Abstract

A new expression for the internal quantum efficiency of a photodiode is presented. It is obtained from the analysis of the photocurrent generated within the diode, considering the power and the cross-sectional diameter of the incident beam. The model explains variations of the internal quantum efficiency with irradiance that are not explained by other existing models, although this experimental fact was already known. The well-known phenomenon of supraresponsivity is also explained with this model. Finally, we show the dependence of the internal quantum efficiency on the variables involved in the model.

© 2005 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 1.
  2. 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]
  3. T. R. Gentile, J. M. Houston, C. L. Cromer, “Realization of a scale of absolute spectral response using the National Institute of Standards and Technology high-accuracy cryogenic radiometer,” Appl. Opt. 35, 4392–4403 (1996).
    [CrossRef] [PubMed]
  4. L. P. Boivin, “Automated absolute and relative spectral linearity measurements on photovoltaic detectors,” Metrologia 30, 355–360 (1993).
    [CrossRef]
  5. A. S. Grove, Physics and Technology of Semiconductor Devices (Wiley, New York, 1967), Chaps. 4–6, pp. 91–207.
  6. R. N. Hall, “Electron-hole recombination in germanium,” Phys. Rev. 87, 387 (1952).
    [CrossRef]
  7. W. Shockley, W. T. Read, “Statistics of the recombination of holes and electrons,” Phys. Rev. 87, 835–842 (1952).
    [CrossRef]
  8. A. R. Schaefer, E. F. Zalewski, J. Geist, “Silicon detector nonlinearity and related effects,” Appl. Opt. 22, 1232–1235 (1983).
    [CrossRef] [PubMed]
  9. L. Werner, J. Fischer, U. Johannsen, J. Hartmann, “Accurate determination of the spectral responsivity of silicon trap detectors between 238 nm and 1015 nm using a laser-based cryogenic radiometer,” Metrologia 37, 279–284 (2000).
    [CrossRef]

2000 (1)

L. Werner, J. Fischer, U. Johannsen, J. Hartmann, “Accurate determination of the spectral responsivity of silicon trap detectors between 238 nm and 1015 nm using a laser-based cryogenic radiometer,” Metrologia 37, 279–284 (2000).
[CrossRef]

1996 (1)

1993 (1)

L. P. Boivin, “Automated absolute and relative spectral linearity measurements on photovoltaic detectors,” Metrologia 30, 355–360 (1993).
[CrossRef]

1983 (1)

1980 (1)

1952 (2)

R. N. Hall, “Electron-hole recombination in germanium,” Phys. Rev. 87, 387 (1952).
[CrossRef]

W. Shockley, W. T. Read, “Statistics of the recombination of holes and electrons,” Phys. Rev. 87, 835–842 (1952).
[CrossRef]

Boivin, L. P.

L. P. Boivin, “Automated absolute and relative spectral linearity measurements on photovoltaic detectors,” Metrologia 30, 355–360 (1993).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 1.

Cromer, C. L.

Fischer, J.

L. Werner, J. Fischer, U. Johannsen, J. Hartmann, “Accurate determination of the spectral responsivity of silicon trap detectors between 238 nm and 1015 nm using a laser-based cryogenic radiometer,” Metrologia 37, 279–284 (2000).
[CrossRef]

Geist, J.

Gentile, T. R.

Grove, A. S.

A. S. Grove, Physics and Technology of Semiconductor Devices (Wiley, New York, 1967), Chaps. 4–6, pp. 91–207.

Hall, R. N.

R. N. Hall, “Electron-hole recombination in germanium,” Phys. Rev. 87, 387 (1952).
[CrossRef]

Hartmann, J.

L. Werner, J. Fischer, U. Johannsen, J. Hartmann, “Accurate determination of the spectral responsivity of silicon trap detectors between 238 nm and 1015 nm using a laser-based cryogenic radiometer,” Metrologia 37, 279–284 (2000).
[CrossRef]

Houston, J. M.

Johannsen, U.

L. Werner, J. Fischer, U. Johannsen, J. Hartmann, “Accurate determination of the spectral responsivity of silicon trap detectors between 238 nm and 1015 nm using a laser-based cryogenic radiometer,” Metrologia 37, 279–284 (2000).
[CrossRef]

Read, W. T.

