Ann. Phys. (Leipzig) 15, No. 7 – 8, 480 – 497 (2006) / DOI 10.1002/andp.200510204 Another century of ellipsometry Mathias Schubert1,2,∗ 1 2 Department of Electrical Engineering and Center for Materials Research and Analysis, 209N Walter Scott Engineering Center, University of Nebraska-Lincoln, Lincoln NE 68588-0511, USA Universität Leipzig, Fakultät für Physik und Geowissenschaften, Institut für Experimentelle Physik II, Linnéstrasse 5, 04103 Leipzig, Germany Received 13 November 2005, revised 1 Februar 2006, accepted 6 February 2006 Published online 26 May 2006 Key words Dielectric function, ellipsometry, Drude model, free-charge-carriers, effective mass, magnetooptic, Landau splitting, EUV mirrors. PACS 42.25.Ja, 42.25.Lc, 42.50.Nn, 78.20.-e, 78.30.-j, 78.40.-q, 78.67.Pt In commemoration of Paul Drude (1863–1906) In commemoration of the inventor – Paul Karl Ludwig Drude – examples of contemporary interest in the vivid ﬁeld of ellipsometry are presented. More than 100 years after his provision of the general concept and the ﬁrst experimental application, ellipsometry has matured as tool-of-excellence in almost all materials research areas. This contribution reviews selected applications in solid-state materials physics and engineering addressing state-of-the-art ellipsometry concepts. Particular emphasis is placed on the generalized ellipsometry framework. Infrared ellipsometry and magnetooptic infrared ellipsometry provide access to yet another important achievement of Paul Drude in optical physics: The free electron model in conductors, but also clearly reveal deviations from thereof within charge-correlated systems, exempliﬁed here by the observation of Landau level splitting in two-dimensional electron gases using synchrotron light source ellipsometry at terahertz frequencies. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Although not called “ellipsometry” at the time, the credit for not only inventing the basic equations but also for carrying out the ﬁrst experiments and simultaneously deriving the ﬁrst optical constants for materials of contemporary interest – which were even highly anisotropic – has without any doubt to be given to Professor Dr. Paul Karl Ludwig Drude (1863–1906).1 Ellipsometry determines the polarization state of a parallel electromagnetic radiation ﬁeld after interaction with a sample.2 Two independent parameters are obtained in one principal measurement, which sufﬁce primarily for calculation of both real and imaginary parts of a dielectric function, or alternatively, for accurate determination of the thickness and of either real or imaginary part of the dielectric function from a single thin layer. Ellipsometry does not contact or damage samples, and – from its very beginning – is an ideal and precise measurement technique for determining optical and hence physical properties of materials at the nano scale. While introduced initially for determination of the optical constants from bulk materials with improved accuracy over intensity measurements [5–7], the original paper also reported on effects of tarnish formation on cleavages of ore minerals (stibnite), which drastically affected the ellipsometric parameters [6]. Descending from Maxwell’s postulates, equations governing the ∗ 1 2 E-mail: Schubert@engr.unl.edu, www: ellipsometry.unl.edu Previous notes on ellipsometry history were given by Rothen [1], Hall [2], Azzam [3], and Vašiček [4]. For convenient mathematical modelling of this polarization transfer, samples with plane-parallel interfaces, i.e., thin ﬁlms or bulk samples with planar surfaces are typically considered in ellipsometry applications. However, the concept of ellipsometry itself, i.e., measurement of the polarization state of an electromagnetic plane wave is not restricted to such sample situations. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Ann. Phys. (Leipzig) 15, No. 7 – 8 (2006) 481 relation between the optical constants of both crystal and over layer and the ellipsometric parameters were laid out by Drude [7–13], and a new measurement technique was born. Today’s ellipsometry is becoming popular in a continuously widening circle of applications because of increasing miniaturization of integrated circuits, breakthroughs in knowledge of biological macromolecules deriving from DNA and protein surface research, materials physics design of thin ﬁlm multilayer surfaces, composite and smart materials, and materials engineering at the nano scale. Contemporary applications cover widest spectral regions from the terahertz domain to ultra short wavelengths, addressing bound and unbound charge excitations in complex layer structures unveiling critical foundation parameters of new materials, or controlling intricate layer structures in real time during growth (e.g., [14–18]). Drude’s work is more often encountered in ellipsometry than in its basic invention. Harmonic oscillator functions accounting for dielectric polarization due to bound charge excitations were introduced by Drude in his “Erklärungssystem” for description of dispersion and absorption observed in the optical constants for various materials [19]. His probably most important contribution to optical physics, the Drude free electron model of conductivity and the Drude relaxor model (a heavily over-damped charge plasma) [20,21], have become standard models in describing dispersion and absorption phenomena caused by free-charge-carriers in optical spectroscopy. A general layout of complex reﬂectance and transmission equations descending from Maxwell’s postulates was basically done already by Drude for arbitrarily layered thin ﬁlm samples – including orthorhombic situations – where necessary room for treatment of arbitrary anisotropy situations is contained. He reported the ﬁrst measurement of optical constants from anisotropic materials, and already indicated a possible use of ellipsometry or related optical techniques for determination of orientation parameters (i.e., the Euler rotation angles) of intrinsic (Cartesian or non-Cartesian) major polarization axes upon determination of optical axes inclinations with a given (Cartesian) laboratory system [13, 22]. Drude also included effects of static external and internal magnetic ﬁelds onto reﬂectance and transmittance characteristics of samples with bound and unbound charge oscillations, and explained observed polarization rotation in magnetic ﬁelds [23]. Without any claim whatsoever regarding completeness, the following sections review modern applications and developments in today’s ellipsometry. As vivid ﬁeld with diverse applications, solid state materials chosen here are to be marked as just one example, where further exciting applications in biology, soft