Abstract

Photothermal beam deflection studies were carried out with GaAs epitaxial double layers grown on semi-insulating GaAs substrates. The impurity densities in thin epitaxial layers were found to influence the effective thermal diffusivity of the entire structure.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  13. M. Bertolotti, R. L. Voti, G. Liakhou, C. Sibilia, “On the photodeflection method applied to low thermal-diffusivity measurements,” Rev. Sci. Instrum. 64, 1576–1583 (1993).
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    [CrossRef]
  16. A. Mandelis, B. S. H. Royce, “Fundamental-mode laser-beam propagation in optically inhomogeneous electrochemical media with chemical species concentration gradients,” Appl. Opt. 23, 2892–2901 (1984).
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    [CrossRef]
  28. Y. S. Ju, K. E. Goodson, “Phonon scattering in silicon films with thickness of order 100 nm,” Appl. Phys. Lett. 74, 3005–3007 (1999).
    [CrossRef]
  29. G. Chen, “Phonon wave heat conduction in thin films and superlattices,” J. Heat Transfer 121, 945–953 (1999).
    [CrossRef]
  30. T. F. Zeng, G. Chen, “Phonon heat conduction in thin films: impacts of thermal boundary resistance and internal heat generation,” J. Heat Transfer 123, 340–347 (2001).
    [CrossRef]
  31. M. Asheghi, Y. K. Leung, S. S. Wong, K. E. Goodson, “Phonon-boundary scattering in thin silicon layers,” Appl. Phys. Lett. 71, 1798–1800 (1997).
    [CrossRef]

2001

T. F. Zeng, G. Chen, “Phonon heat conduction in thin films: impacts of thermal boundary resistance and internal heat generation,” J. Heat Transfer 123, 340–347 (2001).
[CrossRef]

1999

Y. S. Ju, K. E. Goodson, “Phonon scattering in silicon films with thickness of order 100 nm,” Appl. Phys. Lett. 74, 3005–3007 (1999).
[CrossRef]

G. Chen, “Phonon wave heat conduction in thin films and superlattices,” J. Heat Transfer 121, 945–953 (1999).
[CrossRef]

M. Soltanolkotabi, G. L. Bennis, R. Gupta, “Temperature dependence of the thermal diffusivity of GaAs in the 100–305 K range measured by the pulsed photothermal displacement technique,” J. Appl. Phys. 85, 794–798 (1999).
[CrossRef]

C. Wang, A. Mandelis, “Purely thermal-wave photopyroelectric interferometry,” J. Appl. Phys. 85, 8366–8377 (1999).
[CrossRef]

1998

M. Bertolotti, G. L. Liakhou, R. L. Voti, S. Paolini, C. Sibilia, “Analysis of the photothermal deflection technique in the surface reflection scheme: theory and experiment,” J. Appl. Phys. 83, 966–982 (1998).
[CrossRef]

1997

M. Asheghi, Y. K. Leung, S. S. Wong, K. E. Goodson, “Phonon-boundary scattering in thin silicon layers,” Appl. Phys. Lett. 71, 1798–1800 (1997).
[CrossRef]

1996

C. Christofides, F. Diakonos, A. Seas, C. Christou, M. Nestoros, A. Mandelis, “Two-layer model for photomodulated thermoreflectance of semiconductor wafers,” J. Appl. Phys. 80, 1713–1725 (1996).
[CrossRef]

R. E. Wagner, A. Mandelis, “Nonlinear photothermal modulated optical reflectance and photocurrent phenomena in crystalline semiconductors: theoretical,” Semicond. Sci. Technol. 11, 289–299 (1996).
[CrossRef]

1994

A. Salazar, A. S. Lavega, “Thermal-diffusivity measurements using linear relations from photothermal wave experiments,” Rev. Sci. Instrum. 65, 2896–2900 (1994).
[CrossRef]

G. Chen, C. L. Tien, X. Wu, J. S. Smith, “Thermal-diffusivity measurement of GaAs/AlGaAs thin-film structures,” J. Heat. Transfer 116, 325–331 (1994).
[CrossRef]

