Abstract

We describe a microreflectance difference (μRD) spectrometer based on a charge coupled device (CCD), in contrast to most common RD spectrometers that are based on a photomultiplier or a photodiode as a light detector. The advantage of our instrument over others is the possibility to isolate the RD spectrum of specific areas of the sample; thus topographic maps of the surface can be obtained. In our setup we have a maximum spatial resolution of approximately 2.50μm×2.50μm and a spectral range from 1.2 to 5.5eV. To illustrate the performance of the spectrometer, we have measured strains in mechanically polished GaAs (001) single crystals.

© 2009 Optical Society of America

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  1. D. E. Aspnes and A. A. Studna, “Anisotropies in the above-band-gap optical spectra of cubic semiconductors,” Phys. Rev. Lett. 54, 1956-1959 (1985).
    [CrossRef] [PubMed]
  2. P. Weightman, D. S. Martin, R. J. Cole, and T. Farrell, “Reflection anisotropy spectroscopy,” Rep. Prog. Phys. 68, 1251-1341(2005).
    [CrossRef]
  3. G. E. Isted, P. D. Lane, and R. J. Cole, “Effect of thermally induced surface defects on the optical anisotropy of Ag(110),” Phys. Rev. B 79, 205424 (2009).
    [CrossRef]
  4. L. F. Lastras-Martínez and A. Lastras-Martínez, “Dislocation-induced effects in the reflectance-difference spectrum of semi-insulating GaAs (100),” Solid State Commun. 98, 479-483(1996).
    [CrossRef]
  5. L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance anisotropy of GaAs(100): Dislocation-induced piezo-optic effects,” Phys. Rev. B 54, 10726-10735 (1996).
    [CrossRef]
  6. Y. H. Chen, Z. G. Wang, J. J. Qian, and Z. Yang, “Polishing-related optical anisotropy of semi-insulating GaAs studied by reflectance difference spectroscopy,” J. Appl. Phys. 88, 1695-1697 (2000).
    [CrossRef]
  7. L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance-difference spectroscopy of semi-insulating GaAs (110) around the fundamental gap,” Phys. Rev. B 64, 085309 (2001).
    [CrossRef]
  8. A. Lastras-Martínez, R. E. Balderas-Navarro, and L. F. Lastras-Martínez, “Linear electro-optic reflectance modulated spectra of GaAs (001) around E1 and E11,” Thin Solid Films 373, 207-210 (2000).
    [CrossRef]
  9. L. F. Lastras-Martínez, J. M. Flores-Camacho, R. E. Balderas-Navarro, M. Chavira-Rodríguez, A. Lastras-Martínez, and M. Cardona, “Effect of reconstruction-induced strain on the reflectance difference spectroscopy of GaAs (001) around E1 and E11 transitions,” Phys. Rev. B 75, 235315(2007).
    [CrossRef]
  10. D. Rönnow, L. F. Lastras-Martínez, M. Cardona, and P. V. Santos, “Determination of the piezo-optical properties of semiconductors above the fundamental gap by means of reflectance difference spectroscopy,” J. Opt. Soc. Am. A 16, 568-573 (1999).
    [CrossRef]
  11. L. F. Lastras-Martínez, R. E. Balderas-Navarro, M. Chavira-Rodríguez, J. M. Flores-Camacho, and A. Lastras-Martínez, “Strain induced optical anisotropies in zinc blende semiconductor,” Phys. Status Solidi B 240, 500-508 (2003).
    [CrossRef]
  12. L. F. Lastras-Martínez, M. Chavira-Rodríguez, R. E. Balderas-Navarro, J. M. Flores-Camacho, and A. Lastras-Martínez, “Reflectance difference spectroscopy of GaAs (001) under a [110] uniaxial stress,” Phys. Rev. B 70, 035306 (2004).
    [CrossRef]
  13. A. Lastras-Martínez, I. Lara-Velázquez, R. E. Balderas-Navarro, J. Ortega-Gallegos, S. Guel-Sandoval, and L. F. Lastras-Martínez, “Reflectance-difference spectroscopy as an optical probe for in situ determination of doping levels in GaAs,” Phys. Status Solidi C 5, 2565-2568 (2008).
    [CrossRef]
  14. K. Sawano, S. Koh, Y. Shiraki, N. Usami, and K. Nakagawa, “In-plane strain fluctuation in strained-Si/SiGe heterostructures,” Appl. Phys. Lett. 83, 4339-4341 (2003).
    [CrossRef]
  15. P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
    [CrossRef]
  16. G. Irmer and M. Jurisch, “Micro-Raman study of strain fields around dislocations in GaAs,” Phys. Status Solidi A 204, 2309-2318 (2007).
    [CrossRef]
  17. N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
    [CrossRef]
  18. N. Gmeinwieser and U. T. Schwarz, “Strain of single edge dislocations in bulk GaN,” Phys. Status Solidi B 244, 1857-1861 (2007).
    [CrossRef]
  19. B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona, “In-plane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy,” Appl. Phys. Lett. 69, 782-784 (1996).
    [CrossRef]
  20. M. Yamada, “High-sensitivity computer-controlled infrared polariscope,” Rev. Sci. Instrum. 64, 1815-1821 (1993).
    [CrossRef]
  21. H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
    [CrossRef]
  22. L. F. Lastras-Martínez, R. Castro-Garca, R. E. Balderas-Navarro, and A. Lastras-Martínez, “Reflectance difference spectrometer based on the use of a CCD camera,” Proc. SPIE 6422, 64221C (2007).
    [CrossRef]
  23. C. G. Hu, L. D. Sun, Y. N. Li, M. Hohage, J. M. Flores-Camacho, X. T. Hu, and P. Zeppenfeld, “Retardation correction for photoelastic modulator-based multichannel reflectance difference spectroscopy,” J. Opt. Soc. Am. A 25, 1240-1245(2008).
    [CrossRef]
  24. L. F. Lastras-Martínez, A. Lastras-Martínez, and R. E. Balderas-Navarro, “A spectrometer for the measurement of reflectance-difference spectra,” Rev. Sci. Instrum. 64, 2147-2152 (1993).
    [CrossRef]