and liquid matter research are left out for brevity. The selection of subjects may underline the distinction of ellipsometry as matured tool in modern spectroscopy. While many details can be found in recent textbooks on ellipsometry [18,24], short explanations of standard and generalized ellipsometry, and Jones and Stokes vector formalisms are given. Constitutive descriptions of complex matter response are addressed shortly, as well as the nested within “Erklärungssystem” of Drude, intended to provide access to physical material parameters. The optical Hall effect – a new measurement technique based upon the generalized ellipsometry concept and application within external magnetic ﬁelds at long wavelengths – is introduced. Non-classical deviations from the Drude free-charge-carrier conductivity concept is demonstrated by application of the optical Hall effect to two-dimensional carrier gases at low temperatures. A review should not close without future prospects, one of which is not difﬁcult to provide here: Another very interesting and fascinating century of ellipsometry has just started. 2 Ellipsometry Ellipsometry determines the change of the polarization state of an electromagnetic plane wave upon interaction with a sample. As suggested by Drude, this change is cast into the complex-valued scalar ratio ρ relating two linearly-independent electric ﬁeld components √ (Xζ , Xξ ) of polarized electromagnetic plane waves before (A) and after (B) sample interaction (i = −1; Fig. 1 depicts the reﬂection setup, the transmission case is alike.) [6, 18, 24] ρ= www.ann-phys.org Bζ Bξ Aζ / = tan Ψ exp (i∆) . Aξ (1) c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 482 M. Schubert: Another century of ellipsometry P As a sample normal Ap E Bp Bs E A Fig. 1 Deﬁnition of the plane of incidence (p plane) and the incidence angle Φa through the wave vectors of the incident and emerging (reﬂection set up) plane waves. Ap , As , Bp , and Bs , denote the complex amplitudes of the p and s modes before and after reﬂection, respectively. P and A are the azimuth angles of the linear polarizer used, e.g., in the standard arrangement of rotating-analyzer (polarizer) ellipsometer. Two concepts emerge from this deﬁnition: The standard and the generalized ellipsometry situations, which differ by the dependence of ρ on Aζ /Aξ . The concept developed by Drude, and which is now referred to as standard ellipsometry, requires ρ to be independent of Aζ /Aξ , while the general case includes all deviations from this restriction. For convenient connection with appropriate experimental conﬁgurations a coordinate system may now be chosen, and which, following Drude, is given in terms of p and s polarized ﬁelds in Fig. 1, relating both (Xζ , Xξ ) and (Xζ , Xξ ) to the plane of incidence. It is almost needless to state that alternative coordinate choices exist. The result of an ellipsometry measurement is often – still following Drude – presented by real-valued parameters Ψ and ∆, where now tanΨ is deﬁned as the absolute value of the complex ratio, and ∆ denotes the relative phase change of the p and s components of the electric ﬁeld vector in Eq. (1). Measurement of the complex ratio ρ can be addressed within different presentations of the electromagnetic plane wave response.3 Convenient are the Stokes and Jones vector descriptions, allowing to cast Drude’s subsequent equations into short and intuitive forms, leaving sufﬁcient room for required extensions. Such is, depending on the sample properties, i.e., whether a surface reﬂects (or transmits) light upon polarization mode conversion or not, Drude’s parameter set in Eq. (1) must be further expanded into the so-called generalized ellipsometry parameter set. Furthermore, while the Jones approach sufﬁciently frames the required extension for non-depolarizing samples, the Stokes formalism including Mueller matrices must be invoked for cases when either sample or the experimental setup cause partially depolarized light. The Jones vector is the usual arrangement of electric ﬁeld vector amplitudes transverse to propagation direction (suppressing time dependence explicitly and thereby ignoring partial depolarization implicitly). The Jones matrix j = (jζξ ) then relates Jones vectors before and after sample interaction Bp jpp jsp Ap = . (2) Bs jps jss As Herein, case of either reﬂection or transmission is addressed as usual, remaining with the linear p − s polarization system, while changes to (jζξ ) will occur accordingly if conversion is done to other than p − s presentation of ﬁelds before and/or after sample interaction. Notably, off-diagonal elements are nonzero for optical systems that convert p into s waves and vice versa. Owing to the artiﬁcial construction of the Jones vector, which cannot be directly related to physically observable quantities, model assumption have to be implemented in order to connect measurement with the ellipsometric parameters. An alternative description provides the Stokes vector formalism, where real-valued matrix elements connect the Stokes parameters of the electromagnetic plane waves before and after sample interaction, and which can, in principle, be directly measured because the linearly-independent Stokes vector elements are well-deﬁned physically observable quantities. For the p − s system: S0 = Ip + Is , S1 = Ip − Is , S2 = I45 − I−45 , S3 = Iσ+ − Iσ− , where Ip , Is , I45 , I−45 , Iσ+ , and Iσ− denote the intensities for the p-, s-, +45◦ , -45◦ , right-, and left-handed circularly 3 All equations (wavelength, polarization and angle-of-incidence dependencies) governing the boundary conditions of a given sample relating ρ (or subsequent deﬁnitions thereof) with the intrinsic polarizability functions and structure (symmetry, geometry, etc.) are left out in this communication. The interested reader is referred to existing literature, e.g., [24–28]) c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 7 – 8 (2006) 483 polarized light components, respectively [24, 29, 30]. Importantly, S0 is proportional to the total intensity of the light beam, and the inequality S0 ≥ S12 + S22 + S32 gives a measure of the degree of polarization while the equality sign holds for total polarized light only. Arranging the Stokes parameters into a column vector, the Mueller matrix deﬁned thereby then describes the changes of each quantity upon interaction of the electromagnetic plane wave with an optical system4 S0 S1 S2 S3 = output M11 M12 M13 M14 M12 M22 M32 M42 M13 M23 M33 M43 M14 M24 M34 M44 S0 S1 S2 S3 . (3) input The advantage of this concept is the ability to handle situations with partial polarization of the electromagnetic plane wave, particularly when caused by device components’ imperfections [24, 29, 31–34]. 