Invited Article: An integrated mid-infrared, far-infrared, and terahertz optical Hall effect instrument

Invited Article: An integrated mid-infrared, far-infrared, and terahertz optical Hall effect instrument
Invited Article: An integrated mid-infrared, far-infrared, and terahertz optical Hall effect
instrument
P. Kühne, C. M. Herzinger, M. Schubert, J. A. Woollam, and T. Hofmann
Citation: Review of Scientific Instruments 85, 071301 (2014); doi: 10.1063/1.4889920
View online: http://dx.doi.org/10.1063/1.4889920
View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/7?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Detection of pulsed far-infrared and terahertz light with an atomic force microscope
Appl. Phys. Lett. 101, 141117 (2012); 10.1063/1.4757606
Space charge mediated negative differential resistance in terahertz quantum well detectors
J. Appl. Phys. 110, 013714 (2011); 10.1063/1.3605480
Terahertz and infrared photodetection using p-i-n multiple-graphene-layer structures
J. Appl. Phys. 107, 054512 (2010); 10.1063/1.3327441
GaSb homojunctions for far-infrared (terahertz) detection
Appl. Phys. Lett. 90, 111109 (2007); 10.1063/1.2713760
Heterojunction wavelength-tailorable far-infrared photodetectors with response out to 70 m
Appl. Phys. Lett. 78, 2241 (2001); 10.1063/1.1361283
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
REVIEW OF SCIENTIFIC INSTRUMENTS 85, 071301 (2014)
Invited Article: An integrated mid-infrared, far-infrared, and terahertz
optical Hall effect instrument
P. Kühne,1,a) C. M. Herzinger,2,b) M. Schubert,1,c) J. A. Woollam,2,d) and T. Hofmann1,e)
1
Department of Electrical Engineering and Center for Nanohybrid Functional Materials,
University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA
2
J. A. Woollam Co., Inc., 645 M Street, Suite 102, Lincoln, Nebraska 68508-2243, USA
(Received 19 February 2014; accepted 17 May 2014; published online 30 July 2014)
We report on the development of the first integrated mid-infrared, far-infrared, and terahertz optical
Hall effect instrument, covering an ultra wide spectral range from 3 cm−1 to 7000 cm−1 (0.1–210 THz
or 0.4–870 meV). The instrument comprises four sub-systems, where the magneto-cryostat-transfer
sub-system enables the usage of the magneto-cryostat sub-system with the mid-infrared ellipsometer
sub-system, and the far-infrared/terahertz ellipsometer sub-system. Both ellipsometer sub-systems
can be used as variable angle-of-incidence spectroscopic ellipsometers in reflection or transmission
mode, and are equipped with multiple light sources and detectors. The ellipsometer sub-systems are
operated in polarizer-sample-rotating-analyzer configuration granting access to the upper left 3 × 3
block of the normalized 4 × 4 Mueller matrix. The closed cycle magneto-cryostat sub-system provides sample temperatures between room temperature and 1.4 K and magnetic fields up to 8 T, enabling the detection of transverse and longitudinal magnetic field-induced birefringence. We discuss
theoretical background and practical realization of the integrated mid-infrared, far-infrared, and terahertz optical Hall effect instrument, as well as acquisition of optical Hall effect data and the corresponding model analysis procedures. Exemplarily, epitaxial graphene grown on 6H-SiC, a tellurium
doped bulk GaAs sample and an AlGaN/GaN high electron mobility transistor structure are investigated. The selected experimental datasets display the full spectral, magnetic field and temperature
range of the instrument and demonstrate data analysis strategies. Effects from free charge carriers
in two dimensional confinement and in a volume material, as well as quantum mechanical effects
(inter-Landau-level transitions) are observed and discussed exemplarily. © 2014 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4889920]
I. INTRODUCTION
The optical Hall effect (OHE) is a physical phenomenon,
which describes the occurrence of transverse and longitudinal magnetic field-induced birefringence, caused by the nonreciprocal magneto-optic response of electric charge carriers.1
The term OHE is used in analogy to the classic, electrical
Hall effect,2 since the electrical Hall effect and certain cases
of OHE observation can be explained by extensions of the
classic Drude model for the transport of electrons in matter
(metals).3, 4 For the OHE, Drude’s classic model is extended
by a magnetic field and frequency dependency, describing
the electron’s momentum under the influence of the Lorentz
force. As a result an antisymmetric contribution is added to
the dielectric tensor ε(ω), whose sign depends on the type of
the free charge carrier (electron, hole). The non-vanishing offdiagonal elements of the dielectric tensor reflect the magneto-
a) Electronic mail: [email protected] URL: http://ellipsometry.
unl.edu/.
b) Electronic
mail: [email protected] URL: http://www.
jawoollam.com.
c) Electronic mail: [email protected] URL: http://ellipsometry.unl.edu.
d) Electronic
mail: [email protected] URL: http://www.
jawoollam.com.
e) Electronic mail: [email protected] URL: http://ellipsometry.
unl.edu.
0034-6748/2014/85(7)/071301/19/$30.00
optic birefringence, which lead to conversion of p-polarized
into s-polarized electromagnetic waves, and vice versa.
The OHE can be quantified in terms of the Mueller matrix, which characterizes the transformation of an electromagnetic wave’s polarization state.5 Experimentally, the Mueller
matrix is measured by generalized ellipsometry (GE).6–13
During a GE measurement different polarization states of the
incident light are prepared and their change upon reflection
from or transmission through a sample is determined.
An OHE instrument conducts GE measurements on samples in high, quasi-static magnetic fields, and detects the magnetic field induced changes of the Mueller matrix.14 Though
several instruments with partial OHE instrument characteristics were described in the literature, most instruments did
not fulfill all criteria for an OHE instrument. Nederpel and
Martens developed in 1985 a single wavelength (444 nm)
magneto-optical ellipsometer for the visible spectral range,
but the instrument provided only low magnetic fields (B ≤ 50
mT).15 In 2003, Černe et al. presented a magneto-polarimetry
instrument (B ≤ 8 T) for the mid-infrared spectral range
(spectral lines of CO2 laser),16 and in 2004 Padilla et al.
developed a terahertz-visible (6 to 20 000 cm−1 ) magnetoreflectance and -transmittance instrument (B ≤ 9 T).17 While
both instruments provide high magnetic fields, and contain
polarizers and photo-elastic-modulators, these instruments
were not designed to record Mueller matrix data (GE). A THz
85, 071301-1
© 2014 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-2
Kühne et al.
time-domain spectroscopy based instrument capable to record
the complex reflection coefficients at magnetic fields of about
B ≈ 0.5 T was described in 2004 by Ino et al.18 The full
4 × 4 Mueller matrix in the terahertz-mid-infrared spectral
range (20 to 4000 cm−1 ) can be measured by an instrument
described in 2013 by Stanislavchuk et al.,19 but there the instrument is not designed for experiments with the sample exposed to external magnetic fields.
The first full OHE instrument was developed and demonstrated in 2003 by Schubert et al.20 for the far-infrared (FIR)
spectral range (30 to 650 cm−1 ), which provided magnetic
fields up to 6 T and allowed sample temperatures between
4.2 K and room temperature.21 This first OHE instrument has
since been successfully used to determine free charge carrier
properties,20, 22–26 including effective mass parameters for a
variety of material systems.20, 27–30 Later, OHE experiments
were conducted in the terahertz (THz) spectral range,31 but
were limited to room temperature and low magnetic fields
(B ≤ 1.8 T).32–35 Since the magnitude of the OHE depends
on the magnetic field strength, higher magnetic fields facilitate the detection of the OHE. Furthermore, the sensitivity to
the OHE is greatly enhanced by phonon mode coupling36, 37
and Fabry-Pérot interferences.33, 38 Since these effects appear
from the THz to the mid-infrared (MIR) spectral range, depending on the structure and material of the sample, it is necessary to extend the spectral range covered by OHE instrumentation. An OHE instrument for the MIR, for example,
can detect the magneto-optic response of free charge carriers enhanced by phonon modes present in the spectral range
above 600 cm−1 , which applies to many substrate materials,
e.g., SiC,36, 39, 40 Al2 O3 ,30, 41 or GaN,42 as well as to many materials used for thin films, e.g., III-V nitride semiconductors
Al1 − x Gax N,37, 43 Al1 − x Inx N44 or In1 − x Gax N.30, 44 In addition, inter-Landau-level transitions can be studied in the MIR
spectral range45–48 with a MIR OHE instrument.40 The extension to the THz spectral range enables the detection of the
OHE in samples with low carrier concentrations.21, 33 Furthermore, the strongest magneto-optic response can be observed at the cyclotron resonance frequency, which typically
lies in the microwave/THz spectral range for moderate magnetic fields (few Tesla) and effective mass values comparable
to the free electron mass.
In this article, we present an OHE instrument, covering an ultra wide spectral range from 3 cm−1 to 7000 cm−1
(0.1–210 THz or 0.4–870 meV), which combines MIR, FIR,
and THz magneto-optic generalized ellipsometry in a single instrument. This integrated MIR, FIR, and THz OHE
instrument incorporates a commercially available, closed
cycle refrigerated, superconducting 8 T magneto-cryostat
sub-system, with four optical ports, providing sample temperatures between T = 1.4 K and room temperature. The ellipsometer sub-systems were built in-house and operate in
the rotating-analyzer configuration, capable of determining
the normalized upper 3 × 3 block of the sample Mueller
matrix.
The operation of the integrated MIR, FIR, and THz OHE
instrument is demonstrated by three sample systems. Combined experimental data from the MIR, FIR, and THz spectral range of a single epitaxial graphene sample, grown on
Rev. Sci. Instrum. 85, 071301 (2014)
a 6H-SiC substrate by thermal decomposition, are shown.
The MIR OHE data of the same epitaxial graphene sample
are used to demonstrate the operation of the MIR ellipsometer sub-system of the integrated MIR, FIR, and THz OHE
instrument,40 over the full available magnetic field range of
the instrument. The magneto-optic response of free charge
carriers and quantum mechanical inter-Landau-level transitions are observed, and their polarization selection rules obtained therefrom are briefly discussed. A Te-doped, n-type
GaAs substrate serves as a model system for the FIR spectral range of the FIR/THz ellipsometer sub-system. The OHE
signal originating from valence band electrons in a bulk material is discussed, and the concentration, mobility, and effective mass parameters of the valence band electrons are
determined. Finally, OHE data from an AlGaN/GaN high
electron mobility transistor structure (HEMT) from the THz
spectral range of the FIR/THz ellipsometer sub-system are
presented and analyzed.38 The data were recorded at different temperature between T = 1.5 K and room temperature, representing the full sample temperature range of the
instrument.
The manuscript is organized as follows, in Sec. II dielectric and magneto-optic dielectric tensors are introduced, a
brief theoretical overview on Mueller matrices and GE dataacquisition is given, and general GE data analysis procedures
are introduced. Section III gives a detailed description of the
experimental setup, while in Sec. IV data acquisition and data
analysis procedures for OHE data are discussed. Examples of
experimental results, demonstrating the operation of the integrated MIR, FIR, and THz OHE instrument, are presented
and discussed in Sec. V, which is followed by a short summary in Sec. VI. Colleagues who might wish to collaborate
on projects of mutual interest with this instrumentation should
contact Professor Tino Hofmann, [email protected]
II. THEORY
The evaluation of physical relevant parameters from the
OHE requires the experimental observation and quantification
of the OHE, and a physical model to analyze OHE data. Experimentally, the OHE is quantified in terms of the Mueller
matrix MOHE 49, 50 by employing GE. The physical model
which is used to analyze the observed transverse and longitudinal magneto-optic birefringence of the OHE is based
on the magneto-optic dielectric tensor εOHE (B), which is a
function of the slowly varying external magnetic field B. If,
among other parameters, the magneto-optic dielectric tensor
of a sample is known, experimental Mueller matrices MOHE
can be modeled from εOHE (B) using the relation
MOHE (εOHE (B)).
(1)
This relation is in general not invertible analytically, but can
be used to determine the magneto-optic dielectric tensor from
experimental Mueller matrix data through nonlinear model
regression analysis.51 Dielectric tensors, Mueller matrix calculus, generalized ellipsometry including data acquisition, as
well as data analysis will be addressed in this section.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-3
Kühne et al.
