Abstract

The optical Hall effect is a physical phenomenon that describes the occurrence of magnetic-field-induced dielectric displacement at optical wavelengths, transverse and longitudinal to the incident electric field, and analogous to the static electrical Hall effect. The electrical Hall effect and certain cases of the optical Hall effect observations can be explained by extensions of the classic Drude model for the transport of electrons in metals. The optical Hall effect is most useful for characterization of electrical properties in semiconductors. Among many advantages, while the optical Hall effect dispenses with the need of electrical contacts, electrical material properties such as effective mass and mobility parameters, including their anisotropy as well as carrier type and density, can be determined from the optical Hall effect. Measurement of the optical Hall effect can be performed within the concept of generalized ellipsometry at an oblique angle of incidence. In this paper, we review and discuss physical model equations, which can be used to calculate the optical Hall effect in single- and multiple-layered structures of semiconductor materials. We define the optical Hall effect dielectric function tensor, demonstrate diagonalization approaches, and show requirements for the optical Hall effect tensor from energy conservation. We discuss both continuum and quantum approaches, and we provide a brief description of the generalized ellipsometry concept, the Mueller matrix calculus, and a 4×4 matrix algebra to calculate data accessible by experiment. In a follow-up paper, we will discuss strategies and approaches for experimental data acquisition and analysis.

© 2016 Optical Society of America

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  134. Conceptually, a magneto electric optical Hall effect also may exist, where a current driven by the time-harmonic electric field component, under the influence of an external magnetic field, produces in addition to, or separately from a magneto-optic dielectric displacement, a magnetization response.
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  141. A common requirement in theoretical studies of the electromagnetic response of matter consists in the imposition that a specific medium should be Lorentz reciprocal. For a dielectric medium (the magnetic susceptibility tensor being diagonal and unity), this means that the dielectric tensor is equal to its transposed form. The magnetized plasma and more general types of gyrotropic mediums belong to the most prominent representatives of nonreciprocal mediums. A gyrotropic material is a material in which left- and right-rotating elliptical polarizations can propagate at different speeds. The gyrotropic effect caused by a quasi-static magnetic field breaks the time-reversal symmetry as well as the Lorentz reciprocity. For more information see, for example, [102].
  142. Corresponding expressions for arbitrary orientations of the magnetic field are given by EC′=Aφ,θ,ψ(R)EC, with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.
  143. Corresponding transformation matrices for arbitrary orientations of the magnetic field are given by (A(C))=(Aϕ,θ,ψ(R))−1A(C)Aϕ,θ,ψ(R), with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.
  144. For ΨE=π/4 and ΔE=π/2, the elliptic eigensystem is equivalent to the circular eigensystem A(E)=A(C).
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  146. Note that Eq. (17) must hold for any E0; thus, Eq. (17) can be stated for each λk separately.
  147. In the following equation the Einstein notation is used, and the covariance and contravariance are ignored because all coordinate systems are Cartesian. The summation is only executed over pairs of lower indices.
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  158. Note that the constant but generally complex amplitude parameter in Eq. (30) also may be augmented with a frequency-dependent imaginary part in order to represent the effect of an harmonic coupling. See also [150].
  159. The same formalism can be used in case of holes but with a different effective mass parameter.
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  164. Note that electric field vectors E contain four independent pieces of information if the plane wave is fully coherent and time harmonic. Representing time averages over infinite observation times, the four parameters can be used to characterize the electric field amplitude, absolute phase, ellipticity, and orientation of the polarization ellipse.
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  168. Note the four independent pieces of information contained in the Stokes vector. The four parameters can be used to characterize the total light intensity, degree of polarization, ellipticity, and orientation of the polarization ellipse.
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  177. For example, j=2→{k,l,m}={0,1,3}.
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2016 (4)

M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, “Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals,” Phys. Rev. B 93, 125209 (2016).
[Crossref]

A. Parisini and R. Fornari, “Analysis of the scattering mechanisms controlling electron mobility in β−Ga2O3 crystals,” Semicond. Sci. Technol. 31, 035023 (2016).
[Crossref]

B. F. Spencer, W. F. Smith, M. T. Hibberd, P. Dawson, M. Beck, A. Bartels, I. Guiney, C. J. Humphreys, and D. M. Graham, “Terahertz cyclotron resonance spectroscopy of an AlGaN/GaN heterostructure using a high-field pulsed magnet and an asynchronous optical sampling technique,” Appl. Phys. Lett. 108, 212101 (2016).
[Crossref]

J. A. Curtis, T. Tokumoto, A. T. Hatke, J. G. Cherian, J. L. Reno, S. A. McGill, D. Karaiskaj, and D. J. Hilton, “Cyclotron decay time of a two-dimensional electron gas from 0.4 to 100 K,” Phys. Rev. B 93, 155437 (2016).
[Crossref]

2015 (3)

L. Wu, W.-K. Tse, M. Brahlek, C. Morris, R. V. Aguilar, N. Koirala, S. Oh, and N. Armitage, “High-resolution Faraday rotation and electron-phonon coupling in surface states of the bulk-insulating topological insulator Cu0.02Bi2Se3,” Phys. Rev. Lett. 115, 217602 (2015).
[Crossref]

S. Knight, S. Schöche, V. Darakchieva, P. Kühne, J.-F. Carlin, N. Grandjean, C. M. Herzinger, M. Schubert, and T. Hofmann, “Cavity-enhanced optical Hall effect in two-dimensional free charge carrier gases detected at terahertz frequencies,” Opt. Lett. 40, 2688–2691 (2015).
[Crossref]

Z. Jin, A. Tkach, F. Casper, V. Spetter, H. Grimm, A. Thomas, T. Kampfrat, M. Bonn, M. Kläui, and D. Turchinovich, “Accessing the fundamentals of magnetotransport in metals with terahertz probes,” Nat. Phys. 11, 761–766 (2015).
[Crossref]

2014 (5)

P. Kühne, C. M. Herzinger, M. Schubert, J. A. Woollam, and T. Hofmann, “An integrated mid-infrared, far-infrared and terahertz optical Hall effect instrument,” Rev. Sci. Instrum. 85, 071301 (2014).
[Crossref]

M. Higashiwaki, K. Sasaki, A. Kuramata, T. Masui, and S. Yamakoshi, “Development of gallium oxide power devices,” Phys. Stat. Solidi A 211, 21–26 (2014).
[Crossref]

O. Arteaga, M. Baldrìs, J. Antó, A. Canillas, E. Pascual, and E. Bertran, “Mueller matrix microscope with a dual continuous rotating compensator setup and digital demodulation,” Appl. Opt. 53, 2236–2245 (2014).
[Crossref]

Y. Lubashevsky, L. Pan, T. Kirzhner, G. Koren, and N. Armitage, “Optical birefringence and dichroism of cuprate superconductors in the THz regime,” Phys. Rev. Lett. 112, 147001 (2014).
[Crossref]

J. Lloyd-Hughes, “Terahertz spectroscopy of quantum 2D electron systems,” J. Phys. D 47, 374006 (2014).
[Crossref]

2013 (7)

T. Nagashima, M. Tani, and M. Hangyo, “Polarization-sensitive THz-TDS and its application to anisotropy sensing,” J. Infrared Millim. Terahertz Waves 34, 740–775 (2013).
[Crossref]

N. Ubrig, I. Crassee, J. Levallois, I. O. Nedoliuk, F. Fromm, M. Kaiser, T. Seyller, and A. B. Kuzmenko, “Fabry–Perot enhanced Faraday rotation in graphene,” Opt. Express 21, 24736–24741 (2013).
[Crossref]

K. Sasaki, M. Higashiwaki, A. Kuramata, T. Masui, and S. Yamakoshi, “MBE grown Ga2O3 and its power device applications,” J. Cryst. Growth 378, 591–595 (2013).
[Crossref]

M.-H. Kim, T. Tanaka, C. T. Ellis, A. Mukherjee, G. Acbas, I. Ohkubo, H. Christen, D. Mandrus, H. Kontani, and J. Cerne, “Infrared anomalous Hall effect in CaxSr1−xRuO3 films,” Phys. Rev. B 88, 155101 (2013).
[Crossref]

S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, “Electron effective mass in Al0.72Ga0.28N alloys determined by mid-infrared optical Hall effect,” Appl. Phys. Lett. 103, 212107 (2013).
[Crossref]

P. Kühne, V. Darakchieva, R. Yakimova, J. D. Tedesco, R. L. Myers-Ward, C. R. Eddy, D. K. Gaskill, C. M. Herzinger, J. A. Woollam, M. Schubert, and T. Hofmann, “Polarization selection rules for inter-landau-level transitions in epitaxial graphene revealed by the infrared optical Hall effect,” Phys. Rev. Lett. 111, 077402 (2013).
[Crossref]

S. Schöche, T. Hofmann, V. Darakchieva, N. B. Sedrine, X. Wang, A. Yoshikawa, and M. Schubert, “Infrared to vacuum-ultraviolet ellipsometry and optical Hall-effect study of free-charge carrier parameters in Mg-doped InN,” J. Appl. Phys. 113, 013502 (2013).
[Crossref]