2.1 Standard ellipsometry By deﬁnition, in the standard ellipsometry situation Ψ and ∆ do not depend on the polarization state of the incident plane wave. Within the Jones presentation the generic expression is ρ= jp = tan Ψ exp (i∆) . js (4) Here jp and js denote the p- and s-polarized complex reﬂection (“jp,s ” = “rp,s ”) or transmission coefﬁcients (“jp,s ” = “tp,s ”). For the Mueller matrix approach – a nondepolarizing system taken as example – a one-toone relation exists between matrices r and M [24,35]. If the sample is also isotropic, then M11 = M22 = 1, M12 = M21 = − cos 2Ψ, M33 = M44 = sin 2Ψ cos ∆, M34 = −M43 = sin 2Ψ sin ∆, and Mij = 0 2 2 2 otherwise, with the constraint M11 + M33 + M34 = 1 [33], and ρ= M33 + iM34 . 1 − M12 (5) 2.2 Generalized ellipsometry By deﬁnition, in the generalized ellipsometry situation Ψ and ∆ depend on the polarization state of the incident plane wave. This concept is valid within both, the Mueller matrix as well as within the Jones matrix formalism. Within the Jones presentation generic expressions for the generalized ellipsometry parameters are Ψij , ∆ij (“J, j” = “T , t” or “R, r”)5 jpp jps jsp = tan Ψpp exp(i∆pp ), = tan Ψps exp(i∆ps ), = tan Ψsp exp(i∆sp ). jss jpp jss (6) The real-valued quantities Ψpp , Ψps , Ψsp , ∆pp , ∆ps , ∆sp comprise the generalized ellipsometry data presentation in the Jones formalism. While the latter is valid and sufﬁcient for nondepolarized light conditions only (neither sample nor optical ellipsometer components depolarize the light beam), the generalized ellipsometry parameter set comprises then all sixteen elements of the Mueller matrix in case of presence of depolarizing conditions. 4 Sample, mirrors, rotators, optical devices within the light path, and any combinations thereof. 5 This set lacks the light beam’s absolute intensity and the light beam’s absolute phase information contained within the Jones matrix. For acquisition of generalized ellipsometry parameters from anisotropic samples see [24, 36, 37]. Note that the choice of diagonal elements of the Jones matrix for normalization was convenient, but arbitrary [36]. www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 484 M. Schubert: Another century of ellipsometry 2.3 Data acquisition and analysis Substantial information with detailed description of various procedures and techniques in ellipsometry data acquisition can be found in a very recent handbook compilation [18]. Brieﬂy, polarizing elements with variable polarization parameters for generating polarization states (which will then interact with a sample) and for analyzing polarization states (after interaction with a sample) constitute ellipsometer systems. Details shall be omitted here, and the interested reader is referred to existing and extensive literature [18, 24, 26, 29, 38]. Data analysis requires nonlinear regression methods, where measured and calculated ellipsometry spectra are matched as close as possible by varying appropriate physical model parameters. The calculations require setup of models for geometry, layer structure, and polarizability properties of materials involved in the sample of interest, and comprise the actual art in performing ellipsometry. Due to the complexity of this subject, thorough discussion of this issue is beyond the scope of this paper, and referral is made to the literature again. An excellent introduction can be found in the chapter by G. E. Jellison, Jr. in [18]. 3 Constitutive considerations Description for optical polarization response of matter in terms of optical constants begins with considerations of the electromagnetic properties of a given matter system. The interaction of the electromagnetic and H with matter are described by dielectric and magnetic polarizability functions and ﬁelds D ﬁelds E and B, with rules set by Maxwell’s postulates [39]. So called constitutive relations are deﬁned by operations and B with E and H (ε̃0 and µ̃0 are the vacuum permittivity f , g, connecting ﬁelds pairwise, for example D 6 and permeability, respectively) [25], H), B = µ̃0 g (E, H), = ε̃0 f (E, D (7) and which reﬂect symmetry, structure and physical properties of a given matter constitution, and which allow combining externally deﬁned quantities (such as the index of refraction and extinction coefﬁcient for a given propagating polarization mode) with internal structure parameters (electronic eigenmode frequencies and lifetimes, geometry values, chemical composition etc.). The star indicates operator functions in f and g, which will also depend on the domain presentation of Eq. (7), such as the frequency-momentum presentation. The internal structure parameters are equivalent to the model parameters in Drude’s “Erklärungssystem” [23],7 built to explain the temporal and spatial response of the electric charge system to the electromagnetic waves within the given matter constitution. Speciﬁc circumstances require characteristic model descriptions, and the cases of homogeneous dielectric (reciprocity) and homogeneous magnetooptic (non-reciprocity) anisotropy may be taken as examples here. 3.1 Linear dielectric anisotropy = 0) occurs for charge excitations and Linear dielectric anisotropy (ignoring magnetic ﬁeld effects: P (H) with spatial preference directions collinear with major axes a, b, subsequent dielectric polarization P (E) 8 c, a + ρb b · E b + ρc c · E c, + Pb (E) + Pc (E) = ρa a · E = Pa (E) (8) P (E) 6 Functions f , g may be linear or nonlinear in their ﬁeld arguments, rendering linear or non-linear optical properties. Herein we restrict to linear considerations, where inverse functions may further exist. Throughout this contribution we restrict to a basic laboratory system in Cartesian coordinates. 7 A system of equations intended to explain the experimental observations using basic physics argumentations. 8 The intrinsic (bound or unbound) charge polarizations (eigenvectors) set up a spatial non-Cartesian (monoclinic, triclinic), or Cartesian (orthorhombic, tetragonal, hexagonal, trigonal, and cubic) center-of-gravity system, with axes described by vectors a = xax + y ay + z az , b = xbx + y by + z bz , and c = xcx + y cy + z cz . c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 7 – 8 (2006) 485 represented by three complex-valued scalar polarizability functions ρa , ρb , ρc . The dot denotes the scalar product. The complex-valued scalar polarizabilities ρa , ρb , ρc must obey Kramers-Kronig consistency, and correspond to the intrinsic center-of-gravity polarizability system within homogeneously responding matter (e.g., a homogeneous thin ﬁlm). Within the time-frequency domain, the constitutive relations have simple tensorial form deﬁning the dielectric function tensor ε (the magnetic permeability tensor is unity) [26]: = ε̃0 E B = µ̃0 H µ = 1. Consequently, the dielectric function + P (E) = ε̃0 εE, + P (H) = µ̃0 µH, D tensor is symmetric causing reciprocive optical response. Twelve independent parameters (two for each vector a, b, c giving their orientations, and two for each function ρa , ρb , ρc giving their real (refractive) and imaginary (absorptive) parts) are contained within ε, at ﬁrst independent of the electromagnetic wave frequency. The photon energy (or frequency) dependencies of ρa , ρb , ρc require model descriptions, as will be discussed below. 