Rev. Sci. Instrum. 85, 071301 (2014)
A. Magneto-optical dielectric tensors
The optical response of a sample is here described by the
dielectric tensor ε = I + χ, where I and χ denote the 3 × 3
identity matrix and the permittivity tensor of the medium,
respectively. If the dielectric tensor of the sample without a
magnetic field is given by εB=0 = I + χ B=0 and the change
of the dielectric tensor induced by a magnetic field B is given
by ε B = χ B , the magneto-optic dielectric tensor describing
the OHE, can be expressed as
εOHE (B) = I + χ B=0 + χ B .
where n is the charge carrier density. With the Levi-CevitaSymbol ijk ,56 the conductivity tensor σ , the dielectric constant ε0 , and using E = σ −1 j and ε = I + iε1ω σ the permit0
tivity tensor χ = χ B=0 + χ B for charge carriers subject to the
external magnetic field B can be expressed as
(2)
The magneto-optic permittivity of a material within a given
sample described by χ B may originate from the response
of bound and unbound charge carriers subjected to the
magnetic field and the action of the Lorentz force. The
magneto-optic response of a sample subjected to the integrated MIR, FIR, and THz OHE instrument, and which
is addressed in this paper, is represented by a generally
anisotropic and nonreciprocal tensor, i.e., the tensor χ B is
usually not equal to its transposed form.1, 13, 52 Thus, the
corresponding magneto-optic contributions χ + and χ – to
the permittivity tensor χ B , originate from the interaction
of right- and left-handed circularly polarized light with the
sample, respectively.14, 20 Without loss of generality, if the
magnetic field B is pointing in z-direction and the convention N = n + ik is used for the index of refraction,49, 53 the
polarization vector P = ε0 χ E can be described by arranging the electric fields in their circularly polarized eigensystem Ee = (Ex − iEy , Ex + iEy , Ez ) = (E+ , E− , Ez ) by
√
Pe = ε0 χ e Ee = ε0 (χ+ E+ , χ− E− , 0), where i = −1 is the
imaginary unit.21, 54 Transforming Pe back into the laboratory
system the change of the dielectric tensor induced by the magnetic field takes the form:21, 54
⎛
⎞
(χ+ + χ− ) −i(χ+ − χ− ) 0
1⎜
⎟
(χ+ + χ− ) 0 ⎠.
χ B = ⎝ i(χ+ − χ− )
(3)
2
0
0
0
Note, under field inversion B → −B, the polarizabilities for
left- and right-handed circularly polarized light interchange.
χ B is only diagonal if χ + = χ – , and otherwise non-diagonal
with anti-symmetric off diagonal elements.
1. Classic dielectric tensors (Lorentz-Drude model)
Charges carriers, subject to a slowly varying magnetic field obey the classical Newtonian equation of motion
(Lorentz-Drude model)55
mẍ + mγ ẋ + mω02 x = qE + q(ẋ × B),
the charge carrier. With the current density j = nqv Eq. (4)
reads
m 2
1
2
i
ω I − ω I − iωγ j + (B × j) , (5)
E=
nq qω 0
(4)
where m, q, μ = qm−1 γ −1 , x, and ω0 represent the effective
mass tensor, the electric charge, the mobility tensor, the spatial coordinate of the charge carrier, and the eigenfrequency
of the undamped system without external excitation and magnetic field, respectively. For a time harmonic electromagnetic
plane wave with an electric field E → E exp(−iωt) with angular frequency ω, the time derivative of the spacial displacement of the charge carrier ẋ = v therefore also becomes time
harmonic v → v exp(−iωt), where v is the drift velocity of
χik =
−1
nq 2 2
mik ω0 − ω2 − iωγik − iωij k qBj
.
ε0
(6)
For isotropic
can
effective mass tensors the cyclotron frequency ωc = q|B|
m
be defined. For the mass of the vibrating atoms of polar lattice vibrations, the cyclotron frequency is several orders of
magnitude smaller than for effective electron masses, and can
be neglected for the magnetic fields and spectral ranges discussed in this paper. Therefore, the dielectric tensor of polar
lattice vibrations ε L can be approximated using Eq. (6) with
B = 0. When assuming isotropic effective mass and mobility tensors, the result is a simple harmonic oscillator function
with Lorentzian-type broadening.55, 57, 58 For materials with
orthorhombic symmetry and multiple optical excitable lattice
vibrations, the dielectric tensor can be diagonalized to
⎞
⎛ L
εx 0 0
⎟
⎜
(7)
ε L = ⎝ 0 εyL 0 ⎠,
a. Polar lattice vibrations (Lorentz oscillator).
0
0
εzL
where εkL (k = {x, y, z}) is given by59
εkL
= ε∞,k
l
2
ω2 + iωγLO,k ,j − ωLO
,k ,j
j =1
2
ω2 + iωγTO,k ,j − ωTO
,k ,j
,
(8)
where ωLO,k,j , γLO,k,j , ωTO,k,j , and γTO,k,j denote the
k = {x, y, z} component of the frequency and the broadening
values of the j th longitudinal optical (LO) and transverse
optical (TO) phonon modes, respectively, while the index j
runs over l modes. Further details can be found in Refs. 28
and 59–63, and a detailed discussion of the
requirements to
the broadening parameters, such as m εkL ≥ 0, in Ref. 42.
For free
charge carriers no restoring force is present and the eigenfrequency of the system is ω0 = 0. For isotropic effective
mass and conductivity tensors, and magnetic fields aligned
along the z-axis, Eq. (6) can be written in the form εD
OHE (B)
D
+
χ
,
with
the
Drude
permittivity
tensor
= I + χD
B=0
B
for B = 0
b. Free charges carriers (extended Drude model).
χD
B=0 = −
ωp2
I = χ D I,
ω(ω + iγ )
(9)
2
nq
is the plasma frequency, and χ D is perwhere ωp = mε
0
mittivity function of the isotropic Drude dielectric function.
The magneto-optic contribution to the dielectric tensor χ D
B
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-4
Kühne et al.
Rev. Sci. Instrum. 85, 071301 (2014)
for isotropic effective masses and conductivities can be expressed, using Eq. (3), through polarizability functions for
right- and left-handed circularly polarized light
χ± =
where ωc =
q|B|
m
χD
1∓
ω+iγ
ωc
,
(10)
is the isotropic cyclotron frequency.
2. Non-classic dielectric tensors (inter-Landau-level
transitions)
χ LL
B
The permittivity tensor
describing the contribution
of a series of inter-Landau-level transitions to the dielectric
tensor can be approximated by a sum of Lorentz oscillators.
The quantities χ ± in Eq. (3) are then expressed by
Ak
,
(11)
χ± = e±iφ
2
2 − iγ ω
ω
−
ω
k
0,k
k
where Ak , ω0, k , and γ k are amplitude, transition energy, and
broadening parameter of the kth inter-Landau-level transition, respectively, which in general depend on the magnetic
field. The phase factor φ was introduced empirically here
to describe the experimentally observed line shapes of all
Mueller matrix elements. For inter-Landau-level transitions in
graphite or bi-layer graphene we find φ = π /4, and for interLandau-level transitions in single layer graphene φ = 0.
Note that for φ = 0, the polarizabilities for left- and righthanded circularly polarized light are equal (χ + = χ − ), and
ε LL
B is diagonal.
B. Mueller matrix calculus, GE, and data acquisition
1. Stokes vector/Mueller matrix calculus
The real-valued Stokes vector S has four components,64
carries the dimension of an intensity, and can quantify any polarization state of plane electromagnetic waves. If expressed
in terms of the p- and s-coordinate system,65 its individual
components can be defined by S1 = Ip + Is , S2 = Ip − Is ,
S3 = I45 − I−45 , and S4 = Iσ + − Iσ − , with Ip , Is , I45 , I−45 ,
Iσ + , and Iσ − being the intensities for the p-, s-, +45◦ , −45◦ ,
right- and left-handed circularly polarized light components,
respectively.6, 66
The real-valued 4 × 4 Mueller matrix M describes the
change of electromagnetic plane wave properties (intensity,
polarization state), expressed by a Stokes vector S, upon
change of the coordinate system or the interaction with a sample, optical element, or any other matter6, 49
Sj(out)
=
4
Mij Si(in) ,
(j = 1 . . . 4),
(12)
i=1
where S(in) and S(out) denote the Stokes vectors of the electromagnetic plane wave before and after the change of the
coordinate system, or an interaction with a sample, respectively. Note that all Mueller matrix elements of the GE data
discussed in this paper, are normalized by the element M11 ,
!
therefore |Mij | ≤ 1 and M11 = 1.
2. Mueller matrix and OHE data
The Mueller matrix can be decomposed in 4 submatrices, where the matrix elements
of the two off
M13 M14
M31 M32
diagonal-blocks
and
only deM23 M24
M41 M42
viate from zero if p-s-polarization mode conversion appears,
while
the matrix
elements
in the
two on-diagonal-blocks
M11 M12
M33 M34
and
mainly contain informaM21 M22
M43 M44
tion about p-s-polarization mode conserving processes. p-spolarization mode conversion is defined as the transfer of
energy from the p-polarized channel of an electromagnetic
plane wave to the s-polarized channel, or vice versa. Polarization mode conversion can appear when the p-s-coordinate
system is different for S(in) and S(out) ,67 or when a sample
shows birefringence, for example. In particular, polarization
mode conversion appears if the dielectric tensor of a sample
possesses non-vanishing off-diagonal elements. Therefore, in
Mueller matrix data from optically isotropic samples, ideally
all off-diagonal-block elements vanish, while, for example,
magneto-optic birefringence can cause non-zero off-diagonalblock elements in the Mueller matrix.
Here, we define OHE data as Mueller matrix data from
an OHE experiment [Eq. (1)] with magnetic field ±B
M±
OHE = M(ε B=0 + ε ±B ),
(13)
where ε±B = χ B is the magnetic field induced change in
the dielectric tensor.
Furthermore, we define the derived OHE datasets δM±
as difference data between the Mueller matrix datasets, measured at the magnetic field ±B and the corresponding zero
field dataset
δM± = M±
OHE − M0
= M(εB=0 , ε ±B ),
(14)
where M0 = M(ε B=0 ) is the Mueller matrix of the zero field
experiment and M(ε B=0 , ε ±B ) is the magnetic field induced change of the Mueller matrix. This form of presentation is in particular advantageous in case the magnetic field
causes only small changes in the Mueller matrix, and provides improved sensitivity to magnetic field dependent model
parameters during data analysis. Another form of presentation
for derived OHE data is
δM+ ± δM− = M(εB=0 , ε +B )
± M(εB=0 , ε −B ),
(15)
which can be used to inspect symmetry properties of
magneto-optic Mueller matrix data, and can help to improve
the sensitivity to magnetic field dependent model parameters
during data analysis.
3. Mueller matrix data acquisition (GE)
Spectroscopic ellipsometers can be categorized according to their polarization optical components and operation
principles, where different subsets of Mueller matrix elements
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-5
Kühne et al.
Rev. Sci. Instrum. 85, 071301 (2014)
may be accessible.49 For example: (i) rotating analyzer
ellipsometers (RAE)7, 49, 68, 69 in polarizer-sample-rotatinganalyzer (PSAR ) or rotating-polarizer-sample-analyzer
(PR SA) configuration are capable to measure the upper left 3
× 3 block of the Mueller matrix; (ii) rotating compensators
ellipsometers (RCE)49, 68, 70–73 in polarizer-sample-rotatingcompensator-analyzer (PSCR A) or polarizer-rotatingcompensator-sample-analyzer (PCR SA) configuration are
capable to measure the upper left 3 × 4 or 4 × 3 block of the
Mueller matrix, respectively.
The Mueller matrices of a polarizer P, analyzer A, compensator C(δ) with phase shift δ, coordinate rotation along
beam path R (θ ) by an angle θ , and of the sample M are given
by
⎡
⎤
1 1 0 0
⎢
⎥
1⎢1 1 0 0⎥
⎥,
P=A= ⎢
⎥
2⎢
⎣0 0 0 0⎦
0 0 0 0
⎡
1 0
⎢
⎢0 1
C (δ) = ⎢
⎢0 0
⎣
0 0
⎡
1
0
0
cos δ
sin δ
0
⎢
⎢ 0 cos 2θj
R (θ ) = ⎢
⎢ 0 – sin 2θ
⎣
j
0
0
⎡
M11
⎢
⎢ M21
M=⎢
⎢M
⎣ 31
M41
⎤
0
⎥
⎥
⎥,
– sin δ ⎥
⎦
cos δ
0
0
cos 2θj
0
M12
M13
M22
M23
M22
M33
M42
M43
MSE = E
2
a b S
G
Mij,k − Mij,k
1
, (17)
abS − K i=1 j =1 k=1
σMij,k
E
denotes the total number of specwhere S, K, a, b, and σMij,k
E
tral data points, the total number of parameters varied during the non-linear regression process, the number experimentally determined columns and rows of the Mueller matrix,
E
, obtained during the exand the standard deviation of Mij,k
periment, respectively. For fast convergence of the MSE regression, the Levenberg-Marquardt fitting algorithm is used.75
The MSE regression is interrupted when the decrease in the
MSE is smaller than a set threshold and the determined parameters are considered as best model parameters. The sensitivity and possible correlation of the varied parameters is
checked and, if necessary, the model is changed and the process is repeated.76–78
D. THz time-domain spectroscopy based ellipsometry
0
sin 2θj
points, angles of incidence, and magnetic fields. During the
mean square error (MSE) regression, the generated Mueller
G
are compared with the experimental Mueller
matrix data Mij,k
E
matrix data Mij,k and their match is quantified by the MSE
⎤
(16)
⎥
0⎥
⎥,
0⎥
⎦
1
M14
⎤
⎥
M24 ⎥
⎥,
M34 ⎥
⎦
M44
respectively. Execution of the matrix multiplication characteristic for the corresponding ellipsometer type49 shows that,
due to the rotation of optical elements, the measured intensity at the detector is typically sinusoidal. Fourier analysis
of the detector signal provides Fourier coefficients, which
are used to determine the Mueller matrix of the sample (see
Sec. IV A).