2009 (1)

G. E. Isted, P. D. Lane, and R. J. Cole, “Effect of thermally induced surface defects on the optical anisotropy of Ag(110),” Phys. Rev. B 79, 205424 (2009).
[CrossRef]

2008 (2)

A. Lastras-Martínez, I. Lara-Velázquez, R. E. Balderas-Navarro, J. Ortega-Gallegos, S. Guel-Sandoval, and L. F. Lastras-Martínez, “Reflectance-difference spectroscopy as an optical probe for in situ determination of doping levels in GaAs,” Phys. Status Solidi C 5, 2565-2568 (2008).
[CrossRef]

C. G. Hu, L. D. Sun, Y. N. Li, M. Hohage, J. M. Flores-Camacho, X. T. Hu, and P. Zeppenfeld, “Retardation correction for photoelastic modulator-based multichannel reflectance difference spectroscopy,” J. Opt. Soc. Am. A 25, 1240-1245(2008).
[CrossRef]

2007 (4)

L. F. Lastras-Martínez, R. Castro-Garca, R. E. Balderas-Navarro, and A. Lastras-Martínez, “Reflectance difference spectrometer based on the use of a CCD camera,” Proc. SPIE 6422, 64221C (2007).
[CrossRef]

G. Irmer and M. Jurisch, “Micro-Raman study of strain fields around dislocations in GaAs,” Phys. Status Solidi A 204, 2309-2318 (2007).
[CrossRef]

N. Gmeinwieser and U. T. Schwarz, “Strain of single edge dislocations in bulk GaN,” Phys. Status Solidi B 244, 1857-1861 (2007).
[CrossRef]

L. F. Lastras-Martínez, J. M. Flores-Camacho, R. E. Balderas-Navarro, M. Chavira-Rodríguez, A. Lastras-Martínez, and M. Cardona, “Effect of reconstruction-induced strain on the reflectance difference spectroscopy of GaAs (001) around E1 and E11 transitions,” Phys. Rev. B 75, 235315(2007).
[CrossRef]

2006 (1)

H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
[CrossRef]

2005 (2)

P. Weightman, D. S. Martin, R. J. Cole, and T. Farrell, “Reflection anisotropy spectroscopy,” Rep. Prog. Phys. 68, 1251-1341(2005).
[CrossRef]

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

2004 (2)

L. F. Lastras-Martínez, M. Chavira-Rodríguez, R. E. Balderas-Navarro, J. M. Flores-Camacho, and A. Lastras-Martínez, “Reflectance difference spectroscopy of GaAs (001) under a [110] uniaxial stress,” Phys. Rev. B 70, 035306 (2004).
[CrossRef]

P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
[CrossRef]

2003 (2)

K. Sawano, S. Koh, Y. Shiraki, N. Usami, and K. Nakagawa, “In-plane strain fluctuation in strained-Si/SiGe heterostructures,” Appl. Phys. Lett. 83, 4339-4341 (2003).
[CrossRef]

L. F. Lastras-Martínez, R. E. Balderas-Navarro, M. Chavira-Rodríguez, J. M. Flores-Camacho, and A. Lastras-Martínez, “Strain induced optical anisotropies in zinc blende semiconductor,” Phys. Status Solidi B 240, 500-508 (2003).
[CrossRef]

2001 (1)

L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance-difference spectroscopy of semi-insulating GaAs (110) around the fundamental gap,” Phys. Rev. B 64, 085309 (2001).
[CrossRef]

2000 (2)

A. Lastras-Martínez, R. E. Balderas-Navarro, and L. F. Lastras-Martínez, “Linear electro-optic reflectance modulated spectra of GaAs (001) around E1 and E11,” Thin Solid Films 373, 207-210 (2000).
[CrossRef]

