Abstract

We present a practical experimental design for performing photoluminescence (PL) and photoreflectance (PR) measurements of semiconductors with only one PL spectroscopic system. The measurement setup is more cost efficient than typical PL-plus-PR systems. The design of the experimental setup of the PL–PR system is described in detail. Measurements of two actual device structures, a high-electron-mobility transistor (HEMT) and a double heterojunction-bipolar transistor (DHBT), are carried out by using this design. The experimental PL and PR spectra of the HEMT device, as well as polarized-photoreflectance (PPR) spectra of the DHBT structure, are analyzed in detailed and discussed. The experimental analyses demonstrate the well-behaved performance of this PL–PR design.

© 2005 Optical Society of America

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References

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  1. P. Bhattacharya, “Elemental and compound semiconductors,” in Semiconductor Optoelectronic Devices, 2nd ed. (Prentice-Hall, London, 1997), Chap. 1, pp. 2–58.
  2. D. K. Schroder, “Optical characterization,” in Semiconductor Material and Device Characterization1st ed. (Wiley, New York, 1990), Chap. 9, pp. 490–494
  3. M. Cardona, Modulation Spectroscopy (Academic, New York, 1969).
  4. F. H. Pollak and H. Shen, “Modulation spectroscopy of semiconductors: bulk/thin film, microstructures, surfaces/interfaces and devices,” Mater. Sci. Eng. R10, 275–374 (1993).
  5. D. E. Aspnes, “Modulation spectroscopy/electric field effects on the dielectric function of semiconductors,” in Handbook on Semiconductors, Vol. 2., M. Balkanski, ed. (North Holland, New York, 1980), p. 109.
  6. C. H. Ho, “Optical study of the structural change in ReS2 single crystals using polarized thermoreflectance spectroscopy,” Opt. Express 13, 8–19 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-8.
    [Crossref] [PubMed]
  7. C. H. Ho, P. C. Yen, Y. S. Huang, and K. K. Tiong, “Photoreflectance study of the excitonic transitions of rhenium disulphide layer compounds,” Phys. Rev. B 66, 245207 (2002).
    [Crossref]
  8. D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
    [Crossref]
  9. W. C. Hsu, C. M. Chen, and R. T. Hsu, “A δ-doped GaAs/graded InxGa1-xAs pseudomorphic structure grown by low-pressure metal organic chemical vapor deposition,” Appl. Phys. Lett. 59, 1075–1077 (1991).
    [Crossref]
  10. D. Y. Lin, S. H. Liang, Y. S. Huang, K. K. Tiong, F. H. Pollak, and K. R. Evans, “Room-temperature photoreflectance and photoluminescence characterization of the AlGaAs/InGaAs/GaAs pseudomorphic high electron mobility transistor structure with varied quantum well compositional profiles,” J. Appl. Phys. 85, 8235–8241 (1999).
    [Crossref]
  11. Y. Yin, H. Qiang, F. H. Pollak, D. C. Streit, and M. Wojtowicz, “Two-dimensional electron gas effects in the electromodulation spectra of a pseudomorphic Ga0.78Al0.22As/In0.21Ga0.79As/GaAs modulation-doped quantum well structure,” Appl. Phys. Lett. 61, 1579–1581 (1992).
    [Crossref]
  12. H. Shen and F. H. Pollak, “Generalized Franz-Keldysh theory of electromodulation,” Phys. Rev. B 42, 7097–7102 (1990).
    [Crossref]
  13. C. J. Lin, Y. S. Huang, N. Y. Li, P. W. Li, and K. K. Tiong, “Polarized-photoreflectance characterization of an InGaP/InGaAsN/GaAs NpN double-heterojunction bipolar transistor structure,” J. Appl. Phys. 90, 4565–4569 (2001).
    [Crossref]
  14. A. Lindell, M. Pessa, A. Salokatve, F. Bernardini, and M. Paalanen, “Band offsets at the GaInP/GaAs heterojunction,” J. Appl. Phys. 82, 3374–3380 (1997).
    [Crossref]