3.2 Magnetooptic anisotropy Magnetooptic anisotropy is induced by magnetic ﬁelds and the associated Lorentz force acting on the motion of bound and unbound charge carriers under the inﬂuence of the incident electromagnetic ﬁelds. Functions slow . The latter inﬂuences electronic wave f , g in Eqs.(7) now include a slowly varying magnetic ﬁeld H functions and their energy-momentum distribution upon coupling with the charge particle spin and their spatial motions, and affect the electromagnetic ﬁeld exchange with bound and unbound charge systems. Assuming spatially-homogeneous response, and six additional independent parameters to begin with, the magnetooptic dielectric polarizability may be described by the magnetic ﬁeld unit vector h, along which ± propagate, and polarizability functions clockwise (+) or counter clockwise (-) circularly polarized modes E ± slow E ±, ± ) = ρ± H (9) P (E which results in nonsymmetric properties of the dielectric function tensor. For example, if h = (0, 0, z), = searching for the appropriate polarization vector, and arranging the electric ﬁelds into the form E (Ex + iEy , Ex − iEy , Ez ) = (E+ , E− , Ez ), results in P = (− E− , + E+ , 0), and thus in (x, y, z) coordinates 1 i 2 (+ + − ) 2 (+ − − ) 0 ε (ω) = 2i (− − + ) 12 (+ + − ) 0 . (10) 0 0 1 Causality considerations and physical assumptions may provide access to model descriptions for ± which slow = Hh. are proportional to the magnitude and direction of H 3.3 Bound and unbound charge response model Drude provided generic expressions for the photon energy dependencies of the polarizability functions thereby providing access to physical model parameters: Ω = Ω(ω; X1 , X2 , X3 , . . .) where Xi are, e.g., phonon mode frequencies, transition energies or amplitudes, lifetimes etc., and ω is the photon energy. 3.3.1 Bound oscillations The simplest model description for (linear) electromagnetic excitation of bound-charge polarization modes (resonant charge separation with a restoring force) is the solution of the linear second-order differential equation, i.e., the harmonic Lorentzian-damped oscillator function, already employed in Drude’s www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 486 M. Schubert: Another century of ellipsometry “Erklärungssystem” [40] Ω(ω; ωp , ω0 , γ) = (ωp )2 , (ω0 )2 − (ω)2 − iωγ (11) in which ωp is the energy equivalent (amplitude) of the polarizable (bound) charge density with resonance (binding) energy ω0 and the inverse relaxation time (broadening) γ, and where denotes the Planck constant. While Eq. (11) is unable to explain the characteristics of materials at photon energies near their absorption gaps, speciﬁcally for semi- and superconducting materials where more complex functions have to be involved [41],9 for most observable dispersions, i.e., photon energy dependence of the optical constants, this is often the best approximation at hand. 3.3.2 Unbound oscillations Implying linear charge carrier scattering regimes, and retracting the restoring force in Eq. (11) results almost naturally in one of Drude’s most acclaimed contributions to physics: The Drude free electron model (ωp )2 , (12) (ω)2 + iωγ in which the plasma frequency ωp = e ε̃0Nm deﬁnes a new observable quantity (N denotes the charge carrier density, and m their effective mass). At its wavelength equivalent, a strong reﬂectance drop (“plasma edge”) and resonance excitation of polarized interface modes (surface polaritons) occurs, which strongly affect intensity and polarization state of reﬂected or transmitted electromagnetic plane waves [26,46]. The broadening parameter γ is related to the inverse of the momentum relaxation time γ = τ −1 , which can be mµ substituted by the optical free-charge-carrier mobility parameter µopt [41, 47–49]: τ = eopt . Ω(ω; ωp , ω0 = 0, γ) = − 3.3.3 Magnetooptic oscillations Inclusion of the Lorentz force into Eq. (11) results in the magnetooptic Drude model, which provides access to the cyclotron frequency ωc . The magnetic ﬁeld produces the antisymmetric matrix Ξ(h), which occurs in the denominator coupling spatial bound and unbound (ω0 = 0) charge motions to the Lorentz force. In general, the generic expression Ω now renders a nonsymmetric tensor Ω(ω; ωp , ω0 , γ, ωc , h) = (ωp )2 [(ω0 )2 − (ω)2 − iωγ − iωωc Ξ(h)]−1 . (13) 4 Contemporary ellipsometry applications 4.1 Dielectric function: Infrared-to-vacuum ultra violet range Fig. 2 depicts exemplarily ellipsometry and dielectric function spectra for a zincblende semiconductor thin ﬁlm sample.10 The spectrum covers the infrared to the vacuum ultra violet spectral region (ω ≈ 50meV to ω ≈ 9.5eV). Contributions due to the excitation of bound (lattice electrons) and unbound (free carriers) charge modes to the dielectric function can be differentiated and quantiﬁed by model lineshape analysis. Above ≈ 3.5eV the dielectric function reveals here the electronic band-to-band transition energies (E0 , E1 , E2 ) and exciton properties. At smaller wavenumbers, the dielectric function is governed by polar lattice mode excitations, caused here by the Ga-N and Al-N sublattices, and the free-charge-carrier excitations, 9 For related substantial and pioneering ellipsometry work on superconducting materials see papers by Christian Bernhard and 10 coworkers, e.g., [42–45]. ρ is often presented within the so-called pseudodielectric function ε, which is a direct inversion of Ψ, ∆ in Eq. (1) assuming bulk behavior of the sample surface. The two-phase (ambient(εa )-substrate(ε)) model relates ε with the ellipsometric parameters [24]. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 7 – 8 (2006) 487 -1 ω [cm ] -1 Φ = 70° 0 3 5 7 9 ε1 <ε1> E1 6 4 2 0 TO 0 12 10 8 E2 ε2 <ε2> 0 12 4 1 β-Al0.12Ga 0.88N 50 -4 -100 8066 24197 40328 56459 72590 100 8 0.05 0.06 0.07 0.08 0.09 726 GaN E2(AlGaN) E1(AlGaN) E0(AlGaN) E0(GaN) LO(GaN) TO (AlGaN) (AlGaN) -50 645 150 0 AlN TO 0 565 ε2 8 484 E0 β-Al0.12Ga0.88N/β-GaN/GaAs 4 TO(GaN) 50 403 8066 24197 40328 56459 72590 AlN 726 TO Φ = 70° 645 100 0 <ε1> 565 GaN <ε2> 150 484 8 -50 4 ε1 ω [cm ] 403 -100 0.05 0.06 0.07 0.08 0.09 1 3 5 7 9 0 E [eV] E [eV] (a) (b) Fig. 2 (a) Real and imaginary parts of the pseudodielectric function of a cubic (β) Al0.12 Ga0.88 N/GaN layer sequence deposited on GaAs, measured from the infrared to the vacuum-ultra violet spectral regions. The observed phonon mode and interband transition energies are marked by vertical arrows. Solid and dashed arrows refer to the Al0.12 Ga0.88 N layer and the GaN buffer layer, respectively. Note the different scales on the left and right axes. (b) Complex dielectric function of the Al0.12 Ga0.88 N layer determined by various model analysis procedures from the infrared to the vacuum ultra violet spectral regions. Redrawn from Kasic et al. [50, 51]. which are reﬂected by the asymptotic increase (decrease) of the imaginary (real) parts of the dielectric function towards longer wavelengths according to the Drude free electron model. The bound and unbound charge excitations are sensitive to the state of strain [52], the chemical composition [53], and the state-oforder [54], for example. Mobility, concentration, and / or effective mass parameters can be derived from the dielectric function, provided either the effective mass or the concentration parameter is known from a different experiment [26, 55]. 4.2 Thin ﬁlm growth monitoring: Soft x-ray region Bragg mirrors Spectroscopic ellipsometry has undoubtedly emerged as powerful technique for in-situ monitoring of thinﬁlm layer growth [33] with extreme sensitivity to layer thickness on the molecular scale (e.g., gate oxide materials in microelectronics). Respectable reviewing of this matter is far beyond the scope and ability of this author, and the interested reader is referred to recent conference publications [15–17]. The evolution of the ellipsometric parameters versus time easily reveal growth rate, thickness, and optical constants. Nevertheless, a mathematical model must appropriately describe the sample structure. Optical constants, growth rates and layer thickness then follow upon parameter variation during best-model calculations matching the experimental data with calculated spectra as close as possible. The basic concept was already pointed out by Drude when discussing the equations for a thin ﬁlm with optical constants n and k and thickness d covering a supporting material with different optical constants [11]. He noted that the ellipsometry parameters allow access to both n and k, provided the thickness would be known. While from a single measurement all three parameters cannot be obtained alone (implying that those of the supporting material are at hand), a second measurement on the same ﬁlm material but with different thickness, in principle, completely removes the mathematical parameter correlation among them. While monitoring the growth of a ﬁlm within short periods of time, any two subsequent data points at t1 , t2 can be analyzed in terms of n, k and thickness d(t2 )−d(t1 ) d(t2 ) > d(t1 ) providing solid information on process parameter stability and growth rate δd . δt = t2 −t1 Measurement at multiple wavelengths increases sensitivity to d, and spectral information contained in n, k can be further evaluated. Of course, non-idealities such as interface roughness, light path alterations, and device imperfections set limits to accuracy and speed. Nonetheless, its feasibility, easy-to-use installations, www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 488 M. Schubert: Another century of ellipsometry Si/Mo/Si../Mo/Si Si/Mo/Si../Mo/Si 45 λ = 633.5 nm ... Si 80 44 P1 Mo t [min] Im{< ε >} 40 P2 43 Re{< ε >} t [min] 60 Si P1 42 Mo ... 41 50 nm 20 10 20 <ε> (a) 40 10 30 (b) <ε> (c) 20 Fig. 3 (a) Experimental (symbols) and best-model (solid lines) in-situ ellipsometry data versus growth time t for a 50-period Mo/Si 13.4-nm Bragg-mirror deposited in ultra-high vacuum. A mechanical shutter closes during pause times interrupting the growth during target change. (b) Transmission electron microscopy bright-ﬁeld image from a section of the same sample, where the Mo layers are rendered by dark contrast and the Si layers cause bright contrast. (c) Detail indicating pause times (P1, P2) during shutter closure [56]. and today’s availability of fast real-time in-situ systems equipped with multi-channel wavelength read-out continue to attract users. Time-dependent charts of in-situ ellipsometry data are shown in Fig. 3, monitored exemplarily during the growth of an amorphous Mo/Si-multilayer structure for furnishing next-generation soft-x-ray optics. The optical contrast, i.e., the difference between both real and imaginary parts of the Mo/Si layers is very large in the visible region, and warrant sufﬁcient sensitivity to monitor time-resolved growth rates and layer evolution, here (20.0±0.1)Å Mo with δd δt = (6.7 ± 0.1)Å/s followed by 17.4 s pause for changing from the Mo to the Si target, and (48.8±0.5)Å Si with δd δt = (1.69 ± 0.01)Å/s, followed by 28.8 s pause for reverting to the Mo target (n = 4.49 ± 0.02, k = 5.33 ± 0.02 (a-Mo), and n = 4.83 ± 0.02, k = 0.72 ± 0.01 (a-Si)). 4.3 Anisotropic mediums: Orthorhombic ore minerals For symmetrically-dielectric anisotropic mediums measurement of polarization state after Drude [5] (standard ellipsometry) is still valid for appropriately cut and precisely aligned surfaces, but fails for arbitrary orientations (skew cuts from orthorhombic systems), and in general for monoclinic or triclinic systems. Hence, complete and accurate sets of optical constants for biaxial (orthorhombic, monoclinic and triclinic) materials, measured over a range of wavelengths, rarely exist, if at all. Generalized ellipsometry overcomes this limitation and allows accurate and rigorous treatment of orthorhombic, monoclinic or triclinic absorbing materials. Stibnite (Sb2 S3 ) illuminates multiple aspects of the strength of Drude’s ellipsometry technique. Drude was able to determine n and k for an absorbing material, which is even highly anisotropic. Using a natural cleavage of stibnite, which is perpendicular to lattice direction a, he was able to obtain the optical constants for lattice directions b and c, aligning the sample appropriately defacto treating the optical response c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 7 – 8 (2006) 489 6 20° λ = 589 nm 5 75° Sb2S3 180 75° 0 ∆sp [°] Ψsp [°] (313) Sb2S3 Pnma 5 20° 0 4 360 b 75° 75° c 0 n, k 20° ∆ps [°] Ψps [°] 180 10 3 a b 20° 0 180 20° 2 c 65° 70° Φ a = 75° 75° 0 0 180 360 0 180 ϕ0 [°] ϕ0 [°] (a) 120 a 1 ∆pp [°] Ψpp [°] 30 60 360 0 1 2 3 4 Energy [eV] (b) Fig. 4 (a) Generalized ellipsometry data (symbols: experiment; solid lines: best-model) from offcut surface of antimony trisulﬁde (Sb2 S3 , orthorhombic) at sodium light versus in-plane sample