Y. H. Chen, Z. G. Wang, J. J. Qian, and Z. Yang, “Polishing-related optical anisotropy of semi-insulating GaAs studied by reflectance difference spectroscopy,” J. Appl. Phys. 88, 1695-1697 (2000).
[CrossRef]

1999 (1)

1996 (3)

B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona, “In-plane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy,” Appl. Phys. Lett. 69, 782-784 (1996).
[CrossRef]

L. F. Lastras-Martínez and A. Lastras-Martínez, “Dislocation-induced effects in the reflectance-difference spectrum of semi-insulating GaAs (100),” Solid State Commun. 98, 479-483(1996).
[CrossRef]

L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance anisotropy of GaAs(100): Dislocation-induced piezo-optic effects,” Phys. Rev. B 54, 10726-10735 (1996).
[CrossRef]

1993 (2)

M. Yamada, “High-sensitivity computer-controlled infrared polariscope,” Rev. Sci. Instrum. 64, 1815-1821 (1993).
[CrossRef]

L. F. Lastras-Martínez, A. Lastras-Martínez, and R. E. Balderas-Navarro, “A spectrometer for the measurement of reflectance-difference spectra,” Rev. Sci. Instrum. 64, 2147-2152 (1993).
[CrossRef]

1985 (1)

D. E. Aspnes and A. A. Studna, “Anisotropies in the above-band-gap optical spectra of cubic semiconductors,” Phys. Rev. Lett. 54, 1956-1959 (1985).
[CrossRef] [PubMed]

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, “Anisotropies in the above-band-gap optical spectra of cubic semiconductors,” Phys. Rev. Lett. 54, 1956-1959 (1985).
[CrossRef] [PubMed]

Balderas-Navarro, R. E.

A. Lastras-Martínez, I. Lara-Velázquez, R. E. Balderas-Navarro, J. Ortega-Gallegos, S. Guel-Sandoval, and L. F. Lastras-Martínez, “Reflectance-difference spectroscopy as an optical probe for in situ determination of doping levels in GaAs,” Phys. Status Solidi C 5, 2565-2568 (2008).
[CrossRef]

L. F. Lastras-Martínez, R. Castro-Garca, R. E. Balderas-Navarro, and A. Lastras-Martínez, “Reflectance difference spectrometer based on the use of a CCD camera,” Proc. SPIE 6422, 64221C (2007).
[CrossRef]

L. F. Lastras-Martínez, J. M. Flores-Camacho, R. E. Balderas-Navarro, M. Chavira-Rodríguez, A. Lastras-Martínez, and M. Cardona, “Effect of reconstruction-induced strain on the reflectance difference spectroscopy of GaAs (001) around E1 and E11 transitions,” Phys. Rev. B 75, 235315(2007).
[CrossRef]

L. F. Lastras-Martínez, M. Chavira-Rodríguez, R. E. Balderas-Navarro, J. M. Flores-Camacho, and A. Lastras-Martínez, “Reflectance difference spectroscopy of GaAs (001) under a [110] uniaxial stress,” Phys. Rev. B 70, 035306 (2004).
[CrossRef]

L. F. Lastras-Martínez, R. E. Balderas-Navarro, M. Chavira-Rodríguez, J. M. Flores-Camacho, and A. Lastras-Martínez, “Strain induced optical anisotropies in zinc blende semiconductor,” Phys. Status Solidi B 240, 500-508 (2003).
[CrossRef]

A. Lastras-Martínez, R. E. Balderas-Navarro, and L. F. Lastras-Martínez, “Linear electro-optic reflectance modulated spectra of GaAs (001) around E1 and E11,” Thin Solid Films 373, 207-210 (2000).
[CrossRef]

L. F. Lastras-Martínez, A. Lastras-Martínez, and R. E. Balderas-Navarro, “A spectrometer for the measurement of reflectance-difference spectra,” Rev. Sci. Instrum. 64, 2147-2152 (1993).
[CrossRef]

Brüderl, G.

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Cardona, M.

L. F. Lastras-Martínez, J. M. Flores-Camacho, R. E. Balderas-Navarro, M. Chavira-Rodríguez, A. Lastras-Martínez, and M. Cardona, “Effect of reconstruction-induced strain on the reflectance difference spectroscopy of GaAs (001) around E1 and E11 transitions,” Phys. Rev. B 75, 235315(2007).
[CrossRef]

D. Rönnow, L. F. Lastras-Martínez, M. Cardona, and P. V. Santos, “Determination of the piezo-optical properties of semiconductors above the fundamental gap by means of reflectance difference spectroscopy,” J. Opt. Soc. Am. A 16, 568-573 (1999).
[CrossRef]

B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona, “In-plane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy,” Appl. Phys. Lett. 69, 782-784 (1996).
[CrossRef]

Castro-Garca, R.

L. F. Lastras-Martínez, R. Castro-Garca, R. E. Balderas-Navarro, and A. Lastras-Martínez, “Reflectance difference spectrometer based on the use of a CCD camera,” Proc. SPIE 6422, 64221C (2007).
[CrossRef]

Chavira-Rodríguez, M.