2005 (1)

2002 (1)

C. H. Ho, P. C. Yen, Y. S. Huang, and K. K. Tiong, “Photoreflectance study of the excitonic transitions of rhenium disulphide layer compounds,” Phys. Rev. B 66, 245207 (2002).
[Crossref]

2001 (1)

C. J. Lin, Y. S. Huang, N. Y. Li, P. W. Li, and K. K. Tiong, “Polarized-photoreflectance characterization of an InGaP/InGaAsN/GaAs NpN double-heterojunction bipolar transistor structure,” J. Appl. Phys. 90, 4565–4569 (2001).
[Crossref]

1999 (1)

D. Y. Lin, S. H. Liang, Y. S. Huang, K. K. Tiong, F. H. Pollak, and K. R. Evans, “Room-temperature photoreflectance and photoluminescence characterization of the AlGaAs/InGaAs/GaAs pseudomorphic high electron mobility transistor structure with varied quantum well compositional profiles,” J. Appl. Phys. 85, 8235–8241 (1999).
[Crossref]

1997 (1)

A. Lindell, M. Pessa, A. Salokatve, F. Bernardini, and M. Paalanen, “Band offsets at the GaInP/GaAs heterojunction,” J. Appl. Phys. 82, 3374–3380 (1997).
[Crossref]

1996 (1)

D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
[Crossref]

1993 (1)

F. H. Pollak and H. Shen, “Modulation spectroscopy of semiconductors: bulk/thin film, microstructures, surfaces/interfaces and devices,” Mater. Sci. Eng. R10, 275–374 (1993).

1992 (1)

Y. Yin, H. Qiang, F. H. Pollak, D. C. Streit, and M. Wojtowicz, “Two-dimensional electron gas effects in the electromodulation spectra of a pseudomorphic Ga0.78Al0.22As/In0.21Ga0.79As/GaAs modulation-doped quantum well structure,” Appl. Phys. Lett. 61, 1579–1581 (1992).
[Crossref]

1991 (1)

W. C. Hsu, C. M. Chen, and R. T. Hsu, “A δ-doped GaAs/graded InxGa1-xAs pseudomorphic structure grown by low-pressure metal organic chemical vapor deposition,” Appl. Phys. Lett. 59, 1075–1077 (1991).
[Crossref]

1990 (1)

H. Shen and F. H. Pollak, “Generalized Franz-Keldysh theory of electromodulation,” Phys. Rev. B 42, 7097–7102 (1990).
[Crossref]

Aspnes, D. E.

D. E. Aspnes, “Modulation spectroscopy/electric field effects on the dielectric function of semiconductors,” in Handbook on Semiconductors, Vol. 2., M. Balkanski, ed. (North Holland, New York, 1980), p. 109.

Bernardini, F.

A. Lindell, M. Pessa, A. Salokatve, F. Bernardini, and M. Paalanen, “Band offsets at the GaInP/GaAs heterojunction,” J. Appl. Phys. 82, 3374–3380 (1997).
[Crossref]

Bhattacharya, P.

P. Bhattacharya, “Elemental and compound semiconductors,” in Semiconductor Optoelectronic Devices, 2nd ed. (Prentice-Hall, London, 1997), Chap. 1, pp. 2–58.

Cardona, M.

M. Cardona, Modulation Spectroscopy (Academic, New York, 1969).

Chen, C. M.

W. C. Hsu, C. M. Chen, and R. T. Hsu, “A δ-doped GaAs/graded InxGa1-xAs pseudomorphic structure grown by low-pressure metal organic chemical vapor deposition,” Appl. Phys. Lett. 59, 1075–1077 (1991).
[Crossref]

Evans, K. R.