2012 (9)

T. Hofmann, P. Kühne, S. Schöche, J.-T. Chen, U. Forsberg, E. Janzén, N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, and V. Darakchieva, “Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 101, 192102 (2012).
[Crossref]

N. A. Goncharuk, L. Nádvorník, C. Faugeras, M. Orlita, and L. Smrčka, “Infrared magnetospectroscopy of graphite in tilted fields,” Phys. Rev. B 86, 155409 (2012).
[Crossref]

B. N. Szafranek, G. Fiori, D. Schall, D. Neumaier, and H. Kurz, “Current saturation and voltage gain in bilayer graphene field effect transistors,” Nano Lett. 12, 1324–1328 (2012).
[Crossref]

D. Molter, G. Torosyan, G. Ballon, L. Drigo, R. Beigang, and J. Léotin, “Step-scan time-domain terahertz magneto-spectroscopy,” Opt. Express 20, 5993–6002 (2012).
[Crossref]

C. M. Morris, R. V. Aguilar, A. V. Stier, and N. P. Armitage, “Polarization modulation time-domain terahertz polarimetry,” Opt. Express 20, 12303–12317 (2012).
[Crossref]

D. K. George, A. V. Stier, C. T. Ellis, B. D. McCombe, J. Černe, and A. G. Markelz, “Terahertz magneto-optical polarization modulation spectroscopy,” J. Opt. Soc. Am. B 29, 1406–1412 (2012).
[Crossref]

O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration,” Appl. Opt. 51, 6805–6817 (2012).
[Crossref]

M. Neshat and N. P. Armitage, “Terahertz time-domain spectroscopic ellipsometry: instrumentation and calibration,” Opt. Express 20, 29063–29075 (2012).
[Crossref]

T. Morimoto, M. Koshino, and H. Aoki, “Faraday rotation in bilayer and trilayer graphene in the quantum Hall regime,” Phys. Rev. B 86, 155426 (2012).
[Crossref]

2011 (10)

M. Orlita, C. Faugeras, R. Grill, A. Wysmolek, W. Strupinski, C. Berger, W. A. de Heer, G. Martinez, and M. Potemski, “Carrier scattering from dynamical magnetoconductivity in quasi-neutral epitaxial graphene,” Phys. Rev. Lett. 107, 216603 (2011).
[Crossref]

K. Järrendahl and B. Kahr, “Hans Mueller (1900-1965),” Woollam Annual Newsletter 2011(11), 8–9 (2011).

K. Yatsugi, N. Matsumoto, T. Nagashima, and M. Hangyo, “Transport properties of free carriers in semiconductors studied by terahertz time domain magneto-optical ellipsometry,” Appl. Phys. Lett. 98, 212108 (2011).
[Crossref]

B. Kumara and S.-W. Kim, “Recent advances in power generation through piezoelectric nanogenerators,” J. Mater. Chem. 21, 18946–18958 (2011).
[Crossref]

T. Hofmann, D. Schmidt, A. Boosalis, P. Kühne, R. Skomski, C. M. Herzinger, J. A. Woollam, M. Schubert, and E. Schubert, “Thz dielectric anisotropy of metal slanted columnar thin films,” Appl. Phys. Lett. 99, 081903 (2011).
[Crossref]

I. Crassee, J. Levallois, D. van der Marel, A. L. Walter, T. Seyller, and A. B. Kuzmenko, “Multicomponent magneto-optical conductivity of multilayer graphene on SiC,” Phys. Rev. B 84, 035103 (2011).
[Crossref]

S. Schöche, J. Shi, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, W. J. Schaff, L. F. Eastman, M. Schubert, and T. Hofmann, “Terahertz optical-Hall effect characterization of two-dimensional electron gas properties in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 98, 092103 (2011).
[Crossref]

T. Hofmann, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, D. K. Gaskill, J. L. Tedesco, and M. Schubert, “Hole-channel conductivity in epitaxial graphene determined by terahertz optical-Hall effect and midinfrared ellipsometry,” Appl. Phys. Lett. 98, 041906 (2011).
[Crossref]

P. Kühne, T. Hofmann, C. M. Herzinger, and M. Schubert, “Terahertz frequency optical-Hall effect in multiple valley band materials,” Thin Solid Films 519, 2613–2616 (2011).

T. Hofmann, C. M. Herzinger, J. L. Tedesco, D. K. Gaskill, J. A. Woollam, and M. Schubert, “Terahertz ellipsometry and terahertz optical-Hall effect,” Thin Solid Films 519, 2593–2600 (2011).
[Crossref]

2010 (5)

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2010).
[Crossref]

Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, and R. Shimano, “Optical Hall effect in the integer quantum Hall regime,” Phys. Rev. Lett. 104, 256802 (2010).
[Crossref]

T. Hofmann, C. M. Herzinger, A. Boosalis, T. E. Tiwald, J. A. Woollam, and M. Schubert, “Variable-wavelength frequency-domain terahertz ellipsometry,” Rev. Sci. Instrum. 81, 023101 (2010).
[Crossref]

X. Wang, D. J. Hilton, J. L. Reno, D. M. Mittleman, and J. Kono, “Direct measurement of cyclotron coherence times of high-mobility two-dimensional electron gases,” Opt. Express 18, 12354–12361 (2010).
[Crossref]

D. Molter, F. Ellrich, T. Weinland, S. George, M. Goiran, F. Keilmann, R. Beigang, and J. Lèotin, “High-speed terahertz time-domain spectroscopy of cyclotron resonance in pulsed magnetic field,” Opt. Express 18, 26163–26168 (2010).
[Crossref]

2009 (1)

M. Orlita, C. Faugeras, G. Martinez, D. Maude, J. Schneider, M. Sprinkle, C. Berger, W. de Heer, and M. Potemski, “Magneto-transmission of multi-layer epitaxial graphene and bulk graphite: a comparison,” Solid State Commun. 149, 1128–1131 (2009).
[Crossref]

2008 (8)

O. L. Berman, G. Gumbs, and Y. E. Lozovik, “Magnetoplasmons in layered graphene structures,” Phys. Rev. B 78, 085401 (2008).
[Crossref]

M. Koshino and T. Ando, “Magneto-optical properties of multilayer graphene,” Phys. Rev. B 77, 115313 (2008).
[Crossref]

W. Weber, S. Seidl, V. Bel’akov, L. Golub, S. Danilov, E. Ivchenko, W. Prettl, Z. Kvon, H.-I. Cho, J.-H. Lee, and S. Ganicheva, “Magneto-gyrotropic photogalvanic effects in GaN/AlGaN two-dimensional systems,” Solid State Commun. 145, 56–60 (2008).
[Crossref]

T. Hofmann, C. von Middendorff, V. Gottschalch, and M. Schubert, “Optical Hall effect studies on modulation-doped AlxGa1−xAs:Si/GaAs quantum wells,” Phys. Stat. Solidi C 5, 1386–1390 (2008).
[Crossref]

T. Hofmann, V. Darakchieva, B. Monemar, H. Lu, W. Schaff, and M. Schubert, “Optical Hall effect in hexagonal InN,” J. Electron. Mater. 37, 611–615 (2008).
[Crossref]

T. Hofmann, C. M. Herzinger, C. Krahmer, K. Streubel, and M. Schubert, “The optical Hall effect,” Phys. Stat. Solidi A 205, 779–783 (2008).
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Y. Ikebe and R. Shimano, “Characterization of doped silicon in low carrier density region by terahertz frequency Faraday effect,” Appl. Phys. Lett. 92, 012111 (2008).
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M. Orlita, C. Faugeras, P. Plochocka, P. Neugebauer, G. Martinez, D. K. Maude, A.-L. Barra, M. Sprinkle, C. Berger, W. A. de Heer, and M. Potemski, “Approaching the Dirac point in high-mobility multilayer epitaxial graphene,” Phys. Rev. Lett. 101, 267601 (2008).
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2007 (9)

X. Wang, D. J. Hilton, L. Rein, D. M. Mittleman, J. Kono, and J. L. Reno, “Coherent THz cyclotron oscillations in a two-dimensional electron gas,” Opt. Lett. 32, 1845–1847 (2007).
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M. L. Sadowski, G. Martynez, M. Potemski, C. Berger, and W. A. D. Heer, “Magneto-spectroscopy of epitaxial graphene,” Int. J. Mod. Phys. B 21, 1145–1154 (2007).
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A. Geim and K. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
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T. Hofmann, M. Schubert, G. Leibiger, and V. Gottschalch, “Electron effective mass and phonon modes in GaAs incorporating Boron and Indium,” Appl. Phys. Lett. 90, 182110 (2007).
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M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
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H. Sumikura, T. Nagashima, H. Kitahara, and M. Hangyo, “Development of a cryogen-free terahertz time-domain magnetooptical measurement system,” Jpn. J. Appl. Phys. 46, 1739–1744 (2007).
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D. S. L. Abergel and V. I. Fal’ko, “Optical and magneto-optical far-infrared properties of bilayer graphene,” Phys. Rev. B 75, 155430 (2007).
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M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer, “Infrared magnetospectroscopy of two-dimensional electrons in epitaxial graphene,” AIP Conf. Proc. 893, 619–620 (2007).

