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Phase-sensitive time-domain terahertz reflection spectroscopy
A. Pashkin,a) M. Kempa, H. Ne
ˇmec, F. Kadlec, and P. Kuz
ˇel
Institute of Physics, Academy of Sciences of the Czech Republic and Center for Complex Molecular Systems
and Biomolecules, Na Slovance 2, 182 21 Prague 8, Czech Republic
共Received 1 April 2003; accepted 1 August 2003兲
An approach to time-domain terahertz reflection spectroscopy is proposed and demonstrated. It
allows one to obtain very accurately the relative phase of a reflected THz wave form, and
consequently the complex dielectric function can be precisely extracted. The relevant setup was
demonstrated to allow measurements of a variety of samples: we present results for doped silicon
and for ferroelectric SrBi2Ta2O9共bulk ceramics as well as thin film on sapphire substrates兲.
©2003 American Institute of Physics. 关DOI: 10.1063/1.1614878兴
I. INTRODUCTION
Time-domain terahertz transmission spectroscopy
共TDTTS兲has become a standard method for measurements
of complex dielectric constant or conductivity of dielectrics,
semiconductors, and superconductors in the millimeter and
submillimeter spectral range.1The technique requires mea-
surement of the temporal profile of the electric field of a
picosecond terahertz 共THz兲pulse transmitted through an in-
vestigated sample. The complex spectrum of this pulse is
normalized by a reference spectrum 共obtained when the
sample is removed from the THz beam path兲thus yielding
the complex transmittance of the sample. Finally, the com-
plex refractive index in the whole frequency range studied is
obtained through numerical solution of a system of two real
nonlinear equations for the transmittance.2Let us emphasize
the importance of the reference measurement: it ensures the
result is independent of the THz pulse shape as well as of
instrumental functions.
The transmission setup is fully developed and reliable,
but it can be applied only to transparent samples. However,
in the case of samples that are opaque in the THz frequency
range 共thick and/or with high dielectric loss兲, the transmis-
sion geometry is not useful and the use of time-domain THz
reflection spectroscopy 共TDTRS兲is required. Furthermore,
in the case of thin films on thick substrates, TDTTS some-
times does not offer sufficient sensitivity to provide precise
information about the optical constants of a thin film. This is
due to a large difference between the phase change of the
THz signal induced by the thin film and that related to the
substrate.3Thus the evaluation of transmittance related to the
thin film involves large errors. In this case, TDTRS can pro-
vide valuable information about such structures because the
radiation reflected on air–thin film–substrate interfaces is in-
dependent of the substrate thickness.
TDTRS as a spectroscopic method, and in analogy with
TDTTS, requires also a reference measurement which can be
obtained, e.g., using the reflection on a mirror with known
characteristics. The main difficulty in realization of TDTRS
then consists of correct determination of the reflectance
phase which is strongly affected by errors in the relative
position of the sample and reference mirror.4–9 Due to the
very strong dependence of the dielectric function on the re-
flectance phase, mispositioning as small as 1
m can signifi-
cantly influence the dielectric function calculated.10
In this article we present a new approach to TDTRS
designed to provide in many cases easy and accurate mea-
surement of the phase of complex reflectance. We have
tested our setup using different types of samples which were
chosen to illustrate potential application of the method. The
THz spectral range is appropriate for measurement of the
carrier scattering rate and plasma frequency of doped semi-
conductors. The first samples studied were thus two n-type
silicon wafers with different levels of doping. The next
sample was a ferroelectric ceramic, SrBi2Ta2O9共SBT兲,
which is a very good candidate for nonvolatile ferroelectric
memories due to its polarization fatigue-free nature and low
switching voltage.11 It presents a rather strong IR-active soft
phonon mode in the frequency range studied and therefore it
cannot be investigated by TDTTS. The last sample was a thin
film of the same compound deposited on a sapphire sub-
strate; this structure allows direct comparison of the reflec-
tion and transmission measurements.
