Abstract

We demonstrate thermal imaging using a charge-coupled device (CCD) thermoreflectance lock-in technique that achieves a record temperature resolution of 18mK, 44dB below the nominal dynamic range of the camera (from 72 to 116dB) for 105 periods of measurement. We show that the quantization limit of the CCD camera does not set the lower bound on the precision of the technique. We present a theoretical description of the measurement technique, accounting for the effects of noise and nonideal analog-to-digital conversion, resulting in analytic expressions for the probability distribution function of the measured signals, and allowing for explicit calculation of resolution and error bars. The theory is tested against parametrically varied measurements and can be applied to other sampled lock-in measurements. We also experimentally demonstrate sub-quantization-limit imaging on a well-characterized model system, joule heating in a silicon resistor. The accuracy of the resistor thermoreflectance measurement is confirmed by comparing the results with those of a standard 3ω measurement.

© 2007 Optical Society of America

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  1. J. Marek and Y. E. Strausser, "Correlation of thermal-wave imaging to other analysis methods," Appl. Phys. Lett. 44, 1152-1154 (1984).
    [CrossRef]
  2. O. Breitenstein, M. Langenkamp, F. Altmann, D. Katzer, A. Lindner, and H. Eggers, "Microscopic lock-in thermography investigation of leakage sites in integrated circuits," Rev. Sci. Instrum. 71, 4155-4160 (2000).
    [CrossRef]
  3. K. Luo, Z. Shi, J. Varesi, and A. Majumdar, "Sensor nanofabrication, performance, and conduction mechanisms in scanning thermal microscopy," J. Vac. Sci. Technol. B 15, 349-360 (1997).
    [CrossRef]
  4. M. Cardona, Modulation Spectroscopy, Solid-State Physics, Supplement 11 (Academic, 1969).
  5. D. Cahill, K. Goodson, and A. Majumdar, "Thermometry and thermal transport in micro/nanoscale solid-state devices and structures," J. Heat Transfer 124, 223-241 (2002).
    [CrossRef]
  6. J. Christofferson and A. Shakouri, "Thermoreflectance based thermal microscope," Rev. Sci. Instrum. 76, 24903-24909 (2005).
    [CrossRef]
  7. S. Dilhaire, S. Grauby, and W. Claeys, "Calibration procedure for temperature measurements by thermoreflectance under high magnification conditions," Appl. Phys. Lett. 84, 822-824 (2004).
    [CrossRef]
  8. S. Grauby, B. C. Forget, S. Hole, and D. Fournier, "High resolution photothermal imaging of high frequency phenomena using a visible charge coupled device camera associated with a multichannel lock-in scheme," Rev. Sci. Instrum. 70, 3603-3608 (1999).
    [CrossRef]
  9. C. Filloy, G. Tessier, S. Hole, G. Jerosolimski, and D. Fournier, "The contribution of thermoreflectance to high resolution thermal mapping," Sens. Rev. 23, 35-39 (2003).
    [CrossRef]
  10. M. Fujinami, K. Toya, and T. Sawada, "Development of photothermal near-field scanning optical microscope photothermal near-field scanning optical microscope," Rev. Sci. Instrum. 74, 621-623 (2003).
    [CrossRef]
  11. G. Tessier, S. Hole, and D. Fournier, "Ultraviolet illumination thermoreflectance for the temperature mapping of integrated circuits," Opt. Lett. 28, 875-877 (2003).
    [CrossRef] [PubMed]
  12. G. Tessier, S. Hole, S. Grauby, and D. Fournier, "Quantitative thermal imaging by thermoreflectance using a CCD array," Presented at THERMINIC 2000, International Workshop No. 6, Budapest, September 24, 2000.
  13. S. Grauby, S. Dilhaire, S. Jorez, and W. Claeys, "Imaging setup for temperature, topography, and surface displacement measurements of microelectronic devices," Rev. Sci. Instrum. 74, 645-647 (2003).
    [CrossRef]
  14. C.-H. Ho, H.-W. Lee, and Z.-H. Cheng, "Practical thermoreflectance design for optical characterization of layer semiconductors," Rev. Sci. Instrum. 75, 1098-1102 (2004).
    [CrossRef]
  15. J. Christofferson and A. Shakouri, "Camera for thermal imaging of semiconductor devices based on thermoreflectance," in Proceedings of the 20th Annual IEEE Semiconductor Thermal Measurement and Management Symposium (IEEE, 2004), pp. 87-91.
  16. J. Philip and K. Carlsson, "Theoretical investigation of the signal-to-noise ratio in fluorescence lifetime imaging," J. Opt. Soc. Am. A 20, 368-379 (2003).
    [CrossRef]
  17. E. Balestrieri, P. Daponte, and S. Rapuano, "A state of the art on ADC error compensation methods," IEEE Trans. Instrum. Meas. 54, 1-8 (2005).
    [CrossRef]
  18. P. Carbone, "Quantitative criteria for the design of dither-based quantizing systems," IEEE Trans. Instrum. Meas. 46, 656-659 (1997).
    [CrossRef]
  19. P. Carbone and D. Petri, "Effect of additive dither on the resolution of ideal quantizers," IEEE Trans. Instrum. Meas. 43, 389-396 (1994).
    [CrossRef]
  20. L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, "Stochastic resonance," Rev. Mod. Phys. 70, 223-287 (1998).
    [CrossRef]
  21. R. A. Wannamaker, S. P. Lipshitz, J. Vanderkooy, and J. N. Wright, "A theory of nonsubtractive dither," IEEE Trans. Signal Process. 48, 499-516 (2000).
    [CrossRef]
  22. J. Potzick, "Noise averaging and measurement resolution (or "A little noise is a good thing")," Rev. Sci. Instrum. 70, 2038-2040 (1999).
    [CrossRef]
  23. D. Luerssen, J. A. Hudgings, P. M. Mayer, and R. J. Ram, "Nanoscale thermoreflectance with 10mK temperature resolution using stochastic resonance," in Proceedings of the 21st Annual IEEE Semiconductor Thermal Measurement and Management Symposium (IEEE, 2005), pp. 253-258.
  24. S. Inoué and K. R. Spring, Video Microscopy: The Fundamentals, 2nd ed. (Plenum, 1997).
    [CrossRef]
  25. G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
    [CrossRef]
  26. O. Breitenstein and M. Langenkamp, Lock-in Thermography: Basics and use for Functional Diagnostics of Electronic Components, Springer Series in Advanced Microelectronics (Springer, 2003).
  27. W. M. Hubbard, "Approximation of a Poisson distribution by a Gaussian distribution," Proc. IEEE 58, 1374-1375 (1970).
    [CrossRef]
  28. J. Janesick, "CCD transfer method—standard for absolute performance of CCDs and digital CCD camera systems," Proc. SPIE 3019, 70-102 (1997).
    [CrossRef]
  29. Y. Reibel, M. Jung, M. Bouhifd, B. Cunin, and C. Draman, "CCD or CMOS camera noise characterization," Eur. Phys. J.: Appl. Phys. 21, 75-80 (2003).
    [CrossRef]
  30. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, Prentice Hall Signal Processing Series, 2nd ed. (Prentice Hall, 1999).
  31. C. F. Coombs, Electronic Instrument Handbook, 3rd ed. (McGraw-Hill, 2000).
  32. J. C. Mullikin, L. J. V. Vliet, H. Netten, F. R. Boddeke, G. v. d. Feltz, and I. T. Young, "Methods for CCD camera characterization," Proc. SPIE 2173, 73-84 (1994).