M. Krames, O. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” IEEE J. Disp. Technol. 3, 160–175 (2007).
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2006 (6)

O. Morikawa, A. Quema, S. Nashima, H. Sumikura, T. Nagashima, and M. Hangyo, “Faraday ellipticity and Faraday rotation of a doped-silicon wafer studied by terahertz time-domain spectroscopy,” J. Appl. Phys. 100, 033105 (2006).
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M. Schubert, “Another century of ellipsometry,” Ann. Phys. 15, 480–497 (2006).
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T. Hofmann, T. Chavdarov, V. Darakchieva, H. Lu, W. J. Schaff, and M. Schubert, “Anisotropy of the Γ -point effective mass and mobility in hexagonal InN,” Phys. Stat. Solidi C 3, 1854–1857 (2006).
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T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
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T. Hofmann, U. Schade, W. Eberhardt, C. M. Herzinger, P. Esquinazi, and M. Schubert, “Terahertz magnetooptic generalized ellipsometry using synchrotron and black-body radiation,” Rev. Sci. Instrum. 77, 063902 (2006).
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G. E. Jellison, J. D. Hunn, and C. M. Rouleau, “Normal-incidence generalized ellipsometry using the two-modulator generalized ellipsometry microscope,” Appl. Opt. 45, 5479–5488 (2006).
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2005 (1)

T. Hofmann, M. Schubert, C. von Middendorff, G. Leibiger, V. Gottschalch, C. M. Herzinger, A. Lindsay, and E. O’Reilly, “The inertial-mass scale for free-charge-carriers in semiconductor heterostructures,” AIP Conf. Proc. 772, 455–456 (2005).
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2004 (3)

M. Schubert, T. Hofmann, and C. M. Herzinger, “Far-infrared magnetooptic generalized ellipsometry: determination of free-charge-carrier parameters in semiconductor thin film structures,” Thin Solid Films 455–456, 563–570 (2004).
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M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, “Generalized ellipsometry for orthorhombic, absorbing materials: dielectric functions, phonon modes and band-to-band transitions of Sb2S3,” Thin Solid Films 455–456, 619–623 (2004).

Y. Ino, R. Shimano, Y. Svirko, and M. Kuwata-Gonokami, “Terahertz time domain magneto-optical ellipsometry in reflection geometry,” Phys. Rev. B 70, 155101 (2004).
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2003 (5)

M. Schubert, T. Hofmann, and C. 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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T. Hofmann, M. Schubert, C. M. Herzinger, and I. Pietzonka, “Far-infrared-magneto-optic ellipsometry characterization of free-charge-carrier properties in highly disordered n-type Al0.19Ga0.33In0.48P,” Appl. Phys. Lett. 82, 3463–3465 (2003).
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T. Hofmann, M. Grundmann, C. M. Herzinger, M. Schubert, and W. Grill, “Far-infrared magnetooptical generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Mater. Res. Soc. Symp. Proc. 744, M5.32.1–M5.32.16 (2003).

J. F. Wager, “Transparent electronics,” Science 300, 1245–1246 (2003).
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A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
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2002 (4)

R. Shimano, Y. Ino, Y. P. Svirko, and M. Kuwata-Gonokami, “Terahertz frequency Hall measurement by magneto-optical Kerr spectroscopy in InAs,” Appl. Phys. Lett. 81, 199–201 (2002).
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T. Hofmann, V. Gottschalch, and M. Schubert, “Far-infrared dielectric anisotropy and phonon modes in spontaneously CuPt-ordered Ga0.52In0.48P,” Phys. Rev. B 66, 195204 (2002).

T. Hofmann, M. Schubert, and C. M. Herzinger, “Far-infrared magneto-optic generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Proc. SPIE 4779, 90–97 (2002).
[Crossref]

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
[Crossref]

2001 (3)

M. A. Zudov, R. R. Du, J. A. Simmons, and J. L. Reno, “Shubnikov-de Haas-like oscillations in millimeter wave photoconductivity in a high-mobility two-dimensional electron gas,” Phys. Rev. B 64, 201311 (2001).
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Y.-F. Wu, D. Kapolnek, J. Ibbetson, P. Parikh, B. Keller, and U. K. Mishra, “Very-high power density AlGaN/GaN HEMTs,” IEEE Trans. Electron Devices 48, 586–590 (2001).
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L. F. Eastman, V. Tilak, J. Smart, B. Green, E. Chumbes, R. Dimitrov, H. Kim, O. Ambacher, N. Weimann, T. Prunty, M. Murphy, W. J. Schaff, and J. Shealy, “Undoped AlGaN/GaN HEMTs for microwave power amplification,” IEEE Trans. Electron Devices 48, 479–485 (2001).
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2000 (4)

O. Morikawa, M. Tonouchi, and M. Hangyo, “A cross-correlation spectroscopy in subterahertz region using an incoherent light source,” Appl. Phys. Lett. 76, 1519–1521 (2000).
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T. E. Tiwald and M. Schubert, “Measurement of rutile TiO2 dielectric tensor from 0.148 to 33 μm using generalized ellipsometry,” Proc. SPIE 4103, 19–29 (2000).
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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).
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W. Xu, L. Wood, and T. Golding, “Optical degeneracies in anisotropic layered media: Treatment of singularities in a 4 × 4 matrix formalism,” Phys. Rev. B 61, 1740–1743 (2000).
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1999 (1)

O. Morikawa, M. Tonouchi, and M. Hangyo, “Sub-THz spectroscopic system using a multimode laser diode and photoconductive antenna,” Appl. Phys. Lett. 75, 3772–3774 (1999).
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1998 (1)

G. E. Moore, “Cramming more components onto integrated circuits,” Proc. IEEE 86, 82–85 (1998).

1997 (3)

W. Knap, S. Contreras, H. Alause, C. Skierbiszewski, J. Camassel, M. Dyakonov, J. L. Robert, J. Yang, Q. Chen, M. A. Khan, M. L. Sadowski, S. Huant, F. H. Yang, M. Goiran, J. Leotin, and M. S. Shur, “Cyclotron resonance and quantum hall effect studies of the two-dimensional electron gas confined at the GaN/AlGaN interface,” Appl. Phys. Lett. 70, 2123–2125 (1997).
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G. E. Jellison and F. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36, 8184–8189 (1997).
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G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
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1996 (3)

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
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M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
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Y. J. Wang, R. Kaplan, H. K. Ng, K. Doverspike, D. K. Gaskill, T. Ikedo, I. Akasaki, and H. Amono, “Magneto-optical studies of GaN and GaN/AlxGa1−xN: donor Zeeman spectroscopy and two dimensional electron gas cyclotron resonance,” J. Appl. Phys. 79, 8007–8010 (1996).
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1994 (1)

D. Some and A. Nurmikko, “Real-time electron cyclotron oscillations observed by terahertz techniques in semiconductor heterostructures,” Appl. Phys. Lett. 65, 3377–3379 (1994).
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1993 (1)

J. R. Meyer, C. A. Hoffman, F. J. Bartoli, D. A. Arnold, S. Sivananthan, and J. P. Faurie, “Methods for magnetotransport characterization of IR detector materials,” Semicond. Sci. Technol. 8, 805–823 (1993).
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1988 (1)

1978 (1)

1977 (2)

W. W. Toy, M. S. Dresselhaus, and G. Dresselhaus, “Minority carriers in graphite and the H-point magnetoreflection spectra,” Phys. Rev. B 15, 4077–4090 (1977).
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1974 (3)

K. Suzuki and J. C. Hensel, “Quantum resonances in the valence bands of germanium,” Phys. Rev. B 9, 4184–4218 (1974).
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F. Gervais and B. Piriou, “Anharmonicity in several-polar-mode crystals: adjusting phonon self-energy of LO and TO modes in Al2O3 and TiO2 to fit infrared reflectivity,” J. Phys. C 7, 2374–2386 (1974).

