. Curves were fitted to Sn molar fraction comprised between 0% and 20%.

Equations (15)

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P i ( 1 , 2 ) = P i ( 1 , 2 ) d ( D , S O ) [ x ( 1 x ) ] ( P i ( 1 ) P i ( 2 ) ) .
R A R = 2 π V 3 ( 2 π ) 9 [ 1 e ( μ v μ c ) / k B T ] d k 1 d k 2 d k 1 d k 2 | M e e | 2 δ ( k 1 + k 2 k 1 k 2 ) × d E 1 d E 2 d E 1 d E 2 Θ ( E 1 ) Θ ( E 2 ) [ 1 Θ ( E 1 ) ] [ 1 Θ ( E 2 ) ] × Im G l 1 R ( k 1 , E 1 ) Im G l 2 R ( k 2 , E 2 ) Im G l 1 R ( k 1 , E 1 ) Im G l 2 R ( k 2 , E 2 ) .
ε ( q ) = 1 + c 1 | q | c 2 + c 3 ,
R A R = ( n C n + p C p ) ( n p n i 2 ) ,
τ eeh = n p n i 2 n R A R ( eeh ) , τ hhe = n p n i 2 p R A R ( hhe ) .
R R R = 2 e 2 n r ω p h π m 0 2 c 0 3 V ε 0 k v , k c | k v | e ^ P | k c | 2 d E 1 d E 2 Θ ( E 2 ) [ 1 Θ ( E 1 ) ] × δ ( k c k v k ph ) δ ( μ c μ v + E 2 E 1 ω p h ) Im G l v R ( k v , E 1 ) Im G l c R ( k c , E 2 ) .
τ n = n p n i 2 n R R R , τ p = n p n i 2 p R R R .
Im G l i R ( k , E ) = 1 π Im Σ i ( k , E ) [ E E i Re Σ i ( k , E ) ] 2 + [ Im Σ i ( k , E ) ] 2 .
| g AC ( q ) | 2 = Ξ d 2 ω a c ( q ) 2 c l q 4 ( q 2 + λ 2 ) 2 ,
| g NPO ( q ) | 2 = D 2 v s 2 2 c ¯ ω o p ( q ) q 4 ( q 2 + λ 2 ) 2 ,
1 m i = 1 m i ( Σ ) 1 m i ( Δ ) 1 m i ( Λ )
m i = m i 0 + x m i 1
B = R R R n p n i 2 ,
C eeh = R A R ( eeh ) n ( n p n i 2 ) ,
C hhe = R A R ( hhe ) p ( n p n i 2 ) .

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