2005

E. Balestrieri, P. Daponte, and S. Rapuano, "A state of the art on ADC error compensation methods," IEEE Trans. Instrum. Meas. 54, 1-8 (2005).
[CrossRef]

D. Luerssen, J. A. Hudgings, P. M. Mayer, and R. J. Ram, "Nanoscale thermoreflectance with 10mK temperature resolution using stochastic resonance," in Proceedings of the 21st Annual IEEE Semiconductor Thermal Measurement and Management Symposium (IEEE, 2005), pp. 253-258.

J. Christofferson and A. Shakouri, "Thermoreflectance based thermal microscope," Rev. Sci. Instrum. 76, 24903-24909 (2005).
[CrossRef]

2004

S. Dilhaire, S. Grauby, and W. Claeys, "Calibration procedure for temperature measurements by thermoreflectance under high magnification conditions," Appl. Phys. Lett. 84, 822-824 (2004).
[CrossRef]

C.-H. Ho, H.-W. Lee, and Z.-H. Cheng, "Practical thermoreflectance design for optical characterization of layer semiconductors," Rev. Sci. Instrum. 75, 1098-1102 (2004).
[CrossRef]

J. Christofferson and A. Shakouri, "Camera for thermal imaging of semiconductor devices based on thermoreflectance," in Proceedings of the 20th Annual IEEE Semiconductor Thermal Measurement and Management Symposium (IEEE, 2004), pp. 87-91.