II. OVERVIEW OF THE PHASE PROBLEM
Because THz radiation is reflected directly onto the
sample surface, the phase shift induced by the sample is
much smaller than that in the transmission experiment where
it is proportional to the sample thickness. Therefore even
small errors in phase lead to appreciable errors in determina-
tion of the complex refractive index. This is demonstrated in
Fig. 1. The curves in the plane of the complex refractive
index correspond to constant values of the reflectance ampli-
tude and adjacent points correspond to the difference in re-
flectance phase induced by a 1
m shift of the sample for
frequency of 1 THz. The shape of the curve remains un-
changed for other frequencies while the spacing between the
points is proportional to the frequency. For a low absorption
index 共close to the real axis兲the slope of the curves is almost
vertical. Therefore a small phase error leads to large errors in
the calculated absorption index. On the other hand, when the
a兲Electronic mail: pashkin@fzu.cz
REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 74, NUMBER 11 NOVEMBER 2003
47110034-6748/2003/74(11)/4711/7/$20.00 © 2003 American Institute of Physics
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refractive and absorption indices are comparable, the slope is
horizontal and errors in the real part of the refractive index
become dominant. One can also note that the points are more
dense in the case of ppolarization, which implies better sta-
bility with respect to phase errors compared to the case of s
polarization.
To avoid the problem of phase uncertainty in TDTRS,
several different approaches have been used. One of them
consists of substitution of the reference signal by a signal
reflected from the sample under specific conditions. Howells
and Schlie4have investigated the low-temperature dielectric
function of undoped InSb in this way taking as a reference
the wave form obtained at 360 K. They used the fact that the
reflectance of InSb at high temperature is comparable to that
of a silver mirror due to the narrow band gap of the material.
Thrane et al.5have measured the refractive index of liquid
water in a silicon cell using the signal reflected from the
air–silicon interface as a reference and the signal from the
silicon–water interface as a sample wave form. Such meth-
ods make use of specific sample properties and can be ap-
plied only in particular cases. Other methods similar to ellip-
sometry extract the complex dielectric function from the s-
and p-polarized THz signals reflected from the sample at
high angles of incidence.6,7 This approach provides very sat-
isfactory results in some cases. On the other hand, it requires
good quality THz polarizers and, for highly reflective
samples, it is necessary to measure under angles of incidence
close to 90°, which restricts the measurements to only large
enough homogeneous samples. In the case of TDTRS with a
reference mirror, the uncontrollable time shift of the refer-
ence pulse can be a posteriori adjusted to fit some model of
the dielectric response8or to minimize the difference be-
tween the measured and calculated interference pattern in a
silicon slab attached to the sample surface.10 The last method
does not make any assumption about the sample dielectric
behavior model, but is rather difficult to realize because there
should be good optical contact between the sample and the
slab. Recently Hashimshony et al.9have succeeded in per-
forming TDTRS measurements of epitaxial semiconductor
layers using a special sample holder which allowed replacing
the reference mirror by the sample within accuracy of 1
m.
However, this is not an easy task, and in some cases even
this precision is not sufficient for correct determination of the
dielectric function.
III. EXPERIMENTAL SETUP
Figure 2 shows schematically the relevant part of our
experimental setup. THz pulses are emitted by a ZnTe 关011兴
crystal via optical rectification of amplified femtosecond la-
ser pulses 共wavelength 800 nm, repetition rate 1 kHz兲and
focused onto the sample 共or reference mirror兲by an ellipsoi-
dal mirror. The optical sampling pulses have a variable time
delay with respect to THz pulses and serve as electro-optic
detector for the THz wave form. The key idea consists of
making the two beams coincide between the emitter and the
sample, in contrast to the usual arrangement where there is
coincidence between the sample and the sensor. In our ge-
ometry both beams propagate collinearly and reflect from the
sample surface. The sample leans on a flat surface of a cir-
cular metallic aperture at angle
with respect to the incident
beam. The aperture angle
␣
has to be small enough 共
␣
⬍45°
⫺
/2兲to ensure that possible weak reflection of the THz
beam off the aperture falls away from the sensor. To maxi-
mize a clear aperture for the THz beam,
should be kept
small, however, in practice,
needs to be larger than about
10°. In this article we present results obtained for angles of
incidence
⫽12.5° and 45°. This latter geometry 共depicted in
FIG. 1. Complex refractive indices corresponding to the same reflectance
amplitude of 共a兲p- and 共b兲s-polarized waves. Adjacent points correspond to
phase change equivalent to 1
m displacement of the sample. The values are
calculated for angle of incidence of 45° and frequency of 1 THz.