R. F. Wallis, J. J. Brion, E. Burstein, and A. Hartstein, “Theory of surface polaritons in anisotropic dielectric media with application to surface magnetoplasmons in semiconductors,” Phys. Rev. B 9, 3424–3437 (1974).
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1973 (1)

J. J. Brion, R. F. Wallis, A. Hartstein, and E. Burstein, “Interaction of surface magnetoplasmons and surface optical phonons in polar semiconductors,” Surf. Sci. 34, 73–80 (1973).
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1972 (3)

J. J. Brion, R. F. Wallis, A. Hartstein, and E. Burstein, “Theory of magnetoplasmons in semiconductors,” Phys. Rev. Lett. 28, 1455–1458 (1972).
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D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
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B. Rheinländer, H. Neumann, P. Fischer, and G. Kühn, “Anisotropic effective masses of electrons in AlAs,” Phys. Stat. Solidi B 49, K167–K169 (1972).
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1971 (1)

B. Rheinländer and H. Neumann, “Faraday rotation in n-type AlAs,” Phys. Stat. Solidi B 45, K9–K13 (1971).
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1969 (2)

B. Rheinländer, “Der Einfluss der Energieverteilung der Ladungsträger in Halbleitern auf die Bestimmung ihrer effektiven Masse aus der Infrarot-Plasma-Reflexion,” Phys. Lett. 29A, 420–421 (1969).
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R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
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1968 (2)

D. W. Berreman and F. C. Unterwald, “Adjusting poles and zeros of dielectric dispersion to fit reststrahlen of PrCl3 and LaCl3,” Phys. Rev. 174, 791–799 (1968).
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P. R. Schroeder, M. S. Dresselhaus, and A. Javan, “Location of electron and hole carriers in graphite from laser magnetoreflection data,” Phys. Rev. Lett. 20, 1292–1295 (1968).
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1965 (2)

H. H. Tippins, “Optical absorption and photoconductivity in the band edge of β−Ga2O3,” Phys. Rev. 140A, A316–A319 (1965).
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M. E. Brodwin and R. J. Vernon, “Free-carrier magneto-microwave Kerr effect in semiconductors,” Phys. Rev. 140, A1390–A1400 (1965).
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1964 (2)

N. Hindley, “Cyclotron resonance and the free-carrier magneto-optical properties of a semiconductor,” Phys. Stat. Solidi B 7, 67–80 (1964).
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A. S. Barker, “Transverse and longitudinal optic mode study in MgF2 and ZnF2,” Phys. Rev. 136, A1290–A1295 (1964).
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1961 (3)

M. Cardona, “Electron effective masses of InAs and GaAs as a function of temperature and doping,” Phys. Rev. 121, 752–758 (1961).
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E. D. Palik, S. Teitler, and R. F. Wallis, “Free carrier cyclotron resonance, Faraday rotation, and Voigt double refraction in compound semiconductors,” J. Appl. Phys. 32, 2132–2136 (1961).
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G. B. Wright and B. Lax, “Magnetoreflection experiments in intermetallics,” J. Appl. Phys. 32, 2113–2117 (1961).
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1958 (1)

P. Nozières, “Cyclotron resonance in graphite,” Phys. Rev. 109, 1510–1521 (1958).
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1957 (1)

W. G. Spitzer and H. Y. Fan, “Determination of optical constants and carrier effective mass of semiconductors,” Phys. Rev. 106, 882–890 (1957).
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1956 (1)

J. K. Galt, W. A. Yager, and H. W. Dail, “Cyclotron resonance effects in graphite,” Phys. Rev. 103, 1586–1587 (1956).
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1955 (1)

G. Dresselhaus, A. F. Kip, and C. Kittel, “Cyclotron resonance of electrons and holes in silicon and germanium crystals,” Phys. Rev. 98, 368–384 (1955).
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1952 (1)

R. Roy, V. G. Hill, and E. F. Osborn, “Polymorphism of Ga2O3 and the system Ga2O3,” J. Am. Chem. Soc. 74, 719–722 (1952).
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1948 (1)

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1930 (1)

L. Landau, “Diamagnetismus der Metalle,” Z. Phys. 64, 629–637 (1930).

1900 (2)

P. Drude, “Zur Ionentheorie der Metalle,” Physikal. Z. 1, 161–165 (1900).

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1890 (1)

P. Drude, “Bestimmung der optischen Constanten der Metalle,” Ann. Phys. Chem. 275, 481–554 (1890).
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1888 (1)

P. Drude, “Beobachtungen über die Reflexion des Lichtes am Antimonglanz,” Ann. Phys. 270, 489–531 (1888).
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1887 (1)

P. Drude, “Üeber die Gesetze der Reflexion und Brechung des Lichtes an der Grenze absorbirender Krystalle,” Ann. Phys. 268, 584–625 (1887).
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1879 (1)

E. H. Hall, “On a new action of the magnet on electric currents,” Amer. J. Math. 2, 287–292 (1879).
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D. S. L. Abergel and V. I. Fal’ko, “Optical and magneto-optical far-infrared properties of bilayer graphene,” Phys. Rev. B 75, 155430 (2007).
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Acbas, G.

M.-H. Kim, T. Tanaka, C. T. Ellis, A. Mukherjee, G. Acbas, I. Ohkubo, H. Christen, D. Mandrus, H. Kontani, and J. Cerne, “Infrared anomalous Hall effect in CaxSr1−xRuO3 films,” Phys. Rev. B 88, 155101 (2013).
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Agarwal, K. C.

T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
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Aguilar, R. V.

L. Wu, W.-K. Tse, M. Brahlek, C. Morris, R. V. Aguilar, N. Koirala, S. Oh, and N. Armitage, “High-resolution Faraday rotation and electron-phonon coupling in surface states of the bulk-insulating topological insulator Cu0.02Bi2Se3,” Phys. Rev. Lett. 115, 217602 (2015).
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C. M. Morris, R. V. Aguilar, A. V. Stier, and N. P. Armitage, “Polarization modulation time-domain terahertz polarimetry,” Opt. Express 20, 12303–12317 (2012).
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Akasaki, I.

A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
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Y. J. Wang, R. Kaplan, H. K. Ng, K. Doverspike, D. K. Gaskill, T. Ikedo, I. Akasaki, and H. Amono, “Magneto-optical studies of GaN and GaN/AlxGa1−xN: donor Zeeman spectroscopy and two dimensional electron gas cyclotron resonance,” J. Appl. Phys. 79, 8007–8010 (1996).
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W. Knap, S. Contreras, H. Alause, C. Skierbiszewski, J. Camassel, M. Dyakonov, J. L. Robert, J. Yang, Q. Chen, M. A. Khan, M. L. Sadowski, S. Huant, F. H. Yang, M. Goiran, J. Leotin, and M. S. Shur, “Cyclotron resonance and quantum hall effect studies of the two-dimensional electron gas confined at the GaN/AlGaN interface,” Appl. Phys. Lett. 70, 2123–2125 (1997).
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Amano, H.

A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
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L. F. Eastman, V. Tilak, J. Smart, B. Green, E. Chumbes, R. Dimitrov, H. Kim, O. Ambacher, N. Weimann, T. Prunty, M. Murphy, W. J. Schaff, and J. Shealy, “Undoped AlGaN/GaN HEMTs for microwave power amplification,” IEEE Trans. Electron Devices 48, 479–485 (2001).
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Amono, H.

Y. J. Wang, R. Kaplan, H. K. Ng, K. Doverspike, D. K. Gaskill, T. Ikedo, I. Akasaki, and H. Amono, “Magneto-optical studies of GaN and GaN/AlxGa1−xN: donor Zeeman spectroscopy and two dimensional electron gas cyclotron resonance,” J. Appl. Phys. 79, 8007–8010 (1996).
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T. Morimoto, M. Koshino, and H. Aoki, “Faraday rotation in bilayer and trilayer graphene in the quantum Hall regime,” Phys. Rev. B 86, 155426 (2012).
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L. Wu, W.-K. Tse, M. Brahlek, C. Morris, R. V. Aguilar, N. Koirala, S. Oh, and N. Armitage, “High-resolution Faraday rotation and electron-phonon coupling in surface states of the bulk-insulating topological insulator Cu0.02Bi2Se3,” Phys. Rev. Lett. 115, 217602 (2015).
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J. R. Meyer, C. A. Hoffman, F. J. Bartoli, D. A. Arnold, S. Sivananthan, and J. P. Faurie, “Methods for magnetotransport characterization of IR detector materials,” Semicond. Sci. Technol. 8, 805–823 (1993).
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M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
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M. Orlita, C. Faugeras, P. Plochocka, P. Neugebauer, G. Martinez, D. K. Maude, A.-L. Barra, M. Sprinkle, C. Berger, W. A. de Heer, and M. Potemski, “Approaching the Dirac point in high-mobility multilayer epitaxial graphene,” Phys. Rev. Lett. 101, 267601 (2008).
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J. R. Meyer, C. A. Hoffman, F. J. Bartoli, D. A. Arnold, S. Sivananthan, and J. P. Faurie, “Methods for magnetotransport characterization of IR detector materials,” Semicond. Sci. Technol. 8, 805–823 (1993).
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M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
[Crossref]

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J. R. Meyer, C. A. Hoffman, F. J. Bartoli, D. A. Arnold, S. Sivananthan, and J. P. Faurie, “Methods for magnetotransport characterization of IR detector materials,” Semicond. Sci. Technol. 8, 805–823 (1993).
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Hofmann, T.