L. F. Lastras-Martínez, J. M. Flores-Camacho, R. E. Balderas-Navarro, M. Chavira-Rodríguez, A. Lastras-Martínez, and M. Cardona, “Effect of reconstruction-induced strain on the reflectance difference spectroscopy of GaAs (001) around E1 and E11 transitions,” Phys. Rev. B 75, 235315(2007).
[CrossRef]

L. F. Lastras-Martínez, M. Chavira-Rodríguez, R. E. Balderas-Navarro, J. M. Flores-Camacho, and A. Lastras-Martínez, “Reflectance difference spectroscopy of GaAs (001) under a [110] uniaxial stress,” Phys. Rev. B 70, 035306 (2004).
[CrossRef]

L. F. Lastras-Martínez, R. E. Balderas-Navarro, M. Chavira-Rodríguez, J. M. Flores-Camacho, and A. Lastras-Martínez, “Strain induced optical anisotropies in zinc blende semiconductor,” Phys. Status Solidi B 240, 500-508 (2003).
[CrossRef]

Chen, Y. H.

Y. H. Chen, Z. G. Wang, J. J. Qian, and Z. Yang, “Polishing-related optical anisotropy of semi-insulating GaAs studied by reflectance difference spectroscopy,” J. Appl. Phys. 88, 1695-1697 (2000).
[CrossRef]

Clos, R.

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Cole, R. J.

G. E. Isted, P. D. Lane, and R. J. Cole, “Effect of thermally induced surface defects on the optical anisotropy of Ag(110),” Phys. Rev. B 79, 205424 (2009).
[CrossRef]

P. Weightman, D. S. Martin, R. J. Cole, and T. Farrell, “Reflection anisotropy spectroscopy,” Rep. Prog. Phys. 68, 1251-1341(2005).
[CrossRef]

Demangeot, F.

P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
[CrossRef]

Domnich, V.

P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
[CrossRef]

Eberl, K.

B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona, “In-plane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy,” Appl. Phys. Lett. 69, 782-784 (1996).
[CrossRef]

Eichler, St.

H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
[CrossRef]

Farrell, T.

P. Weightman, D. S. Martin, R. J. Cole, and T. Farrell, “Reflection anisotropy spectroscopy,” Rep. Prog. Phys. 68, 1251-1341(2005).
[CrossRef]

Flores-Camacho, J. M.

C. G. Hu, L. D. Sun, Y. N. Li, M. Hohage, J. M. Flores-Camacho, X. T. Hu, and P. Zeppenfeld, “Retardation correction for photoelastic modulator-based multichannel reflectance difference spectroscopy,” J. Opt. Soc. Am. A 25, 1240-1245(2008).
[CrossRef]

L. F. Lastras-Martínez, J. M. Flores-Camacho, R. E. Balderas-Navarro, M. Chavira-Rodríguez, A. Lastras-Martínez, and M. Cardona, “Effect of reconstruction-induced strain on the reflectance difference spectroscopy of GaAs (001) around E1 and E11 transitions,” Phys. Rev. B 75, 235315(2007).
[CrossRef]

L. F. Lastras-Martínez, M. Chavira-Rodríguez, R. E. Balderas-Navarro, J. M. Flores-Camacho, and A. Lastras-Martínez, “Reflectance difference spectroscopy of GaAs (001) under a [110] uniaxial stress,” Phys. Rev. B 70, 035306 (2004).
[CrossRef]

L. F. Lastras-Martínez, R. E. Balderas-Navarro, M. Chavira-Rodríguez, J. M. Flores-Camacho, and A. Lastras-Martínez, “Strain induced optical anisotropies in zinc blende semiconductor,” Phys. Status Solidi B 240, 500-508 (2003).
[CrossRef]

Frandon, J.

P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
[CrossRef]

Geiler, H. D.

H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
[CrossRef]

Gmeinwieser, N.

N. Gmeinwieser and U. T. Schwarz, “Strain of single edge dislocations in bulk GaN,” Phys. Status Solidi B 244, 1857-1861 (2007).
[CrossRef]

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Gogotsi, Y.

P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
[CrossRef]

Gottfriedsen, P.

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Guel-Sandoval, S.

A. Lastras-Martínez, I. Lara-Velázquez, R. E. Balderas-Navarro, J. Ortega-Gallegos, S. Guel-Sandoval, and L. F. Lastras-Martínez, “Reflectance-difference spectroscopy as an optical probe for in situ determination of doping levels in GaAs,” Phys. Status Solidi C 5, 2565-2568 (2008).
[CrossRef]

Härle, V.

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Hohage, M.

Hu, C. G.

Hu, X. T.

Irmer, G.

G. Irmer and M. Jurisch, “Micro-Raman study of strain fields around dislocations in GaAs,” Phys. Status Solidi A 204, 2309-2318 (2007).
[CrossRef]

Isted, G. E.