1993

M. Bertolotti, R. L. Voti, G. Liakhou, C. Sibilia, “On the photodeflection method applied to low thermal-diffusivity measurements,” Rev. Sci. Instrum. 64, 1576–1583 (1993).
[CrossRef]

1991

A. Salazar, A. S. Lavega, J. Fernandez, “Thermal-diffusivity measurements in solids by the mirage technique—experimental results,” J. Appl. Phys. 69, 1216–1223 (1991).
[CrossRef]

1986

D. Fournier, C. Boccara, A. Skumanich, N. M. Amer, “Photothermal investigation of transport in semiconductors: theory and experiment,” J. Appl. Phys. 59, 787–795 (1986).
[CrossRef]

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

1984

1983

K. R. Grice, L. J. Inglehart, L. O. Favro, P. K. Kuo, R. L. Thomas, “Thermal wave imaging of closed cracks in opaque solids,” J. Appl. Phys. 54, 6245–6255 (1983).
[CrossRef]

1981

L. C. Aamodt, J. C. Murphy, “Photothermal measurements using a localized excitation source,” J. Appl. Phys. 52, 4903–4914 (1981).
[CrossRef]

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333–1344 (1981).
[CrossRef] [PubMed]

1980

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the ‘mirage effect,’” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Aamodt, L. C.

L. C. Aamodt, J. C. Murphy, “Photothermal measurements using a localized excitation source,” J. Appl. Phys. 52, 4903–4914 (1981).
[CrossRef]

Adachi, S.

S. Adachi, Physical Properties of III–V Semiconductor Compounds (Wiley, New York, 1992).
[CrossRef]

Amer, N. M.

D. Fournier, C. Boccara, A. Skumanich, N. M. Amer, “Photothermal investigation of transport in semiconductors: theory and experiment,” J. Appl. Phys. 59, 787–795 (1986).
[CrossRef]

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333–1344 (1981).
[CrossRef] [PubMed]

Asheghi, M.

M. Asheghi, Y. K. Leung, S. S. Wong, K. E. Goodson, “Phonon-boundary scattering in thin silicon layers,” Appl. Phys. Lett. 71, 1798–1800 (1997).
[CrossRef]

Badoz, J.

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the ‘mirage effect,’” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Bennis, G. L.

M. Soltanolkotabi, G. L. Bennis, R. Gupta, “Temperature dependence of the thermal diffusivity of GaAs in the 100–305 K range measured by the pulsed photothermal displacement technique,” J. Appl. Phys. 85, 794–798 (1999).
[CrossRef]

Bertolotti, M.

M. Bertolotti, G. L. Liakhou, R. L. Voti, S. Paolini, C. Sibilia, “Analysis of the photothermal deflection technique in the surface reflection scheme: theory and experiment,” J. Appl. Phys. 83, 966–982 (1998).
[CrossRef]

M. Bertolotti, R. L. Voti, G. Liakhou, C. Sibilia, “On the photodeflection method applied to low thermal-diffusivity measurements,” Rev. Sci. Instrum. 64, 1576–1583 (1993).
[CrossRef]

Bhandari, C. M.

C. M. Bhandari, D. M. Rowe, Thermal Conduction in Semiconductors (Wiley, New York, 1988).

Bialkowski, S. E.

S. E. Bialkowski, Photothermal Spectroscopy Method for Chemical Analysis (Wiley, New York, 1996).

Boccara, A. C.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333–1344 (1981).
[CrossRef] [PubMed]

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the ‘mirage effect,’” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Boccara, C.

D. Fournier, C. Boccara, A. Skumanich, N. M. Amer, “Photothermal investigation of transport in semiconductors: theory and experiment,” J. Appl. Phys. 59, 787–795 (1986).
[CrossRef]

Chen, G.

T. F. Zeng, G. Chen, “Phonon heat conduction in thin films: impacts of thermal boundary resistance and internal heat generation,” J. Heat Transfer 123, 340–347 (2001).
[CrossRef]

G. Chen, “Phonon wave heat conduction in thin films and superlattices,” J. Heat Transfer 121, 945–953 (1999).
[CrossRef]

G. Chen, C. L. Tien, X. Wu, J. S. Smith, “Thermal-diffusivity measurement of GaAs/AlGaAs thin-film structures,” J. Heat. Transfer 116, 325–331 (1994).
[CrossRef]

Christofides, C.