C. Data analysis
Ellipsometry is an indirect experimental technique.
Therefore, in general, ellipsometric data analysis invokes
model calculations to determine physical parameters in dielectric tensors or the thickness of layers, for instance.74
Sequences of homogeneous layers with smooth and parallel interfaces are assumed in order to calculate the propagation of light through a layered sample, by the 4 × 4 matrix formalism.8, 13, 49 To best match the generated data with
experimental results, parameters with significance are varied
and Mueller matrix data are calculated for all spectral data
Beside THz frequency-domain spectroscopy based ellipsometry discussed in this paper, ellipsometry and magnetooptic ellipsometry can be conducted using THz time-domain
spectroscopy (THz-TDS).79 Typically, THz-TDS is based on
the Fourier transformation of the time resolved signal of ultrashort (picosecond) laser pulses, revealing the THz spectrum. THz ellipsometry based on THz-TDS was reported
by the Hangyo-group in 200180–82 and in 2012 by Neshat
et al.83 THz-TDS magneto-ellipsometry measurements were
reported by Ino et al.,18 but external magnetic fields were limited to B ≈ 0.5 T and Mueller matrix capabilities of the instrument were not demonstrated.
III. INTEGRATED MIR, FIR, AND THZ OHE
INSTRUMENT
Figure 1 shows (side view) the integrated MIR, FIR, and
THz OHE instrument with its four sub-systems: (A) the MIR
ellipsometer sub-system, (B) the FIR/THz ellipsometer subsystem, (C) the magneto-cryostat sub-system, and (D) the
magneto-cryostat transfer sub-system. In order to utilize the
magneto-cryostat sub-system with the MIR or the FIR/THz
ellipsometer sub-system the magneto-cryostat transfer subsystem was installed. The integrated MIR, FIR, and THz OHE
instrument contains multiple light sources and detectors, and
covers a spectral range from 3 cm−1 to 7000 cm−1 (0.1–
210 THz or 0.4–870 meV). Both ellipsometer sub-systems
can be operated without the magneto-cryostat sub-system
in a variable angle of incidence ellipsometry mode84 (a
= 30◦ . . . 90◦ ). Figure 2 shows a schematic overview (top
view) of all major components in the integrated MIR, FIR,
and THz OHE instrument.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-6
Kühne et al.
Rev. Sci. Instrum. 85, 071301 (2014)
FIG. 1. Technical drawing (side view) of the integrated MIR, FIR and THz OHE instrument. The instrument has four sub-systems, (i) the MIR ellipsometer
sub-system, (ii) the FIR/THz ellipsometer sub-system, (iii) the magneto-cryostat sub-system, and (iv) the magneto-cryostat transfer sub-system. The magnetocryostat transfer sub-system holds both ellipsometers and serves as a translation system for the magneto-cryostat sub-system, which can be used with the
ellipsometer sub-systems. The total dimensions of the integrated MIR, FIR, and THz OHE instrument are 160 cm × 450 cm × 115 cm (h × w × d).
A. MIR ellipsometer sub-system
The upper part of Fig. 2 shows a schematic drawing (top
view) of the optical configuration of the MIR sub-system
of the integrated MIR, FIR, and THz OHE instrument. The
MIR ellipsometer sub-system is composed of (i) the MIR
source unit, (ii) the polarization state preparation unit, (iii) the
MIR goniometer unit, and (iv) the polarization state detection
unit. To minimize absorption due to water vapor, the complete beam path of the MIR ellipsometer sub-system is purged
with dried air.85 Due to the high magnetic stray-fields (see
Fig. 4), all opto-mechanical components in the polarization
state preparation and detection units were designed and manufactured without ferromagnetic materials (with exception of
the stepper motors).
The MIR source unit of the MIR ellipsometer sub-system
is a Bruker Vertex 70 Fourier-transform-infrared spectrometer (Fig. 2: MIR-FTIR) with a silicon carbide globar light
source (spectral range 580–7000 cm−1 ). After being collimated, the light beam passes the interferometer (potassium
bromide (KBr) beam splitter), is reflected by a plane mirror,
and exits the MIR source unit.
The beam enters the polarization state preparation unit.
Inside the polarization state preparation unit the beam passes
a beam steering plane mirror assembly (m1 ), a source selection and beam focusing assembly (R1 ), and a rotation stage
assembly (P1 -b1 -St1 ). The beam steering plane mirror assembly (m1 ) is composed of an opto-mechanic mount and a plane
first surface gold mirror.86 The source selection and beam focusing assembly (R1 ) comprises two sub-assemblies (detailed
drawing: Fig. 3(a)), the rotatable plane mirror sub-assembly87
(stepper motor, axis extension, mechanic mount for axis extension, opto-mechanic mount for mirror, plane first surface
gold mirror86 ) and the beam focusing off-axis paraboloid
stage sub-assembly (opto-mechanic mount for paraboloid,
gold surface 90◦ off-axis paraboloid with an effective focal
length of fe = 350 mm88 ), which focuses the beam onto the
sample position. The focused beam then reaches the rotation
stage assembly (P1 -b1 -St1 ), which has a nominal angular resolution of 0.045◦ . The rotation stage assembly (detailed drawing: Fig. 3(b)) contains a KRS-5 substrate based wire grid
polarizer (P1 ), which is mounted in a hollow tube that is fitted into two polymer bearings with glass balls and held by an
aluminum block. A 48-tooth polymer gear is mounted to the
hollow tube, and is connected via a Kevlar timing belt (b1 ) to
a 12-tooth polymer gear on a stainless steel shaft (gear ratio:
1:4), leading to the stepper motor (St1 ). After passing the rotation stage assembly, the polarized and focused beam leaves
the polarization state preparation unit.
The beam is then reflected by, or transmitted through
the sample (S1 ). The sample can be mounted on a sample
holder, attached to the MIR goniometer unit (G1 ) (commercially available 2-circle goniometer 415, Huber Diffraktionstechnik), or inside the magneto-cryostat sub-system (M1).
If the magneto-cryostat sub-system is used, reflection type
measurements can only be conducted at a a = 45◦ angle
of incidence. A detailed description of the magneto-cryostat
sub-system, its sample mount, and the optical window configuration is given in Sec. III C.
The beam then enters the polarization state detection
unit, which is mounted to the rotatable arm of the MIR
goniometer unit (G1 ). The polarization state detection unit
contains a rotation stage assembly (A1 -b2 -St2 ), a beam collimation and detector selection assembly (R2 ), and three
beam focusing/detection assemblies (o1 -MCT, o2 -DTGS1 and
o3 -B1 ). The rotation stage assembly (A1 -b2 -St2 ) is equivalent
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
Kühne et al.
071301-7
Rev. Sci. Instrum. 85, 071301 (2014)
A:
B:
b:
BWO:
Analyzer
Bolometer
Timing Belt
Backward Wave
MIR-FTIR
Oscillator Source
Ch
Chopper Wheel
DTGS: DTGS Detector
St1
FTIR: Bruker V70 FTIR
g:
Golay Cell
In
Detector
G:
Goniometer Unit
R1
In:
Input Port (for
m2
m1
additional light
b1 P1
sources)
MCT: HgCdTe Detector
MCT
m:
Plane Mirror
o1
M1
o2
M1,M2: Closed-Cycle 8T
A
Magneto-Cryostat
R2 DTGS1
S1
in Position 1&2
St2
o:
90° First Surface
b2
Gold Off-AxisG1
o3
Paraboloid
B1
P:
Polarizer
PR:
Polarization State
Rotator
R:
Source/Detector
0
10 20 30 40cm
Selection, Beam
Focusing
Assembly
S:
Sample
MIR
sub-system
o7
B2
St5
G2
b5
g
DTGS2
S2
R4
A2
m7
o6
M2
o5
m6
0
10
20
30
40cm
FIR/THz
subsystem
b4
P2
m5
R3
b3
St4
St3
FIR-FTIR
m4
FIG. 3. (a) Technical drawing of the source selection and beam focusing/beam collimation and detector selection assemblies (R1 − 4 in Fig. 2).
Each source selection and beam focusing/beam collimation and detector
selection assembly is composed of a rotatable plane mirror sub-assembly
[plane first surface gold mirror (mr ), opto-mechanic mount for mirror
(Tm ), opto-mechanic mount for axis (TA ), axis with stepper motor and encoder wheel (Str )] and a collimating off-axis paraboloid stage sub-assembly
[opto-mechanic mount for paraboloid (Tp ), 90◦ first surface gold off-axisparaboloid (or )]. (b) Technical drawing of the rotation stage assemblies for
polarizers and analyzers (here depicted: rotation stage assembly for polarizer of the FIR/THz ellipsometer sub-system, P2 -b4 -St4 in Fig. 2). The stages
comprise an aluminum frame, two plastic bearings with glass balls (Ba), a
shaft which can hold up to two wire grid polarizers (P2 ), two gears [12 teeth
(G12 ) and 48 teeth (G48 )], a Kevlar timing belt (b4 ) and an approx. 30 cm
long stainless steel shaft to a stepper motor with encoder wheel (St4 ).
m3
Ch
PR
BWO
o4
FIG. 2. Schematic drawing (top view) of the in-house built, variable angleof-incidence spectroscopic ellipsometer sub-systems, used for magneto-optic
measurements in the wavelength range from 3 to 7000 cm−1 . In the top part
the Fourier-transform-infrared spectroscopy based MIR ellipsometer subsystem is depicted while the lower part shows the combined FIR/THz ellipsometer sub-system. The closed-cycle 8 T magneto-cryostat sub-system can
be moved between the two ellipsometer sub-systems (M1 or M2) utilizing
the magneto-cryostat transfer sub-system (not depicted).
to the one in the polarization state preparation unit (P1 -b1 -St1 ,
Fig. 3(b)), but uses an approx. 35 cm stainless steel shaft
(located under the aluminum base plate of the polarization
state detection unit) to connect the stepper motor St2 with
the polymer timing belt b2 . Further, in the polarization state
detection unit the KRS-5 substrate based wire grid polarizer serves as the analyzer (A1 ) of the MIR ellipsometer
sub-system. The beam collimation and detector selection assembly (R2 ) is composed of two sub-assemblies: the rotatable plane mirror sub-assembly and the collimating off-
axis paraboloid stage sub-assembly (gold surface 90◦ off-axis
paraboloid, fe = 350 mm). Both sub-assemblies are equivalent to the source selection and beam focusing assembly in
the polarization state preparation unit (Fig. 3(a)), but are used
in reverse order to first collimate and then redirect the beam
to one of the three beam focusing/detection assemblies using the rotatable mirror.86 Each beam focusing/detection assembly contains a beam focusing off-axis paraboloid stage
sub-assembly and a detector sub-assembly. All detector subassemblies contain an opto-mechanic mount and a detector.
The beam focusing off-axis paraboloid stage sub-assemblies
(o1−3 ) are composed of an opto-mechanic mount and a 90◦
off-axis paraboloid with an uncoated gold surface. The focal lengths of the off-axis paraboloids are matched to the
corresponding detector. The off-axis paraboloids (o1 , o2 ) for
the liquid nitrogen cooled HgCdTe-detector sub-assembly
(MCT) and the pyroelectric, solid state deuterated triglycine
sulfate detector sub-assembly (DTGS1 ), each have an effective focal length of 38 mm (1.5 in). The off-axis paraboloid
o3 for the liquid helium cooled bolometer detector subassembly (B1 )89, 90 has an effective focal length of 190.5 mm
(7.5 in). The signal of the selected detector is fed back into
the MIR-FTIR-spectrometer to record interferograms. For
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-8
Kühne et al.
FIG. 4. Technical drawing of the FIR/THz ellipsometer sub-system of the
integrated MIR, FIR, and THz OHE instrument (MIR ellipsometer subsystem not shown) and the magneto-cryostat sub-system. A cutout view of
the magneto-cryostat sub-system (blue cylinder, top, center) shows the superconducting magnet coils, the variable temperature inset (VTI), and the
sample. The three cutout prolate ellipsoids (green) represent the spacial positions at which the magnetic stray field is less than 0.1, 0.025, and 0.01 T. The
beam path is indicated in red.
more information on the data acquisition and processing see
Sec. II C.