orientation ϕ0 [57]. (b) Refractive indices (na , nb , nc ) and extinction coefﬁcients (ka , kb , kc ) of Sb2 S3 for polarizations along crystallographic axes a, b and c. The ﬁrst available data of nb and nc were reported by Drude (1888; triangles) [6], later extended by Müller (1903; squares) [58] and Tyndall (1923; crosses). Complete data including polarizations along a and optical axes orientations were obtained from generalized ellipsometry measurements on arbitrarily cut surfaces (2004; solid circles). Solid lines are best-match harmonic oscillator functions to the polarizability spectra [59]. isotropic. Thereby he reported in 1888 the ﬁrst ellipsometry application, and ﬁrst data for stibnite using sodium light [6]. Then Drude clearly noticed that stibnite on its natural cleavage forms tarnish, identiﬁed upon change of the ellipsometry parameters over extended observation periods, caused by the growth of a thin ﬁlm with the sample exposed to natural environment, and with different optical properties depending on the surface orientation. Hence, the strength of ellipsometry to monitor thin ﬁlm optical properties (and later on so importantly to control ﬁlm growth on the atomic layer scale) was impressively demonstrated. It became also implicitly clear, that with appropriate calculation schemes and with further progress of ellipsometry it should be possible that both, optical constants and spatial orientations of the high-symmetry lattice axes can be determined. In particular, at the time Drude was avoiding the situation when polarization-mode coupling would occur, i.e., when crystallographic axes would not be perfectly aligned with the laboratory system, which is a weakness of Drude’s (standard) ellipsometry. Hence, for Drude’s approach the crystallographic orientations of a given anisotropic specimen needed to be known. The latter is not a prerequisite anymore within the generalized ellipsometry approach, and both, optical constants and spatial orientation of all lattice axes can be obtained simultaneously, and in principle without any prior knowledge (Fig. 4). www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 490 M. Schubert: Another century of ellipsometry λ2 0.0 λ1 10 λ2 ny 4 0.5 Cos∆sp Ψsp [°] 20 nx 2 -0.5 λ1 λ1 0 λ2 0.5 50 0.0 λ1 25 λ1 1500 Wavelength [nm] z Si/Si y λ2 λ1 0.5 0.0 10 λ2 0 λ1 180 ϕ0 [°] 360 0 180 ϕ0[°] Cos∆pp Ψpp [°] 1450 kx k z x -0.5 λ2 20 1400 Cos∆ps Ψps [°] λ2 nz ky -0.5 360 (a) (b) Fig. 5 (a) Experimental and best-model calculated generalized ellipsometry data for a silicon-on-silicon nano-chevron structure deposited by glancing-angle-ion-beam (GLAD) deposition (Fig. 5b). Data were taken at ten wavelengths (λ1 = 1380 nm . . . λ2 = 1542 nm) versus sample rotation ϕ0 . 4.3. (b) Upper panel: optical constants nx , ny , nz , and kx , ky , kz obtained from the best-model calculation, inherent to the orthorhombic polarizability system attached to the columns as indicated. Lower panel: Transmission electron microscopy contrast image of the nano-chevron structure. 4.4 Complex anisotropic mediums: Sculptured thin ﬁlms A new class of design materials emerges upon sculpturing solid-state materials in thin ﬁlms, for which the generalized ellipsometry approach is mandatory in any attempt to characterize the intrinsic polarizability values and axes system. Such designs involve physical deposition techniques in three-dimensional growth regimes, where, depending on growth parameters and appropriate substrate rotation, e.g., “zigzag” pattern, chevrons, “S”-shapes or helices can be deposited [56, 60–65]. Design dimensions can be well within the nanometer region, setting the stage for new physics and novel applications [66]. Due to the complexity of such ﬁlms, optical characterization by ellipsometry is a challenge, and sound description, i.e., appropriate mathematical models (“Erklärungssystem”), for ellipsometry spectra of complex nano-structured thin ﬁlms are rare so far [67–69]. The generalized ellipsometry approach is well suited for nondestructive characterization, and best-model calculations concordant with nano-structure geometries can be found. Fig. 5a depicts experimental and best-model calculated data for a silicon-on-silicon nano-chevron structure deposited by glancing-angle-ion-beam (GLAD) deposition (Dr. Eva Schubert, Leibnitz Institute for Surface Modiﬁcation e.V.). The structure consists of 5 sequences with silicon columns subtending alternating angles with the sample normal. The shape of the columns causes form birefringence, and which can be well described by an orthorhombic (hence optically biaxial) symmetrically-dielectric anisotropy throughout the entire structure with alternating in-plane but common out-of-plane orientations. The resulting spectra for nx , ny , nz , and kx , c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 7 – 8 (2006) 491 0.2 0.0 -0.4 M31 δM31 BInGaAs/GaAs 0.1 M21 δM21 -0.1 M22 δM32 1.000 0.975 -0.4 M12 0.00 -0.15 -0.6 M13 GaAs BInGaAs:Si M23 * -0.8 ** -0.1 GaAs BInGaAs:Si * 0.2 δM23 0.1 M33 -1.0 100 δM12 δM22 0.1 M32 -0.2 -0.1 200 300 400 -1 ω [cm ] (a) 500 600 100 ** δM13 δM33 200 300 400 -1 ω [cm ] (b) 500 600 Fig. 6 (a) Generalized ellipsometry data (symbols: experiment; solid lines: best-model) for a 1288nm-thick silicon doped n-type B0.03 In0.06 Ga0.91As thin ﬁlm on undoped GaAs substrate. The angle of incidence is 45◦ . The sample is isotropic. The bands of total reﬂection for GaAs and B0.03 In0.06 Ga0.91As:Si are indicated by brackets. The GaAs-band extends from its TO (267 cm−1 : solid vertical bar) to its LO frequency (292 cm−1 : dotted vertical bar). For the B0.03 In0.06 Ga0.91As:Si this band includes the GaAs-like band (ωTO =267 cm−1 : solid vertical bar; ωLO =289 cm−1 ), which extends due to LPP mode coupling (ωLPP+ =303 cm−1 : dotted vertical bar), and In-related (273 cm−1 :*) and Si-related (355 cm−1 :**) impurity modes. (b) Magnetic-ﬁeldinduced optical non-reciprocity of the free-charge-carrier response within the thin ﬁlm rendered by differences between spectra obtained within magnetic ﬁelds of µ0 H = −3, and +3T. A strong chiral resonance occurs near ωLPP+ . The best-model calculations provide the effective mass parameter of the charge carriers in the B0.03 In0.06 Ga0.91As layer. The sign of the difference spectra reveal that the free-charge-carriers are electrons [70]. ky , kz and the coordinate system are depicted in Fig. 5b. The same model provides orientation description of the columns in each sublayer as well as the sublayer thickness values. Further details of the sample shown here will be given somewhere else. 4.5 The optical Hall effect: “Weighing” free-charge-carriers According to Drude’s free electron model, the dielectric