G. E. Isted, P. D. Lane, and R. J. Cole, “Effect of thermally induced surface defects on the optical anisotropy of Ag(110),” Phys. Rev. B 79, 205424 (2009).
[CrossRef]

Jurisch, M.

G. Irmer and M. Jurisch, “Micro-Raman study of strain fields around dislocations in GaAs,” Phys. Status Solidi A 204, 2309-2318 (2007).
[CrossRef]

H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
[CrossRef]

Karge, H.

H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
[CrossRef]

Koh, S.

K. Sawano, S. Koh, Y. Shiraki, N. Usami, and K. Nakagawa, “In-plane strain fluctuation in strained-Si/SiGe heterostructures,” Appl. Phys. Lett. 83, 4339-4341 (2003).
[CrossRef]

Koopmans, B.

B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona, “In-plane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy,” Appl. Phys. Lett. 69, 782-784 (1996).
[CrossRef]

Kretzer, U.

H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
[CrossRef]

Krost, A.

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Krtschil, A.

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Kuball, M.

P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
[CrossRef]

Lane, P. D.

G. E. Isted, P. D. Lane, and R. J. Cole, “Effect of thermally induced surface defects on the optical anisotropy of Ag(110),” Phys. Rev. B 79, 205424 (2009).
[CrossRef]

Lara-Velázquez, I.

A. Lastras-Martínez, I. Lara-Velázquez, R. E. Balderas-Navarro, J. Ortega-Gallegos, S. Guel-Sandoval, and L. F. Lastras-Martínez, “Reflectance-difference spectroscopy as an optical probe for in situ determination of doping levels in GaAs,” Phys. Status Solidi C 5, 2565-2568 (2008).
[CrossRef]

Lastras-Martínez, A.

A. Lastras-Martínez, I. Lara-Velázquez, R. E. Balderas-Navarro, J. Ortega-Gallegos, S. Guel-Sandoval, and L. F. Lastras-Martínez, “Reflectance-difference spectroscopy as an optical probe for in situ determination of doping levels in GaAs,” Phys. Status Solidi C 5, 2565-2568 (2008).
[CrossRef]

L. F. Lastras-Martínez, R. Castro-Garca, R. E. Balderas-Navarro, and A. Lastras-Martínez, “Reflectance difference spectrometer based on the use of a CCD camera,” Proc. SPIE 6422, 64221C (2007).
[CrossRef]

L. F. Lastras-Martínez, J. M. Flores-Camacho, R. E. Balderas-Navarro, M. Chavira-Rodríguez, A. Lastras-Martínez, and M. Cardona, “Effect of reconstruction-induced strain on the reflectance difference spectroscopy of GaAs (001) around E1 and E11 transitions,” Phys. Rev. B 75, 235315(2007).
[CrossRef]

L. F. Lastras-Martínez, M. Chavira-Rodríguez, R. E. Balderas-Navarro, J. M. Flores-Camacho, and A. Lastras-Martínez, “Reflectance difference spectroscopy of GaAs (001) under a [110] uniaxial stress,” Phys. Rev. B 70, 035306 (2004).
[CrossRef]

L. F. Lastras-Martínez, R. E. Balderas-Navarro, M. Chavira-Rodríguez, J. M. Flores-Camacho, and A. Lastras-Martínez, “Strain induced optical anisotropies in zinc blende semiconductor,” Phys. Status Solidi B 240, 500-508 (2003).
[CrossRef]

L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance-difference spectroscopy of semi-insulating GaAs (110) around the fundamental gap,” Phys. Rev. B 64, 085309 (2001).
[CrossRef]

A. Lastras-Martínez, R. E. Balderas-Navarro, and L. F. Lastras-Martínez, “Linear electro-optic reflectance modulated spectra of GaAs (001) around E1 and E11,” Thin Solid Films 373, 207-210 (2000).
[CrossRef]

L. F. Lastras-Martínez and A. Lastras-Martínez, “Dislocation-induced effects in the reflectance-difference spectrum of semi-insulating GaAs (100),” Solid State Commun. 98, 479-483(1996).
[CrossRef]

L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance anisotropy of GaAs(100): Dislocation-induced piezo-optic effects,” Phys. Rev. B 54, 10726-10735 (1996).
[CrossRef]

L. F. Lastras-Martínez, A. Lastras-Martínez, and R. E. Balderas-Navarro, “A spectrometer for the measurement of reflectance-difference spectra,” Rev. Sci. Instrum. 64, 2147-2152 (1993).
[CrossRef]

Lastras-Martínez, L. F.