B. FIR/THz ellipsometer sub-system
The lower part of Fig. 2 shows a schematic drawing (top
view) of the optical configuration of the FIR/THz ellipsometer sub-system of the integrated MIR, FIR, and THz OHE instrument. The FIR/THz ellipsometer sub-system can be divided into five units: (i) the FIR source unit, (ii) the THz
source unit, (iii) the polarization state preparation unit, (iv) the
FIR/THz goniometer unit, and (v) the polarization state detection unit. For measurements in the FIR spectral range, the
FIR/THz ellipsometer sub-system is operated in analyzer-step
mode,66 while for measurements in the THz spectral range the
FIR/THz ellipsometer sub-system is operated in continuously
rotating analyzer mode. To minimize absorption due to water
vapor, the complete beam path of the FIR/THz ellipsometer
sub-system can be purged with dried air.85 Due to the high
magnetic stray-fields (see Fig. 4), all opto-mechanical components in the THz source unit, polarization state preparation
unit and polarization state detection unit were designed and
manufactured without ferromagnetic materials (with exception of the stepper motors and the THz source).
The FIR source unit of the FIR/THz ellipsometer subsystem is a Bruker Vertex V-70 FTIR-spectrometer (Fig. 2:
FIR-FTIR). The spectrometer is equipped with a silicon beam
splitter, but is otherwise identical to the spectrometer used as
the MIR source unit in the MIR ellipsometer sub-system.
The THz source unit comprises five assemblies: the THz
source and THz-beam collimation assembly (Fig. 2: BWOo4 ), the optical chopper assembly (Ch), the beam steering
plane mirror assembly (m3 ), the polarization state rotator assembly (PR – b3 -St3 ), and the beam steering plane mirror
assembly (m4 ). The THz source and THz-beam collimation
assembly contains a THz source sub-assembly and a THzbeam collimating off-axis paraboloid stage sub-assembly. The
THz source sub-assembly is composed of an opto-mechanic
Rev. Sci. Instrum. 85, 071301 (2014)
mount and the backward wave oscillator (BWO) THz source
(Microtech). The BWO source emits THz radiation with a
high brilliance (bandwidth ∼2 MHz) and a high output power
(∼0.1–0.01 W). The THz radiation is almost perfectly linearly polarized and the orientation of the polarization is fixed
in space relative to the BWO housing. The base frequency
range of the BWO is 107–177 GHz, which can be converted
to higher frequency bands using GaAs Schottky diode frequency multipliers. The spectral range accessible by the BWO
can be expanded to 220–350 GHz (× 2 multiplier), 330–525
GHz (× 3 multiplier), 650–1040 GHz (× 2 and × 3 multiplier), and 980–1580 GHz (double × 3 multiplier). Further
details on the BWO based THz source and THz ellipsometry
are given in Ref. 31 and references therein. The THz radiation from the BWO is collimated by the THz-beam collimating off-axis paraboloid stage sub-assembly, composed of
an opto-mechanic mount and a 90◦ off-axis paraboloid (o4 )
with an uncoated gold surface and an effective focal length
fe = 60 mm. The THz-beam then reaches the optical chopper assembly (Ch), which contains an opto-mechanic mount
and a 3 bladed optical chopper, driven by a linear motor.
The 3 bladed optical chopper is rotated with a frequency of
fc = 3.8 Hz, resulting in an optical chopping frequency of
fo = 11.4 Hz, which is close to the optimal frequency response of the Golay cell detector (fopt ∼ 12–15 Hz). After
interaction with the THz-beam steering plane mirror assembly (m3 ) (opto-mechanic mount and plane first surface gold
mirror86 ), the THz-beam is redirected to the polarization state
rotator assembly (PR). The polarization state rotator assembly
(Fig. 5) is an odd-bounce image rotation system.91 The
polarization state rotator is designed to rotate the polarization
FIG. 5. Technical drawing of the polarization state rotator assembly, used in
the THz source unit of the FIR/THz ellipsometer sub-system to pre-align the
polarization direction of the linearly polarized THz-beam with the polarizing
axis of the polarizer in the polarization state preparation unit. The odd-bounce
image rotation system91 is composed of a frame with an opto-mechanic aluminum mount (F), a stepper motor with encoder wheel (St3 ), a 12 teeth gear
(G12 ), connected by a Kevlar timing belt (b3 ) to a 48 teeth gear (G48 ). The 48
teeth gear is glued to a rotatable PEEK cage (PR) which comprises 3 optomechanic mounts with plane first surface gold mirrors (mr ),86 and is mounted
into the aluminum mount (F) by two plastic bearings with glass balls (Ba).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-9
Kühne et al.
state of an incoming electromagnetic beam azimuthally in a
non-deviating, non-displacing fashion (with respect to the incoming electromagnetic beam direction), and is used to prealign the polarization direction of the THz-beam with the polarizing axis of the wire-grid polarizer in the polarization state
preparation unit.92 The polarization state rotator assembly is
composed of a stepper motor (St3 ) with a 12-tooth polymer
gear, which is connected to a 48-tooth polymer gear (gear ratio: 1:4) via a Kevlar timing belt (b3 ), rotating a PEEK cage,
(PR) which contains three opto-mechanic mounts with plane
first surface gold mirrors86 (rotation axis parallel to the incoming and outgoing THz-beam). After reflection on a THzbeam steering plane mirror assembly (m4 ) (opto-mechanic
mount and plane first surface gold mirror86 ), the THz-beam
leaves the THz source unit.
The polarization state preparation unit contains four assemblies: a FIR-beam steering plane mirror assembly (m6 ),
a THz-beam steering plane mirror assembly (m5 ), a source
selection and beam focusing assembly (R3 ), and a rotation
stage assembly (P2 -b4 -St4 ). The FIR- and THz-beam steering
plane mirror assemblies (m6,5 ) are identical and both are composed of an opto-mechanic mount and a plane first surface
gold mirror.86 The plane first surface gold mirrors redirect,
depending on the selected spectral range of the FIR/THz ellipsometer sub-system, the FIR- or THz-beam to the source selection and beam focusing assembly. The source selection and
beam focusing assembly (R3 ) comprises two sub-assemblies,
the rotatable plane mirror sub-assembly and the beam focusing off-axis paraboloid stage sub-assembly, which are equivalent to those in the source selection and beam focusing assembly in the polarization state preparation unit of the MIR
ellipsometer sub-system (Fig. 3(a)). Depending on the orientation of the plane first surface gold mirror86 in the rotatable plane mirror sub-assembly, either the FIR- or THz
beam is directed to beam focusing off-axis paraboloid stage
sub-assembly (gold surface 90◦ off-axis paraboloid, fe = 350
mm). The focused beam is then routed through the rotation
stage assembly, which contains two polyethylene substrate
based wire-grid polarizers (P2 ), but is otherwise identical to
the rotation stage assembly in the polarization state preparation unit of the MIR ellipsometer sub-system (Fig. 3(b)). The
beam then leaves the polarization state preparation unit.
The beam is then reflected by, or transmitted through the
sample (S2 ). The sample can be mounted on a sample holder,
attached to the FIR/THz goniometer unit (G2 ) (commercially
available, 2-circle goniometer 415, Huber Diffraktionstechnik), or inside the magneto-cryostat sub-system (M2). If the
magneto-cryostat sub-system is used, reflection type measurements can only be conducted at a = 45◦ angle of incidence.
A detailed description of the magneto-cryostat sub-system, its
sample mount, and the optical window configuration is given
in Sec. III C.
The beam then enters the polarization state detection
unit, which comprises a rotation stage assembly (A2 -b5 -St5 ),
a beam collimation and detector selection assembly (R4 ),
and three beam focusing/detection assemblies (m7 -o5 -g, o6 DTGS2 , and o7 -B2 ). The beam is routed through the rotation
stage assembly for the analyzer of the FIR/THz ellipsometer
sub-system (A2 -b5 -St5 ). This assembly is similar to the ro-
Rev. Sci. Instrum. 85, 071301 (2014)
tation stage assembly in the polarization state detection unit
of the MIR ellipsometer sub-system (P1 -b1 -St1 , Fig. 3(b)),
but contains two polyethylene substrate based wire-grid polarizers (A2 ) and uses an approx. 35 cm stainless steel shaft
(located between the aluminum base plate of the polarization state detection unit and the bolometer B2 ) to connect
the stepper motor St5 with the polymer timing belt b5 . The
rotation stage assembly for the analyzer of the FIR/THz ellipsometer sub-system can be operated in step mode for FIR
measurements66 or in continuous rotation mode for THz measurements. The beam is then collimated (gold surface 90◦
off-axis paraboloid, fe = 350 mm) and redirected to the selected detector by the beam collimation and detector selection
assembly (R4 ), identical to the beam collimation and detector selection assembly in the polarization state detection unit
of the MIR ellipsometer sub-system (Fig. 3(a)). The beam
focusing and Golay-cell-detector assembly (m7 -o5 -g) contains a beam steering plane mirror sub-assembly (m7 ) (optomechanic mount, plane first surface gold mirror86 ), a beam
focusing off-axis paraboloid stage sub-assembly (o5 ) (optomechanic mount, gold surface 90◦ off-axis paraboloid, fe
= 60 mm), and a Golay-cell detector sub-assembly (g) (optomechanic mount, Golay-cell detector with diamond windows
from Pye Unicam Ltd, model IR50). The beam focusing
and DTGS detector assembly (o6 -DTGS2 ) comprises a beam
focusing off-axis paraboloid stage sub-assembly (o6 ) (optomechanic mount, gold surface 90◦ off-axis paraboloid, fe
= 38 mm), and a solid state deuterated triglycine sulfate detector sub-assembly (DTGS2 ) (opto-mechanic mount, Bruker
Vertex V-70 DTGS detector). Alternatively, the beam focusing and bolometer detector assembly (o7 -B2 ), composed of
a beam focusing off-axis paraboloid stage sub-assembly (o7 )
(opto-mechanic mount, gold surface 90◦ off-axis paraboloid,
fe = 190.5 mm) and the bolometer detector sub-assembly
(B2 ) (opto-mechanic mount; commercially available, liquid
helium cooled bolometer detector, Infrared Laboratories Inc.),
can be used. For THz measurements, the bolometer or the golay cell detector can be chosen, while for FIR measurements
only the bolometer or the DTGS detector provides frequency
responses fast enough to record interferograms.
C. Magneto-cryostat sub-system
The central piece of equipment of the integrated MIR,
FIR, and THz OHE instrument is the commercially available,
superconducting, closed cycle magneto-cryostat sub-system
(7T-SpectromagPT, Oxford Instruments) with four optical
ports (Fig. 4). The design of the integrated MIR, FIR, and THz
OHE instrument allows the usage of the magneto-cryostat
sub-system with the MIR and the FIR/THz ellipsometer
sub-system by employing the magneto-cryostat transfer subsystem. The magneto-cryostat sub-system can be subdivided
into the magnet head, the primary cooling cycle, the secondary cooling cycle, the sample holder, and the optical
windows.
The magnet head contains two magnet coils, which are
mounted around the sample position and are fabricated in
a split-coil pair design. In a spherical volume of 10 mm
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-10
Kühne et al.
diameter around the sample, magnetic fields up to B = 8 T
with an inhomogeneity of less than 0.3% can be achieved.
The magnetic field can be reversed and points towards one
of the optical windows. Therefore, for reflection type OHE
measurements, the magnetic field lies within the plane of innormal.
cidence and forms an angle of 45◦ with the sample √
This leads to a magnetic field component Bc = |B|/ 2 perpendicular to the sample surface.
The primary cooling cycle comprises a pulse tube cooler
(SRP-082, SHI Cryogenics), high pressure helium lines and
a helium compressor (F-70, Sumitomo Heavy Industries).
Ultra-high-purity helium (UHP-He) gas at high pressure is
provided by the helium compressor, and guided by a high
pressure helium line to the pulse tube cooler. The pulse tube
cooler is thermally coupled to the superconducting magnet
coils and allows to cool the magnet coils to temperatures of
T ≈ 3.1 K. The pulse tube cooler also pre-cools the UHPHe in the secondary cooling cycle. The UHP-He gas is then
guided back to the helium compressor by a high pressure helium line, and is reused.
The secondary cooling cycle uses UHP-He gas to cool
the sample. The UHP-He gas in the closed cycle, is circulated by an oil-free, dry scroll pump (XDS-10, Edwards).
When the sample is at base temperature, the UHP-He gas
leaves the outlet of the scroll pump at a pressure of ∼0.5
bars, and is pumped through a zeolite and a liquid nitrogen
trap, in order to extract possible contaminants leaking into
the closed cycle. The UHP-He gas then passes a distiller spiral, which is thermally coupled to the pulse tube cooler (primary cooling cycle), and condenses the UHP-He gas into liquid helium (LHe). The LHe flow is controlled by a needle
valve, and the LHe is then injected into a heat exchanger.
The heat exchanger is attached to the double-walled, hollow
variable temperature inset (VTI) cryostat (Fig. 6). Finally,
the scroll pump reduces the gas pressure above the LHe to
1.5-3 mbar and thereby cools the VTI to a minimal temper-
FIG. 6. Technical drawing of the sample (S) and its holder, including the
sample heater, capsuled in a copper block (H). The sample is thermally coupled to the variable temperature inset (VTI) by a static exchange gas (UHPHe) surrounding the sample. The inner, 0.35 mm thick, diamond windows
(W) are wedged and were glued to their stainless steel window frames (C) by
a two component epoxy (G).