2003

J. Philip and K. Carlsson, "Theoretical investigation of the signal-to-noise ratio in fluorescence lifetime imaging," J. Opt. Soc. Am. A 20, 368-379 (2003).
[CrossRef]

G. Tessier, S. Hole, and D. Fournier, "Ultraviolet illumination thermoreflectance for the temperature mapping of integrated circuits," Opt. Lett. 28, 875-877 (2003).
[CrossRef] [PubMed]

C. Filloy, G. Tessier, S. Hole, G. Jerosolimski, and D. Fournier, "The contribution of thermoreflectance to high resolution thermal mapping," Sens. Rev. 23, 35-39 (2003).
[CrossRef]

M. Fujinami, K. Toya, and T. Sawada, "Development of photothermal near-field scanning optical microscope photothermal near-field scanning optical microscope," Rev. Sci. Instrum. 74, 621-623 (2003).
[CrossRef]

S. Grauby, S. Dilhaire, S. Jorez, and W. Claeys, "Imaging setup for temperature, topography, and surface displacement measurements of microelectronic devices," Rev. Sci. Instrum. 74, 645-647 (2003).
[CrossRef]

O. Breitenstein and M. Langenkamp, Lock-in Thermography: Basics and use for Functional Diagnostics of Electronic Components, Springer Series in Advanced Microelectronics (Springer, 2003).

Y. Reibel, M. Jung, M. Bouhifd, B. Cunin, and C. Draman, "CCD or CMOS camera noise characterization," Eur. Phys. J.: Appl. Phys. 21, 75-80 (2003).
[CrossRef]

2002

D. Cahill, K. Goodson, and A. Majumdar, "Thermometry and thermal transport in micro/nanoscale solid-state devices and structures," J. Heat Transfer 124, 223-241 (2002).
[CrossRef]

2000

G. Tessier, S. Hole, S. Grauby, and D. Fournier, "Quantitative thermal imaging by thermoreflectance using a CCD array," Presented at THERMINIC 2000, International Workshop No. 6, Budapest, September 24, 2000.

C. F. Coombs, Electronic Instrument Handbook, 3rd ed. (McGraw-Hill, 2000).

O. Breitenstein, M. Langenkamp, F. Altmann, D. Katzer, A. Lindner, and H. Eggers, "Microscopic lock-in thermography investigation of leakage sites in integrated circuits," Rev. Sci. Instrum. 71, 4155-4160 (2000).
[CrossRef]

R. A. Wannamaker, S. P. Lipshitz, J. Vanderkooy, and J. N. Wright, "A theory of nonsubtractive dither," IEEE Trans. Signal Process. 48, 499-516 (2000).
[CrossRef]

1999

J. Potzick, "Noise averaging and measurement resolution (or "A little noise is a good thing")," Rev. Sci. Instrum. 70, 2038-2040 (1999).
[CrossRef]

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, Prentice Hall Signal Processing Series, 2nd ed. (Prentice Hall, 1999).

S. Grauby, B. C. Forget, S. Hole, and D. Fournier, "High resolution photothermal imaging of high frequency phenomena using a visible charge coupled device camera associated with a multichannel lock-in scheme," Rev. Sci. Instrum. 70, 3603-3608 (1999).
[CrossRef]

1998

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, "Stochastic resonance," Rev. Mod. Phys. 70, 223-287 (1998).
[CrossRef]

1997

J. Janesick, "CCD transfer method—standard for absolute performance of CCDs and digital CCD camera systems," Proc. SPIE 3019, 70-102 (1997).
[CrossRef]

K. Luo, Z. Shi, J. Varesi, and A. Majumdar, "Sensor nanofabrication, performance, and conduction mechanisms in scanning thermal microscopy," J. Vac. Sci. Technol. B 15, 349-360 (1997).
[CrossRef]

P. Carbone, "Quantitative criteria for the design of dither-based quantizing systems," IEEE Trans. Instrum. Meas. 46, 656-659 (1997).
[CrossRef]

S. Inoué and K. R. Spring, Video Microscopy: The Fundamentals, 2nd ed. (Plenum, 1997).
[CrossRef]

1994

P. Carbone and D. Petri, "Effect of additive dither on the resolution of ideal quantizers," IEEE Trans. Instrum. Meas. 43, 389-396 (1994).
[CrossRef]

J. C. Mullikin, L. J. V. Vliet, H. Netten, F. R. Boddeke, G. v. d. Feltz, and I. T. Young, "Methods for CCD camera characterization," Proc. SPIE 2173, 73-84 (1994).