A. Lastras-Martínez, I. Lara-Velázquez, R. E. Balderas-Navarro, J. Ortega-Gallegos, S. Guel-Sandoval, and L. F. Lastras-Martínez, “Reflectance-difference spectroscopy as an optical probe for in situ determination of doping levels in GaAs,” Phys. Status Solidi C 5, 2565-2568 (2008).
[CrossRef]

L. F. Lastras-Martínez, R. Castro-Garca, R. E. Balderas-Navarro, and A. Lastras-Martínez, “Reflectance difference spectrometer based on the use of a CCD camera,” Proc. SPIE 6422, 64221C (2007).
[CrossRef]

L. F. Lastras-Martínez, J. M. Flores-Camacho, R. E. Balderas-Navarro, M. Chavira-Rodríguez, A. Lastras-Martínez, and M. Cardona, “Effect of reconstruction-induced strain on the reflectance difference spectroscopy of GaAs (001) around E1 and E11 transitions,” Phys. Rev. B 75, 235315(2007).
[CrossRef]

L. F. Lastras-Martínez, M. Chavira-Rodríguez, R. E. Balderas-Navarro, J. M. Flores-Camacho, and A. Lastras-Martínez, “Reflectance difference spectroscopy of GaAs (001) under a [110] uniaxial stress,” Phys. Rev. B 70, 035306 (2004).
[CrossRef]

L. F. Lastras-Martínez, R. E. Balderas-Navarro, M. Chavira-Rodríguez, J. M. Flores-Camacho, and A. Lastras-Martínez, “Strain induced optical anisotropies in zinc blende semiconductor,” Phys. Status Solidi B 240, 500-508 (2003).
[CrossRef]

L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance-difference spectroscopy of semi-insulating GaAs (110) around the fundamental gap,” Phys. Rev. B 64, 085309 (2001).
[CrossRef]

A. Lastras-Martínez, R. E. Balderas-Navarro, and L. F. Lastras-Martínez, “Linear electro-optic reflectance modulated spectra of GaAs (001) around E1 and E11,” Thin Solid Films 373, 207-210 (2000).
[CrossRef]

D. Rönnow, L. F. Lastras-Martínez, M. Cardona, and P. V. Santos, “Determination of the piezo-optical properties of semiconductors above the fundamental gap by means of reflectance difference spectroscopy,” J. Opt. Soc. Am. A 16, 568-573 (1999).
[CrossRef]

L. F. Lastras-Martínez and A. Lastras-Martínez, “Dislocation-induced effects in the reflectance-difference spectrum of semi-insulating GaAs (100),” Solid State Commun. 98, 479-483(1996).
[CrossRef]

L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance anisotropy of GaAs(100): Dislocation-induced piezo-optic effects,” Phys. Rev. B 54, 10726-10735 (1996).
[CrossRef]

L. F. Lastras-Martínez, A. Lastras-Martínez, and R. E. Balderas-Navarro, “A spectrometer for the measurement of reflectance-difference spectra,” Rev. Sci. Instrum. 64, 2147-2152 (1993).
[CrossRef]

Lell, A.

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Li, Y. N.

Martin, D. S.

P. Weightman, D. S. Martin, R. J. Cole, and T. Farrell, “Reflection anisotropy spectroscopy,” Rep. Prog. Phys. 68, 1251-1341(2005).
[CrossRef]

Nakagawa, K.

K. Sawano, S. Koh, Y. Shiraki, N. Usami, and K. Nakagawa, “In-plane strain fluctuation in strained-Si/SiGe heterostructures,” Appl. Phys. Lett. 83, 4339-4341 (2003).
[CrossRef]

Ortega-Gallegos, J.

A. Lastras-Martínez, I. Lara-Velázquez, R. E. Balderas-Navarro, J. Ortega-Gallegos, S. Guel-Sandoval, and L. F. Lastras-Martínez, “Reflectance-difference spectroscopy as an optical probe for in situ determination of doping levels in GaAs,” Phys. Status Solidi C 5, 2565-2568 (2008).
[CrossRef]

Pinquier, C.

P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
[CrossRef]

Puech, P.

P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
[CrossRef]

Qian, J. J.

Y. H. Chen, Z. G. Wang, J. J. Qian, and Z. Yang, “Polishing-related optical anisotropy of semi-insulating GaAs studied by reflectance difference spectroscopy,” J. Appl. Phys. 88, 1695-1697 (2000).
[CrossRef]

Richards, B.

B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona, “In-plane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy,” Appl. Phys. Lett. 69, 782-784 (1996).
[CrossRef]

Rönnow, D.

Santos, P.

B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona, “In-plane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy,” Appl. Phys. Lett. 69, 782-784 (1996).
[CrossRef]

Santos, P. V.

Sawano, K.

K. Sawano, S. Koh, Y. Shiraki, N. Usami, and K. Nakagawa, “In-plane strain fluctuation in strained-Si/SiGe heterostructures,” Appl. Phys. Lett. 83, 4339-4341 (2003).
[CrossRef]

Scheffer-Czygan, M.

H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
[CrossRef]

Schwarz, U. T.

N. Gmeinwieser and U. T. Schwarz, “Strain of single edge dislocations in bulk GaN,” Phys. Status Solidi B 244, 1857-1861 (2007).
[CrossRef]

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Shiraki, Y.