C. Christofides, F. Diakonos, A. Seas, C. Christou, M. Nestoros, A. Mandelis, “Two-layer model for photomodulated thermoreflectance of semiconductor wafers,” J. Appl. Phys. 80, 1713–1725 (1996).
[CrossRef]

Christou, C.

C. Christofides, F. Diakonos, A. Seas, C. Christou, M. Nestoros, A. Mandelis, “Two-layer model for photomodulated thermoreflectance of semiconductor wafers,” J. Appl. Phys. 80, 1713–1725 (1996).
[CrossRef]

Dargys, A.

A. Dargys, J. Kundroats, Handbook on Physical Properties of Ge, Si, GaAs, and InP (Science and Encyclopedia Publishers, Vilnius, Lithuania, 1994).

Diakonos, F.

C. Christofides, F. Diakonos, A. Seas, C. Christou, M. Nestoros, A. Mandelis, “Two-layer model for photomodulated thermoreflectance of semiconductor wafers,” J. Appl. Phys. 80, 1713–1725 (1996).
[CrossRef]

Favro, L. D.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

Favro, L. O.

K. R. Grice, L. J. Inglehart, L. O. Favro, P. K. Kuo, R. L. Thomas, “Thermal wave imaging of closed cracks in opaque solids,” J. Appl. Phys. 54, 6245–6255 (1983).
[CrossRef]

Fernandez, J.

A. Salazar, A. S. Lavega, J. Fernandez, “Thermal-diffusivity measurements in solids by the mirage technique—experimental results,” J. Appl. Phys. 69, 1216–1223 (1991).
[CrossRef]

Fournier, D.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

D. Fournier, C. Boccara, A. Skumanich, N. M. Amer, “Photothermal investigation of transport in semiconductors: theory and experiment,” J. Appl. Phys. 59, 787–795 (1986).
[CrossRef]

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333–1344 (1981).
[CrossRef] [PubMed]

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the ‘mirage effect,’” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Goodson, K. E.

Y. S. Ju, K. E. Goodson, “Phonon scattering in silicon films with thickness of order 100 nm,” Appl. Phys. Lett. 74, 3005–3007 (1999).
[CrossRef]

M. Asheghi, Y. K. Leung, S. S. Wong, K. E. Goodson, “Phonon-boundary scattering in thin silicon layers,” Appl. Phys. Lett. 71, 1798–1800 (1997).
[CrossRef]

Grice, K. R.

K. R. Grice, L. J. Inglehart, L. O. Favro, P. K. Kuo, R. L. Thomas, “Thermal wave imaging of closed cracks in opaque solids,” J. Appl. Phys. 54, 6245–6255 (1983).
[CrossRef]

Gupta, R.

M. Soltanolkotabi, G. L. Bennis, R. Gupta, “Temperature dependence of the thermal diffusivity of GaAs in the 100–305 K range measured by the pulsed photothermal displacement technique,” J. Appl. Phys. 85, 794–798 (1999).
[CrossRef]

Inglehart, L. J.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

K. R. Grice, L. J. Inglehart, L. O. Favro, P. K. Kuo, R. L. Thomas, “Thermal wave imaging of closed cracks in opaque solids,” J. Appl. Phys. 54, 6245–6255 (1983).
[CrossRef]

Jackson, W. B.

Ju, Y. S.

Y. S. Ju, K. E. Goodson, “Phonon scattering in silicon films with thickness of order 100 nm,” Appl. Phys. Lett. 74, 3005–3007 (1999).
[CrossRef]

Kim, D. S.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

Kundroats, J.

A. Dargys, J. Kundroats, Handbook on Physical Properties of Ge, Si, GaAs, and InP (Science and Encyclopedia Publishers, Vilnius, Lithuania, 1994).

Kuo, P. K.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

K. R. Grice, L. J. Inglehart, L. O. Favro, P. K. Kuo, R. L. Thomas, “Thermal wave imaging of closed cracks in opaque solids,” J. Appl. Phys. 54, 6245–6255 (1983).
[CrossRef]

Lavega, A. S.