1992

G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
[CrossRef]

1984

J. Marek and Y. E. Strausser, "Correlation of thermal-wave imaging to other analysis methods," Appl. Phys. Lett. 44, 1152-1154 (1984).
[CrossRef]

1970

W. M. Hubbard, "Approximation of a Poisson distribution by a Gaussian distribution," Proc. IEEE 58, 1374-1375 (1970).
[CrossRef]

Altmann, F.

O. Breitenstein, M. Langenkamp, F. Altmann, D. Katzer, A. Lindner, and H. Eggers, "Microscopic lock-in thermography investigation of leakage sites in integrated circuits," Rev. Sci. Instrum. 71, 4155-4160 (2000).
[CrossRef]

Balestrieri, E.

E. Balestrieri, P. Daponte, and S. Rapuano, "A state of the art on ADC error compensation methods," IEEE Trans. Instrum. Meas. 54, 1-8 (2005).
[CrossRef]

Boddeke, F. R.

J. C. Mullikin, L. J. V. Vliet, H. Netten, F. R. Boddeke, G. v. d. Feltz, and I. T. Young, "Methods for CCD camera characterization," Proc. SPIE 2173, 73-84 (1994).

Bouhifd, M.

Y. Reibel, M. Jung, M. Bouhifd, B. Cunin, and C. Draman, "CCD or CMOS camera noise characterization," Eur. Phys. J.: Appl. Phys. 21, 75-80 (2003).
[CrossRef]

Breitenstein, O.

O. Breitenstein and M. Langenkamp, Lock-in Thermography: Basics and use for Functional Diagnostics of Electronic Components, Springer Series in Advanced Microelectronics (Springer, 2003).

O. Breitenstein, M. Langenkamp, F. Altmann, D. Katzer, A. Lindner, and H. Eggers, "Microscopic lock-in thermography investigation of leakage sites in integrated circuits," Rev. Sci. Instrum. 71, 4155-4160 (2000).
[CrossRef]

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, Prentice Hall Signal Processing Series, 2nd ed. (Prentice Hall, 1999).

Busse, G.

G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
[CrossRef]

Cahill, D.

D. Cahill, K. Goodson, and A. Majumdar, "Thermometry and thermal transport in micro/nanoscale solid-state devices and structures," J. Heat Transfer 124, 223-241 (2002).
[CrossRef]

Carbone, P.

P. Carbone, "Quantitative criteria for the design of dither-based quantizing systems," IEEE Trans. Instrum. Meas. 46, 656-659 (1997).
[CrossRef]

P. Carbone and D. Petri, "Effect of additive dither on the resolution of ideal quantizers," IEEE Trans. Instrum. Meas. 43, 389-396 (1994).
[CrossRef]

Cardona, M.

M. Cardona, Modulation Spectroscopy, Solid-State Physics, Supplement 11 (Academic, 1969).

Carlsson, K.

Cheng, Z.-H.

C.-H. Ho, H.-W. Lee, and Z.-H. Cheng, "Practical thermoreflectance design for optical characterization of layer semiconductors," Rev. Sci. Instrum. 75, 1098-1102 (2004).
[CrossRef]

Christofferson, J.

J. Christofferson and A. Shakouri, "Thermoreflectance based thermal microscope," Rev. Sci. Instrum. 76, 24903-24909 (2005).
[CrossRef]

J. Christofferson and A. Shakouri, "Camera for thermal imaging of semiconductor devices based on thermoreflectance," in Proceedings of the 20th Annual IEEE Semiconductor Thermal Measurement and Management Symposium (IEEE, 2004), pp. 87-91.

Claeys, W.

S. Dilhaire, S. Grauby, and W. Claeys, "Calibration procedure for temperature measurements by thermoreflectance under high magnification conditions," Appl. Phys. Lett. 84, 822-824 (2004).
[CrossRef]

S. Grauby, S. Dilhaire, S. Jorez, and W. Claeys, "Imaging setup for temperature, topography, and surface displacement measurements of microelectronic devices," Rev. Sci. Instrum. 74, 645-647 (2003).
[CrossRef]

Coombs, C. F.

C. F. Coombs, Electronic Instrument Handbook, 3rd ed. (McGraw-Hill, 2000).

Cunin, B.

Y. Reibel, M. Jung, M. Bouhifd, B. Cunin, and C. Draman, "CCD or CMOS camera noise characterization," Eur. Phys. J.: Appl. Phys. 21, 75-80 (2003).
[CrossRef]

Daponte, P.

E. Balestrieri, P. Daponte, and S. Rapuano, "A state of the art on ADC error compensation methods," IEEE Trans. Instrum. Meas. 54, 1-8 (2005).
[CrossRef]

Dilhaire, S.

