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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 20, Iss. 2 — Feb. 1, 2003
  • pp: 347–356

Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs

Mathias Schubert, Tino Hofmann, and Craig M. Herzinger  »View Author Affiliations

JOSA A, Vol. 20, Issue 2, pp. 347-356 (2003)

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We report for the first time on the application of generalized ellipsometry at far-infrared wavelengths (wave numbers from 150 cm−1 to 600 cm−1) for measurement of the anisotropic dielectric response of doped polar semiconductors in layered structures within an external magnetic field. Upon determination of normalized Mueller matrix elements and subsequent derivation of the normalized complex Jones reflection matrix r of an n-type doped GaAs substrate covered by a highly resistive GaAs layer, the spectral dependence of the room-temperature magneto-optic dielectric function tensor of n-type GaAs with free-electron concentration of 1.6× 1018 cm−3 at the magnetic field strength of 2.3 T is obtained on a wavelength-by-wavelength basis. These data are in excellent agreement with values predicted by the Drude model. From the magneto-optic generalized ellipsometry measurements of the layered structure, the free-carrier concentration, their optical mobility, the effective-mass parameters, and the sign of the charge carriers can be determined independently, which will be demonstrated. We propose magneto-optic generalized ellipsometry as a novel approach for exploration of free-carrier parameters in complex organic or inorganic semiconducting material heterostructures, regardless of the anisotropic properties of the individual constituents.

© 2003 Optical Society of America

OCIS Codes
(260.1180) Physical optics : Crystal optics
(260.2130) Physical optics : Ellipsometry and polarimetry
(260.3090) Physical optics : Infrared, far

Mathias Schubert, Tino Hofmann, and Craig M. Herzinger, "Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs," J. Opt. Soc. Am. A 20, 347-356 (2003)

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  1. The effective-mass concept addressed here descends from the similarity with the Newton force equation (acceleration of a body with mass m) and the acceleration experienced by a Bloch electron due to an external force. The inverse tensor obtained thereby depends on the curvature of the plots of electron energetic states versus electron momentum, which is diagonal by a suitable choice of axes. Different experiments require different concepts, resulting in definition of the effective conductivity mass, the density-of-states effective mass, the Hall effective mass, or the cyclotron effective mass, all of which are not discussed here. For the material investigated here (GaAs), the response of the zinc-blended, Γ-point conduction band (single-species, i.e., single-valley) Bloch electrons studied at infrared wavelengths is on a time scale much smaller than the average time between scattering events of the free electrons. One may also refer to the effective mass here as an (infrared) optical effective mass. See also Refs. 2 and 3.
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  14. Restrictions apply to highly conductive layers that are optically thick, such as highly doped semiconductor substrates, or metal films several hundreds of nanometers thick. As long as the layer with high free-carrier concentration is passing electromagnetic radiation on to the next constituent, the ellipsometric parameters will contain information about the buried layers.
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  23. M. Schubert and W. Dollase, “Generalized ellipsometry for biaxial absorbing minerals: determination of crystal orientation and optical constants from Sb2S3,” Opt. Lett. 27, 2073–2075 (2002).
  24. M. Schubert, “Theory and application of generalized ellipsometry,” in Handbook of Ellipsometry, G. E. Irene and H. W. Tompkins, eds. (to be published).
  25. M. Schubert, B. Rheinländer, B. Johs, and J. A. Woollam, “Application of generalized ellipsometry to complex optical systems,” in Polarimetry and Ellipsometry, M. Pluta and T. R. Wolinsky, eds., Proc. SPIE 3094, 255–265 (1997).
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  28. This set comprises six real-valued quantities out of the eight possible values contained within the Jones matrix—lacking the light beam’s absolute intensity and absolute phase information. For a definition of the Jones matrix elements, see Refs. 15, 17, 18, 24 25 26 27, and references therein.
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  31. M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
  32. T. E. Tiwald and M. Schubert, “Measurement of rutile TiO2 from 0.148 to 33 μm using generalized ellipsometry,” in Optical Diagnostic Methods For Inorganic Materials II, L. M. Hanssen, ed., Proc. SPIE 4103, 19–29 (2000).
  33. R. Henn, C. Bernhard, A. Wittlin, M. Cardona, and S. Uchida, “Far infrared ellipsometry using synchrotron radiation: the out-of-plane response of La2−xSrxCuO4,” Thin Solid Films 313–314, 643–648 (1998).
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  36. T. E. Tiwald, J. A. Woollam, St. Zollner, J. Christiansen, R. B. Gregory, T. Wetteroth, and S. R. Wilson, “Carrier concentration and lattice absorption in bulk and epitaxial silicon carbide determined using infrared ellipsometry,” Phys. Rev. B 60, 11464–11474 (1999).
  37. M. Schubert, T. E. Tiwald, and C. M. Herzinger, “Infrared dielectric anisotropy and phonon modes of sapphire,” Phys. Rev. B 61, 8187–8201 (2000).
  38. A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, and T. E. Tiwald, “Free-carrier and phonon properties of n- and p-type hexagonal GaN films measured by infrared ellipsometry,” Phys. Rev. B 62, 7365–7377 (2000).
  39. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1985).
  40. In view of the generalized ellipsometry applications at oblique incidence, MO setups will be addressed as follows: the polar MO (PMO) setup with B parallel to the sample normal, the longitudinal MO (LMO) setup with B parallel to the sample surface and parallel to the plane of incidence, and the transverse MO (TMO) setup with B perpendicular to the sample normal and perpendicular to the plane of incidence. In the MO literature, different terms are in use: Near-normal-incidence reflection-type Kerr-effect measurements are referred to as transverse, longitudinal, and polar configurations, in conceptual agreement with the above notation. Faraday- and Voigt-effect measurements address transmission-type linear polarization rotation measurements in the above PMO and mixed LMO–TMO configurations, respectively. See Ref. 3 or Ref. 41. These configurations mostly result from the simplicity of the corresponding equations, which describe the polarized light reflection and transmission situations for the anisotropic materials. These requirements are now dispensed with because of the availability of explicit solutions for light propagation in arbitrarily nonsymmetric (MO) dielectric materials.21
  41. M. Mansuripur, The Physical Principles of Magneto-Optical Recording (Cambridge U. Press, Cambridge, UK, 1995).
  42. E. D. Palik, “Gallium arsenide (GaAs),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1998), Vol. I; pp. 429–444.
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  45. St. Zollner, J. P. Carrejo, T. E. Tiwald, and J. A. Woollam, “The origin of the Berreman effect in SiC homostructures,” Phys. Status Solidi B 208, R3–R4 (1998).
  46. J. Humlíček, “Infrared spectroscopy of LiF on Ag and Si,” Phys. Status Solidi B 215, 155–159 (1999).
  47. M. Schubert, B. Rheinländer, C. Cramer, H. Schmiedel, J. A. Woollam, B. Johs, and C. M. Herzinger, “Generalized transmission ellipsometry for twisted biaxial dielectric media: application to chiral liquid crystals,” J. Opt. Soc. Am. A 13, 1930–1940 (1996).
  48. A. Raymond, J. L. Robert, and C. Bernard, “The electron effective mass in heavily doped GaAs,” J. Phys. C 12, 2289–2293 (1979).

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