A. Salazar, A. S. Lavega, “Thermal-diffusivity measurements using linear relations from photothermal wave experiments,” Rev. Sci. Instrum. 65, 2896–2900 (1994).
[CrossRef]

A. Salazar, A. S. Lavega, J. Fernandez, “Thermal-diffusivity measurements in solids by the mirage technique—experimental results,” J. Appl. Phys. 69, 1216–1223 (1991).
[CrossRef]

Leung, Y. K.

M. Asheghi, Y. K. Leung, S. S. Wong, K. E. Goodson, “Phonon-boundary scattering in thin silicon layers,” Appl. Phys. Lett. 71, 1798–1800 (1997).
[CrossRef]

Liakhou, G.

M. Bertolotti, R. L. Voti, G. Liakhou, C. Sibilia, “On the photodeflection method applied to low thermal-diffusivity measurements,” Rev. Sci. Instrum. 64, 1576–1583 (1993).
[CrossRef]

Liakhou, G. L.

M. Bertolotti, G. L. Liakhou, R. L. Voti, S. Paolini, C. Sibilia, “Analysis of the photothermal deflection technique in the surface reflection scheme: theory and experiment,” J. Appl. Phys. 83, 966–982 (1998).
[CrossRef]

Lin, M. J.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

Mandelis, A.

C. Wang, A. Mandelis, “Purely thermal-wave photopyroelectric interferometry,” J. Appl. Phys. 85, 8366–8377 (1999).
[CrossRef]

R. E. Wagner, A. Mandelis, “Nonlinear photothermal modulated optical reflectance and photocurrent phenomena in crystalline semiconductors: theoretical,” Semicond. Sci. Technol. 11, 289–299 (1996).
[CrossRef]

C. Christofides, F. Diakonos, A. Seas, C. Christou, M. Nestoros, A. Mandelis, “Two-layer model for photomodulated thermoreflectance of semiconductor wafers,” J. Appl. Phys. 80, 1713–1725 (1996).
[CrossRef]

A. Mandelis, B. S. H. Royce, “Fundamental-mode laser-beam propagation in optically inhomogeneous electrochemical media with chemical species concentration gradients,” Appl. Opt. 23, 2892–2901 (1984).
[CrossRef] [PubMed]

Murphy, J. C.

L. C. Aamodt, J. C. Murphy, “Photothermal measurements using a localized excitation source,” J. Appl. Phys. 52, 4903–4914 (1981).
[CrossRef]

Nestoros, M.

C. Christofides, F. Diakonos, A. Seas, C. Christou, M. Nestoros, A. Mandelis, “Two-layer model for photomodulated thermoreflectance of semiconductor wafers,” J. Appl. Phys. 80, 1713–1725 (1996).
[CrossRef]

Paolini, S.

M. Bertolotti, G. L. Liakhou, R. L. Voti, S. Paolini, C. Sibilia, “Analysis of the photothermal deflection technique in the surface reflection scheme: theory and experiment,” J. Appl. Phys. 83, 966–982 (1998).
[CrossRef]

Reyes, C. B.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

Rowe, D. M.

C. M. Bhandari, D. M. Rowe, Thermal Conduction in Semiconductors (Wiley, New York, 1988).

Royce, B. S. H.

Salazar, A.

A. Salazar, A. S. Lavega, “Thermal-diffusivity measurements using linear relations from photothermal wave experiments,” Rev. Sci. Instrum. 65, 2896–2900 (1994).
[CrossRef]

A. Salazar, A. S. Lavega, J. Fernandez, “Thermal-diffusivity measurements in solids by the mirage technique—experimental results,” J. Appl. Phys. 69, 1216–1223 (1991).
[CrossRef]

Seas, A.

C. Christofides, F. Diakonos, A. Seas, C. Christou, M. Nestoros, A. Mandelis, “Two-layer model for photomodulated thermoreflectance of semiconductor wafers,” J. Appl. Phys. 80, 1713–1725 (1996).
[CrossRef]

Sell, J. A.