S. Dilhaire, S. Grauby, and W. Claeys, "Calibration procedure for temperature measurements by thermoreflectance under high magnification conditions," Appl. Phys. Lett. 84, 822-824 (2004).
[CrossRef]

S. Grauby, S. Dilhaire, S. Jorez, and W. Claeys, "Imaging setup for temperature, topography, and surface displacement measurements of microelectronic devices," Rev. Sci. Instrum. 74, 645-647 (2003).
[CrossRef]

Draman, C.

Y. Reibel, M. Jung, M. Bouhifd, B. Cunin, and C. Draman, "CCD or CMOS camera noise characterization," Eur. Phys. J.: Appl. Phys. 21, 75-80 (2003).
[CrossRef]

Eggers, H.

O. Breitenstein, M. Langenkamp, F. Altmann, D. Katzer, A. Lindner, and H. Eggers, "Microscopic lock-in thermography investigation of leakage sites in integrated circuits," Rev. Sci. Instrum. 71, 4155-4160 (2000).
[CrossRef]

Feltz, G. v. d.

J. C. Mullikin, L. J. V. Vliet, H. Netten, F. R. Boddeke, G. v. d. Feltz, and I. T. Young, "Methods for CCD camera characterization," Proc. SPIE 2173, 73-84 (1994).

Filloy, C.

C. Filloy, G. Tessier, S. Hole, G. Jerosolimski, and D. Fournier, "The contribution of thermoreflectance to high resolution thermal mapping," Sens. Rev. 23, 35-39 (2003).
[CrossRef]

Forget, B. C.

S. Grauby, B. C. Forget, S. Hole, and D. Fournier, "High resolution photothermal imaging of high frequency phenomena using a visible charge coupled device camera associated with a multichannel lock-in scheme," Rev. Sci. Instrum. 70, 3603-3608 (1999).
[CrossRef]

Fournier, D.

C. Filloy, G. Tessier, S. Hole, G. Jerosolimski, and D. Fournier, "The contribution of thermoreflectance to high resolution thermal mapping," Sens. Rev. 23, 35-39 (2003).
[CrossRef]

G. Tessier, S. Hole, and D. Fournier, "Ultraviolet illumination thermoreflectance for the temperature mapping of integrated circuits," Opt. Lett. 28, 875-877 (2003).
[CrossRef] [PubMed]

G. Tessier, S. Hole, S. Grauby, and D. Fournier, "Quantitative thermal imaging by thermoreflectance using a CCD array," Presented at THERMINIC 2000, International Workshop No. 6, Budapest, September 24, 2000.

S. Grauby, B. C. Forget, S. Hole, and D. Fournier, "High resolution photothermal imaging of high frequency phenomena using a visible charge coupled device camera associated with a multichannel lock-in scheme," Rev. Sci. Instrum. 70, 3603-3608 (1999).
[CrossRef]

Fujinami, M.

M. Fujinami, K. Toya, and T. Sawada, "Development of photothermal near-field scanning optical microscope photothermal near-field scanning optical microscope," Rev. Sci. Instrum. 74, 621-623 (2003).
[CrossRef]

Gammaitoni, L.

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, "Stochastic resonance," Rev. Mod. Phys. 70, 223-287 (1998).
[CrossRef]

Goodson, K.

D. Cahill, K. Goodson, and A. Majumdar, "Thermometry and thermal transport in micro/nanoscale solid-state devices and structures," J. Heat Transfer 124, 223-241 (2002).
[CrossRef]

Grauby, S.

S. Dilhaire, S. Grauby, and W. Claeys, "Calibration procedure for temperature measurements by thermoreflectance under high magnification conditions," Appl. Phys. Lett. 84, 822-824 (2004).
[CrossRef]

S. Grauby, S. Dilhaire, S. Jorez, and W. Claeys, "Imaging setup for temperature, topography, and surface displacement measurements of microelectronic devices," Rev. Sci. Instrum. 74, 645-647 (2003).
[CrossRef]

G. Tessier, S. Hole, S. Grauby, and D. Fournier, "Quantitative thermal imaging by thermoreflectance using a CCD array," Presented at THERMINIC 2000, International Workshop No. 6, Budapest, September 24, 2000.

S. Grauby, B. C. Forget, S. Hole, and D. Fournier, "High resolution photothermal imaging of high frequency phenomena using a visible charge coupled device camera associated with a multichannel lock-in scheme," Rev. Sci. Instrum. 70, 3603-3608 (1999).
[CrossRef]

Hanggi, P.