K. Sawano, S. Koh, Y. Shiraki, N. Usami, and K. Nakagawa, “In-plane strain fluctuation in strained-Si/SiGe heterostructures,” Appl. Phys. Lett. 83, 4339-4341 (2003).
[CrossRef]

Studna, A. A.

D. E. Aspnes and A. A. Studna, “Anisotropies in the above-band-gap optical spectra of cubic semiconductors,” Phys. Rev. Lett. 54, 1956-1959 (1985).
[CrossRef] [PubMed]

Sun, L. D.

Usami, N.

K. Sawano, S. Koh, Y. Shiraki, N. Usami, and K. Nakagawa, “In-plane strain fluctuation in strained-Si/SiGe heterostructures,” Appl. Phys. Lett. 83, 4339-4341 (2003).
[CrossRef]

Wagner, M.

H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
[CrossRef]

Wang, Z. G.

Y. H. Chen, Z. G. Wang, J. J. Qian, and Z. Yang, “Polishing-related optical anisotropy of semi-insulating GaAs studied by reflectance difference spectroscopy,” J. Appl. Phys. 88, 1695-1697 (2000).
[CrossRef]

Wegscheider, W.

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Weightman, P.

P. Weightman, D. S. Martin, R. J. Cole, and T. Farrell, “Reflection anisotropy spectroscopy,” Rep. Prog. Phys. 68, 1251-1341(2005).
[CrossRef]

Weimar, A.

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

Yamada, M.

M. Yamada, “High-sensitivity computer-controlled infrared polariscope,” Rev. Sci. Instrum. 64, 1815-1821 (1993).
[CrossRef]

Yang, Z.

Y. H. Chen, Z. G. Wang, J. J. Qian, and Z. Yang, “Polishing-related optical anisotropy of semi-insulating GaAs studied by reflectance difference spectroscopy,” J. Appl. Phys. 88, 1695-1697 (2000).
[CrossRef]

Zeppenfeld, P.

Appl. Phys. Lett. (2)

K. Sawano, S. Koh, Y. Shiraki, N. Usami, and K. Nakagawa, “In-plane strain fluctuation in strained-Si/SiGe heterostructures,” Appl. Phys. Lett. 83, 4339-4341 (2003).
[CrossRef]

B. Koopmans, B. Richards, P. Santos, K. Eberl, and M. Cardona, “In-plane optical anisotropy of GaAs/AlAs multiple quantum wells probed by microscopic reflectance difference spectroscopy,” Appl. Phys. Lett. 69, 782-784 (1996).
[CrossRef]

J. Appl. Phys. (3)

N. Gmeinwieser, P. Gottfriedsen, U. T. Schwarz, W. Wegscheider, R. Clos, A. Krtschil, A. Krost, A. Weimar, G. Brüderl, A. Lell, and V. Härle, “Local strain and potential distribution induced by single dislocations in GaN,” J. Appl. Phys. 98, 116102 (2005).
[CrossRef]

P. Puech, F. Demangeot, J. Frandon, C. Pinquier, M. Kuball, V. Domnich, and Y. Gogotsi, “GaN nanoindentation: a micro-Raman spectroscopy study of local strain fields,” J. Appl. Phys. 96, 2853-2856 (2004).
[CrossRef]

Y. H. Chen, Z. G. Wang, J. J. Qian, and Z. Yang, “Polishing-related optical anisotropy of semi-insulating GaAs studied by reflectance difference spectroscopy,” J. Appl. Phys. 88, 1695-1697 (2000).
[CrossRef]

J. Opt. Soc. Am. A (2)

Mater. Sci. Semicond. Process. (1)

H. D. Geiler, H. Karge, M. Wagner, St. Eichler, M. Jurisch, U. Kretzer, and M. Scheffer-Czygan, “Photoelastic characterization of residual stress in GaAs-wafers,” Mater. Sci. Semicond. Process. 9, 345-350 (2006).
[CrossRef]

Phys. Rev. B (5)

L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance anisotropy of GaAs(100): Dislocation-induced piezo-optic effects,” Phys. Rev. B 54, 10726-10735 (1996).
[CrossRef]

L. F. Lastras-Martínez and A. Lastras-Martínez, “Reflectance-difference spectroscopy of semi-insulating GaAs (110) around the fundamental gap,” Phys. Rev. B 64, 085309 (2001).
[CrossRef]

L. F. Lastras-Martínez, J. M. Flores-Camacho, R. E. Balderas-Navarro, M. Chavira-Rodríguez, A. Lastras-Martínez, and M. Cardona, “Effect of reconstruction-induced strain on the reflectance difference spectroscopy of GaAs (001) around E1 and E11 transitions,” Phys. Rev. B 75, 235315(2007).
[CrossRef]

G. E. Isted, P. D. Lane, and R. J. Cole, “Effect of thermally induced surface defects on the optical anisotropy of Ag(110),” Phys. Rev. B 79, 205424 (2009).
[CrossRef]