M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, “Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals,” Phys. Rev. B 93, 125209 (2016).
[Crossref]

S. Knight, S. Schöche, V. Darakchieva, P. Kühne, J.-F. Carlin, N. Grandjean, C. M. Herzinger, M. Schubert, and T. Hofmann, “Cavity-enhanced optical Hall effect in two-dimensional free charge carrier gases detected at terahertz frequencies,” Opt. Lett. 40, 2688–2691 (2015).
[Crossref]

P. Kühne, C. M. Herzinger, M. Schubert, J. A. Woollam, and T. Hofmann, “An integrated mid-infrared, far-infrared and terahertz optical Hall effect instrument,” Rev. Sci. Instrum. 85, 071301 (2014).
[Crossref]

P. Kühne, V. Darakchieva, R. Yakimova, J. D. Tedesco, R. L. Myers-Ward, C. R. Eddy, D. K. Gaskill, C. M. Herzinger, J. A. Woollam, M. Schubert, and T. Hofmann, “Polarization selection rules for inter-landau-level transitions in epitaxial graphene revealed by the infrared optical Hall effect,” Phys. Rev. Lett. 111, 077402 (2013).
[Crossref]

S. Schöche, T. Hofmann, V. Darakchieva, N. B. Sedrine, X. Wang, A. Yoshikawa, and M. Schubert, “Infrared to vacuum-ultraviolet ellipsometry and optical Hall-effect study of free-charge carrier parameters in Mg-doped InN,” J. Appl. Phys. 113, 013502 (2013).
[Crossref]

S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, “Electron effective mass in Al0.72Ga0.28N alloys determined by mid-infrared optical Hall effect,” Appl. Phys. Lett. 103, 212107 (2013).
[Crossref]

T. Hofmann, P. Kühne, S. Schöche, J.-T. Chen, U. Forsberg, E. Janzén, N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, and V. Darakchieva, “Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 101, 192102 (2012).
[Crossref]

S. Schöche, J. Shi, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, W. J. Schaff, L. F. Eastman, M. Schubert, and T. Hofmann, “Terahertz optical-Hall effect characterization of two-dimensional electron gas properties in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 98, 092103 (2011).
[Crossref]

T. Hofmann, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, D. K. Gaskill, J. L. Tedesco, and M. Schubert, “Hole-channel conductivity in epitaxial graphene determined by terahertz optical-Hall effect and midinfrared ellipsometry,” Appl. Phys. Lett. 98, 041906 (2011).
[Crossref]

T. Hofmann, C. M. Herzinger, J. L. Tedesco, D. K. Gaskill, J. A. Woollam, and M. Schubert, “Terahertz ellipsometry and terahertz optical-Hall effect,” Thin Solid Films 519, 2593–2600 (2011).
[Crossref]

T. Hofmann, D. Schmidt, A. Boosalis, P. Kühne, R. Skomski, C. M. Herzinger, J. A. Woollam, M. Schubert, and E. Schubert, “Thz dielectric anisotropy of metal slanted columnar thin films,” Appl. Phys. Lett. 99, 081903 (2011).
[Crossref]

P. Kühne, T. Hofmann, C. M. Herzinger, and M. Schubert, “Terahertz frequency optical-Hall effect in multiple valley band materials,” Thin Solid Films 519, 2613–2616 (2011).

T. Hofmann, C. M. Herzinger, A. Boosalis, T. E. Tiwald, J. A. Woollam, and M. Schubert, “Variable-wavelength frequency-domain terahertz ellipsometry,” Rev. Sci. Instrum. 81, 023101 (2010).
[Crossref]

T. Hofmann, C. von Middendorff, V. Gottschalch, and M. Schubert, “Optical Hall effect studies on modulation-doped AlxGa1−xAs:Si/GaAs quantum wells,” Phys. Stat. Solidi C 5, 1386–1390 (2008).
[Crossref]

T. Hofmann, V. Darakchieva, B. Monemar, H. Lu, W. Schaff, and M. Schubert, “Optical Hall effect in hexagonal InN,” J. Electron. Mater. 37, 611–615 (2008).
[Crossref]

T. Hofmann, C. M. Herzinger, C. Krahmer, K. Streubel, and M. Schubert, “The optical Hall effect,” Phys. Stat. Solidi A 205, 779–783 (2008).
[Crossref]

T. Hofmann, M. Schubert, G. Leibiger, and V. Gottschalch, “Electron effective mass and phonon modes in GaAs incorporating Boron and Indium,” Appl. Phys. Lett. 90, 182110 (2007).
[Crossref]

T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
[Crossref]

T. Hofmann, U. Schade, W. Eberhardt, C. M. Herzinger, P. Esquinazi, and M. Schubert, “Terahertz magnetooptic generalized ellipsometry using synchrotron and black-body radiation,” Rev. Sci. Instrum. 77, 063902 (2006).
[Crossref]

T. Hofmann, T. Chavdarov, V. Darakchieva, H. Lu, W. J. Schaff, and M. Schubert, “Anisotropy of the Γ -point effective mass and mobility in hexagonal InN,” Phys. Stat. Solidi C 3, 1854–1857 (2006).
[Crossref]

T. Hofmann, M. Schubert, C. von Middendorff, G. Leibiger, V. Gottschalch, C. M. Herzinger, A. Lindsay, and E. O’Reilly, “The inertial-mass scale for free-charge-carriers in semiconductor heterostructures,” AIP Conf. Proc. 772, 455–456 (2005).
[Crossref]

M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, “Generalized ellipsometry for orthorhombic, absorbing materials: dielectric functions, phonon modes and band-to-band transitions of Sb2S3,” Thin Solid Films 455–456, 619–623 (2004).

M. Schubert, T. Hofmann, and C. M. Herzinger, “Far-infrared magnetooptic generalized ellipsometry: determination of free-charge-carrier parameters in semiconductor thin film structures,” Thin Solid Films 455–456, 563–570 (2004).
[Crossref]

T. Hofmann, M. Grundmann, C. M. Herzinger, M. Schubert, and W. Grill, “Far-infrared magnetooptical generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Mater. Res. Soc. Symp. Proc. 744, M5.32.1–M5.32.16 (2003).

M. Schubert, T. Hofmann, and C. 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).
[Crossref]

T. Hofmann, M. Schubert, C. M. Herzinger, and I. Pietzonka, “Far-infrared-magneto-optic ellipsometry characterization of free-charge-carrier properties in highly disordered n-type Al0.19Ga0.33In0.48P,” Appl. Phys. Lett. 82, 3463–3465 (2003).
[Crossref]

T. Hofmann, M. Schubert, and C. M. Herzinger, “Far-infrared magneto-optic generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Proc. SPIE 4779, 90–97 (2002).
[Crossref]

M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
[Crossref]

T. Hofmann, V. Gottschalch, and M. Schubert, “Far-infrared dielectric anisotropy and phonon modes in spontaneously CuPt-ordered Ga0.52In0.48P,” Phys. Rev. B 66, 195204 (2002).

T. Hofmann, “Far-infrared spectroscopic ellipsometry on semiconductor heterostructures,” Ph.D. thesis (University of Leipzig, 2004).