W. Shockley, W. T. Read, “Statistics of the recombination of holes and electrons,” Phys. Rev. 87, 835–842 (1952).
[CrossRef]

Schaefer, A. R.

Shockley, W.

W. Shockley, W. T. Read, “Statistics of the recombination of holes and electrons,” Phys. Rev. 87, 835–842 (1952).
[CrossRef]

Werner, L.

L. Werner, J. Fischer, U. Johannsen, J. Hartmann, “Accurate determination of the spectral responsivity of silicon trap detectors between 238 nm and 1015 nm using a laser-based cryogenic radiometer,” Metrologia 37, 279–284 (2000).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 1.

Zalewski, E. F.

Appl. Opt. (3)

Metrologia (2)

L. Werner, J. Fischer, U. Johannsen, J. Hartmann, “Accurate determination of the spectral responsivity of silicon trap detectors between 238 nm and 1015 nm using a laser-based cryogenic radiometer,” Metrologia 37, 279–284 (2000).
[CrossRef]

L. P. Boivin, “Automated absolute and relative spectral linearity measurements on photovoltaic detectors,” Metrologia 30, 355–360 (1993).
[CrossRef]

Phys. Rev. (2)

R. N. Hall, “Electron-hole recombination in germanium,” Phys. Rev. 87, 387 (1952).
[CrossRef]

W. Shockley, W. T. Read, “Statistics of the recombination of holes and electrons,” Phys. Rev. 87, 835–842 (1952).
[CrossRef]

Other (2)

A. S. Grove, Physics and Technology of Semiconductor Devices (Wiley, New York, 1967), Chaps. 4–6, pp. 91–207.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 1.

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Figures (11)

Fig. 1
Fig. 1

Structure of a photodiode PN junction.

Fig. 2
Fig. 2

Upper and lower limits of the internal quantum efficiency for a specific photodiode.

Fig. 3
Fig. 3

Relative variation of the internal quantum efficiency for a specific photodiode.

Fig. 4
Fig. 4

ɛf and ɛr as a function of M.

Fig. 5
Fig. 5

Internal quantum efficiency of five hypothetical photodiodes whose parameters are shown in Table 1.

Fig. 6
Fig. 6

Spectral responsivity of the five hypothetical photodiodes whose parameters are shown in Table 1.

Fig. 7
Fig. 7

Spectral variation of the internal quantum efficiency of photodiode 1 in Table 1 with the incident power. The beam diameter used is 3 mm.

Fig. 8
Fig. 8

Spectral variation of the internal quantum efficiency of photodiode 1 in Table 1 with the beam diameter. The beam power used is 0.1 mW.

Fig. 9
Fig. 9

Spectral variation of the internal quantum efficiency of photodiode 1 in Table 1 with the irradiance.

Fig. 10
Fig. 10

Internal quantum efficiency as a function of irradiance.

Fig. 11
Fig. 11

Absorption coefficient of silicon.

Tables (1)

Tables Icon

Table 1 Internal Parameters of the Five Hypothetical Photodiodes Used in This Study

Equations (48)