L. F. Lastras-Martínez, M. Chavira-Rodríguez, R. E. Balderas-Navarro, J. M. Flores-Camacho, and A. Lastras-Martínez, “Reflectance difference spectroscopy of GaAs (001) under a [110] uniaxial stress,” Phys. Rev. B 70, 035306 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

D. E. Aspnes and A. A. Studna, “Anisotropies in the above-band-gap optical spectra of cubic semiconductors,” Phys. Rev. Lett. 54, 1956-1959 (1985).
[CrossRef] [PubMed]

Phys. Status Solidi A (1)

G. Irmer and M. Jurisch, “Micro-Raman study of strain fields around dislocations in GaAs,” Phys. Status Solidi A 204, 2309-2318 (2007).
[CrossRef]

Phys. Status Solidi B (2)

N. Gmeinwieser and U. T. Schwarz, “Strain of single edge dislocations in bulk GaN,” Phys. Status Solidi B 244, 1857-1861 (2007).
[CrossRef]

L. F. Lastras-Martínez, R. E. Balderas-Navarro, M. Chavira-Rodríguez, J. M. Flores-Camacho, and A. Lastras-Martínez, “Strain induced optical anisotropies in zinc blende semiconductor,” Phys. Status Solidi B 240, 500-508 (2003).
[CrossRef]

Phys. Status Solidi C (1)

A. Lastras-Martínez, I. Lara-Velázquez, R. E. Balderas-Navarro, J. Ortega-Gallegos, S. Guel-Sandoval, and L. F. Lastras-Martínez, “Reflectance-difference spectroscopy as an optical probe for in situ determination of doping levels in GaAs,” Phys. Status Solidi C 5, 2565-2568 (2008).
[CrossRef]

Proc. SPIE (1)

L. F. Lastras-Martínez, R. Castro-Garca, R. E. Balderas-Navarro, and A. Lastras-Martínez, “Reflectance difference spectrometer based on the use of a CCD camera,” Proc. SPIE 6422, 64221C (2007).
[CrossRef]

Rep. Prog. Phys. (1)

P. Weightman, D. S. Martin, R. J. Cole, and T. Farrell, “Reflection anisotropy spectroscopy,” Rep. Prog. Phys. 68, 1251-1341(2005).
[CrossRef]

Rev. Sci. Instrum. (2)

M. Yamada, “High-sensitivity computer-controlled infrared polariscope,” Rev. Sci. Instrum. 64, 1815-1821 (1993).
[CrossRef]

L. F. Lastras-Martínez, A. Lastras-Martínez, and R. E. Balderas-Navarro, “A spectrometer for the measurement of reflectance-difference spectra,” Rev. Sci. Instrum. 64, 2147-2152 (1993).
[CrossRef]

Solid State Commun. (1)

L. F. Lastras-Martínez and A. Lastras-Martínez, “Dislocation-induced effects in the reflectance-difference spectrum of semi-insulating GaAs (100),” Solid State Commun. 98, 479-483(1996).
[CrossRef]

Thin Solid Films (1)

A. Lastras-Martínez, R. E. Balderas-Navarro, and L. F. Lastras-Martínez, “Linear electro-optic reflectance modulated spectra of GaAs (001) around E1 and E11,” Thin Solid Films 373, 207-210 (2000).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Optical agreement of the μRD spectrometer used in this work. The polarized and unpolarized spectra are taken by turning the PEM off and on, respectively. The RD signal is obtained by subtracting numerically both spectra. The working spectral range is 1.1 5.5 eV .

Fig. 2
Fig. 2

RD spectra obtained by using a CCD- and PM-based spectrometers. The CCD spectrum was multiplied by 4 according to Eq. (1). The PM spectrum was multiplied by 2 2 to obtain the peak to peak amplitude of the signal. The solid line is the best fit obtained by using Eqs. (2, 3). Note that although the CCD spectrum is noisier, both spectrometers measure essentially the same line shape and amplitude.

Fig. 3
Fig. 3

Spacial distribution of the defects density on the surface of GaAs ( 001 ) after a mechanical polishing along the [ 110 ] direction. The surface was generated after a linear interpolation of the points obtained by fitting the amplitude of 4096 ( 64 × 64 ) spectra. Note the symmetry of the distribution oriented along the linear defects. The spatial resolution is 20 μm . The grid is spaced 90 μm . The performance of the instrument is excellent.

Fig. 4
Fig. 4

Inset: Image of the right bottom square indicated in Fig. 3 amplified by a factor of 2. The spatial resolution is 10 μm , and the grid spacing is 45 μm . The profile obtained along the diagonal line is indicated in the inset, from point A to point B. The amplitude is normalized to the amplitude obtained in the spectrum of Fig. 3.

Equations (3)

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

Δ R R = 1 4 R [ 110 ] R [ 1 1 ¯ 0 ] R [ 110 ] ,
μ ( Δ R R ) = 2 l 2 p ρ Re [ 1 n r d r d n Δ ε ] ,
Δ ε = ± 4 D 5 Δ 1 6 e 22 ε + D 1 5 2 2 e 22 d ε d E ,

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