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, "Stochastic resonance," Rev. Mod. Phys. 70, 223-287 (1998).
[CrossRef]

Ho, C.-H.

C.-H. Ho, H.-W. Lee, and Z.-H. Cheng, "Practical thermoreflectance design for optical characterization of layer semiconductors," Rev. Sci. Instrum. 75, 1098-1102 (2004).
[CrossRef]

Hole, S.

G. Tessier, S. Hole, and D. Fournier, "Ultraviolet illumination thermoreflectance for the temperature mapping of integrated circuits," Opt. Lett. 28, 875-877 (2003).
[CrossRef] [PubMed]

C. Filloy, G. Tessier, S. Hole, G. Jerosolimski, and D. Fournier, "The contribution of thermoreflectance to high resolution thermal mapping," Sens. Rev. 23, 35-39 (2003).
[CrossRef]

G. Tessier, S. Hole, S. Grauby, and D. Fournier, "Quantitative thermal imaging by thermoreflectance using a CCD array," Presented at THERMINIC 2000, International Workshop No. 6, Budapest, September 24, 2000.

S. Grauby, B. C. Forget, S. Hole, and D. Fournier, "High resolution photothermal imaging of high frequency phenomena using a visible charge coupled device camera associated with a multichannel lock-in scheme," Rev. Sci. Instrum. 70, 3603-3608 (1999).
[CrossRef]

Hubbard, W. M.

W. M. Hubbard, "Approximation of a Poisson distribution by a Gaussian distribution," Proc. IEEE 58, 1374-1375 (1970).
[CrossRef]

Hudgings, J. A.

D. Luerssen, J. A. Hudgings, P. M. Mayer, and R. J. Ram, "Nanoscale thermoreflectance with 10mK temperature resolution using stochastic resonance," in Proceedings of the 21st Annual IEEE Semiconductor Thermal Measurement and Management Symposium (IEEE, 2005), pp. 253-258.

Inoué, S.

S. Inoué and K. R. Spring, Video Microscopy: The Fundamentals, 2nd ed. (Plenum, 1997).
[CrossRef]

Janesick, J.

J. Janesick, "CCD transfer method—standard for absolute performance of CCDs and digital CCD camera systems," Proc. SPIE 3019, 70-102 (1997).
[CrossRef]

Jerosolimski, G.

C. Filloy, G. Tessier, S. Hole, G. Jerosolimski, and D. Fournier, "The contribution of thermoreflectance to high resolution thermal mapping," Sens. Rev. 23, 35-39 (2003).
[CrossRef]

Jorez, S.

S. Grauby, S. Dilhaire, S. Jorez, and W. Claeys, "Imaging setup for temperature, topography, and surface displacement measurements of microelectronic devices," Rev. Sci. Instrum. 74, 645-647 (2003).
[CrossRef]

Jung, M.

Y. Reibel, M. Jung, M. Bouhifd, B. Cunin, and C. Draman, "CCD or CMOS camera noise characterization," Eur. Phys. J.: Appl. Phys. 21, 75-80 (2003).
[CrossRef]

Jung, P.

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, "Stochastic resonance," Rev. Mod. Phys. 70, 223-287 (1998).
[CrossRef]

Karpen, W.

G. Busse, D. Wu, and W. Karpen, "Thermal wave imaging with phase sensitive modulated thermography," J. Appl. Phys. 71, 3962-3965 (1992).
[CrossRef]

Katzer, D.

O. Breitenstein, M. Langenkamp, F. Altmann, D. Katzer, A. Lindner, and H. Eggers, "Microscopic lock-in thermography investigation of leakage sites in integrated circuits," Rev. Sci. Instrum. 71, 4155-4160 (2000).
[CrossRef]

Langenkamp, M.

O. Breitenstein and M. Langenkamp, Lock-in Thermography: Basics and use for Functional Diagnostics of Electronic Components, Springer Series in Advanced Microelectronics (Springer, 2003).

O. Breitenstein, M. Langenkamp, F. Altmann, D. Katzer, A. Lindner, and H. Eggers, "Microscopic lock-in thermography investigation of leakage sites in integrated circuits," Rev. Sci. Instrum. 71, 4155-4160 (2000).
[CrossRef]

Lee, H.-W.

C.-H. Ho, H.-W. Lee, and Z.-H. Cheng, "Practical thermoreflectance design for optical characterization of layer semiconductors," Rev. Sci. Instrum. 75, 1098-1102 (2004).
[CrossRef]

Lindner, A.