D. Y. Lin, S. H. Liang, Y. S. Huang, K. K. Tiong, F. H. Pollak, and K. R. Evans, “Room-temperature photoreflectance and photoluminescence characterization of the AlGaAs/InGaAs/GaAs pseudomorphic high electron mobility transistor structure with varied quantum well compositional profiles,” J. Appl. Phys. 85, 8235–8241 (1999).
[Crossref]

Ho, C. H.

C. H. Ho, “Optical study of the structural change in ReS2 single crystals using polarized thermoreflectance spectroscopy,” Opt. Express 13, 8–19 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-8.
[Crossref] [PubMed]

C. H. Ho, P. C. Yen, Y. S. Huang, and K. K. Tiong, “Photoreflectance study of the excitonic transitions of rhenium disulphide layer compounds,” Phys. Rev. B 66, 245207 (2002).
[Crossref]

Hsu, R. T.

W. C. Hsu, C. M. Chen, and R. T. Hsu, “A δ-doped GaAs/graded InxGa1-xAs pseudomorphic structure grown by low-pressure metal organic chemical vapor deposition,” Appl. Phys. Lett. 59, 1075–1077 (1991).
[Crossref]

Hsu, W. C.

W. C. Hsu, C. M. Chen, and R. T. Hsu, “A δ-doped GaAs/graded InxGa1-xAs pseudomorphic structure grown by low-pressure metal organic chemical vapor deposition,” Appl. Phys. Lett. 59, 1075–1077 (1991).
[Crossref]

Huang, Y. S.

C. H. Ho, P. C. Yen, Y. S. Huang, and K. K. Tiong, “Photoreflectance study of the excitonic transitions of rhenium disulphide layer compounds,” Phys. Rev. B 66, 245207 (2002).
[Crossref]

C. J. Lin, Y. S. Huang, N. Y. Li, P. W. Li, and K. K. Tiong, “Polarized-photoreflectance characterization of an InGaP/InGaAsN/GaAs NpN double-heterojunction bipolar transistor structure,” J. Appl. Phys. 90, 4565–4569 (2001).
[Crossref]

D. Y. Lin, S. H. Liang, Y. S. Huang, K. K. Tiong, F. H. Pollak, and K. R. Evans, “Room-temperature photoreflectance and photoluminescence characterization of the AlGaAs/InGaAs/GaAs pseudomorphic high electron mobility transistor structure with varied quantum well compositional profiles,” J. Appl. Phys. 85, 8235–8241 (1999).
[Crossref]

D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
[Crossref]

Li, N. Y.

C. J. Lin, Y. S. Huang, N. Y. Li, P. W. Li, and K. K. Tiong, “Polarized-photoreflectance characterization of an InGaP/InGaAsN/GaAs NpN double-heterojunction bipolar transistor structure,” J. Appl. Phys. 90, 4565–4569 (2001).
[Crossref]

Li, P. W.

C. J. Lin, Y. S. Huang, N. Y. Li, P. W. Li, and K. K. Tiong, “Polarized-photoreflectance characterization of an InGaP/InGaAsN/GaAs NpN double-heterojunction bipolar transistor structure,” J. Appl. Phys. 90, 4565–4569 (2001).
[Crossref]

Liang, S. H.

D. Y. Lin, S. H. Liang, Y. S. Huang, K. K. Tiong, F. H. Pollak, and K. R. Evans, “Room-temperature photoreflectance and photoluminescence characterization of the AlGaAs/InGaAs/GaAs pseudomorphic high electron mobility transistor structure with varied quantum well compositional profiles,” J. Appl. Phys. 85, 8235–8241 (1999).
[Crossref]

Lin, C. J.

C. J. Lin, Y. S. Huang, N. Y. Li, P. W. Li, and K. K. Tiong, “Polarized-photoreflectance characterization of an InGaP/InGaAsN/GaAs NpN double-heterojunction bipolar transistor structure,” J. Appl. Phys. 90, 4565–4569 (2001).
[Crossref]

Lin, D. Y.