Rev. Sci. Instrum. 85, 071301 (2014)
ature of T = 1.4 K. The sample is thermally coupled to the
VTI by a static exchange gas (UHP-He), and a resistive heater
which allows to warm the sample up to room temperature,
without bringing the temperature of the superconducting magnet coils above the critical temperature.
The sample holder of the magneto-cryostat sub-system
(Fig. 6) can hold up to two samples at a time. If two samples are mounted, the optimal sample size is 0.5 × 12 ×
12 mm3 , while the maximum sample size is 1 × 30 × 30
mm3 (only one sample can be mounted). The sample position is adjustable in the vertical direction (±15 mm) and rotationally around the vertical axis of the VTI by 360◦ (angle
of incidence alignment). All other degrees of freedom necessary for sample alignment (linear motion, rotation, tip/tilt)
can only be accessed by moving the sample together with
the magneto-cryostat sub-system. The magneto-cryostat subsystem can be moved parallel to the incoming beam, using
the magneto-cryostat transfer sub-system. The alignment perpendicular to the incoming beam and the rotational alignment of the magneto-cryostat sub-system is achieved by sliding the magneto-cryostat sub-system in the magneto-cryostat
holding frame of the magneto-cryostat transfer sub-system
(see Sec. III D). Four screws in the same magneto-cryostat
holding frame of the magneto-cryostat transfer sub-system
are used for tip/tilt alignment. After successful alignment, in
order to minimize motion, the magneto-cryostat sub-system
is clamped to the magneto-cryostat holding frame of the
magneto-cryostat transfer sub-system.
When the light beam is routed through the magnet head,
it passes a set of exterior and interior optical windows.
For measurements in the FIR/THz spectral range, the exterior optical windows are made of 0.27 mm thick homopolypropylene films, while for the MIR spectral range potassium bromide (KBr) windows are used. All four exterior
windows are purged on the exterior side with dried air, to
prevent condensation of moisture from the ambient air. The
exterior optical windows of the magnet can be exchanged and
arranged for both transmission- and reflection-type measurements. The latter window configuration allows reflection-type
measurements over the full spectral range of the OHE instrument by simply moving the magneto-cryostat between the
ellipsometer sub-systems without warming up the superconducting coils or the sample. The interior four optical windows
on the VTI are made of polished diamond, grown by chemical vapor deposition (CVD), with a thickness of 0.35 mm, a
diameter of 14 mm, and an average surface roughness of Ra
≤ 15 nm (arithmetic average). To reduce Fabry-Pérot interferences in the interior optical windows the interior windows
were wedged by an angle of ∼0.5◦ . The interior windows
were glued into stainless steel window frames by a cryogenic
two component epoxy, leaving a clear aperture of ∼12 mm
(see cutout in Fig. 6).
In general, optical windows affect experimental ellipsometry data, especially if the optical window material is birefringent. Therefore all optical windows were characterized by
transmission GE on a commercial MIR ellipsometer (J. A.
Woollam Co., Inc.). No birefringence was observed in the
MIR spectral range. Nevertheless, through mounting of the
optical windows, and in particular through stress due to
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-11
Kühne et al.
Rev. Sci. Instrum. 85, 071301 (2014)
the vacuum in the magneto-cryostat sub-system, strain induced birefringence cannot be completely excluded and therefore window effects are included in the analysis of the OHE
data (see Sec. IV B).
The magneto-cryostat transfer sub-system (Fig. 1) contains the magneto-cryostat transfer frame, the MIR source
unit frame and the FIR/THz source unit frame. The magnetocryostat transfer frame comprises the magneto-cryostat transfer assembly, the MIR goniometer unit platform assembly, and FIR/THz goniometer unit platform assembly. The
magneto-cryostat transfer assembly is built from aluminum
extrusions. The grooves of the aluminum extrusions are used
as guides for a Teflon roll based rail sub-assembly. On top of
the Teflon roll based rail sub-assembly the magneto-cryostat
holding frame (black aluminum extrusions in Figs. 1 and
4) is mounted, which is used for alignment purposes (see
Sec. III C). The MIR and FIR/THz goniometer unit platform assemblies are mounted into the magneto-cryostat transfer frame and are platforms for the MIR and FIR/THz goniometer unit of the ellipsometer sub-systems. The MIR and
FIR/THz source unit frames are built from aluminum extrusions, equipped with a 19-in. rack mounting system, and provide platforms for the source units of the MIR and FIR/THz
ellipsometer sub-systems, respectively.
IV. OHE DATA ACQUISITION AND ANALYSIS
A. OHE data acquisition
All spectroscopic ellipsometer sub-systems of the integrated MIR, FIR, and THz OHE instrument are operated in
a PSAR configuration, capable of measuring the upper left
3 × 3 block of the Mueller matrix. Note that this does not affect the ability to determine certain sample properties related
to anisotropy, which can be obtained from the off-diagonalblock elements M13 , M23 , M31 , and M32 . For ellipsometers
based on rotating analyzers with lossless and ideal polarizing optical elements, the stokes vector of a beam of light at
the detector ID can be described within the Mueller matrix
formalism by49
(18)
where IS is the
stokes vector of the light leaving the source,
and P, A, R θj , and M are the Mueller matrices of a polarizer, analyzer, coordinate rotation by the angle θ j (polarizer:
j = P; analyzer: j = A) and sample, respectively.
If the analyzer is rotated with a constant angular frequency ωA , and both light source and detector exhibit no polarization dependency, i.e., the source emits unpolarized light
with intensity I0S and the detector is only sensitive to the total
intensity I0D ,93 the ratio of these quantities is
I0D
1
= [λ(M, θP ) + α(M, θP ) cos(2ωA t)
S
4
I0
+ β(M, θP ) sin(2ωA t)],
λ(M, θP ) = M11 + M12 cos(2θP ) + M13 sin(2θP ),
α(M, θP ) = M21 + M22 cos(2θP ) + M23 sin(2θP ),
(20)
β(M, θP ) = M31 + M32 cos(2θP ) + M33 sin(2θP ) .
D. Magneto-cryostat transfer sub-system
ID = R(−θA )AR(θA )MR(−θP )PR(θP )IS ,
with the time harmonic Fourier coefficients
(19)
Individual Mueller matrix elements Mij are determined from
Fourier coefficients measured at different polarizer orientations θP on the input side (see subroutine Mueller matrix regression in Fig. 7). The complete sequence of operations executed by the OHE instrument during the GE and OHE data
acquisition is summarized in Fig. 7.
Since the intensity of light at the detector and the Fourier
coefficients (employed to determine the Mueller matrix elements) depend on the absolute angular positions of all rotating optical elements [Eqs. (19) and (20)], the knowledge of
these absolute angular positions is crucial for the operation of
every ellipsometer. Furthermore, all spectroscopic ellipsometers have to account for non-ideal polarization characteristics
of sources and detectors, as well as commonly present nonidealities of MIR and FIR optical elements (such as polarizers), in their calibration routine. The absolute positions of the
optical components used in the OHE instrument, with respect
to the p- and s-coordinate system (the s-axis is parallel to the
goniometer axis, the p-axis is perpendicular to the goniometer
axis and to the beam), as well as the non-idealities of the optical elements are carefully calibrated prior to the OHE data
acquisition. All ellipsometer configurations described here
are calibrated by an algorithm using the regression calibration method for rotating analyzer element ellipsometers as
described in Ref. 78. This algorithm determines the Fourier
coefficients λ(M, θP ), α(M, θP ), and β(M, θP ) [Eq. (20)] for
multiple azimuths θP of the polarizer, using different isotropic
samples depending on the spectral range and the ellipsometer
sub-system for which the calibration is conducted. The calibration of the THz ellipsometer uses Fourier coefficients from
measurements on a moderately doped silicon substrate with a
thermally grown silicon dioxide surface layer.94 The calibration of the MIR and FIR ellipsometers use Fourier coefficient
data acquired in a straight-through configuration (“sample”:
air), and data from reflection type, off-sample measurements
on gold coated mirrors, and a moderately doped silicon substrate with a thermally grown silicon dioxide surface layer
at multiple angles of incidence.94 For each ellipsometer subsystem an optical model similar to Eq. (18), but including
a model description for non-idealities of the polarizing elements, is then fit to the experimental Fourier coefficient data
using the Levenberg-Marquardt algorithm75 for nonlinear regression. During the calibration the polarizer and analyzer azimuthal offsets, the ellipsometric parameters for the isotropic
calibration sample, and non-idealities95 of the polarizing elements of the corresponding ellipsometer sub-system are determined. Further details of the calibration data acquisition and
analysis are described in Ref. 78.
The data acquisition times for a single OHE dataset (single magnetic field strength) vary depending on the spectral
resolution, the intensity of the light reaching the detector and
the type of detector that is used. For each dataset presented
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-12
Kühne et al.
Rev. Sci. Instrum. 85, 071301 (2014)
FIG. 7. Flowchart of the OHE-data acquisition process. (a) Shows the OHE-data acquisition process for all spectroscopic ellipsometer sub-systems.
(b) Summarizes the subroutine for MIR/FIR data acquisition, (c) subroutine for THz data acquisition and (d) summarizes the Mueller matrix regression process
(subroutine of the MIR/FIR and THz data acquisition).
in Sec. V the data acquisition times were approximately 6 h
(MIR), 2 h (FIR), and 5 h (THz).
B. OHE data analysis
During OHE data analysis, additional non-idealities,
not included in the ellipsometer calibration, introduced by
the magneto-cryostat sub-system have to be considered. In
particular, three effects are considered: (i) in- and-out-ofplane anisotropy in the optical window,96, 97 (ii) changes
in the alignment of mirrors and/or off-axis paraboloids in
the polarization state preparation and detection units due to
magnetic forces on ferromagnetic components (ellipsometric
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-13
Kühne et al.
Rev. Sci. Instrum. 85, 071301 (2014)
coordinate system change), and (iii) imperfect sample alignment (ellipsometric coordinate system and angle-of-incidence
change). In order to account for sample misalignment in the
model-based data analysis described in Sec. II C, the angle
of incidence and the sample tilt angle98 are included as model
parameters. In order to model the combined non-idealities due
to strain induced birefringence in the optical windows and
minute changes in the alignment of optical elements on the
input (output) side, an additional Mueller matrix Min (Mout )
was included in the model-based data analysis (Sec. II C). The
Mueller matrices Min and Mout are assumed to have no dispersion (wavelength independent). The best-model Mueller matrix Mbest , used for MSE regression as described in Sec. II C
reads
Mbest = Mout Mmod Min ,
(21)
where Mmod represents the sample Mueller matrix calculated
by the 4 × 4 matrix formalism.99
For the analysis of OHE data, two strategies exploiting either model-free or model-based approaches may be
used. Model-free analysis provides semi-quantitative results,
by studying trends in amplitudes or spectral positions of features in OHE or derived OHE data vs. the magnitude of the
magnetic field |B|. The model-free analysis can provide insight into the symmetry properties of magneto-optic dielectric tensors,40 and an example for the model-free analysis of
derived OHE data δM+ vs. |B| is given in Sec. V A for the
case of epitaxial graphene.
The model-based data analysis approach provides more
quantitative parameters than the model-free data analysis approach, and can be used to determine model parameters such
as the free charge carrier concentration or the effective mass
parameters. During the data analysis the GE, OHE, and derived OHE datasets are analyzed simultaneously to determine
physical model parameters of εB=0 and ε B by a single, consistent optical model (Fig. 8).
r In the first step, M = M(ε ) (GE data obtained at B
0
B=0
= 0 T) is analyzed only. During the analysis, all model
parameters independent of the magnetic field, such as
those describing, for example, the polar lattice resonances or layer thickness are varied until the calculated
data match the measured data as closely as possible.
r In the second step, OHE data M± for B = 0 T is
OHE
included in the analysis and all model parameters dependent on the magnetic field, such as, for example,
effective mass or inter-Landau-level transition parameters are varied. In addition, derived OHE data δM±
and/or δM+ ± δM− can be included into the simultaneously analyzed dataset. These derived OHE datasets
do not increase the amount of information collected
during the experiment, but help (a) to visualize magnetic field induced changes, and (b) improve the sensitivity to magnetic-field dependent model parameters
during OHE data analysis.100
The parameters determined in the first analysis step are
used as starting values for the second analysis step and are
varied if necessary. Note that an optical model, describing the
GE dataset (B = 0 T) sufficiently well, might not be capable of
FIG. 8. Flowchart of model-based analysis of OHE data. In a first step only
experimental data for B = 0 T is analyzed and only best-model parameters
independent on the magnetic field are obtained. Their values are then used as
starting parameters in a second step, where the whole OHE dataset is utilized
for analysis, and best-model parameters dependent on the magnetic field are
determined.
describing the complete dataset including OHE and/or derived OHE data correctly. Sensitivity to any model parameter
being varied in a given data analysis is provided when after
reaching the minimum MSE criterion, any small variation in
any of the currently varied parameters increases the current
MSE value. In particular, no sensitivity is obtained when any
small parameter variation does not change the current MSE
value. Such decisions may be obtained from inspecting the
uncertainty limit on each parameter provided by the numerical
regression algorithm. In case of no sensitivity to one or more
parameters, the optical model has to be adjusted, and the data
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
Kühne et al.