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A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
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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).
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W. Knap, S. Contreras, H. Alause, C. Skierbiszewski, J. Camassel, M. Dyakonov, J. L. Robert, J. Yang, Q. Chen, M. A. Khan, M. L. Sadowski, S. Huant, F. H. Yang, M. Goiran, J. Leotin, and M. S. Shur, “Cyclotron resonance and quantum hall effect studies of the two-dimensional electron gas confined at the GaN/AlGaN interface,” Appl. Phys. Lett. 70, 2123–2125 (1997).
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M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, “Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals,” Phys. Rev. B 93, 125209 (2016).
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S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, “Electron effective mass in Al0.72Ga0.28N alloys determined by mid-infrared optical Hall effect,” Appl. Phys. Lett. 103, 212107 (2013).
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T. Hofmann, P. Kühne, S. Schöche, J.-T. Chen, U. Forsberg, E. Janzén, N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, and V. Darakchieva, “Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 101, 192102 (2012).
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S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A. Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, “Electron effective mass in Al0.72Ga0.28N alloys determined by mid-infrared optical Hall effect,” Appl. Phys. Lett. 103, 212107 (2013).
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Y.-F. Wu, D. Kapolnek, J. Ibbetson, P. Parikh, B. Keller, and U. K. Mishra, “Very-high power density AlGaN/GaN HEMTs,” IEEE Trans. Electron Devices 48, 586–590 (2001).
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M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E. Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase, and C. M. Herzinger, “Generalized ellipsometry of complex mediums in layered systems,” Proc. SPIE 4806, 264–276 (2002).
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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).
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Y.-F. Wu, D. Kapolnek, J. Ibbetson, P. Parikh, B. Keller, and U. K. Mishra, “Very-high power density AlGaN/GaN HEMTs,” IEEE Trans. Electron Devices 48, 586–590 (2001).
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T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
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M. Schubert, R. Korlacki, S. Knight, T. Hofmann, S. Schöche, V. Darakchieva, E. Janzén, B. Monemar, D. Gogova, Q.-T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, K. Goto, A. Kuramata, S. Yamakoshi, and M. Higashiwaki, “Anisotropy, phonon modes, and free charge carrier parameters in monoclinic β-gallium oxide single crystals,” Phys. Rev. B 93, 125209 (2016).
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S. Knight, S. Schöche, V. Darakchieva, P. Kühne, J.-F. Carlin, N. Grandjean, C. M. Herzinger, M. Schubert, and T. Hofmann, “Cavity-enhanced optical Hall effect in two-dimensional free charge carrier gases detected at terahertz frequencies,” Opt. Lett. 40, 2688–2691 (2015).
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A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
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T. Hofmann, C. M. Herzinger, C. Krahmer, K. Streubel, and M. Schubert, “The optical Hall effect,” Phys. Stat. Solidi A 205, 779–783 (2008).
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A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher, D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi, M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I. Akasaki, and G. Wagner, “Phonons and free-carrier properties of binary, ternary, and quaternary group-III Nitride layers measured by infrared spectroscopic ellipsometry,” Phys. Stat. Solidi C 0, 1750–1769 (2003).
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P. Kühne, V. Darakchieva, R. Yakimova, J. D. Tedesco, R. L. Myers-Ward, C. R. Eddy, D. K. Gaskill, C. M. Herzinger, J. A. Woollam, M. Schubert, and T. Hofmann, “Polarization selection rules for inter-landau-level transitions in epitaxial graphene revealed by the infrared optical Hall effect,” Phys. Rev. Lett. 111, 077402 (2013).
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S. Schöche, T. Hofmann, V. Darakchieva, N. B. Sedrine, X. Wang, A. Yoshikawa, and M. Schubert, “Infrared to vacuum-ultraviolet ellipsometry and optical Hall-effect study of free-charge carrier parameters in Mg-doped InN,” J. Appl. Phys. 113, 013502 (2013).
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S. Schöche, J. Shi, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, W. J. Schaff, L. F. Eastman, M. Schubert, and T. Hofmann, “Terahertz optical-Hall effect characterization of two-dimensional electron gas properties in AlGaN/GaN high electron mobility transistor structures,” Appl. Phys. Lett. 98, 092103 (2011).
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T. Hofmann, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, D. K. Gaskill, J. L. Tedesco, and M. Schubert, “Hole-channel conductivity in epitaxial graphene determined by terahertz optical-Hall effect and midinfrared ellipsometry,” Appl. Phys. Lett. 98, 041906 (2011).
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T. Hofmann, C. M. Herzinger, J. L. Tedesco, D. K. Gaskill, J. A. Woollam, and M. Schubert, “Terahertz ellipsometry and terahertz optical-Hall effect,” Thin Solid Films 519, 2593–2600 (2011).
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T. Hofmann, D. Schmidt, A. Boosalis, P. Kühne, R. Skomski, C. M. Herzinger, J. A. Woollam, M. Schubert, and E. Schubert, “Thz dielectric anisotropy of metal slanted columnar thin films,” Appl. Phys. Lett. 99, 081903 (2011).
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P. Kühne, T. Hofmann, C. M. Herzinger, and M. Schubert, “Terahertz frequency optical-Hall effect in multiple valley band materials,” Thin Solid Films 519, 2613–2616 (2011).

T. Hofmann, C. M. Herzinger, A. Boosalis, T. E. Tiwald, J. A. Woollam, and M. Schubert, “Variable-wavelength frequency-domain terahertz ellipsometry,” Rev. Sci. Instrum. 81, 023101 (2010).
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T. Hofmann, C. von Middendorff, V. Gottschalch, and M. Schubert, “Optical Hall effect studies on modulation-doped AlxGa1−xAs:Si/GaAs quantum wells,” Phys. Stat. Solidi C 5, 1386–1390 (2008).
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T. Hofmann, C. M. Herzinger, C. Krahmer, K. Streubel, and M. Schubert, “The optical Hall effect,” Phys. Stat. Solidi A 205, 779–783 (2008).
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T. Hofmann, M. Schubert, G. Leibiger, and V. Gottschalch, “Electron effective mass and phonon modes in GaAs incorporating Boron and Indium,” Appl. Phys. Lett. 90, 182110 (2007).
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T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, “Conduction-band electron effective mass in Zn0.87Mn0.13Se measured by terahertz and far-infrared magnetooptic ellipsometry,” Appl. Phys. Lett. 88, 042105 (2006).
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M. Schubert, “Another century of ellipsometry,” Ann. Phys. 15, 480–497 (2006).
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T. Hofmann, U. Schade, W. Eberhardt, C. M. Herzinger, P. Esquinazi, and M. Schubert, “Terahertz magnetooptic generalized ellipsometry using synchrotron and black-body radiation,” Rev. Sci. Instrum. 77, 063902 (2006).
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T. Hofmann, T. Chavdarov, V. Darakchieva, H. Lu, W. J. Schaff, and M. Schubert, “Anisotropy of the Γ -point effective mass and mobility in hexagonal InN,” Phys. Stat. Solidi C 3, 1854–1857 (2006).
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T. Hofmann, M. Schubert, C. von Middendorff, G. Leibiger, V. Gottschalch, C. M. Herzinger, A. Lindsay, and E. O’Reilly, “The inertial-mass scale for free-charge-carriers in semiconductor heterostructures,” AIP Conf. Proc. 772, 455–456 (2005).
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M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, “Generalized ellipsometry for orthorhombic, absorbing materials: dielectric functions, phonon modes and band-to-band transitions of Sb2S3,” Thin Solid Films 455–456, 619–623 (2004).

T. Hofmann, M. Grundmann, C. M. Herzinger, M. Schubert, and W. Grill, “Far-infrared magnetooptical generalized ellipsometry determination of free-carrier parameters in semiconductor thin film structures,” Mater. Res. Soc. Symp. Proc. 744, M5.32.1–M5.32.16 (2003).

M. Schubert, T. Hofmann, and C. 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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T. Hofmann, M. Schubert, C. M. Herzinger, and I. Pietzonka, “Far-infrared-magneto-optic ellipsometry characterization of free-charge-carrier properties in highly disordered n-type Al0.19Ga0.33In0.48P,” Appl. Phys. Lett. 82, 3463–3465 (2003).
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T. Hofmann, V. Gottschalch, and M. Schubert, “Far-infrared dielectric anisotropy and phonon modes in spontaneously CuPt-ordered Ga0.52In0.48P,” Phys. Rev. B 66, 195204 (2002).

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M. Schubert, “Theory and application of generalized ellipsometry,” in Handbook of Ellipsometry, E. Irene and H. Tompkins, eds. (William Andrew, 2004).

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T. Hofmann, T. Chavdarov, V. Darakchieva, H. Lu, W. J. Schaff, and M. Schubert, “Anisotropy of the Γ -point effective mass and mobility in hexagonal InN,” Phys. Stat. Solidi C 3, 1854–1857 (2006).
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Proc. IEEE (1)

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Proc. SPIE (3)

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Rev. Sci. Instrum. (3)

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T. Hofmann, U. Schade, W. Eberhardt, C. M. Herzinger, P. Esquinazi, and M. Schubert, “Terahertz magnetooptic generalized ellipsometry using synchrotron and black-body radiation,” Rev. Sci. Instrum. 77, 063902 (2006).
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P. Kühne, C. M. Herzinger, M. Schubert, J. A. Woollam, and T. Hofmann, “An integrated mid-infrared, far-infrared and terahertz optical Hall effect instrument,” Rev. Sci. Instrum. 85, 071301 (2014).
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P. Kühne, T. Hofmann, C. M. Herzinger, and M. Schubert, “Terahertz frequency optical-Hall effect in multiple valley band materials,” Thin Solid Films 519, 2613–2616 (2011).

T. Hofmann, C. M. Herzinger, J. L. Tedesco, D. K. Gaskill, J. A. Woollam, and M. Schubert, “Terahertz ellipsometry and terahertz optical-Hall effect,” Thin Solid Films 519, 2593–2600 (2011).
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M. Schubert, T. Hofmann, and C. M. Herzinger, “Far-infrared magnetooptic generalized ellipsometry: determination of free-charge-carrier parameters in semiconductor thin film structures,” Thin Solid Films 455–456, 563–570 (2004).
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Other (44)

P. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, 1999).

Note that electric field vectors E contain four independent pieces of information if the plane wave is fully coherent and time harmonic. Representing time averages over infinite observation times, the four parameters can be used to characterize the electric field amplitude, absolute phase, ellipticity, and orientation of the polarization ellipse.

W. S. Weiglhofer and A. Lakhtakia, Introduction to Complex Mediums for Optics and Electromagnetics (SPIE, 2003).

B. Rheinländer, “Infrarot-Faraday-Effekt an Halbleitern,” Master’s thesis (Universität Leipzig, 1965).

The electrical Hall effect is well known and has been described in many textbooks. It is beyond the scope of this paper to provide an in-depth review of the electrical Hall effect. The primary limitation in the electrical Hall effect is the physical requirement of ohmic contacts. Due to the fact that proper electric contact formation requires precise knowledge of surface potential functions for any given material and appropriate technological procedures, it is commonly difficult to provide equal contacts to multiple layered structures. Usually, contacts are made to the surface or bottom layer. Often, the actual passage, which the driving currents will take within the sample structure, is difficult to ascertain and hampers accurate data analysis.

H. Mueller, “Memorandum on the polarization optics of the photoelastic shutter,” Report of the OSRD project (Massachusetts Institute of Technology, 1943).

E. Hecht, Optics (Addison-Wesley, 1987).

M. Schubert, “Generalized ellipsometry,” in Introduction to Complex Mediums for Optics and Electromagnetics, W. S. Weiglhofer and A. Lakhtakia, eds. (SPIE, 2003), pp. 677–710.

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A. Gerrard and J. Burch, Introduction to Matrix Methods in Optics, Dover Books on Physics (Dover, 1994).

D. Goldstein, Polarized Light, 3rd ed. (CRC Press, 2011).

Note the four independent pieces of information contained in the Stokes vector. The four parameters can be used to characterize the total light intensity, degree of polarization, ellipticity, and orientation of the polarization ellipse.

A. Röseler, Infrared Spectroscopic Ellipsometry (Akademie-Verlag, 1990).

P. J. C. Kühne, “The optical Hall effect in three- and two-dimensional materials,” Ph.D. dissertation (University of Nebraska, 2014).

Note that the constant but generally complex amplitude parameter in Eq. (30) also may be augmented with a frequency-dependent imaginary part in order to represent the effect of an harmonic coupling. See also [150].

The same formalism can be used in case of holes but with a different effective mass parameter.

The transition from the dielectric function tensor of a given anisotropic material into the four indices of refraction is discussed in previous work [124,129].

W. Kleber and H.-J. Bautsch, Einführung in die Kristallographie (de Gruyter, 2010).

T. Hofmann, “Far-infrared spectroscopic ellipsometry on semiconductor heterostructures,” Ph.D. thesis (University of Leipzig, 2004).

A common requirement in theoretical studies of the electromagnetic response of matter consists in the imposition that a specific medium should be Lorentz reciprocal. For a dielectric medium (the magnetic susceptibility tensor being diagonal and unity), this means that the dielectric tensor is equal to its transposed form. The magnetized plasma and more general types of gyrotropic mediums belong to the most prominent representatives of nonreciprocal mediums. A gyrotropic material is a material in which left- and right-rotating elliptical polarizations can propagate at different speeds. The gyrotropic effect caused by a quasi-static magnetic field breaks the time-reversal symmetry as well as the Lorentz reciprocity. For more information see, for example, [102].

Corresponding expressions for arbitrary orientations of the magnetic field are given by EC′=Aφ,θ,ψ(R)EC, with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.

Corresponding transformation matrices for arbitrary orientations of the magnetic field are given by (A(C))=(Aϕ,θ,ψ(R))−1A(C)Aϕ,θ,ψ(R), with the Euler angles given by B|B|=Aϕ,θ,ψ(R)(0,0,1)T.

For ΨE=π/4 and ΔE=π/2, the elliptic eigensystem is equivalent to the circular eigensystem A(E)=A(C).

L. Landau and E. Lifšic, Elektrodynamik der Kontinua, Lehrbuch der theoretischen Physik (Akademie Verlag, 1990).

Note that Eq. (17) must hold for any E0; thus, Eq. (17) can be stated for each λk separately.

In the following equation the Einstein notation is used, and the covariance and contravariance are ignored because all coordinate systems are Cartesian. The summation is only executed over pairs of lower indices.

C. Kittel, Introduction to Solid State Physics (Wiley, 2009).

Conceptually, a magneto electric optical Hall effect also may exist, where a current driven by the time-harmonic electric field component, under the influence of an external magnetic field, produces in addition to, or separately from a magneto-optic dielectric displacement, a magnetization response.

The dielectric tensor is considered nonlocal in time but local in space, that is, frequency dependent but not wave vector dependent. A charged compressible fluid model resulting in a dielectric tensor for a nonlocal spatial response is described by Weiglhofer. In principle, the optical Hall effect should be observable in semiconductors with very large carrier concentrations where nonlocal spatial effects may need to be considered.

C. Klingshirn, Semiconductor Optics (Springer-Verlag, 1995).

J. T. Devreese, ed., Theoretical Aspects and New Developments in Magneto-Optics (Springer, 1979).

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N. Miura, Physics of Semiconductors in High Magnetic Fields (University, 2008).

H. Fujiwara, Spectroscopic Ellipsometry (Wiley, 2007).

R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1984).

M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures: Phonons, Plasmons and Polaritons, Vol. 209 of Springer Tracts in Modern Physics (Springer, 2004).

S. Borkar, “The exascale challenge,” in International Symposium on VLSI Design Automation and Test (2010), pp. 2–3.

http://www.intel.com/content/www/us/en/silicon-innovations/intel-14nm-technology.html .

L. T. Berger and K. Iniewski, Smart Grid Applications, Communications, and Security (Wiley, 2012).

M. Groover, Automation, Production Systems, and Computer-Integrated Manufacturing (Prentice-Hall, 2007).

https://newsroom.intel.com/news-releases/intel-and-micron-produce-breakthrough-memory-technology/ .

For example, explicit expressions for the complex Fresnel transmission coefficients can be found in [45].

M. Schubert, “Theory and application of generalized ellipsometry,” in Handbook of Ellipsometry, E. Irene and H. Tompkins, eds. (William Andrew, 2004).

For example, j=2→{k,l,m}={0,1,3}.

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of the electrical Hall effect in a thin conducting sheet. Free charge carriers produce a transverse voltage by charge separation under the influence of the Lorentz force due to a magnetic field B when driven by a DC current through the sheet. Accordingly, the longitudinal voltage required to drive the DC current differs with and without the magnetic field. Note that, ideally, the sheet should be infinitesimally thin [107]. The electric Hall voltage is characteristic of the sheet material and depends on the free charge carrier types and density properties that constitute the current leading to the Hall voltage. Note that transport must occur homogeneously across the entire sheet, which makes analyses of the electric Hall voltage measured across complex sheet structures with multiple constituents difficult.

Fig. 2.
Fig. 2.

Schematic of the optical Hall effect in a conducting sample consisting of single or multiple conducting layers and sheets, in reflection configuration. Free charge carriers produce a dielectric polarization following the electric field of an incident electromagnetic field (analogous to the longitudinal Hall voltage), here for example parallel to the surface. The induced polarization P FCC produces P Hall due to the Lorentz force, oriented perpendicular to B and the incident electric field vector (analogous to the transverse Hall voltage). P FCC + P Hall are the source of the reflected light and contain a small circular polarization component, which provides information on the type of charge carrier, its density, mobility, and effective mass properties [1]. The physical motion of the charges remains local within the lattice of the material, describing pathways that depend on the Fermi velocity, their average scattering time, the frequency of the incident light, and the magnetic field direction. If multiple layers are thin enough against the skin depth at long wavelengths, light interacts with multiple layers and reveals, for example, free carrier properties within buried layers otherwise inaccessible to direct electrical measurements. Hence, the optical Hall effect can be measured across complex sheet and layer structures.

Fig. 3.
Fig. 3.

Left-handed [ E + , Figs. 3(a) and 3(c)] and right-handed [ E , Figs. 3(b) and 3(d)] circularly polarized electromagnetic plane waves interact with a dielectrically polarizable material under the influence of an external quasi-static magnetic field B . The field B is collinear with the wave propagation direction. The displacement field phasors P ± are proportional to complex-valued, frequency-dependent response functions χ ± ( B ) . Symmetry requires switch of indices upon reversal of the magnetic field: χ ± ( B ) = χ ( B ) . The latter statement originates from the assumption that P ± does not depend on propagation direction of E ± but only on the course of the electric field phasor at a given plane within the material. Functions χ ± then determine the symmetric and antisymmetric parts in the optical Hall effect tensor [Eq. (11)].

Fig. 4.
Fig. 4.

Wave vector k in of the incoming electromagnetic plane wave and the sample normal n define the angle of incidence Φ and the plane of incidence. The amplitudes of the electric field of the incoming E in and the reflected E out plane wave can be decomposed into complex field amplitudes E p in , E s in , E p out , and E s out , where the indices p and s stand for parallel and perpendicular to the plane of incidence, respectively.

Fig. 5.
Fig. 5.

Schematic presentation, under the incoming angle Φ , of the electromagnetic wave E I , and the reflected E R , transmitted E T , and backward-traveling electromagnetic waves E B used in the 4 × 4 matrix formalism. The medium into which the wave is reflected (transmitted) is labeled R (T). Between the media R and T, n slabs of parallel layers with homogenous optical properties may be located. Backward-traveling waves E B in the medium T are permitted. Plane ( x , y ) is parallel to the interfaces/surfaces; z points into the surface. The surface is the interface against which the incoming beam is directed.