D. Y. Lin, S. H. Liang, Y. S. Huang, K. K. Tiong, F. H. Pollak, and K. R. Evans, “Room-temperature photoreflectance and photoluminescence characterization of the AlGaAs/InGaAs/GaAs pseudomorphic high electron mobility transistor structure with varied quantum well compositional profiles,” J. Appl. Phys. 85, 8235–8241 (1999).
[Crossref]

D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
[Crossref]

Lin, F. C.

D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
[Crossref]

Lindell, A.

A. Lindell, M. Pessa, A. Salokatve, F. Bernardini, and M. Paalanen, “Band offsets at the GaInP/GaAs heterojunction,” J. Appl. Phys. 82, 3374–3380 (1997).
[Crossref]

Maracas, G. N.

D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
[Crossref]

Mathine, D. L.

D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
[Crossref]

Paalanen, M.

A. Lindell, M. Pessa, A. Salokatve, F. Bernardini, and M. Paalanen, “Band offsets at the GaInP/GaAs heterojunction,” J. Appl. Phys. 82, 3374–3380 (1997).
[Crossref]

Pessa, M.

A. Lindell, M. Pessa, A. Salokatve, F. Bernardini, and M. Paalanen, “Band offsets at the GaInP/GaAs heterojunction,” J. Appl. Phys. 82, 3374–3380 (1997).
[Crossref]

Pollak, F. H.

D. Y. Lin, S. H. Liang, Y. S. Huang, K. K. Tiong, F. H. Pollak, and K. R. Evans, “Room-temperature photoreflectance and photoluminescence characterization of the AlGaAs/InGaAs/GaAs pseudomorphic high electron mobility transistor structure with varied quantum well compositional profiles,” J. Appl. Phys. 85, 8235–8241 (1999).
[Crossref]

D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
[Crossref]

F. H. Pollak and H. Shen, “Modulation spectroscopy of semiconductors: bulk/thin film, microstructures, surfaces/interfaces and devices,” Mater. Sci. Eng. R10, 275–374 (1993).

Y. Yin, H. Qiang, F. H. Pollak, D. C. Streit, and M. Wojtowicz, “Two-dimensional electron gas effects in the electromodulation spectra of a pseudomorphic Ga0.78Al0.22As/In0.21Ga0.79As/GaAs modulation-doped quantum well structure,” Appl. Phys. Lett. 61, 1579–1581 (1992).
[Crossref]

H. Shen and F. H. Pollak, “Generalized Franz-Keldysh theory of electromodulation,” Phys. Rev. B 42, 7097–7102 (1990).
[Crossref]

Qiang, H.

D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
[Crossref]

Y. Yin, H. Qiang, F. H. Pollak, D. C. Streit, and M. Wojtowicz, “Two-dimensional electron gas effects in the electromodulation spectra of a pseudomorphic Ga0.78Al0.22As/In0.21Ga0.79As/GaAs modulation-doped quantum well structure,” Appl. Phys. Lett. 61, 1579–1581 (1992).
[Crossref]

Salokatve, A.

A. Lindell, M. Pessa, A. Salokatve, F. Bernardini, and M. Paalanen, “Band offsets at the GaInP/GaAs heterojunction,” J. Appl. Phys. 82, 3374–3380 (1997).
[Crossref]

Schroder, D. K.

D. K. Schroder, “Optical characterization,” in Semiconductor Material and Device Characterization1st ed. (Wiley, New York, 1990), Chap. 9, pp. 490–494

Shen, H.

F. H. Pollak and H. Shen, “Modulation spectroscopy of semiconductors: bulk/thin film, microstructures, surfaces/interfaces and devices,” Mater. Sci. Eng. R10, 275–374 (1993).

H. Shen and F. H. Pollak, “Generalized Franz-Keldysh theory of electromodulation,” Phys. Rev. B 42, 7097–7102 (1990).
[Crossref]

Streit, D. C.