071301-14
0.04
Rev. Sci. Instrum. 85, 071301 (2014)
-
1
M11
Experimental data from:
-
M12
2
-
M13
1
0.00
THz sub-system
FIR sub-system
MIR sub-system
-0.04
-0.08
1
0.04
3
-
M21
2
2
-
M22
1
-
M23
1
0.00
0.15
2
-0.04
2
0.00
3
-0.08
-
M31
1
0.04
-
M32
1
0.00
1
2
-
M33
0.15
2
-0.04
0.00
2
-0.08
10
100
1000
10
100
1000
10
100
3
1000
-1
wavenumber [cm ]
FIG. 9. Wide spectral range OHE experiment: derived OHE data δM− from 7.5 cm−1 to 4000−1 (0.22–120 THz or 0.9–500 meV) for epitaxial graphene on
1 : Below 500-600 cm−1 the experimental data is dominated by Fabry-Pérot interferences in the 6H-SiC substrate (indicated by the
6H-SiC at B = −4 T. 2 : The 6H-SiC substrate phonon mode greatly enhances
background color). Fabry-Pérot interferences typically enhanced the sensitivity to the OHE greatly. 3 : Single layer graphene inter-Landau-level transitions are only observed in the on-diagonal block
the magnitude of the OHE signal from free charge carriers. −
−
and δM32
.
elements of the Mueller matrix. Note the different y-axis scale for the matrix elements δM23
analysis procedure has to be repeated. In order to increase the
sensitivity to model parameters of interest, to reduce the correlation between parameters or to improve the overall description of experimental data, a modification or simplification of
the optical model can be considered. Further, the expansion
of the experimental dataset by including data from multiple
angles of incidence or magnetic fields, as well as using other
experimental methods to determine and fix model parameters,
can help to increase the sensitivity to the varied model parameters of interest. If no sensitivity to a given parameter may be
obtained, this parameter may need to be held constant during the regression analysis, and determined by other methods.
Additional information on the data analysis strategies can be
found in Ref. 20.
V. RESULTS AND DISCUSSIONS
In this section, experimental data from the integrated
MIR, FIR, and THz OHE instrument is presented. First, a
combined dataset of derived OHE data from the MIR, FIR,
and THz spectral range from 7.5 cm−1 to 4000−1 (0.22–120
THz or 0.9–500 meV), and a magnetic field of |B| =4 T,
for an epitaxial graphene sample grown on 6H-SiC is shown
(Fig. 9). In the spectral range below approximately 500-600
cm−1 , the experimental data reveal an OHE signal enhanced
by Fabry-Pérot interferences in the 6H-SiC. An OHE signal, enhanced by coupling with the 6H-SiC phonon mode,
is observed near 1000 cm−1 . Furthermore, between 500 and
4000 cm−1 inter-Landau-level transitions in single layer
graphene are observed in the on-diagonal block elements of
the Mueller matrix.
Then, derived OHE data from the individual MIR, FIR,
and THz spectral ranges of the integrated MIR, FIR, and
THz OHE instrument, and the corresponding best-model
calculated data are shown exemplarily. We present results
from OHE experiments on an epitaxial graphene sample
grown on 6H-SiC, a Te doped n-type GaAs substrate, and
an AlGaN/GaN high electron mobility transistor structure
(HEMT), representing the MIR, FIR, and THz spectral range
of the integrated MIR, FIR, and THz OHE instrument, respectively. The selected experimental datasets demonstrate the full
spectral, magnetic field and temperature range of the integrated MIR, FIR, and THz OHE instrument, as well as analysis strategies. Effects from free charge carriers in bulk and
in two dimensional confinement as well as quantum mechanical effects (inter-Landau-level transitions) are observed and
discussed.
A. The MIR optical Hall effect: Graphene
Exemplarily, epitaxial graphene on 6H-SiC was investigated to demonstrate the MIR ellipsometer sub-system of the
OHE instrument (Sec. III A). The epitaxial graphene sample
was grown in an argon atmosphere at 1400 ◦ C, by sublimating
Si from the polar c-face (0001̄) of a semi-insulating 6H-SiC
substrate. From previous measurements on C-face 4H-SiC,101
the number of graphene layers is estimated to be 10–20. Further details on growth conditions are beyond the scope of this
manuscript, and can be found in Ref. 102.
The OHE experiment was conducted at an angle of incidence of a = 45◦ , while the magnetic field was aligned
along the reflected beam. Experimental data were recorded
in the spectral range from 600 to 4000 cm−1 with a spectral
resolution of 1 cm−1 , using the HgCdTe detector, while the
sample was kept at a temperature of T = 1.5 K.
Figure 10 shows derived OHE data δM+ (green, dotted
line) for B√= 8 T (effective field parallel to sample normal
Bc = |B|/ 2 ∼ 5.66 T) and results from best model calculations (red, solid line). The model-free analysis approach
provides valuable information. The OHE data shows several
resonances, which can be divided in three groups: a peak
near ν = 1000 cm−1 labeled FCC (triangles, pink), a set of
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-15
Kühne et al.
Rev. Sci. Instrum. 85, 071301 (2014)
FIG. 10. Derived OHE data δM+ (green, dotted line) and results from best
model calculations (red, solid line) for epitaxial graphene at a = 45◦ ,
T = 1.5 K, and B = 8 T (effective field parallel to sample normal Bc = B/
√
2 ≈ 5.66 T). The magneto-optical signal near ν = 1000 cm−1 labeled
FCC (triangles, pink) is assigned to free charge carriers. The features labeled LLBLG (rectangles, blue) and the set of peaks labeled LLSLG (diamonds, orange) are inter-Landau-level transitions in multi-layer and singlelayer graphene, respectively. The contributions of LLSLG are limited to the
on-diagonal-block elements of the Mueller matrix. Therefore, the polarization selection rules for LLSLG are polarization conserving - while the processes leading to FCC and LLBLG are polarization mixing.40 Reprinted with
permission from P. Kühne et al. Phys. Rev. Lett. 111, 077402 (2013). Copyright 2013, American Physical Society.
features labeled LLBLG (rectangles, blue) and a set of peaks labeled LLSLG (diamonds, orange). While the resonances labeled
FCC and LLBLG are present in all Mueller matrix elements,
the features labeled LLSLG are only present in the on-diagonalblock elements of the Mueller matrix. Since non-vanishing
off-diagonal-block elements in the Mueller matrix are inherently tied to non-vanishing off-diagonal elements in the underlying dielectric tensor, the contribution to the dielectric
tensor, which describe the features labeled LLSLG , has to be
diagonal (εxy = εyx = 0). Furthermore, the magneto-optic contributions of LLSLG to the permittivity tensor in its representation for circularly polarized light [Eq. (3)] must satisfy χ +
= χ – . In other words, the physical processes leading to the fingerprints labeled LLSLG are polarization-conserving, while the
processes leading to FCC and LLBLG are polarization-mixing
(polarization selection rules). The origin of the individual processes can be determined by field-dependent measurements.
Figure 11(c) displays the magnetic field dependence of a rep+
resentative on-diagonal-block Mueller matrix element δM33
of the derived OHE dataset for B = 1 . . . 8 T in 0.1 T increments (the corresponding representative off-diagonal-block
Mueller matrix element is omitted here and the interested
reader is referred to Ref. 40). At a first glance it can be
noted that the resonance labeled FCC increases in amplitude while its spectral position is not affected. This indicates that the resonance labeled FCC is the SiC phonon
mode (best model parameter for Eq. (8): l = 1, ε∞,x,y
= 6.00, γLO,x,y = 4.74 cm−1 , ωLO,x,y = 972.72 cm−1 , γTO,x,y
= 1.18 cm−1 , ωTO,x,y = 799.31 cm−1 , ε∞,z = 5.84, γLO,z
= 2.64 cm−1 , ωLO,z = 966.99 cm−1 , γTO,z = 0.50 cm−1 , and
FIG. 11. Data for epitaxial graphene at T = 1.5 K and a = 45◦ . (a) Experimental GE data for B = 0 T (black, dotted line) and OHE data for B
= 8 T (red, solid line), and (b) derived OHE data δM+ (difference between
the two datasets from (a)) are plotted. Note that the experimental data range
for normalized Mueller matrix data is Mij = −1 . . . 1, while, here, the OHE
only leads to very subtle changes |δMij | ≤ 0.05 with respect to the GE data.
+
(c) Plot of representative on-diagonal-block Mueller matrix element δM33
of the derived OHE dataset for B = 1 . . . 8 T, showing SiC substrate phonon
mode (near 1000 cm−1 ) enhanced free charge carrier magneto-optic response
(triangles, pink), near-linear-B multi-layer √
graphene inter-Landau-level transitions (rectangles, blue), and the typical B-dependency of inter-Landaulevel transitions in single layer graphene (diamonds, orange). Further information can be found in Ref. 40. Reprinted with permission from P. Kühne
et al. Phys. Rev. Lett. 111, 077402 (2013). Copyright 2013, American Physical Society.
ωTO,z = 798.73 cm−1 ) enhanced, magneto-optic fingerprint
from free charge carriers. In contrast, the spectral positions of
the resonances labeled LLSLG shift to higher energies and their
amplitudes increase with increasing magnetic
√ field strength.
A detailed analysis of the peaks reveals a B-dependence
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-16
Kühne et al.
Rev. Sci. Instrum. 85, 071301 (2014)
FIG. 12. Experimental data (dotted line, green) and best-model calculations (solid line, red) for a GaAs substrate in the FIR spectral range. Non magnetic (B = 0)
T GE data (a) possesses only non-vanishing on-diagonal-block elements, while in (b) the derived OHE data δM+ − δM− (for B = 8 T) only off-diagonal-block
elements are different from zero, indicating the anti-symmetry of the magneto-optic dielectric tensor. The asymmetry between corresponding matrix elements
with interchanged indices is caused by the asymmetric orientation of the magnetic field (with respect to the sample normal).
of their spectral positions, indicative of inter-Landau-level
transitions in the Dirac-type band structure of single-layer
graphene with the Fermi level close to the charge neutrality point.103 Landau level energies√in single layered graphene
LL
(n) = sign(n)E0 |n| with E0 = c̃ 2¯e|Bc |,
are given by ESLG
where c̃ is the average velocity of Dirac fermions in graphene.
Optical selection rules for transitions between levels n and n
demand that |n | = |n| ± 1. The Fermi velocity is determined
as c̃ = (1.01 ± 0.01) × 106 m/s, in very good agreement with
Refs. 103–108. The analysis of the magnetic field dependence of the resonances labeled LLBLG reveals their origin
from the bi-layer branch of inter-Landau-level transitions in
ABA-stacked (Bernal) multi-layered graphene. Inter-Landaulevel transitions from bi-layer and tri-layer graphene were
identified. For further details, the interested reader is referred
to Ref. 40.
B. The FIR optical Hall effect: GaAs
The n-type GaAs substrate investigated here, was moderately doped with tellurium (N ≈ 1018 cm−3 ) and is opaque
to FIR radiation. All Measurements were conducted at an
angle of incidence of a = 45◦ , while the magnetic field
was parallel to the reflected beam. Figure 12 shows experimental data from the FIR spectral range (50–650 cm−1 )
obtained at a sample temperature of T = 300 K. Figures 12(a) and 12(b) display GE data (B = 0 T) and derived OHE data δM+ − δM− , respectively, for B = 8 T.
In the absence of a magnetic field, GaAs is optically
isotropic, i.e., for B = 0 T all off-diagonal-block elements
(M13 , M23 , M31 , and M32 ) in (a) are zero. In contrast, due
to the underlying magneto-optic dielectric tensor symmetry, all on-diagonal-block elements in (b) are zero, indicating the anti-symmetry of the off-diagonal elements of the
dielectric tensor, as well as their sign change under field inversion. Note that for a bulk material and a field orientation that
is not perpendicular to the sample surface, all off-diagonal elements of the dielectric tensor are different from zero. This
also leads to the asymmetry between corresponding Mueller
matrix elements with interchanged indices (ij and ji).
The optical model for the GaAs substrate consists of a
single, semi-infinite layer which contains two contributions,
the optical response of polar lattice vibrations and free charge
carriers described by the Drude model. Due to the crystal
symmetry of GaAs, the dielectric tensor for the lattice vibration [Eqs. (8) and (7)] has identical diagonal elements,
e.g., εxL = εyL = εzL and possesses only one TO-LO resonance
in the FIR spectral range. Best model parameter for the lattice vibrations were determined as ε∞ = (9.269 ± 0.009),
ωTO = (268.24 ± 0.06) cm−1 , ωLO = (290.96 ± 0.07) cm−1 ,
and γTO = γLO = (1.9 ± 0.2) cm−1 . The model parameters
for the free-charge-carrier contribution log N = (17.926
± 0.001) cm−3 , μ = (1789 ± 11) cm2 /Vs, and m = (0.0738
± 0.0001) me are in good agreement with results in previous
publications.109, 110 We note that Humlíček et al. reported on
the occurrence of a spectral feature near the LO phonon mode
of GaAs (Berreman mode), which is due to the existence of a
surface free charge carrier depletion layer.111 The same mode
was observed and discussed in n-type GaN, for example, by
Kasic et al.42 See also Chap. 4 in Ref. 13. A similar observation is expected for the GaAs sample studied here, however,
due to the small angle of incidence and comparatively low
carrier concentration of the GaAs sample, the effect of the
surface depletion layer spectral resonance is small, and below
our current signal-to-noise level.