function of materials with free-charge-carriers, doped semiconductors for example, provides access to the two coupled quantities N/m and N µopt by virtue of the plasma frequency and plasma broadening parameters. The quantities N , m and µopt , however, are very important for materials physics understanding and semiconductor device design, for example. The effective mass concept addresses one of the fundamental physical material properties in semiconductors. This concept descends from the Newton force equation (acceleration of a body with inertial mass m) and the acceleration experienced by a Bloch electron due to an external force. The thereby obtained inverse effective www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 492 M. Schubert: Another century of ellipsometry mass tensor depends on the curvature of the plots of the electron energetic states versus electron momentum 2 1 diagrams E(k), and is diagonal by a suitable choice of axes ( m ) = 12 ( ∂∂k2 )E(k). For example, in nondegenerate zincblende-structure semiconductors (e.g., many of the III-V compounds), the long-wavelength response of the electrons in the Γ-point spherical conduction bands provide access to the (isotropic) effective mass parameter, often addressed as (infrared) optical effective mass, which is furthermore, and within simple k · p schemes, approximately proportional to the band-gap energy E0 [41]. In order to break the above mentioned correlation, N obtained from electrical Hall-effect measurements can be combined, e.g., with ωp and γp obtained from infrared ellipsometry results and m and µopt can be calculated for thin layers [55]. However, electrical contacts – needed for this procedure – potentially affect the free-chargecarrier properties upon surface state formation and Fermi level pinning. Moreover, if the material of interest is part of a complex layer structure the Hall-effect interpretation is difﬁcult, if not impossible at all. Extending Drude’s ellipsometry framework, magnetooptic generalized ellipsometry at long wavelengths allows for non-contact and precise determination of m, N , and µopt in such layer structures, dispensing with the need for electrical contacts [70–73]. The optical response of such systems is anisotropic, and the use of the generalized ellipsometry concept is mandatory in general.11 A simple sketch of the “Erklärungssystem” is given here. The conductivity tensor σ (in SI units and Cartesian coordinates) descends from the equation of motion for a free-charge-carrier (a single species is considered only) with charge q (q = +|e| for free holes, and q = -|e| for free electrons; me is the free electron mass) subjected to an external magnetic ﬁeld slow = H(hx , hy , hz ) = Hh H me 1 + µ0 H v × h , γp + ∂t v = E (14) q m where v = (∂t x, ∂t y, ∂t z) denotes the carrier velocity. The corresponding dielectric function tensor is nonsymmetric (assuming spherical conduction bands) 1 2 2 ε (ω) = −ωp ω + iωγp 0 0 0 1 0 0 0 0 − iω hz 1 −hy −hz 0 hx −1 hy −hx ωc , 0 (15) µ0 H denotes the cyclotron frequency. The sign of the charge as well as the direction where ωc = q m em of Hslow determines the sign of the non-reciprocive part in ε. The independent parameter gained from measuring ε is now ωc , which provides the required additional information in order to determine N , m and µopt independently. With the appropriate optical technique to measure ε – generalized ellipsometry – the optical Hall effect is thereby established. In principle, and together with appropriate functions describing polar lattice excitations, the tensor above is sufﬁcient to explain and model the generalized ellipsometry parameters of doped semiconductor thin ﬁlms brought into magnetic ﬁelds, speciﬁcally at infrared and far infrared wavelengths.12 Fig. 6 depicts data from a semiconductor thin ﬁlm structure placed inside a split-coil superconducting magneto-cryostat [70]. Data analysis provided for the quaternary layer ωp = 753.5 cm−1 , γp = 113.5 cm−1 , and ωc = 30.1 cm−1 , or equivalently N = 5.9 × 1017 cm−3 , µopt = 885 cm2 /(Vs), and m = 0.093. 11 For applications of generalized ellipsometry to magnetic materials see R. Rauer, G. Neuber, J. Kunze, J. Bäckström, M. Rübhausen, Sci. Instr. 76, 023910 (2005), or A. Berger, M. R. Pufall, Appl. Phys. Lett. 71, 965, (1997). Rev. 1 and γp may potentially possess tensorial character, and hence the quantities ωp2 and ωc receive tensor character m as well. The response due to polar lattice vibrations must be further taken into account, and which causes the so-called longitudinal plasmon phonon (LPP) modes. Detailed analysis of their magnetooptic response reveals further ﬁne structure due to coupling with the cyclotron motion and interface polariton excitations [26, 41, 74, 75]. 12 Note that c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 7 – 8 (2006) 493 0.08 13 B = 2.83 T 11 9 12 8 10 7 6 5 B = 2.12 T B = 1.77 T 0.06 ωLandau [eV] ωLandau [eV] MM2323/M ∆ / M1111 B = 2.47 T n=4 0.04 B = 1.42 T B = 1.06 T 0 100 200 300 400 -1 ω [cm ] (a) 500 600 0.02 1.0 1.5 2.0 2.5 3.0 3.5 B [T] (b) Fig. 7 (a) Terahertz (black symbols) and far-infrared range (gray symbols) magnetic-ﬁeld induced nonreciprocal anisotropy in highly-oriented pyrolytic graphite (HOPG) rendered by the Muller matrix element M23 normalized to M11 . The component of the magnetic ﬁeld B parallel to the graphite c axis is indicated. A conventional black-body emitter (silicon carbide globar) and the high-brilliant infrared radiation at the IRIS beam line at BESSY were utilized as independent sources for the far-infrared and terahertz region, respectively. Landau quantization of the two-dimensional electron density within the graphene layers at 4.5K causes intraband transitions between Landau splitted electron levels (vertical arrows), separated by multiple amounts of the cyclotron frequency ωc . (b) A fan of transition energies between individual Landau levels versus applied magnetic ﬁeld parallel to the lattice c axis, which matches the observed resonances in the far-infrared region. See text for further detail. 4.6 The optical Hall effect in quantum regimes: Landau level transitions The electronic states of two-dimensionally conﬁned charge carrier systems split into the so-called Landau levels if subjected to external magnetic ﬁelds with direction parallel to the conﬁnement direction. Provided the temperature of the free-charge-carrier system is smaller than the Landau level spacing, which is equivalent to their cyclotron resonance energy ωc , transitions between individual Landau levels can be revealed optically by ellipsometry measurements at low temperatures and very long wavelengths. This is shown here – exemplarily as well as for the ﬁrst time – for an often investigated two-dimensional charge system: Highly oriented pyrolytic graphite (HOPG) [76]. Speciﬁcally, a synchrotron source with high-brilliant terahertz radiation was used as the light source in addition to a thermal radiation source for this generalized ellipsometry experiment [73, 77, 78]. Graphite remains a challenging material with many physical properties to be explored and explained [79]. Yet as one of the best studied materials concerning its electronic and optical properties, many questions remain open for this semimetal or semiconductor with zero band gap, which is thought to be endowed with unique massless linear dispersion relation for the conduction band [80]. www.ann-phys.org c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 494 M. Schubert: Another century of ellipsometry Recently, observation of plateaus in Hall resistance data for a HOPG sample in the quasi-quantum limit suggested Quantum-Hall effect characteristics of the two-dimensional electron system [81], which forms with high electron mobility at low temperatures. Ferromagnetism-like signals and defect-related magnetism in graphite was recently reported [82]. Reentrant magnetic-ﬁeld driven metal-insulator transitions were also reported [83]. It is of interest here to draw an “Erklärungssystem” for the optical polarization response of circularly polarized Landau transitions. The Landau transitions occur for circular polarizations and propagation di slow ||c. Transitions rections parallel c, i.e., perpendicular to the graphite planes, and only in presence of H for electrons (holes) may occur for a given ﬁeld orientation with polarizability e (h ), where each species responds to either one of the polarization modes. The series of transitions may at ﬁrst hand be approximated by sums of harmonic Lorentzian-damped oscillator functions e,h = ν (ωLandau,ν )2 Aν , − (ω)2 − iωγν (16) where Aν , ωLandau,ν , and γν are amplitude, energy and broadening of transition number ν, which may all in general differ for electrons and holes. Fig. 7(a) presents generalized ellipsometry spectra of the normalized Mueller matrix element M23 , taken from a natural cleavage of a HOPG sample under 45◦ angle of incidence and various external magnetic ﬁeld strengths. While HOPG renders an optically uniaxial material, the Muller matrix element M23 depicted here vanishes at zero ﬁeld as there is no polarizationmode coupling to be observable owing to this sample orientation. However, with the magnetic ﬁeld turned on, the optical response becomes anisotropic and non-reciprocive, and the Mueller matrix element M23 taken at opposite ﬁeld directions are equal, except for their sign. The broad resonance near ω ≈ 150 cm−1 , which evolves with increasing ﬁeld, can be explained by coupling of the Landau level transitions with an isotropic electronic interband transition. The ﬁne-structure oscillations, which are superimposed onto all spectra, increase in amplitude and period with increasing ﬁeld, and correspond to the circularly polarized Landau Level transitions with almost perfectly equal polarizabilities for both ﬁeld orientations, as will be discussed elsewhere in more detail. The oscillations can be well explained by Eqs. (10) and (16), where − = h = 0 and e = + was implemented. At least addressing here the conduction band curvature of HOPG near the Fermi energy level, the periods within the far-infrared region can be well explained by ωLandau,ν = ωc (ν + ν0 ), ωc ≈ eµ0 H , ν = 1, 2, . . . m(1 + βµ0 H) (17) Eq. (17) matches excellently with the observed transition energies in Fig. 7(b) when ωc = (3.27±0.02)meV at µ0 H = 1T , ν0 = 0.849 ± 0.006, and where the effective mass parameter was adopted with a linear ﬁeld dependence m = mH=0 (1 + βµ0 H), with mH=0 = 0.0354 ± 0.0002 and β = (0.034 ± 0.002)T −1 . H is the slow along the c axis in this experiment. While both the non-zero ν0 and the ﬁeld-dependence component of H of the effective mass parameter provide indications for coupling phenomena between intra- and interband transitions, the band dispersion at zero ﬁeld is seen parabolic in this energy region, with linearly increasing parabolicity parameter m by ≈ 10% for ﬁelds up to 3T , and no indication seems to exist for linear, i.e., massless carrier dispersion in graphite in the quantum regime. 5 Summary In commemoration of Paul Karl Ludwig Drude – the inventor of ellipsometry – insight into state-of-the art ellipsometry characterization approaches were given by surveying selected applications in today’s science and technology. While ellipsometry is a vast topic the author makes no claim to have presented a complete overview in such a limited space. The selection of subjects covered here thus have undoubtedly reﬂected a personal, yet perhaps interesting and illuminating, viewpoint. Future evolution and maybe even revolution c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Leipzig) 15, No. 7 – 8 (2006) 495 are easy to imagine, for example in methodology development by spectral range extension into soft x-ray regions (x-ray ellipsometry), temporal (micro-, nano- or pico-second ellipsometry) and spatial (far- and near-ﬁeld ellipsometry) resolution enhancement. Interesting results are expected from optical Hall effect measurement on semiconductors, superconductors, and charge correlated systems. Acknowledgements I acknowledge exciting collaboration with Tino Hofmann, Alexander Kasic, Eva Schubert, Craig M. Herzinger, Horst Neumann, Wayne Dollase, Gunnar Leibiger, Volker Gottschalch, Bernd Rheinländer, Marius Grundmann and John A. Woollam. I thank Pablo Esquinazi and Dimitri Basov for the joint efforts on the two-dimensional electron gas project. I am grateful to Gerald Wagner for preparing the TEM view graphs in Figs. 3 and 5b. I am grateful to the editors of this dedicated volume, especially to Martin Dressel, for their invitation to set forth my personal view on modern ellipsometry applications in commemoration of Paul Drude’s physics contributions. Finally I acknowledge support from the Deutsche Forschungsgemeinschaft within grants SCHUH1338/3-1 and SCHUH1338/4-1,2, and startup funds provided by the University of Nebraska-Lincoln and the J.A. Woollam Foundation. References [1] A. Rothen, in: Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Krüger (National Bureau of Standards Miscellaneous Publications, Washington D.C., 1963), pp. 7–24. [2] A. C. Hall, in: Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland Publishing Company, Amsterdam, 1969), pp. 1–13. [3] R. M.A. Azzam, in: Ellipsometry, edited by N. M. Bashara and R. M.A. Azzam (North-Holland Publishing Company, Amsterdam, 1976), pp. 6–18. 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