Y. Yin, H. Qiang, F. H. Pollak, D. C. Streit, and M. Wojtowicz, “Two-dimensional electron gas effects in the electromodulation spectra of a pseudomorphic Ga0.78Al0.22As/In0.21Ga0.79As/GaAs modulation-doped quantum well structure,” Appl. Phys. Lett. 61, 1579–1581 (1992).
[Crossref]

Tiong, K. K.

C. H. Ho, P. C. Yen, Y. S. Huang, and K. K. Tiong, “Photoreflectance study of the excitonic transitions of rhenium disulphide layer compounds,” Phys. Rev. B 66, 245207 (2002).
[Crossref]

C. J. Lin, Y. S. Huang, N. Y. Li, P. W. Li, and K. K. Tiong, “Polarized-photoreflectance characterization of an InGaP/InGaAsN/GaAs NpN double-heterojunction bipolar transistor structure,” J. Appl. Phys. 90, 4565–4569 (2001).
[Crossref]

D. Y. Lin, S. H. Liang, Y. S. Huang, K. K. Tiong, F. H. Pollak, and K. R. Evans, “Room-temperature photoreflectance and photoluminescence characterization of the AlGaAs/InGaAs/GaAs pseudomorphic high electron mobility transistor structure with varied quantum well compositional profiles,” J. Appl. Phys. 85, 8235–8241 (1999).
[Crossref]

Wojtowicz, M.

Y. Yin, H. Qiang, F. H. Pollak, D. C. Streit, and M. Wojtowicz, “Two-dimensional electron gas effects in the electromodulation spectra of a pseudomorphic Ga0.78Al0.22As/In0.21Ga0.79As/GaAs modulation-doped quantum well structure,” Appl. Phys. Lett. 61, 1579–1581 (1992).
[Crossref]

Yen, P. C.

C. H. Ho, P. C. Yen, Y. S. Huang, and K. K. Tiong, “Photoreflectance study of the excitonic transitions of rhenium disulphide layer compounds,” Phys. Rev. B 66, 245207 (2002).
[Crossref]

Yin, Y.

Y. Yin, H. Qiang, F. H. Pollak, D. C. Streit, and M. Wojtowicz, “Two-dimensional electron gas effects in the electromodulation spectra of a pseudomorphic Ga0.78Al0.22As/In0.21Ga0.79As/GaAs modulation-doped quantum well structure,” Appl. Phys. Lett. 61, 1579–1581 (1992).
[Crossref]

Appl. Phys. Lett. (2)

W. C. Hsu, C. M. Chen, and R. T. Hsu, “A δ-doped GaAs/graded InxGa1-xAs pseudomorphic structure grown by low-pressure metal organic chemical vapor deposition,” Appl. Phys. Lett. 59, 1075–1077 (1991).
[Crossref]

Y. Yin, H. Qiang, F. H. Pollak, D. C. Streit, and M. Wojtowicz, “Two-dimensional electron gas effects in the electromodulation spectra of a pseudomorphic Ga0.78Al0.22As/In0.21Ga0.79As/GaAs modulation-doped quantum well structure,” Appl. Phys. Lett. 61, 1579–1581 (1992).
[Crossref]

J. Appl. Phys. (4)

C. J. Lin, Y. S. Huang, N. Y. Li, P. W. Li, and K. K. Tiong, “Polarized-photoreflectance characterization of an InGaP/InGaAsN/GaAs NpN double-heterojunction bipolar transistor structure,” J. Appl. Phys. 90, 4565–4569 (2001).
[Crossref]

A. Lindell, M. Pessa, A. Salokatve, F. Bernardini, and M. Paalanen, “Band offsets at the GaInP/GaAs heterojunction,” J. Appl. Phys. 82, 3374–3380 (1997).
[Crossref]

D. Y. Lin, F. C. Lin, Y. S. Huang, H. Qiang, F. H. Pollak, D. L. Mathine, and G. N. Maracas, “Piezoreflectance and photoreflectance study of GaAs/AlGaAs digital alloy compositional graded structures” J. Appl. Phys. 79, 460–466 (1996).
[Crossref]

