Abstract

We propose a new silicon photodiode model optimized for high-accuracy measurement usage. The new model differs from previous models in that the contribution to the quantum efficiency from the diode front region is described by an integral transform of the equilibrium minority carrier concentration. This description is accurate as long as the recombination of excess minority carriers in the front region occurs only at the front surface and the diode is operating linearly.

© 1989 Optical Society of America

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References

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  1. H. J. Hovel, “Solar Cells,” in Semiconductors and Semimetals, 11, R. K. Willardson, A. C. Beers, Eds. (Academic, New York, 1975), p. 15.
  2. PC-1D is available from Iowa State University Research Foundation (ISURF), Ames, IA 50011, copyright 1985.
  3. Reference to a commercial product is included for completeness of exposition, and constitutes neither an endorsement by U.S. National Institute of Standards and Technology nor representation that the product so referenced is the best available for the purpose.
  4. E. F. Zalewski, J. Geist, “Silicon Photodiode Absolute Spectral Response Self-Calibration,” Appl. Opt. 19, 1214–1216 (1980).
    [CrossRef] [PubMed]
  5. N. Fox, “Uses of a Cryogenic Radiometer in Absolute Radiometry,” in Proceedings, Advances in Absolute Radiometry, P. V. Foukal, Ed. (Atmospheric Environmental Research, Inc., Cambridge, MA, 1985), pp. 56–59.
  6. T. E. Hansen, “Silicon UV-Photodiode Using Natural Inversion Layers,” Phys. Scr. 18, 471–475 (1978).
    [CrossRef]
  7. W. Budde, “Multidecade Linearity Measurements on Si Photodiodes,” Appl. Opt. 18, 1555–1558 (1979).
    [CrossRef] [PubMed]
  8. J. Geist, E. F. Zalewski, A. R. Schaefer, “Spectral Response Self-Calibration and Interpolation of Silicon Photodiodes,” Appl. Opt. 19, 3795–3799 (1980).
    [CrossRef] [PubMed]
  9. F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The Near Ultraviolet Quantum Yield of Silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
    [CrossRef]
  10. J. Verdebout, “Semiquantitative Model for the Oxide Bias Experiment and Its Application to the Study of p+nn+ Photodiode Degradation,” Appl. Opt. 23, 4339–4344 (1984).
    [CrossRef] [PubMed]
  11. J. Geist, “Silicon Photodiode Quantum Efficiency Models,” J. Appl. Phys. 51, 3993–3995 (1980).
    [CrossRef]
  12. J. R. Lowney, J. Geist, “Comparison of Models of the Built-in Electric Field in Silicon at High Donor Densities,” J. Appl. Phys. 55, 3624–3627 (1984).
    [CrossRef]
  13. F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1965).
  14. J. Geist, A. Migdall, H. P. Baltes, “Analytic Representation of the Silicon Absorption Coefficient in the Indirect Transition Region,” Appl. Opt. 27, 3777–3779 (1988).
    [CrossRef] [PubMed]
  15. H. S. Bennett, “Hole and Electron Mobilities in Heavily Doped Silicon: Comparison of Theory and Experiment,” Solid-State Electron. 26, 1157–1166 (1983).
    [CrossRef]
  16. J. R. Lowney, H. S. Bennett, “Limitations of Isotropic Theory of Band-Gap Narrowing in Si and Ge Devices,” in IEEE 1987 Bipolar Circuits and Technology Meeting, 157–159, Jopke Janice, Ed., Cat. No. 87CH2509-8 (IEEEPiscataway, NJ 08854, 1987).
  17. J. Del Alamo, S. Swirhun, R. Swanson, in “Measurement and Modeling of Minority Carrier Transport in Heavily Doped Silicon,” Solid-State Electron. 28, 47–54 (1985).
    [CrossRef]
  18. J. W. Slotboom, H. C. de Graaff, “Measurements of Band-Gap Narrowing in Si Bipolar Transistors,” Solid-State Electron. 19, 857–862 (1976).
    [CrossRef]

1988 (1)

1985 (1)

J. Del Alamo, S. Swirhun, R. Swanson, in “Measurement and Modeling of Minority Carrier Transport in Heavily Doped Silicon,” Solid-State Electron. 28, 47–54 (1985).
[CrossRef]

1984 (2)

J. R. Lowney, J. Geist, “Comparison of Models of the Built-in Electric Field in Silicon at High Donor Densities,” J. Appl. Phys. 55, 3624–3627 (1984).
[CrossRef]

J. Verdebout, “Semiquantitative Model for the Oxide Bias Experiment and Its Application to the Study of p+nn+ Photodiode Degradation,” Appl. Opt. 23, 4339–4344 (1984).
[CrossRef] [PubMed]