FIG. 2. Schemematic of the TDTRS experimental setup 共see the text for
details兲.
4712 Rev. Sci. Instrum., Vol. 74, No. 11, November 2003 Pashkin
et al.
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Fig. 2兲is of particular interest since it is suitable for mea-
surements in standard cryostats with perpendicular windows.
The signal reflected is detected using the electro-optic effect
by a ZnTe 关011兴sensor which is placed directly after the
sample. Measured THz wave forms are normalized by the
voltage on a reference photodiode which is proportional to
the intensity of the sampling beam. In this way, the differ-
ence in optical reflectance between the reference mirror and
the sample is taken into account. Most measurements were
performed using p-polarized THz pulses, however, a reflec-
tivity experiment with s-polarized radiation has been also
tested.
Our arrangement is based on the setup used by Li et al.12
who took advantage of the possibility to easily change the
incidence angle to perform TDTRS of a thin film near the
Brewster angle. In our setup, a fixed angle of incidence and
reflection is used and the focusing mirror after the sample is
absent. We benefit from the major feature that displacement
of the sample changes the length of the optical path by pre-
cisely the same amount for both beams, and produces no
phase change in the measured THz wave form. To illustrate
this, we have performed measurements of the THz signal for
different positions of a gold mirror, shifting it in the way
shown by the bold arrow in Fig. 2. It has been found that
evena1mmshift from the initial position in both directions
does not change the THz wave form. Figure 3 shows the
phase differences between THz pulses measured with the
mirror shifted 10
m and 1 mm. It can be seen in Fig. 3 that
the phase error does not depend on the mirror shift. The
limiting factor for phase reproducibility is then the temporal
stability of the whole setup 共including the beam-pointing sta-
bility of the laser source兲rather than precise positioning of
the sample. Similarly, the setup described is not sensitive to
errors in the relative angular alignment of the sample and
reference mirror. The absence of a focusing mirror after the
sample allows us to avoid problems due to, e.g., possible
lower optical quality of the mirror surface or deviation of its
shape from the ideal one. In fact, focusing of the THz beam
onto the sensor is not necessary since standard THz experi-
ments offer a very good signal-to-noise ratio nowadays.
Suitable samples for measurement have to fulfill the fol-
lowing requirements: 共i兲have an optically flat surface to al-
low nondiffusive 共specular兲reflection of the sampling beam
and 共ii兲the absence of secondary reflections of the sampling
beam from the rear of the sample. According to our experi-
ence, the majority of crystalline and ceramics samples can be
polished with sufficient precision to satisfy the former con-
dition. The latter one is critical for optically transparent
samples where an echo of the sampling beam reflected from
the back side of the sample adds a systematic error to the
reference photodiode voltage and is responsible for several
replicas of the THz pulse in the measured wave form. Para-
site reflections can be unambiguously detected through a rep-
lica that occurs at time ⌬tbefore the main THz pulse:
⌬t⫽2n2d
c
冑
n2⫺sin2
,共1兲
where nis the optical refractive index of the sample and dits
thickness 共see Fig. 4兲.
This situation particularly occurs in thin films deposited
on optically transparent substrates or in dielectric single
crystals. In these cases, special precautions have to be taken:
共i兲roughening or blackening of the back surface of the
sample; 共ii兲spatial filtering of the sampling beam after the
sensor.
IV. RESULTS AND DISCUSSION
A. Doped silicon
Moderately or highly doped silicon crystals are of par-
ticular interest as test samples for the TDTRS.8,10 They have
noticeable dispersion of the complex conductivity in the THz
frequency range and knowledge of their dc conductivity pro-
vides a good possibility to verify the model fits of TDTRS
FIG. 3. Measured phase difference introduced by 10
m共closed squares兲
and1mm共open circles兲shifts of the reference mirror in the setup in Fig. 2
with
⫽45° and ppolarization. The classical setup, where only the THz
beam reflects off the sample, requires the sample surface to be positioned
within 1
m of the reference mirror surface in order to fit the phase inside
the area between the solid lines.