D. Y. Lin, S. H. Liang, Y. S. Huang, K. K. Tiong, F. H. Pollak, and K. R. Evans, “Room-temperature photoreflectance and photoluminescence characterization of the AlGaAs/InGaAs/GaAs pseudomorphic high electron mobility transistor structure with varied quantum well compositional profiles,” J. Appl. Phys. 85, 8235–8241 (1999).
[Crossref]

Mater. Sci. Eng. (1)

F. H. Pollak and H. Shen, “Modulation spectroscopy of semiconductors: bulk/thin film, microstructures, surfaces/interfaces and devices,” Mater. Sci. Eng. R10, 275–374 (1993).

Opt. Express (1)

Phys. Rev. B (2)

C. H. Ho, P. C. Yen, Y. S. Huang, and K. K. Tiong, “Photoreflectance study of the excitonic transitions of rhenium disulphide layer compounds,” Phys. Rev. B 66, 245207 (2002).
[Crossref]

H. Shen and F. H. Pollak, “Generalized Franz-Keldysh theory of electromodulation,” Phys. Rev. B 42, 7097–7102 (1990).
[Crossref]

Other (4)

D. E. Aspnes, “Modulation spectroscopy/electric field effects on the dielectric function of semiconductors,” in Handbook on Semiconductors, Vol. 2., M. Balkanski, ed. (North Holland, New York, 1980), p. 109.

P. Bhattacharya, “Elemental and compound semiconductors,” in Semiconductor Optoelectronic Devices, 2nd ed. (Prentice-Hall, London, 1997), Chap. 1, pp. 2–58.

D. K. Schroder, “Optical characterization,” in Semiconductor Material and Device Characterization1st ed. (Wiley, New York, 1990), Chap. 9, pp. 490–494

M. Cardona, Modulation Spectroscopy (Academic, New York, 1969).

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Figures (5)

Fig. 1.
Fig. 1.

Experimental setup of the PL–PR system used for PL and PR measurements of semiconductors.

Fig. 2.
Fig. 2.

Representative energy band diagram of a selective sample of InxGa1−xAs/GaAs graded-channel HEMT.

Fig. 3.
Fig. 3.

Experimental PR and PL spectra of the InxGa1−xAs/GaAs graded-channel HEMT device.

Fig. 4.
Fig. 4.

Structure of the device’s epilayers for an InGaP/InGaAsN/GaAs DHBT sample.

Fig. 5.
Fig. 5.

PPR spectra of the InGaP/InGaAsN/GaAs DHBT device with E‖[110] and E‖[11̄0] polarizations.

Tables (2)

Tables Icon

Table 1. Specifications and Measurement Conditions of the Testing Samples Used in the PL and PR Experiments

Tables Icon

Table 2. Experimental Values of the Intersubband Transition Energies from the InxGa1-xAs Graded-Channel Layer of a HEMT Structure with 2D Sheet Density Ns Obtained by PR Measurement

Equations (10)

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

Δ R R = α ( ε 1 , ε 2 ) Δ ε 1 + β ( ε 1 , ε 2 ) Δ ε 2 ,
ε 2 = j D j { Im [ ln ( E E j ( m n ) + i Γ j ) ] } · ( 1 f e j ) ,
f e j = { 1 + exp [ ( ( λ E λ E j ( m n ) E ¯ j ( m ) ) k T ] } 1 ,
E ¯ j ( m ) = E F E m , j C ,
λ = m h * m e * + m h * ,
N s = 0 ρ 2 D ( E , Γ ) · { exp [ ( E E F ) k T ] + 1 } 1 d E ,
ρ 2 D ( E , Γ ) = ( m e * / π 2 ) m { 1 2 + ( 1 π ) tan 1 [ ( E E m C ) Γ m ] } .
n π = ( 4 / 3 ) [ ( E n E 0 ) Θ ] 3 2 + χ ,
( Θ ) 3 = q 2 2 F 2 2 μ ,
1 μ = 1 m e * + 1 m h * .

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