1983 (2)

H. S. Bennett, “Hole and Electron Mobilities in Heavily Doped Silicon: Comparison of Theory and Experiment,” Solid-State Electron. 26, 1157–1166 (1983).
[CrossRef]

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The Near Ultraviolet Quantum Yield of Silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

1980 (3)

1979 (1)

1978 (1)

T. E. Hansen, “Silicon UV-Photodiode Using Natural Inversion Layers,” Phys. Scr. 18, 471–475 (1978).
[CrossRef]

1976 (1)

J. W. Slotboom, H. C. de Graaff, “Measurements of Band-Gap Narrowing in Si Bipolar Transistors,” Solid-State Electron. 19, 857–862 (1976).
[CrossRef]

Baltes, H. P.

Bennett, H. S.

H. S. Bennett, “Hole and Electron Mobilities in Heavily Doped Silicon: Comparison of Theory and Experiment,” Solid-State Electron. 26, 1157–1166 (1983).
[CrossRef]

J. R. Lowney, H. S. Bennett, “Limitations of Isotropic Theory of Band-Gap Narrowing in Si and Ge Devices,” in IEEE 1987 Bipolar Circuits and Technology Meeting, 157–159, Jopke Janice, Ed., Cat. No. 87CH2509-8 (IEEEPiscataway, NJ 08854, 1987).

Budde, W.

de Graaff, H. C.

J. W. Slotboom, H. C. de Graaff, “Measurements of Band-Gap Narrowing in Si Bipolar Transistors,” Solid-State Electron. 19, 857–862 (1976).
[CrossRef]

Del Alamo, J.

J. Del Alamo, S. Swirhun, R. Swanson, in “Measurement and Modeling of Minority Carrier Transport in Heavily Doped Silicon,” Solid-State Electron. 28, 47–54 (1985).
[CrossRef]

Farmer, A. J. D.

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The Near Ultraviolet Quantum Yield of Silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

Fox, N.

N. Fox, “Uses of a Cryogenic Radiometer in Absolute Radiometry,” in Proceedings, Advances in Absolute Radiometry, P. V. Foukal, Ed. (Atmospheric Environmental Research, Inc., Cambridge, MA, 1985), pp. 56–59.

Geist, J.

J. Geist, A. Migdall, H. P. Baltes, “Analytic Representation of the Silicon Absorption Coefficient in the Indirect Transition Region,” Appl. Opt. 27, 3777–3779 (1988).
[CrossRef] [PubMed]

J. R. Lowney, J. Geist, “Comparison of Models of the Built-in Electric Field in Silicon at High Donor Densities,” J. Appl. Phys. 55, 3624–3627 (1984).
[CrossRef]

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The Near Ultraviolet Quantum Yield of Silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

J. Geist, E. F. Zalewski, A. R. Schaefer, “Spectral Response Self-Calibration and Interpolation of Silicon Photodiodes,” Appl. Opt. 19, 3795–3799 (1980).
[CrossRef] [PubMed]

J. Geist, “Silicon Photodiode Quantum Efficiency Models,” J. Appl. Phys. 51, 3993–3995 (1980).
[CrossRef]

E. F. Zalewski, J. Geist, “Silicon Photodiode Absolute Spectral Response Self-Calibration,” Appl. Opt. 19, 1214–1216 (1980).
[CrossRef] [PubMed]

Hansen, T. E.

T. E. Hansen, “Silicon UV-Photodiode Using Natural Inversion Layers,” Phys. Scr. 18, 471–475 (1978).
[CrossRef]

Hovel, H. J.

H. J. Hovel, “Solar Cells,” in Semiconductors and Semimetals, 11, R. K. Willardson, A. C. Beers, Eds. (Academic, New York, 1975), p. 15.

Lowney, J. R.

J. R. Lowney, J. Geist, “Comparison of Models of the Built-in Electric Field in Silicon at High Donor Densities,” J. Appl. Phys. 55, 3624–3627 (1984).
[CrossRef]

J. R. Lowney, H. S. Bennett, “Limitations of Isotropic Theory of Band-Gap Narrowing in Si and Ge Devices,” in IEEE 1987 Bipolar Circuits and Technology Meeting, 157–159, Jopke Janice, Ed., Cat. No. 87CH2509-8 (IEEEPiscataway, NJ 08854, 1987).

Migdall, A.

Riesz, F.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1965).

Schaefer, A. R.

Slotboom, J. W.

J. W. Slotboom, H. C. de Graaff, “Measurements of Band-Gap Narrowing in Si Bipolar Transistors,” Solid-State Electron. 19, 857–862 (1976).
[CrossRef]

Swanson, R.