Equations (60)

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D = ϵ 0 E + P = ϵ 0 E + χ E = ϵ 0 ( I + χ ) E = ϵ 0 ϵ E ,
ϵ = I + χ = I + k χ k ,
ϵ ( x , y , z ) = ( ϵ x x ϵ x y ϵ x z ϵ y x ϵ y y ϵ y z ϵ z x ϵ z y ϵ z z ) = I + ( χ x x χ x y χ x z χ y x χ y y χ y z χ z x χ z y χ z z ) .
ϵ i j = δ i j + 1 2 ( χ i j + χ j i ) + 1 2 ( χ i j χ j i ) ,
ϵ ( ± B ) = I + χ B = 0 + χ ± B .
ϵ x , y , z = A 1 ϵ ξ , η , ς A = I + k A 1 ( χ ξ k 0 0 0 χ η k 0 0 0 χ ς k ) A ,
A ϕ , θ , ψ ( R ) = ( cos ψ sin ψ 0 sin ψ cos ψ 0 0 0 1 ) ( 1 0 0 0 cos θ sin θ 0 sin θ cos θ ) ( cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ) .
P C = ( χ + 0 0 0 χ 0 0 0 1 ) E C .
E C = ( E + E E z ) = ( 1 2 ( E x i E y ) 1 2 ( E x + i E y ) E z ) .
A ( C ) = 1 2 ( 1 i 0 1 i 0 0 0 2 ) .
χ ( ± B ) = 1 2 ( ( χ + + χ ) i ( χ + χ ) 0 ± i ( χ + χ ) ( χ + + χ ) 0 0 0 0 ) .
A ( E ) = 1 a a 1 ( a 1 0 a 1 1 0 0 0 a a 1 ) ,
ϵ Ξ k = λ k Ξ k , k = 1,2 , 3 ;
E G = a 1 Ξ 1 + a 2 Ξ 2 + a 3 Ξ 3 .
A ( G ) = 1 det ( A ( G ) ) ( Ξ 1 , x Ξ 1 , y Ξ 1 , z Ξ 2 , x Ξ 2 , y Ξ 2 , z Ξ 3 , x Ξ 3 , y Ξ 3 , z ) .
0 = det ( ϵ ) λ 1 λ 2 λ 3 = det ( A ϵ A 1 ) λ 1 λ 2 λ 3 = det ( A ) det ( ϵ ) det ( A 1 ) λ 1 λ 2 λ 3 = det ( ϵ ) λ 1 λ 2 λ 3 .
u t = R e [ div S ] = E D ˙ * + E * D ˙ + H B ˙ * + H * B ˙ = ω i E ϵ * E * i E * ϵ E + i H μ * H * i H * H = ω I m [ E 0 * ϵ E 0 ] + ω I m [ H 0 * μ H 0 ] 0 ,
I m [ λ k ] 0 ,
m x ¨ + m γ x ˙ + m ω 0 2 x = q E + q ( x ˙ × B ) ,
μ opt = q m 1 γ 1 .
E = 1 N q [ i m q ω ( ω 0 2 ω 2 I i ω γ ) j + ( B × j ) ] ,
χ i k L D = N q 2 ϵ 0 [ m i k ( ω 0 , i k 2 ω 2 i ω γ i k ) i ω ϵ i j k q B j ] 1 .
ϵ L = I + χ L = ( ϵ x L 0 0 0 ϵ y L 0 0 0 ϵ z L ) .
ϵ k L = 1 + χ k L = ϵ , k j = 1 l ω 2 + i ω γ LO , k , j ω LO , k , j 2 ω 2 + i ω γ TO , k , j ω TO , k , j 2 ,
χ D = χ B = 0 D + χ ± B D ,
χ B = 0 D = ω p 2 ω ( ω + i γ ) I = χ D I ,
ω p = N q 2 m ϵ 0 ,
χ ± = χ D 1 ω c ω + i γ ,
ω c = q | B | m .
χ ± = e ± i ϕ k A k ω 0 , k 2 ω 2 i γ k ω ,
E Graphite L L ( n , B ) = ω c ( n + n 0 ) ,
E SLG LL ( n , B ) = sign ( n ) E 0 | n | ,
E N B L G LL ( n , μ , B ) = sign ( n ) 1 2 [ ( λ N γ ) 2 + ( 2 | n | + 1 ) E 0 2 + μ ( λ N γ ) 4 + 2 ( 2 | n | + 1 ) E 0 2 ( λ N γ ) 2 + E 0 4 ] 1 / 2 ,
E out = JE in .
J = ( j p p j p s j s p j s s ) .
r p p = ( E p out E p in ) E s in = 0 , r p s = ( E s out E p in ) E s in = 0 , r s p = ( E p out E s in ) E p in = 0 , r s s = ( E s out E s in ) E p in = 0 .
( S 1 S 2 S 3 S 4 ) = ( E p E p * + E s E s * E p E p * E s E s * E p E s * + E p * E s i ( E p E s * E p * E s ) ) = ( I p + I s I p I s I + 45 I 45 I σ + I σ )
S out = MS in ,
M = ( M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ) .
M i j = 1 2 Tr ( J σ i J σ j ) ,
σ 1 = ( 1 0 0 1 ) , σ 2 = ( 1 0 0 1 ) ,
σ 3 = ( 0 1 1 0 ) , σ 4 = ( 0 i i 0 ) .
M is = ( 1 2 ( r p p r p p * + r s s r s s * ) 1 2 ( r p p r p p * r s s r s s * ) 0 0 1 2 ( r p p r p p * r s s r s s * ) 1 2 ( r p p r p p * + r s s r s s * ) 0 0 0 0 R e ( r p p r s s * ) I m ( r p p r s s * ) 0 0 I m ( r p p r s s * ) R e ( r p p r s s * ) ) ,
M an = ( 1 2 ( r p s r p s * + r s p r s p * ) 1 2 ( r p s r p s * r s p r s p * ) R e ( r p p r p s * + r s s r s p * ) I m ( r p p r p s * r s s r s p * ) 1 2 ( r p s r p s * r s p r s p * ) 1 2 ( r p s r p s * + r s p r s p * ) R e ( r p p r p s * r s s r s p * ) I m ( r p p r p s * + r s s r s p * ) R e ( r p p r s p * + r s s r p s * ) R e ( r p p r s p * r s s r p s * ) R e ( r p s r s p * ) I m ( r p s r s p * ) I m ( r p p r s p * r s s r p s * ) I m ( r p p r s p * + r s s r p s * ) I m ( r p s r s p * ) R e ( r p s r s p * ) ) .
M B = M ( ϵ B 0 ( k ) ) .
φ F K = 1 2 arctan [ M 31 B + M 32 B cos [ 2 β ] + M 33 B sin [ 2 β ] tan [ 2 β ] ( M 21 B + M 22 B cos [ 2 β ] + M 23 B sin [ 2 β ] ) M 21 B + M 22 B cos [ 2 β ] + M 23 B sin [ 2 β ] ± tan [ 2 β ] ( M 31 B + M 32 B cos [ 2 β ] + M 33 B sin [ 2 β ] ) ] ,
Ψ z = i ω c Δ Ψ ,
( E p I E s I E p R E s R ) = L ( E p T E s T E p B E s B ) ,
L = L R 1 ( k = 1 m L P k ) L T .
Δ = ( k x ϵ z x ϵ z z k x ϵ z y ϵ z z 0 1 k x 2 ϵ z z 0 0 1 0 ϵ y x + ϵ y z ϵ z x ϵ z z k x 2 ϵ y y + ϵ y z ϵ z y ϵ z z 0 k x ϵ y z ϵ z z ϵ x x ϵ x z ϵ z x ϵ z z ϵ x y ϵ x z ϵ z y ϵ z z 0 k x ϵ x z ϵ z z ) ,
L R 1 = 1 2 ( 0 1 ( n R cos Φ ) 1 0 0 1 ( n R cos Φ ) 1 0 ( cos Φ ) 1 0 0 n R 1 ( cos Φ ) 1 0 0 n R 1 ) ,
L T = ( 0 0 cos Φ T cos Φ T 1 1 0 0 n T cos Φ T n T cos Φ T 0 0 0 0 n T n T ) .
cos Φ T = 1 ( n R / n T ) 2 sin 2 Φ .
L P k = exp ( i ω c Δ k d k ) = j = 0 3 β j k Δ k j .
β n = j = 0 3 α n exp ( i ω q j ( d ) / c ) ( q j q k ) ( q j q l ) ( q j q m ) ,
α 0 = q k q l q m , α 1 = q k q l + q k q m + q l q m , α 2 = ( q k + q l + q m ) , α 3 = 1 ,
r p p = L 11 L 43 L 13 L 41 L 11 L 33 L 13 L 31 , r p s = L 33 L 41 L 31 L 43 L 11 L 33 L 13 L 31 , r s p = L 11 L 23 L 13 L 21 L 11 L 33 L 13 L 31 , r s s = L 33 L 21 L 31 L 23 L 11 L 33 L 13 L 31 .
φ = 1 2 arctan [ M 23 M 33 ] .
φ = ± 1 2 arctan [ 2 ω c R e [ χ B eff ( 1 + n T i χ LL * ) ] γ | χ B eff | 2 | 1 + n T + i ( ( ω + i γ ) χ B eff + χ LL ) | 2 | χ B eff | 2 ] ,
χ B eff = ω c ω + i γ ( ω + i γ ) 2 ω c 2 , χ B = 0 D = ω p 2 c 1 ( ω + i γ ) 2 ω c 2 .

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