FIG. 4. THz wave forms obtained in the reflection setup for SBT thin film
on a sapphire substrate. Solid line: Sample with a blackened rear surface;
dotted line: sample without this treatment. The replica near ⫺6 ps indicates
the presence of the sampling beam reflection at the rear surface. The differ-
ence in amplitude of the main pulses 共near 0 ps delay兲is due to the change
in sampling beam intensity and indicates the extent of error that would result
of the parasite reflection were not removed.
4713Rev. Sci. Instrum., Vol. 74, No. 11, November 2003 Terahertz reflection spectroscopy
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spectra. Our experimental arrangement requires the optical
sampling beam to reflect from the sample surface which
seems to be undesirable for measurements of semiconduc-
tors, because of photoexcitation of additional carriers in the
sample. It means that the density of photocarriers should be
always carefully estimated during evaluation of these experi-
ments. Taking into account the sampling beam cross section
共approximately 0.5 mm2兲, optical absorption length in silicon
共10
m at 800 nm兲, and sampling pulse energy 共0.5 nJ兲we
obtain photocarrier density n0⬇2⫻1014 cm⫺3which is two
orders of magnitude smaller than the impurity concentration
in our samples. If the photocarriers are long lived, which is
the case for silicon, the laser repetition rate should be taken
into account. Let us consider a sequence of pulses with the
time separation T共1 ms in our case兲; then the density of
carriers can be found from the following relation:
n共t兲⫽n0e⫺共t/
兲⫹n0e⫺共t⫹T/
兲⫹n0e⫺共t⫹2T/
兲⫹¯
⫽n0e⫺共t/
兲
1⫺e⫺共T/
兲,共2兲
where
is the carrier lifetime. Here we neglect the diffusion
process which additionally decreases the carrier density. If
T⬎
ln 2 共which is the case for moderately and highly doped
silicon where
is smaller than 1 ms兲, then n(t) is increased
compared to the carrier density created by a single sampling
pulse by a factor smaller than 2. Therefore the influence of
the sampling beam can be neglected for our experimental
conditions.
In the case of direct gap semiconductors 共GaAs, InP,
etc.兲, the absorption length is smaller than in indirect semi-
conductors such as silicon and n0can be higher by an order
of magnitude or more. However, due to the direct character
of the band gap, the probability of carrier recombination is
also higher, so the carrier lifetime
is appreciably smaller
and possibly
ⰆT. One can take advantage of this to adopt
the experimental approach described below. The excess car-
rier density in the sample just before the arrival of the next
sampling pulse n(T) becomes much smaller than n0:
n共T兲⫽n0
eT/
⫺1.共3兲
To achieve a THz pulse that reflects off the sample in such
conditions, it is sufficient to introduce an appropriate small
delay between the THz and sampling pulses after their re-
flection from the sample in order to make the optical path of
the THz beam between the sample and the sensor longer than
that of the sampling beam. Then, the detection system pro-
cesses information about the THz field unaffected by the
photoexcitation, since it reflects from the sample surface be-
fore the sampling pulse. For this one can insert, e.g., a 共0001兲
sapphire slab just before the sensor 共sapphire is transparent
to both optical and THz radiation and induces a pulse
walkoff of 4.5 ps/mm兲. In order to test this approach we have
put a 0.5 mm thick sapphire plate into optical contact with
the ZnTe sensor and we have significantly increased the sam-
pling pulse energy 共up to 5 nJ兲. No change in reflectance
spectra has been observed compared to those obtained with
0.5 nJ pulse energy.
We present here a measurement of complex reflectance
spectra of two n-type phosphorus doped silicon wafers sup-
plied by ON Semiconductor-Terosil, Roz
ˇnov pod Rad-
hos
ˇte
ˇm, with specifications
dc⫽0.128 ⍀cm 共sample I兲and
dc⫽0.153 ⍀cm 共sample II兲with possible deviations less
than 25%. The experimental data shown in Fig. 5共a兲were
obtained with an angle of incidence
⫽12.5° and p-polarized
radiation with the 0.5 mm sapphire delay plate. The ampli-
tude and phase of the measured reflectance were fitted using
the Drude model.10 Figure 5共b兲shows the dielectric function
and conductivity calculated from the experimental data and
those using the Drude model. The fits yield two independent
parameters: the free-electron mobility
and concentration
Nc共the dielectric constant of undoped silicon
⑀
Si⫽11.66 is
taken as a fixed value兲. From these values, the dc resistivity
can be calculated using the formula
dc⫽(
Nce0)⫺1, where
e0is the elementary charge. One finds
⫽1160 cm2/V s,
Nc⫽4.0⫻1016 cm⫺3, and
dc⫽0.135 ⍀cm for the sample I
and
⫽1270 cm2/V s, Nc⫽2.8⫻1016 cm⫺3, and
dc
FIG. 5. 共a兲Complex reflectance at
⫽12.5°, ppolarization and 共b兲dielec-
tric function and conductivity of
doped silicon samples. Points: Experi-
mental data; lines: fits using the Drude
model.