J. Del Alamo, S. Swirhun, R. Swanson, in “Measurement and Modeling of Minority Carrier Transport in Heavily Doped Silicon,” Solid-State Electron. 28, 47–54 (1985).
[CrossRef]

Swirhun, S.

J. Del Alamo, S. Swirhun, R. Swanson, in “Measurement and Modeling of Minority Carrier Transport in Heavily Doped Silicon,” Solid-State Electron. 28, 47–54 (1985).
[CrossRef]

Sz.-Nagy, B.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1965).

Verdebout, J.

Wilkinson, F. J.

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The Near Ultraviolet Quantum Yield of Silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

Zalewski, E. F.

Appl. Opt. (5)

J. Appl. Phys. (3)

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The Near Ultraviolet Quantum Yield of Silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

J. Geist, “Silicon Photodiode Quantum Efficiency Models,” J. Appl. Phys. 51, 3993–3995 (1980).
[CrossRef]

J. R. Lowney, J. Geist, “Comparison of Models of the Built-in Electric Field in Silicon at High Donor Densities,” J. Appl. Phys. 55, 3624–3627 (1984).
[CrossRef]

Phys. Scr. (1)

T. E. Hansen, “Silicon UV-Photodiode Using Natural Inversion Layers,” Phys. Scr. 18, 471–475 (1978).
[CrossRef]

Solid-State Electron. (3)

J. Del Alamo, S. Swirhun, R. Swanson, in “Measurement and Modeling of Minority Carrier Transport in Heavily Doped Silicon,” Solid-State Electron. 28, 47–54 (1985).
[CrossRef]

J. W. Slotboom, H. C. de Graaff, “Measurements of Band-Gap Narrowing in Si Bipolar Transistors,” Solid-State Electron. 19, 857–862 (1976).
[CrossRef]

H. S. Bennett, “Hole and Electron Mobilities in Heavily Doped Silicon: Comparison of Theory and Experiment,” Solid-State Electron. 26, 1157–1166 (1983).
[CrossRef]

Other (6)

J. R. Lowney, H. S. Bennett, “Limitations of Isotropic Theory of Band-Gap Narrowing in Si and Ge Devices,” in IEEE 1987 Bipolar Circuits and Technology Meeting, 157–159, Jopke Janice, Ed., Cat. No. 87CH2509-8 (IEEEPiscataway, NJ 08854, 1987).

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1965).

H. J. Hovel, “Solar Cells,” in Semiconductors and Semimetals, 11, R. K. Willardson, A. C. Beers, Eds. (Academic, New York, 1975), p. 15.

PC-1D is available from Iowa State University Research Foundation (ISURF), Ames, IA 50011, copyright 1985.

Reference to a commercial product is included for completeness of exposition, and constitutes neither an endorsement by U.S. National Institute of Standards and Technology nor representation that the product so referenced is the best available for the purpose.

N. Fox, “Uses of a Cryogenic Radiometer in Absolute Radiometry,” in Proceedings, Advances in Absolute Radiometry, P. V. Foukal, Ed. (Atmospheric Environmental Research, Inc., Cambridge, MA, 1985), pp. 56–59.

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Figures (6)

Fig. 1
Fig. 1

Equilibrium electron and hole concentrations n0(x) and p0(x) as a function of depth in the photodiode described in Table I according to the PC-1D2,3 numerical solar cell model.

Fig. 2
Fig. 2

Carrier mobility at thermal equilibrium as a function of depth in the front region of the photodiode described in Table I according to the PC-1D2,3 numerical solar cell model.

Fig. 3
Fig. 3

Front region equilibrium electric field at thermal equilibrium as a function of depth in the photodiode described in Table I according to the PC-1D2,3 numerical solar cell model.

Fig. 4
Fig. 4

Comparison of the shape of the quantum deficiency curve for the diode described in Table I when the recombination losses are entirely due to Auger recombination in the volume of the front region (Auger) with the shape for the same diode when the recombination losses are entirely due to surface recombination (Surface). The recombination losses were set artificially large and adjusted to make the curves cross at λ = 400 nm.

Fig. 5
Fig. 5

Doping and carrier concentrations, and partition into regions in a typical high-quality planar silicon photodiode. The point x1 is the point where the excess photogenerated carrier concentration is assumed to be zero, and is used to define the location of the interface between the front and depletion regions.

Fig. 6
Fig. 6

Comparison of the quantum deficiency curve calculated for the photodiode described in Table I by to PC-1D with S = 1.4 × 104 cm/s for the PC-1D default band-gap narrowing model (labeled PC1D Default), and for the Slotboom and de Graaff band-gap narrowing model (labeled PC1D SDG) with the value calculated from Eqs. (31)(33) using the equilibrium carrier concentration and diffusion coefficient calculated from PC-1D using the same parameters used for the curve labeled Default.

Tables (2)

Tables Icon

Table I Nominal Front-Region Diode Parameters Characteristic of the Hamamatsu 1337 Photodiode

Tables Icon

Table II Patch for 700-ppm Error In PC-1D Version 1.1a

Equations (36)

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d 2 ϕ ( x ) d x 2 = ( q 0 ) [ p ( x ) n ( x ) + N D + ( x ) N A ( x ) ] ,
J n ( x ) = q ( μ n ( x ) n ( x ) d ϕ ( x ) d x D n ( x ) d n ( x ) d x ) ,
J p ( x ) = q ( μ p ( x ) p ( x ) d ϕ ( x ) d x + D p ( x ) d p ( x ) d x ) ,
d J n ( x ) d x = q [ R n ( x ) G n ( x ) ] ,
d J p ( x ) d x = q [ R p ( x ) G p ( x ) ] ,
J T ( x ) = J n ( x ) + J p ( x ) .
J T ( x ) = J n ( x ) = J p ( x ) = 0 ,
n ̂ 0 ( x ) = n 0 ( x ) / n 0 ( 0 ) ,
p ̂ 0 ( x ) = p 0 ( x ) / p 0 ( 0 ) .
d ϕ 0 ( x ) d x = ( D p ( x ) μ p ( x ) ) ( 1 p ̂ 0 ( x ) ) d p ̂ 0 ( x ) d x .
p ( x ) = p 0 ( x ) + p ̂ 0 ( x ) q x x 1 d s J p ( s ) p ̂ 0 ( s ) D p ( s ) = p ̂ 0 ( x ) ( p 0 ( x 1 ) p ̂ 0 ( x 1 ) + 1 q x x 1 d s J p ( s ) p ̂ 0 ( s ) D p ( s ) )
d ϕ ( x ) / d x = d ϕ 0 ( x ) / d x .
d p ( x ) d x = d p ̂ 0 ( x ) d x p ( x ) p ̂ 0 ( x ) 1 q J p ( x ) D p ( x ) ,
d p ( x ) d x = ( μ p ( x ) D p ( x ) ) p ( x ) d ϕ 0 ( x ) d x J p ( x ) q D p ( x ) ,
d J n ( x ) d x = J a ( x ) ,
d J p ( x ) d x = J a ( x ) .
a ( x ) = α exp ( α x ) ,
A ( x ) = 1 0 x a ( s ) d s .
A ( x ) = exp ( α x ) .
J n ( x ) = x 1 x d J n ( s ) d s d s = J [ A ( x ) A ( x 1 ) ] + J n ( x 1 ) ,
J p ( x ) = 0 x d J p ( s ) d s = J [ 1 A ( x ) ] J i ,
J T ( x ) = J i + J [ 1 A ( x 1 ) ] + J n ( x 1 ) ,
J i = q S p [ p ( 0 ) p 0 ( 0 ) ] .
p ( 0 ) = p 0 ( 0 ) + 1 q 0 x 1 d s J p ( s ) p ̂ 0 ( s ) D p ( s ) .
J i = Γ p 0 x 1 d s J p ( s ) p ̂ 0 ( s ) D ̂ p ( s ) .
Γ p = S p / D p ( 0 ) ,
D ̂ p ( x ) = D p ( x ) / D p ( 0 ) .
J i = J [ Γ p I p { 1 } Γ p I p { A } ] / [ 1 + Γ p I p { 1 } ] ,
J T = J ( [ 1 + Γ p I p { A } ] [ 1 + Γ p I p { 1 } ] A ( x 1 ) ) + J n ( x 1 ) ,
I p { A } = 0 x 1 d s A ( s ) p ̂ 0 ( s ) D ̂ p ( s ) ,
Q D = Γ m [ I m { 1 } + I m { A } ] [ 1 + Γ m I m { 1 } ] + A ( x 1 ) J M ( x 1 ) J ,
I m { A } = 0 x 1 d s A ( s ) m ̂ 0 ( s ) D ̂ m ( s ) ,
( x c x 1 ) / [ m ̂ 0 ( x 1 ) D ̂ m ( x 1 ) ] < β I m ( 0 ) ,
Δ ( x ) = d ϕ ( u ) / d u 0 x d ϕ 0 ( u ) / d u 0 x .
Δ p ( x ) = 0 x d u p ̂ 0 ( u ) 0 u x 1 d s J p ( s ) p ̂ 0 ( s ) D p ( s ) ,
Δ ( x ) O [ J q x 1 2 / ( 0 k T μ ) ] ,

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