J. A. Sell, Photothermal Investigations of Solids and Fluids (Academic, Boston, Mass., 1989).

Sibilia, C.

M. Bertolotti, G. L. Liakhou, R. L. Voti, S. Paolini, C. Sibilia, “Analysis of the photothermal deflection technique in the surface reflection scheme: theory and experiment,” J. Appl. Phys. 83, 966–982 (1998).
[CrossRef]

M. Bertolotti, R. L. Voti, G. Liakhou, C. Sibilia, “On the photodeflection method applied to low thermal-diffusivity measurements,” Rev. Sci. Instrum. 64, 1576–1583 (1993).
[CrossRef]

Skumanich, A.

D. Fournier, C. Boccara, A. Skumanich, N. M. Amer, “Photothermal investigation of transport in semiconductors: theory and experiment,” J. Appl. Phys. 59, 787–795 (1986).
[CrossRef]

Smith, J. S.

G. Chen, C. L. Tien, X. Wu, J. S. Smith, “Thermal-diffusivity measurement of GaAs/AlGaAs thin-film structures,” J. Heat. Transfer 116, 325–331 (1994).
[CrossRef]

Soltanolkotabi, M.

M. Soltanolkotabi, G. L. Bennis, R. Gupta, “Temperature dependence of the thermal diffusivity of GaAs in the 100–305 K range measured by the pulsed photothermal displacement technique,” J. Appl. Phys. 85, 794–798 (1999).
[CrossRef]

Thomas, R. L.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
[CrossRef]

K. R. Grice, L. J. Inglehart, L. O. Favro, P. K. Kuo, R. L. Thomas, “Thermal wave imaging of closed cracks in opaque solids,” J. Appl. Phys. 54, 6245–6255 (1983).
[CrossRef]

Tien, C. L.

G. Chen, C. L. Tien, X. Wu, J. S. Smith, “Thermal-diffusivity measurement of GaAs/AlGaAs thin-film structures,” J. Heat. Transfer 116, 325–331 (1994).
[CrossRef]

Voti, R. L.

M. Bertolotti, G. L. Liakhou, R. L. Voti, S. Paolini, C. Sibilia, “Analysis of the photothermal deflection technique in the surface reflection scheme: theory and experiment,” J. Appl. Phys. 83, 966–982 (1998).
[CrossRef]

M. Bertolotti, R. L. Voti, G. Liakhou, C. Sibilia, “On the photodeflection method applied to low thermal-diffusivity measurements,” Rev. Sci. Instrum. 64, 1576–1583 (1993).
[CrossRef]

Wagner, R. E.

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P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, A. C. Boccara, N. Yacoubi, “Mirage effect measurement of thermal diffusivity. I. Experimental,” Can. J. Phys. 64, 1165–1167 (1986).
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[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the probe-beam skimming PTD configuration; y, z, the transverse and the vertical offsets, respectively.

Fig. 2
Fig. 2

Schematic view of the experimental setup: M, mirror; C, chopper; L1, L2, lenses; Q, cuvette; S, sample; OF, optical fiber; MC, monochromator; PMT, photomultiplier tube.

Fig. 3
Fig. 3

Variation of PTD signal phase with pump–probe offset for sample 1 (substrate-side illumination).

Fig. 4
Fig. 4

Variation of PTD signal phase with pump–probe offset for sample 1 (film-side illumination).

Tables (2)

Tables Icon

Table 1 Structure, Properties, and Growth Conditions of the Doped GaAs Epitaxial Layers upon the Semi-Insulating GaAs Substrate

Tables Icon

Table 2 Thermal Diffusivity Values of GaAs Multilayers Evaluated from the Photothermal Beam Deflection Measurements

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

ddsnodrods=nr, t,
drods=1nopath nr, tds,
drods=ϕt=1nonTpath Tr, tds,
ϕt=-1πndndT0sinδyA exp-β0zδdδ×expjωt, z>0,
ϕn=-1πndndT0cosδyA exp-β0zβ0dδ×expjωt z>0,
m=1μs=πf/αs1/2.
Λbulk=3kbulk/Cν,

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