C. The THz optical Hall effect: HEMT
The HEMT structure investigated here was grown on a
semi-insulating 4H-SiC substrate by metal-organic chemical
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-17
Kühne et al.
vapor deposition at temperatures of 1050 ◦ C.112, 113 First, a
nominally 100 nm thick AlN nucleation layer was grown, followed by a 1800 nm thick GaN buffer and a 20 nm thick
Al0.25 Ga0.75 N electron barrier layer.
Figure 13 displays the temperature dependence of nonvanishing off-diagonal-block Mueller matrix elements of de+
−
+
−
− δM31
and δM32
− δM32
for B = 3 T
rived OHE data δM31
in the spectral range from 0.22 to 0.32 THz. The experimental OHE data are depicted as dotted lines (green), while bestmodel calculated OHE data are plotted as solid lines (red)
for different temperatures between 1.5 K and 300 K. The observed OHE is caused by free charge carriers in the high mobility channel of the HEMT structure, which is enhanced by
the Fabry-Pérot interference in the (356 ± 1) μm thick SiC
substrate. The optical model is composed of a SiC substrate
layer, a AlN nucleation layer, a GaN buffer layer, a layer for
the GaN HEMT channel and a AlGaN layer. The thickness
of the GaN HEMT layer was set to d = 1 nm and not varied
during data analysis. The HEMT layer thickness was used to
calculate the sheet charge density Ns = Nd, where N is the
bulk charge density. Derived THz OHE data were analyzed
simultaneously with GE data recorded with a commercial
MIR ellipsometer.38 During model-based OHE data analysis
the parameters for the sheet charge density Ns , the mobility
μ and the effective mass parameter m were determined. The
sheet carrier density was found to be constant within the error limits at a value of Ns = Nd = (5 ± 1) × 1012 cm−2 for
all temperatures. The mobility and the effective mass parameters were determined as m = (0.22 ± 0.01)m0 (m = (0.36
± 0.03)m0 ) and μ = (7800 ± 410) cm2 /Vs (μ = (1711 ±
150) cm2 /Vs) at T = 1.5 K (T = 300 K), where m0 is the
mass of the free electron. Further details are omitted here and
the interested reader is referred to an in depth discussion in
Ref. 38.
VI. SUMMARY
In this article, we have given an overview of theoretical
and experimental aspects of the OHE, and have successfully
demonstrated the operation of an integrated MIR, FIR,
and THz optical Hall effect instrument. Two in-house built
ellipsometers sub-systems, operating in the rotating-analyzer
configuration, were employed to determine the upper 3 ×
3 block of the normalized Mueller matrix in an ultra wide
spectral range from 3 cm−1 to 7000 cm−1 (0.1–210 THz or
0.4–870 meV). Different aspects of the integrated MIR, FIR,
and THz optical Hall effect instrument, such as the closedcycle, superconducting 8 T magneto-cryostat sub-system, and
the optical setup of the integrated MIR, FIR, and THz optical
Hall effect instrument, as well as the data acquisition and
data analysis strategies were discussed in detail. For demonstration purposes of the operation of the integrated MIR, FIR,
and THz optical Hall effect instrument in the MIR, FIR, and
THz spectral range, three sample systems were studied: an
epitaxial graphene sample (MIR), a GaAs substrate (FIR),
and a AlGaN/GaN HEMT structure (THz). The presented
data covered the full magnetic field range (B = 0. . . 8 T) and
temperature range (T = 1.5. . . 300 K) of the system. We have
demonstrated that the integrated MIR, FIR, and THz optical
Rev. Sci. Instrum. 85, 071301 (2014)
FIG. 13. AlGaN/GaN HEMT structure:38 experimental (dotted lines, green)
+
and best-model calculated (solid lines, red) OHE data for B = 3 T (δM31
−
+
−
− δM31 , left panel; δM32 − δM32 , right panel) between 0.22 and 0.32 THz.
The spectra were obtained for temperatures ranging from 1.5 to 300 K, at
an angle of incidence a = 45◦ , and show the Fabry-Pérot interference enhanced magneto-optic response of free charge carriers. The spectra for T =
75 K and above are stacked by 0.1. Reprinted with permission from T. Hofmann et al., Appl. Phys. Lett. 101, 192102 (2012). Copyright 2012 American
Institute of Physics.
Hall effect instrument can be used to determine the parameters of free charge carriers in bulk, and in two-dimensional
confinement. Furthermore, it was shown that the integrated
MIR, FIR, and THz optical Hall effect instrument is capable
of investigating the symmetry properties of magneto-optic,
dielectric tensors of quantized systems, and was able to determine the polarization selection rules of inter-Landau-level
transitions. The integrated MIR, FIR, and THz optical Hall
effect instrument has a wide range of applications, from
determining free charge carrier properties contact-free, over
the access to free charge carrier properties in buried layers,
to magneto-optic quantum phenomena, polarization selection
rules, to symmetry properties of magneto-optic dielectric
tensors.
ACKNOWLEDGMENTS
This is the first instrument of its kind; the system as
a whole and its constituent sub-systems are, to the best of
our knowledge, not available in any commercial product or
in any other research laboratory. This project was primarily supported by the National Science Foundation (Grant No.
MRI DMR-0922937). Further support came from the Army
Research Office (D. Woolard, Contract No. W911NF-09-C0097), the National Science Foundation (Grant Nos. MRSEC DMR-0820521, DMR-0907475, and EPS-1004094), the
University of Nebraska-Lincoln, the J.A. Woollam Foundation and the J. A. Woollam Company. The authors thank
Dr. D. K. Gaskill and his workgroup at the U. S. Naval Research Laboratory, Washington, D.C. for providing epitaxial
graphene samples. The authors would like to acknowledge Dr.
V. Darakchieva and Dr. E. Janzén and their workgroups at the
University of Linköping, Sweden for providing the HEMT
structure samples. Thanks to C. Rice, C. Briley, S. M. Slone,
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-18
Kühne et al.
and A. Boosalis for their help building and testing mechanical
and electrical components. Special thanks to the instrument
shop at the department of physics at UNL, and in particular to
Les Marquart for his invaluable advice.
1A
very common requirement in theoretical studies of the electromagnetic
response of matter consists in the imposition that a specific medium should
be Lorentz-reciprocal. For a dielectric medium (the magnetic susceptibility tensor being diagonal and unity), this means that the dielectric tensor
is equal to its transposed form. The magnetized plasma, and more general types of gyrotropic mediums belong to the most prominent representatives of nonreciprocal mediums. A gyrotropic material is a material
in which left- and right-rotating elliptical polarizations can propagate at
different speeds. The gyrotropic effect caused by a quasi-static magnetic
field breaks the time reversal symmetry as well as the Lorentz reciprocity.
See, for example, W. S. Weiglhofer, Constitutive Characterization of Simple and Complex Mediums, in W. S. Weiglhofer and A. Lakhtakia, Introduction to Complex Mediums for Optics and Electromagnetics, SPIE,
Bellingham, 2003.
2 E. H. Hall, Am. J. Math. 2, 287 (1879).
3 P. Drude, Phys. Z. 1, 161 (1900).
4 P. Drude, Ann. Phys. 306, 566 (1900).
5 E. Hecht, Optics (Addison-Wesley, Reading, MA, 1987).
6 R. M. Azzam and N. M. Bashara, Ellipsometry and Polarized Light
(North-Holland Publ. Co., Amsterdam, 1984).
7 M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger,
J. Opt. Soc. Am. A 13, 875 (1996).
8 M. Schubert, Phys. Rev. B 53, 4265 (1996).
9 T. E. Tiwald and M. Schubert, Proc. SPIE 4103, 19 (2000).
10 M. Schubert, A. Kasic, T. Hofmann, V. Gottschalch, J. Off, F. Scholz, E.
Schubert, H. Neumann, I. J. Hodgkinson, M. D. Arnold, W. A. Dollase,
and C. M. Herzinger, Proc. SPIE 4806, 264 (2002).
11 M. Schubert, in Introduction to Complex Mediums for Optics and Electromagnetics, edited by W. S. Weiglhofer and A. Lakhtakia (SPIE, Bellingham, WA, 2004), pp. 677–710.
12 Handbook of Ellipsometry, edited by H. Thompkins and E. A. Irene
(William Andrew Publishing, Highland Mills, 2004).
13 M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures:
Phonons, Plasmons and Polaritons, Springer Tracts in Modern Physics,
Vol. 209 (Springer, Berlin, 2004).
14 T. Hofmann, C. M. Herzinger, C. Krahmer, K. Streubel, and M. Schubert,
Phys. Status Solidi A 205, 779 (2008).
15 P. Q. J. Nederpel and J. W. D. Martens, Rev. Sci. Instrum. 56, 687
(1985).
16 J. Černe, D. C. Schmadel, L. B. Rigal, and H. D. Drew, Rev. Sci. Instrum.
74, 4755 (2003).
17 W. J. Padilla, Z. Q. Li, K. S. Burch, Y. S. Lee, K. J. Mikolaitis, and D. N.
Basov, Rev. Sci. Instrum. 75, 4710 (2004).
18 Y. Ino, R. Shimano, Y. Svirko, and M. Kuwata-Gonokami, Phys. Rev. B
70, 155101 (2004).
19 T. N. Stanislavchuk, T. D. Kang, P. D. Rogers, E. C. Standard, R. Basistyy,
A. M. Kotelyanskii, G. Nita, T. Zhou, G. L. Carr, M. Kotelyanskii, and A.
A. Sirenko, Rev. Sci. Instrum. 84, 023901 (2013).
20 M. Schubert, T. Hofmann, and C. M. Herzinger, J. Opt. Soc. Am. A 20,
347 (2003).
21 T. Hofmann, U. Schade, W. Eberhardt, C. M. Herzinger, P. Esquinazi, and
M. Schubert, Rev. Sci. Instrum. 77, 063902 (2006).
22 T. Hofmann, M. Schubert, and C. M. Herzinger, Proc. SPIE 4779, 90
(2002).
23 T. Hofmann, V. Gottschalch, and M. Schubert, MRS Proc. 744
(2002).
24 T. Hofmann, M. Schubert, C. M. Herzinger, and I. Pietzonka, Appl. Phys.
Lett. 82, 3463 (2003).
25 M. Schubert, T. Hofmann, and C. M. Herzinger, Thin Solid Films 455456, 563 (2004).
26 S. Schöche, T. Hofmann, V. Darakchieva, N. B. Sedrine, X. Wang, A.
Yoshikawa, and M. Schubert, J. Appl. Phys. 113, 013502 (2013).
27 T. Hofmann, T. Chavdarov, V. Darakchieva, H. Lu, W. J. Schaff, and M.
Schubert, Phys. Status Solidi C 3, 1854 (2006).
28 T. Hofmann, U. Schade, K. C. Agarwal, B. Daniel, C. Klingshirn, M. Hetterich, C. M. Herzinger, and M. Schubert, Appl. Phys. Lett. 88, 042105
(2006).
Rev. Sci. Instrum. 85, 071301 (2014)
29 T.
Hofmann, M. Schubert, G. Leibiger, and V. Gottschalch, Appl. Phys.
Lett. 90, 182110 (2007).
30 T. Hofmann, V. Darakchieva, B. Monemar, H. Lu, W. Schaff, and M. Schubert, J. Electron. Mater. 37, 611 (2008).
31 T. Hofmann, C. M. Herzinger, A. Boosalis, T. E. Tiwald, J. A. Woollam,
and M. Schubert, Rev. Sci. Instrum. 81, 023101 (2010).
32 T. Hofmann, C. M. Herzinger, J. L. Tedesco, D. K. Gaskill, J. A. Woollam,
and M. Schubert, Thin Solid Films 519, 2593 (2011), 5th International
Conference on Spectroscopic Ellipsometry (ICSE-V).
33 P. Kühne, T. Hofmann, C. M. Herzinger, and M. Schubert, Thin Solid
Films 519, 2613 (2011), 5th International Conference on Spectroscopic
Ellipsometry (ICSE-V).
34 T. Hofmann, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam, D.
K. Gaskill, J. L. Tedesco, and M. Schubert, Appl. Phys. Lett. 98, 041906
(2011).
35 S. Schöche, J. Shi, A. Boosalis, P. Kühne, C. M. Herzinger, J. A. Woollam,
W. J. Schaff, L. F. Eastman, M. Schubert, and T. Hofmann, Appl. Phys.
Lett. 98, 092103 (2011).
