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Another century of ellipsometry.

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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=
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Bζ Bξ
Aζ
/
= tan Ψ exp (i∆) .
Aξ
(1)
c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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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
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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].
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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
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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].
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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,
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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
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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).
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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 ,
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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
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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
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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].
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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
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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.
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