O. Breitenstein, M. Langenkamp, F. Altmann, D. Katzer, A. Lindner, and H. Eggers, "Microscopic lock-in thermography investigation of leakage sites in integrated circuits," Rev. Sci. Instrum. 71, 4155-4160 (2000).
[CrossRef]

Lipshitz, S. P.

R. A. Wannamaker, S. P. Lipshitz, J. Vanderkooy, and J. N. Wright, "A theory of nonsubtractive dither," IEEE Trans. Signal Process. 48, 499-516 (2000).
[CrossRef]

Luerssen, D.

D. Luerssen, J. A. Hudgings, P. M. Mayer, and R. J. Ram, "Nanoscale thermoreflectance with 10mK temperature resolution using stochastic resonance," in Proceedings of the 21st Annual IEEE Semiconductor Thermal Measurement and Management Symposium (IEEE, 2005), pp. 253-258.

Luo, K.

K. Luo, Z. Shi, J. Varesi, and A. Majumdar, "Sensor nanofabrication, performance, and conduction mechanisms in scanning thermal microscopy," J. Vac. Sci. Technol. B 15, 349-360 (1997).
[CrossRef]

Majumdar, A.

D. Cahill, K. Goodson, and A. Majumdar, "Thermometry and thermal transport in micro/nanoscale solid-state devices and structures," J. Heat Transfer 124, 223-241 (2002).
[CrossRef]

K. Luo, Z. Shi, J. Varesi, and A. Majumdar, "Sensor nanofabrication, performance, and conduction mechanisms in scanning thermal microscopy," J. Vac. Sci. Technol. B 15, 349-360 (1997).
[CrossRef]

Marchesoni, F.

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, "Stochastic resonance," Rev. Mod. Phys. 70, 223-287 (1998).
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J. C. Mullikin, L. J. V. Vliet, H. Netten, F. R. Boddeke, G. v. d. Feltz, and I. T. Young, "Methods for CCD camera characterization," Proc. SPIE 2173, 73-84 (1994).

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D. Luerssen, J. A. Hudgings, P. M. Mayer, and R. J. Ram, "Nanoscale thermoreflectance with 10mK temperature resolution using stochastic resonance," in Proceedings of the 21st Annual IEEE Semiconductor Thermal Measurement and Management Symposium (IEEE, 2005), pp. 253-258.

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J. Potzick, "Noise averaging and measurement resolution (or "A little noise is a good thing")," Rev. Sci. Instrum. 70, 2038-2040 (1999).
[CrossRef]

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G. Tessier, S. Hole, S. Grauby, and D. Fournier, "Quantitative thermal imaging by thermoreflectance using a CCD array," Presented at THERMINIC 2000, International Workshop No. 6, Budapest, September 24, 2000.

D. Luerssen, J. A. Hudgings, P. M. Mayer, and R. J. Ram, "Nanoscale thermoreflectance with 10mK temperature resolution using stochastic resonance," in Proceedings of the 21st Annual IEEE Semiconductor Thermal Measurement and Management Symposium (IEEE, 2005), pp. 253-258.

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J. C. Mullikin, L. J. V. Vliet, H. Netten, F. R. Boddeke, G. v. d. Feltz, and I. T. Young, "Methods for CCD camera characterization," Proc. SPIE 2173, 73-84 (1994).

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Figures (5)

Fig. 1
Fig. 1

Thermoreflectance measurement setup. (a) Measurement configuration relies on a CCD camera whose image acquisition is phased locked to a source of thermal excitation in the device of interest. A LED provides a source of illumination, and a microscope objective is used to obtain the desired spatial resolution. (b) Calibration configuration is used to obtain the thermoreflectance coefficients for a material of interest whose temperature is modulated on a Peltier stage.

Fig. 2
Fig. 2

Thermoreflectance measurements (gray) and theory (black) plotted against the number of iterations for thermal signals (a) above and (b) below the quantization threshold. For each measurement, 25 pixels are used to calculate the mean response and the 1-sigma error bars. Solid curve, theoretically predicted mean response; dotted curves, theoretically predicted 1-sigma error limits. Theory and experiment agree quantitatively in both cases.

Fig. 3
Fig. 3

For a resistor, the amplitude of the temperature change Δ T is expected to be proportional to the electrical input power P. Data follow the expected linear trend, confirming that temperature levels smaller than one quantization bit ( 2.93 K ) can be measured reliably. Inset, thermoreflectance image of the p-doped Si resistor on the Si substrate, along with the thermoreflectance profile across the resistor.

Fig. 4
Fig. 4

Average value (dotted black line) of the measured temperature profile (solid black curve) is close to an independent measurement of the resistor’s temperature using the 3 ω method. Shaded gray box indicates the ± one sigma confidence interval for the 3 ω measurement. Inset, micrograph of the region of interest, where the dark vertical stripe is the resistor. Two thermoreflectance coefficients (for doped and undoped Si) were separately measured to reconstruct the temperature profile inside and outside the resistor.

Fig. 5
Fig. 5

Example of an imperfect A-to-D quantizer. Presence of unequal bin widths simulates the quantizer’s nonlinearity error. In this case, a DNL of 0.35 quantization steps has been assumed. Top inset, plot of the DNL error over 256 bins of the A-to-D converter; bottom inset, similar plot of the INL error.

Equations (21)

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Δ R R = 1 R R T Δ T κ Δ T .
I k ( x , y ) = i = 1 N ( 4 T ( 4 i + R ) T 4 ( 4 i + k + 1 ) T 4 ( c ( x , y ) + Δ ( x , y ) cos ( ω t + ϕ ( x , y ) + ψ ) ) d t ) + d ( x , y ) , k { 1 , 2 , 3 , 4 } .
Δ R R ideal = π 2 ( I 1 I 3 ) 2 + ( I 2 I 4 ) 2 I 1 + I 2 + I 3 + I 4 = Δ c ,
ϕ ideal = arctan ( I 1 I 2 I 3 + I 4 I 1 + I 2 I 3 I 4 ) ψ = ϕ .
σ d = α c + β .
μ k = N ( c 4 Δ π 2 ( cos ( π k 2 π 4 ) sin ( ϕ + ψ ) + sin ( π k 2 π 4 ) cos ( ϕ + ψ ) ) ) .
Δ R R [ π 2 I 1 I 3 k μ k ] 2 + [ π 2 I 2 I 4 k μ k ] 2 = A 2 + B 2 .
σ 2 = π 2 2 2 N σ d 2 ( 4 N c ) 2 = π 2 16 σ d 2 N c 2 .
p r ( r ) = r σ 2 exp ( r 2 + μ A 2 + μ B 2 2 σ 2 ) I 0 ( r μ A 2 + μ B 2 σ 2 ) ,
Δ R R ideal = μ A 2 + μ B 2 = Δ c ,
E [ Δ R R ] = 2 σ exp ( μ A 2 + μ B 2 2 σ 2 ) [ 1 2 π ( 1 + 1 2 μ A 2 + μ B 2 σ 2 ) exp ( μ A 2 + μ B 2 4 σ 2 ) I 0 ( μ A 2 + μ B 2 4 σ 2 ) + π ( μ A 2 + μ B 2 ) 4 σ 2 exp ( μ A 2 + μ B 2 4 σ 2 ) I 1 ( μ A 2 + μ B 2 4 σ 2 ) ] ,
E [ ( Δ R R ) 2 ] = 2 σ 2 + μ A 2 + μ B 2 .
σ Δ R R ( c , N ) = E [ ( Δ R R ) 2 ] E [ Δ R R ] 2 .
ε offset E [ Δ R R ] Δ R R ideal = E [ Δ R R ] μ A 2 + μ B 2 .
σ Δ R R ( c , N ) = E [ ( Δ R R ) 2 ] E [ Δ R R ] 2 = π 4 2 π 2 1 N 1 c ( α c + β ) .
Y A B , p Y = N ( μ A μ B , 2 σ 2 ) = N ( μ Y , σ ϕ 2 ) ,
X A + B , p X = N ( μ A + μ B , 2 σ 2 ) = N ( μ X , σ ϕ 2 ) ,
ϕ = arctan ( Y X ) Ψ = arctan ( I 1 I 2 I 3 + I 4 I 1 + I 2 I 3 I 4 ) Ψ .
p X , Y ( X , Y ) = 1 2 π σ 2 exp ( ( X μ X ) 2 + ( Y μ Y ) 2 2 σ 2 ) .
p r , ϕ ( r , ϕ ) = r 2 π σ 2 exp ( 1 2 σ 2 [ ( r cos ϕ μ X ) 2 + ( r sin ϕ μ Y ) 2 ] ) .
p ϕ = 1 2 π exp ( μ X 2 + μ Y 2 2 σ ϕ 2 ) + μ X 2 + μ Y 2 cos ( ϕ θ ) 2 2 π σ ϕ exp ( ( μ X 2 + μ Y 2 ) sin 2 ( ϕ θ ) 2 σ ϕ 2 ) ( 1 + erf ( μ X 2 + μ Y 2 cos ( ϕ θ ) 2 σ ϕ ) ) .

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