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η e = h c q n λ ( i P ) ,
η i = η e 1 - ρ ( λ ) ,
R = i P ,
R = q n λ h c η e .
η i = 0 H P ( x ) θ ( λ ) exp ( - α x ) d x 0 H exp ( - α x ) d x ,
η i = α 0 H P ( x ) exp ( - α x ) d x ,
η i ( λ ) = P + 1 - P α ( λ ) T { 1 - exp [ - α ( λ ) T ] } - h α ( λ ) L 2 × exp [ - α ( λ ) H ] ,
η i ( λ ) = P f + 1 - P f α ( λ ) f { 1 - exp [ - α ( λ ) f ] } - 1 - P r α ( λ ) ( D - f ) { exp [ - α ( λ ) f ] - exp [ - α ( λ D ) ] } - P r exp [ - α ( λ ) h ] ,
η i = i i ϕ .
G L ( x ) = G L 0 exp ( - α a a ) exp ( - α x ) .
i A = q S 0 D G L ( x ) d x = q S G L 0 exp ( - α a a ) α × [ 1 - exp ( - α D ) ] ,
p n ( x ) t = D p 2 p n x 2 + G L ( x ) - U ( x ) ,
- D p 2 p n x 2 = G L ( x ) - U ( x )
i B = q S D H ( - D p 2 p n x 2 ) d x = q S D H [ G L 0 exp ( - α a a ) exp ( - α x ) - U ( x ) ] d x .
i B = q S { G L 0 exp ( - α a a ) α [ exp ( - α D ) - exp ( - α H ) ] - D H U ( x ) d x } .
i = q S { G L 0 exp ( - α a a ) α [ 1 - exp ( - α H ) ] - D H U ( x ) d x } .
I = D H U ( x ) d x .
U = σ v th N t n p - n i 2 n + p + 2 n i cosh ( E t - E i k B T ) ,
p n ( x ) p n 0 + τ p G L 0 exp ( - α a a ) exp ( - α x ) ,
n n ( x ) N d + τ p G L 0 exp ( - α a a ) exp ( - α x ) ,
N d p n 0 = n i 2 ,
τ p = 1 σ v th N t ,
N d p n 0 + 2 n i cosh ( E t - E i k B T ) ,
U ( x ) = G L 0 exp ( - α a a ) 1 + τ p G L 0 N d exp ( - α x - α a a ) exp ( α x ) + 2 τ p G L 0 N d exp ( - α a a ) .
U ( x ) d x = - 1 4 N d α τ p ln [ 1 + 2 τ p G L 0 exp ( - α x - α a a ) N d ] - G L 0 exp ( - α x - α a a ) 2 α .
D H U ( x ) d x = 1 4 N d α τ p ln [ N d + 2 τ p G L 0 exp ( - α D - α a a ) N d + 2 τ p G L 0 exp ( - α H - α a a ) ] - G L 0 exp ( - α a a ) [ exp ( - α H ) - exp ( - α D ) ] 2 α .
i = exp ( - α a a ) [ i ϕ ( 1 - δ out - 1 2 δ ) - 1 2 δ Ξ × ln ( 1 + Ξ i ϕ 1 + Ξ i ϕ δ out δ ) ] ,
i ϕ 0 + q S G L 0 exp ( - α x ) d x = q S G L 0 α
Ξ = ξ M ξ R ,
ξ M 2 τ p α exp ( - α a a ) q N d ,
ξ R δ S ,
δ exp ( - α D ) - exp ( - α H ) = ϕ R ϕ S C + ϕ R ,
δ out exp ( - α H )
i = exp ( - α a a ) { i ϕ [ 1 - δ out - 1 2 ( δ r + δ f ) ] - 1 2 δ r Ξ r × ln ( 1 + Ξ r i ϕ 1 + Ξ r i ϕ δ out δ r ) - 1 2 δ f Ξ f × ln [ 1 + Ξ f i ϕ 1 + Ξ f i ϕ ( 1 - δ f ) δ f ] } ,
δ f 1 - exp ( - α f ) ,
δ r exp ( - α D ) - exp ( - α H ) ,
i ϕ = q n h c ( 1 - ρ ) P λ .
η i = exp ( - α a a ) ( 1 - δ out - δ r 2 [ 1 + 1 M r × ln ( 1 + M r 1 + M r δ out δ r ) ] - δ f 2 { 1 + 1 M f × ln [ 1 + M f 1 + M f ( 1 - δ f ) δ f ] } ) ,
M r = q n h c ( 1 - ρ ) P λ Ξ r ,             M f = q n h c ( 1 - ρ ) P λ Ξ f .
M r = q n h c ( 1 - ρ ) P λ 2 τ p α exp ( - α a a ) q N δ r S .
P a = exp ( - α a a ) ( 1 - ρ ) P ,
E γ = h c n λ ,
N γ = P a δ r E γ S ( H - D ) ,
M r = 2 α ( H - D ) N γ τ p N ,
ɛ r = 1 M r ln ( 1 + M r 1 + M r δ out δ r )
ɛ f = 1 M f ln [ 1 + M f 1 + M f ( 1 - δ f ) δ f ] .
η i = exp ( - α a a ) { 1 - δ out - δ r 2 [ 1 + ɛ r ( λ , T , P , S ) ] - δ f 2 [ 1 + ɛ f ( λ , T , P , S ) ] } ,
ρ = 0.3 + 0.08 exp ( λ - 4 × 10 - 7 7 × 10 - 8 ) ,

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