36 P. Kühne, A. Boosalis, C. M. Herzinger, L. O. Nyakiti, V. D. Wheeler, R.
L. Myers-Ward, C. R. J. Eddy, D. K. Gaskill, M. Schubert, and T. Hofmann, MRS Proc. 1505 (2013).
37 S. Schöche, P. Kühne, T. Hofmann, M. Schubert, D. Nilsson, A.
Kakanakova-Georgieva, E. Janzén, and V. Darakchieva, Appl. Phys. Lett.
103, 212107 (2013).
38 T. Hofmann, P. Kühne, S. Schöche, J.-T. Chen, U. Forsberg, E. Janzén,
N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, and V.
Darakchieva, Appl. Phys. Lett. 101, 192102 (2012).
39 T. E. Tiwald, J. A. Woollam, S. Zollner, J. Christiansen, R. B. Gregory,
T. Wetteroth, S. R. Wilson, and A. R. Powell, Phys. Rev. B 60, 11464
(1999).
40 P. Kühne, V. Darakchieva, R. Yakimova, J. D. Tedesco, R. L. Myers-Ward,
C. R. Eddy, D. K. Gaskill, C. M. Herzinger, J. A. Woollam, M. Schubert,
and T. Hofmann, Phys. Rev. Lett. 111, 077402 (2013).
41 M. Schubert, T. E. Tiwald, and C. M. Herzinger, Phys. Rev. B 61, 8187
(2000).
42 A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, and T. E. Tiwald, Phys.
Rev. B 62, 7365 (2000).
43 A. Kasic, M. Schubert, T. Frey, U. Köhler, D. J. As, and C. M. Herzinger,
Phys. Rev. B 65, 184302 (2002).
44 A. Kasic, M. Schubert, J. Off, B. Kuhn, F. Scholz, S. Einfeldt, T. Böttcher,
D. Hommel, D. J. As, U. Köhler, A. Dadgar, A. Krost, Y. Saito, Y. Nanishi,
M. R. Correia, S. Pereira, V. Darakchieva, B. Monemar, H. Amano, I.
Akasaki, and G. Wagner, Phys. Status Solidi C 0, 1750 (2003).
45 P. R. Schroeder, M. S. Dresselhaus, and A. Javan, Phys. Rev. Lett. 20,
1292 (1968).
46 N. A. Goncharuk, L. Nádvorník, C. Faugeras, M. Orlita, and L. Smrčka,
Phys. Rev. B 86, 155409 (2012).
47 M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer,
AIP Conf. Proc. 893, 619 (2007).
48 M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer,
Solid State Commun. 143, 123 (2007).
49 H. Fujiwara, Spectroscopic Ellipsometry (John Wiley & Sons, New York,
2007).
50 D. Goldstein and D. Goldstein, Polarized Light, Revised and Expanded,
Optical Engineering (Marcel Dekker/Taylor & Francis, 2011).
51 Numerical Data and Functional Relationships in Science and Technology,
edited by U. Rössler (Springer, Berlin, 1999), Vol. III/41B.
52 T. Hofmann, “Far-infrared spectroscopic ellipsometry on AIII BV semiconductor heterostructures,” Ph.D. thesis (University of Leipzig, 2004).
53 R. H. Muller, Surf. Sci. 16, 14 (1969).
54 M. Schubert, Ann. Phys. 15, 480 (2006).
55 P. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin,
1999).
56 In the following equation the Einstein notation is used, and the covariance
and contravariance is ignored since all coordinate systems are Cartesian
(The summation is only executed over pairs of lower indices).
57 C. Pidgeon, in Handbook on Semiconductors, edited by M. Balkanski
(North-Holland, Amsterdam, 1980).
58 C. Kittel, Introduction to Solid States Physics (Wiley, New York, 1986).
59 A. S. Barker, Phys. Rev. 136, A1290 (1964).
60 M. Schubert, T. Hofmann, C. M. Herzinger, and W. Dollase, Thin Solid
Films 455–456, 619 (2004), 3rd International Conference on Spectroscopic Ellipsometry.
61 D. W. Berreman and F. C. Unterwald, Phys. Rev. 174, 791 (1968).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
071301-19
62 F.
Kühne et al.
Gervais and B. Piriou, J. Phys. C: Solid State Phys. 7, 2374 (1974).
Hofmann, V. Gottschalch, and M. Schubert, Phys. Rev. B 66, 195204
(2002).
64 Four independent components are needed to quantify all properties of polarized light: total intensity, degree of polarization, ellipticity and orientation of the polarization ellipse.
65 The letters p and s stand for “parallel” and “senkrecht” (German for parallel and perpendicular, respectively), and refer to the directions with respect
to the plane of incidence.
66 A. Röseler, Infrared Spectroscopic Ellipsometry (Akademie-Verlag,
Berlin, 1990).
67 For example, fully p-polarized light becomes fully s-polarized light after
a 90◦ rotation of the coordinate system around the beam path.
68 P. Hauge, Surf. Sci. 96, 108 (1980).
69 R. W. Collins, Rev. Sci. Instrum. 61, 2029 (1990).
70 P. S. Hauge and F. H. Dill, Opt. Commun. 14, 431 (1975).
71 P. S. Hauge, Surf. Sci. 56, 148 (1976).
72 D. E. Aspnes and P. S. Hauge, J. Opt. Soc. Am. 66, 949 (1976).
73 P. S. Hauge, J. Opt. Soc. Am. 68, 1519 (1978).
74 G. E. Jellison, Jr., Thin Solid Films 313–314, 33 (1998).
75 W. H. Press, Numerical Recipes: The Art of Scientific Computing, 3rd ed.
(Cambridge University Press, 2007).
76 C. M. Herzinger, H. Yao, P. G. Snyder, F. G. Celii, Y. C. Kao, B. Johs, and
J. A. Woollam, J. Appl. Phys. 77, 4677 (1995).
77 C. M. Herzinger, P. G. Snyder, B. Johs, and J. A. Woollam, J. Appl. Phys.
77, 1715 (1995).
78 B. Johs, Thin Solid Films 234, 395 (1993).
79 S. Nishizawa, K. Sakai, M. Hangyo, T. Nagashima, M. Takeda, K.
Tominaga, A. Oka, K. Tanaka, and O. Morikawa, in Terahertz Optoelectronics, Topics in Applied Physics, edited by K. Sakai (Springer,
Berlin/Heidelberg, 2005), Vol. 97, pp. 203–270.
80 T. Nagashima and M. Hangyo, Appl. Phys. Lett. 79, 3917 (2001).
81 N. Matsumoto, T. Fujii, K. Kageyama, H. Takagi, T. Nagashima, and M.
Hangyo, Jpn. J. Appl. Phys. 48, 09KC11 (2009).
82 N. Matsumoto, T. Hosokura, T. Nagashima, and M. Hangyo, Opt. Lett. 36,
265 (2011).
83 M. Neshat and N. P. Armitage, Opt. Express 20, 29063 (2012).
84 The angle of incidence is defined as the angle between the surface normal
of the sample and the incoming beam.
85 With exception of the space between the polarization preparation unit, the
sample and the polarization detection unit, which is either filled with ambient air, or, if the magneto-cryostat sub-system is used, filled with ultra
high purity helium gas or is under vacuum.
86 In house Gold sputtered mirrors (d
gold ≈ 150 − 200 nm) on commercially
available optical glass substrates (dglass = 3 mm).
87 The rotatable mirror is designed to switch to an alternative input source
(In) (currently unused).
88 The effective focal length f is the distance between the focal point and
e
the center of the off-axis paraboloid mirror. For 90◦ off-axis paraboloids
the effective focal length is twice the focal length fe = 2f .
89 N. A. Pankratov, Y. V. Kulikov, and N. V. Shchetinina, Sov. J. Opt. Technol. 45, 435 (1978).
90 N. A. Pankratov, Y. V. Kulikov, and Y. I. Polushkin, Sov. J. Opt. Technol.
50, 251 (1983).
91 C. Herzinger, S. Green, and B. Johs, “Odd bounce image rotation system
in ellipsometer systems,” U.S. patent 6,795,184 (2004).
92 The stepper motors of the rotation stage in the polarization state preparation unit of the FIR/THz ellipsometer sub-system and the polarization
state rotator assembly are operated in tandem mode.
93 In this case the stokes vectors are given by IS T = I S , 0, 0, 0 and
0
D T
D
I
= I0 , 0, 0, 0 .
63 T.
Rev. Sci. Instrum. 85, 071301 (2014)
94 All
stepper motors in the integrated MIR, FIR, and THz OHE instrument
are equipped with optical encoder wheels, which can be used to reference
the absolute angular position of each rotating optical element with respect
to the p- and s-coordinate system. This reference is used for all rotating
optical elements of all ellipsometers with the exception of the continuously rotating analyzer of the THz ellipsometer. Here the software uses the
stepper motor driving electric pulses to keep track of the position of the
continuously rotating analyzer. Therefore, the absolute angular position of
the analyzer of the THz ellipsometer, determined during the calibration,
is lost every time the THz ellipsometer is powered off, and the calibration
has to be repeated after every power up.
95 Parametrized models for optical elements (such as polarizers, mirrors, detectors, etc.) and their imperfections (such as degree of depolarization,
spectral dispersion, misalignment, etc.) are based on chain multiplication
of Mueller matrices. During the calibration procedure model parameters
are determined and stored, and later used in the data acquisition procedure
in order to determine the Mueller matrix elements of the sample. For an
in-depth discussion of parametrized models for optical elements the reader
is referred to the literature, e.g., Refs. 49 and 50.
96 B. Johs and C. Herzinger, “Methods for uncorrelated evaluation of parameters in parameterized mathematical model equations for window retardance, in ellipsometer and polarimeter systems,” U.S. patent 6,034,777
(2000).
97 J. M. M. de Nijs and A. van Silfhout, J. Opt. Soc. Am. A 5, 773 (1988).
98 The sample tilt angle is defined as the angle by which the sample is rotated
along the intersection of the plane of incidence and the sample surface.
99 The variation of the best-model parameters M of the in- and output matriij
ces M in and M out is smaller than 0.02 over the full magnetic field range.
100 Including derived OHE datasets in the data analysis is in particular useful
if magneto-optic effects only lead to a subtle change of the Mueller matrix
with respect to the Mueller matrix obtained at B = 0 T, as, for example, in
case of inter-Landau-level transitions in epitaxial graphene, discussed in
Sec. V A.
101 A. Boosalis, T. Hofmann, V. Darakchieva, R. Yakimova, and M. Schubert,
Appl. Phys. Lett. 101, 011912 (2012).
102 J. L. Tedesco, G. G. Jernigan, J. C. Culbertson, J. K. Hite, Y. Yang, K. M.
Daniels, R. L. Myers-Ward, C. R. Eddy, J. A. Robinson, K. A. Trumbull,
M. T. Wetherington, P. M. Campbell, and D. K. Gaskill, Appl. Phys. Lett.
96, 222103 (2010).
103 M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer,
Phys. Rev. Lett. 97, 266405 (2006).
104 M. Orlita, C. Faugeras, J. Borysiuk, J. M. Baranowski, W. Strupiński, M.
Sprinkle, C. Berger, W. A. de Heer, D. M. Basko, G. Martinez, and M.
Potemski, Phys. Rev. B 83, 125302 (2011).
105 M. Orlita, C. Faugeras, J. M. Schneider, G. Martinez, D. K. Maude, and
M. Potemski, Phys. Rev. Lett. 102, 166401 (2009).
106 E. A. Henriksen, Z. Jiang, L.-C. Tung, M. E. Schwartz, M. Takita, Y.-J.
Wang, P. Kim, and H. L. Stormer, Phys. Rev. Lett. 100, 087403 (2008).
107 M. Orlita, C. Faugeras, R. Grill, A. Wysmolek, W. Strupinski, C. Berger,
W. A. de Heer, G. Martinez, and M. Potemski, Phys. Rev. Lett. 107,
216603 (2011).
108 M. Orlita, C. Faugeras, P. Plochocka, P. Neugebauer, G. Martinez, D.
K. Maude, A.-L. Barra, M. Sprinkle, C. Berger, W. A. de Heer, and M.
Potemski, Phys. Rev. Lett. 101, 267601 (2008).
109 A. Raymond, J. L. Robert, and C. Bernard, J. Phys. C: Solid State Phys.
12, 2289 (1979).
110 J. S. Blakemore, J. Appl. Phys. 53, R123 (1982).
111 J. Humlíček, R. Henn, and M. Cardona, Appl. Phys. Lett. 69, 2581 (1996).
112 U. Forsberg, A. Lundskog, A. Kakanakova-Georgieva, R. Ciechonski, and
E. Janzén, J. Cryst. Growth 311, 3007 (2009).
113 A. Kakanakova-Georgieva, U. Forsberg, I. G. Ivanov, and E. Janzén, J.
Cryst. Growth 300, 100 (2007).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
129.93.8.42 On: Wed, 30 Jul 2014 14:56:40
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement