Abstract

An analytical theory based on a model of the internal quantum efficiency of the silicon photodiode and the temperature dependence of the absorption coefficient of silicon is developed to predict the temperature dependence of the spectral responsivity in the wavelength range from 400 to 1020 nm near room temperature. The influence of the various parameters of the model on the temperature coefficient is investigated theoretically. Comparison of measurements and prediction shows good agreement. The developed model permits further improvement of accuracy in spectral-responsivity scales based on silicon photodiodes, especially in the near-infrared wavelength region.

© 2001 Optical Society of America

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References

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  1. B. K. Tsai, D. P. DeWitt, F. J. Lovas, K. G. Kreider, C. W. Meyer, and D. W. Allen, “Chamber radiation effects on calibration of radiation thermometers with a thin-film thermocouple test wafer,” in Proceedings of TempMeko, J. F. Dubbeldam and M. J. de Groot, eds. (Edauw & Johannissen, Delft, The Netherlands) (1999), pp. 726–731.
  2. N. Fox, “Radiometry with cryogenic radiometers and semiconductor photodiodes,” Metrologia 32, 535–543 (1995/96).
    [CrossRef]
  3. M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
    [CrossRef]
  4. J. Geist, E. F. Zalewski, and A. R. Schaefer, “Spectral response self-calibration and interpolation of silicon photodiodes,” Appl. Opt. 19, 3795–3799 (1980).
    [CrossRef] [PubMed]
  5. L. Werner, J. Fischer, U. Johannsen, and 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]
  6. N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197–202 (1991).
    [CrossRef]
  7. J. Fischer, J. Seidel, and B. Wende, “The double-heatpipe black body: a radiance and irradiance standard for accurate infrared calibrations in remote sensing,” Metrologia 35, 441–445 (1998).
    [CrossRef]
  8. J. Heller, J. W. Bartha, C. C. Poon, and A. C. Tam, “Temperature dependence of the reflectivity of silicon with surface oxides at wavelengths of 633 nm and 1047 nm,” Appl. Phys. Lett. 75, 43–45 (1999).
    [CrossRef]
  9. R. Goebel, M. Stock, and R. Köhler, “Report on the International Comparison of Cryogenic Radiometers Based on Transfer Detectors,” (Bureau International des Poinds et Mesures, Sèvres, France, 1999).
  10. K. P. Birch and M. J. Downs, “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
    [CrossRef]
  11. T. R. Gentile, J. M. Houston, and 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]
  12. S. M. Sze, Physics of Semiconductor Devices, 2nd ed. (Wiley, New York, 1981).
  13. G. E. Jellison and F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
    [CrossRef]
  14. H. A. Weakliem and D. Redfield, “Temperature dependence of the optical properties of silicon,” J. Appl. Phys. 50, 1491–1493 (1979).
    [CrossRef]
  15. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).
  16. Lei Fu and J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
    [CrossRef]

2000 (1)

L. Werner, J. Fischer, U. Johannsen, and 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]

1999 (1)

J. Heller, J. W. Bartha, C. C. Poon, and A. C. Tam, “Temperature dependence of the reflectivity of silicon with surface oxides at wavelengths of 633 nm and 1047 nm,” Appl. Phys. Lett. 75, 43–45 (1999).
[CrossRef]

1998 (1)

J. Fischer, J. Seidel, and B. Wende, “The double-heatpipe black body: a radiance and irradiance standard for accurate infrared calibrations in remote sensing,” Metrologia 35, 441–445 (1998).
[CrossRef]

1996 (1)

1995 (1)

N. Fox, “Radiometry with cryogenic radiometers and semiconductor photodiodes,” Metrologia 32, 535–543 (1995/96).
[CrossRef]

1993 (3)

M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
[CrossRef]

K. P. Birch and M. J. Downs, “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

Lei Fu and J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
[CrossRef]

1991 (1)

N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197–202 (1991).
[CrossRef]

1982 (1)

G. E. Jellison and F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
[CrossRef]

1980 (1)

1979 (1)

H. A. Weakliem and D. Redfield, “Temperature dependence of the optical properties of silicon,” J. Appl. Phys. 50, 1491–1493 (1979).
[CrossRef]

Bartha, J. W.

J. Heller, J. W. Bartha, C. C. Poon, and A. C. Tam, “Temperature dependence of the reflectivity of silicon with surface oxides at wavelengths of 633 nm and 1047 nm,” Appl. Phys. Lett. 75, 43–45 (1999).
[CrossRef]

Birch, K. P.

K. P. Birch and M. J. Downs, “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

Cromer, C. L.

Downs, M. J.

K. P. Birch and M. J. Downs, “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

Fischer, J.

L. Werner, J. Fischer, U. Johannsen, and 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]

J. Fischer, J. Seidel, and B. Wende, “The double-heatpipe black body: a radiance and irradiance standard for accurate infrared calibrations in remote sensing,” Metrologia 35, 441–445 (1998).
[CrossRef]

M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
[CrossRef]

Lei Fu and J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
[CrossRef]

Fox, N.

N. Fox, “Radiometry with cryogenic radiometers and semiconductor photodiodes,” Metrologia 32, 535–543 (1995/96).
[CrossRef]

Fox, N. P.

N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197–202 (1991).
[CrossRef]

Friedrich, R.

M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
[CrossRef]

Fu, Lei

Lei Fu and J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
[CrossRef]

Geist, J.

Gentile, T. R.

Hartmann, J.

L. Werner, J. Fischer, U. Johannsen, and 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]

Heller, J.

J. Heller, J. W. Bartha, C. C. Poon, and A. C. Tam, “Temperature dependence of the reflectivity of silicon with surface oxides at wavelengths of 633 nm and 1047 nm,” Appl. Phys. Lett. 75, 43–45 (1999).
[CrossRef]

Houston, J. M.

Jellison, G. E.

G. E. Jellison and F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
[CrossRef]

Johannsen, U.

L. Werner, J. Fischer, U. Johannsen, and 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]

Jung, H. J.

M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
[CrossRef]

Modine, F. A.

G. E. Jellison and F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
[CrossRef]

Poon, C. C.

J. Heller, J. W. Bartha, C. C. Poon, and A. C. Tam, “Temperature dependence of the reflectivity of silicon with surface oxides at wavelengths of 633 nm and 1047 nm,” Appl. Phys. Lett. 75, 43–45 (1999).
[CrossRef]

Redfield, D.

H. A. Weakliem and D. Redfield, “Temperature dependence of the optical properties of silicon,” J. Appl. Phys. 50, 1491–1493 (1979).
[CrossRef]

Schaefer, A. R.

Seidel, J.

J. Fischer, J. Seidel, and B. Wende, “The double-heatpipe black body: a radiance and irradiance standard for accurate infrared calibrations in remote sensing,” Metrologia 35, 441–445 (1998).
[CrossRef]

Stock, M.

M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
[CrossRef]

Tam, A. C.

J. Heller, J. W. Bartha, C. C. Poon, and A. C. Tam, “Temperature dependence of the reflectivity of silicon with surface oxides at wavelengths of 633 nm and 1047 nm,” Appl. Phys. Lett. 75, 43–45 (1999).
[CrossRef]

Thornagel, R.

M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
[CrossRef]

Ulm, G.

M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
[CrossRef]

Weakliem, H. A.

H. A. Weakliem and D. Redfield, “Temperature dependence of the optical properties of silicon,” J. Appl. Phys. 50, 1491–1493 (1979).
[CrossRef]

Wende, B.

J. Fischer, J. Seidel, and B. Wende, “The double-heatpipe black body: a radiance and irradiance standard for accurate infrared calibrations in remote sensing,” Metrologia 35, 441–445 (1998).
[CrossRef]

M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
[CrossRef]

Werner, L.

L. Werner, J. Fischer, U. Johannsen, and 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]

Zalewski, E. F.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

G. E. Jellison and F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982).
[CrossRef]

J. Heller, J. W. Bartha, C. C. Poon, and A. C. Tam, “Temperature dependence of the reflectivity of silicon with surface oxides at wavelengths of 633 nm and 1047 nm,” Appl. Phys. Lett. 75, 43–45 (1999).
[CrossRef]

J. Appl. Phys. (1)

H. A. Weakliem and D. Redfield, “Temperature dependence of the optical properties of silicon,” J. Appl. Phys. 50, 1491–1493 (1979).
[CrossRef]

Metrologia (7)

Lei Fu and J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
[CrossRef]

K. P. Birch and M. J. Downs, “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

N. Fox, “Radiometry with cryogenic radiometers and semiconductor photodiodes,” Metrologia 32, 535–543 (1995/96).
[CrossRef]

M. Stock, J. Fischer, R. Friedrich, H. J. Jung, R. Thornagel, G. Ulm, and B. Wende, “Present state of the comparison between radiometric scales based on three primary standards,” Metrologia 30, 439–449 (1993).
[CrossRef]

L. Werner, J. Fischer, U. Johannsen, and 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]

N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197–202 (1991).
[CrossRef]

J. Fischer, J. Seidel, and B. Wende, “The double-heatpipe black body: a radiance and irradiance standard for accurate infrared calibrations in remote sensing,” Metrologia 35, 441–445 (1998).
[CrossRef]

Other (4)

R. Goebel, M. Stock, and R. Köhler, “Report on the International Comparison of Cryogenic Radiometers Based on Transfer Detectors,” (Bureau International des Poinds et Mesures, Sèvres, France, 1999).

B. K. Tsai, D. P. DeWitt, F. J. Lovas, K. G. Kreider, C. W. Meyer, and D. W. Allen, “Chamber radiation effects on calibration of radiation thermometers with a thin-film thermocouple test wafer,” in Proceedings of TempMeko, J. F. Dubbeldam and M. J. de Groot, eds. (Edauw & Johannissen, Delft, The Netherlands) (1999), pp. 726–731.

S. M. Sze, Physics of Semiconductor Devices, 2nd ed. (Wiley, New York, 1981).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

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Figures (5)

Fig. 1
Fig. 1

Schematic view of a reflection-trap detector; also shown is the path of the light inside the trap.

Fig. 2
Fig. 2

Spatial distribution of the collection efficiency inside a silicon photodiode used for the developed model.

Fig. 3
Fig. 3

Temperature coefficient for the absorption coefficient of silicon according to three different models. Circles, model I;12 squares, model II;13 solid curve model III.14

Fig. 4
Fig. 4

Temperature coefficient of the spectral responsivity as a function of the wavelength. A set of reference parameters was obtained by fitting the measured spectral responsivity curve of a silicon trap detector. For each graph, all parameters except the one indicated were kept constant at their reference value.

Fig. 5
Fig. 5

Measured relative change in spectral responsivity (circles) of a silicon-trap detector as a function of wavelength together with theoretical predictions from the various models for the temperature dependence of the absorption coefficient of silicon: open circles, model I;12 squares, model II;13 solid curve, model III;14 crosses, model III together with the temperature coefficient of the trap reflectivity Rtrap.

Tables (3)

Tables Icon

Table 1 Relative Change of Reflectivity of a Single Silicon Photodiode with Temperature

Tables Icon

Table 2 Trap Reflectivity and Temperature Coefficient of (1=Rtrap) Calculated by Using Rtrap=R5 and the Values of Table 1 for R

Tables Icon

Table 3 Parameters Obtained by Fitting the Spectral Responsivity Versus Wavelength Curve of a Silicon Trap Detector in Accordance with Eq. (4)a

Equations (16)

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s(λ, T)=(1-R)ληi(λ, T)enairhc.
dssdT=-dR(1-R)dT+dηiηidT+dnairnairdT.
dnairnairdT=-10-8PPa×0.009721+10-80.601-0.00972t°CPPa-0.00366161+0.0036610t°C.
ηi(λ, T)=Pf+1-Pfα(λ ,T)t×{1-exp[-α(λ, T)t]}-1-Pbα(λ, T)(d-t)×{exp[-α(λ, T)t]-exp[-α(λ, T)d]}+Pb(r-1)exp[-α(λ, T)h],
dηidT=-1-Pfα(λ, T)2t{1-exp[-α(λ, T)t]}+1-Pfα(λ, T)exp[-α(λ, T)t]+1-Pbα(λ, T)2(d-t)×{exp[-α(λ, T)t]-exp[-α(λ, T)d]}-1-Pbα(λ, T)(d-t){-t exp[-α(λ, T)t]+d exp[-α(λ, T)d]}-hPb×(r-1)exp[-α(λ, T)h]dαdT.
Egap(T)=Egap(0 K)-aT2T+b,
α(T)[hν-Egap(T)]γ.
1αdαdT=22aTT+b-aT2(T+b)2hν-Egap(0 K)-aT2T+b.
α(T)=α0(λ)exp[T/T0(λ)];
1αdαdT=1T0.
α(E, T2)=α(E+ΔE, T1),
E=E+ΔE,ΔE=c(E)(T2T1),
c(E)=1.2×10-3 eV/K forE>1.7 eV,
c(E)=1.3EeV-1.010-3 eV/K
for1.1 eV<E<1.7 eV.
dαdT=α(E+ΔE, T2)-α(E, T1)T2-T1.

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