4714 Rev. Sci. Instrum., Vol. 74, No. 11, November 2003 Pashkin
et al.
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⫽0.175 ⍀cm for the sample II. The values of the electron
mobilities are consistent with those published previously for
similar samples.8,10 The dc resistivity values match the sup-
plier’s specifications.
B. SBT ceramics
SBT is a promising material for application in ferroelec-
tric memories and has been extensively investigated espe-
cially during the last few years. A study of IR reflectance
revealed a rather strong polar phonon mode below 30 cm⫺1
at room temperature.13 However, this frequency range is
barely accessible for Fourier transform IR 共FTIR兲spectros-
copy 共the signal from the source is weak兲and the measured
power reflectance allows one to obtain the complex permit-
tivity only by fitting with a model dielectric function. There-
fore direct measurement of the complex permittivity can be
useful for correction and improvement of FTIR data.
We have studied the reflectance of SBT ceramics using
three different arrangements: 共i兲12.5° incidence and ppolar-
ization, 共ii兲45° incidence and ppolarization, and 共iii兲45°
incidence and spolarization. The measured complex reflec-
tance and calculated permittivity of SBT ceramics together
with a fit of FTIR reflectance are presented in Fig. 6. From
the fit we have found the soft-mode frequency
0
⫽28 cm⫺1, damping
␥
0⫽12 cm⫺1, and dielectric strength
⌬
⑀
0⫽81. It has to be pointed out that the peak in the relative
phase which occurs near 40 cm⫺1for SBT corresponds to the
frequency of a longitudinal phonon mode, while the imagi-
nary part of the permittivity 共dielectric loss兲peaks at the
position of transverse resonance at
0. One can see that
TDTRS is able to reproduce correctly the mode structure at
higher frequencies and brings valuable information down to
at least 10 cm⫺1. The complex permittivities measured in
different arrangements are in agreement with each other,
which demonstrates the reliability of the technique pre-
sented.
C. SBT thin film
A 5.5
m thick SBT film on a 共0001兲sapphire substrate
has been characterized in reflection as well as in transmission
geometry. For the transmission measurements, the THz pulse
transmitted through the bare sapphire substrate was used as a
reference and the complex permittivity was numerically cal-
culated in a standard way. The reflection measurement was
performed using p-polarized THz pulses 45° incident on the
sample with a blackened back surface to avoid the above-
mentioned multiple reflection of the optical sampling beam
inside the sapphire substrate 共see Fig. 4兲. The complex re-
flectance was calculated taking into account only the THz
pulse reflected from the front surface of the sample. Fabry–
Pe
´rot interference inside the substrate was cut off 共time win-
dowing兲. An additional correction was made to take into ac-
count multiple reflections of the sampling beam inside the
film. Usually the thickness of thin films is smaller than 1
m
and the delay of the sampling beam echoes is negligible
compared to the duration of the sampling pulse 共typically
50–100 fs兲. In our case 共film thickness d⫽5.5
m) special
care has to be taken in order to deconvolute the influence of
Fabry–Pe
´rot reflections of the sampling beam inside the film.
The time delay of the sampling pulse needed for its propa-
gation back and forth through the film can be calculated
using Eq. 共1兲. We deduced the optical refractive index of
SBT from
⑀
⬁obtained by FTIR measurements on SBT ce-
ramics: n⫽2.45; the corresponding time delay is ⌬t
⫽94 fs. Thus the sampling pulse is divided into a sequence
of pulses with decreasing amplitude which are separated in
time. The detected THz wave form can be written in the form
of
y共t兲⫽y0共t兲⫹ay0共t⫹⌬t兲⫹¯
1⫹a⫹¯,共4兲
where y0(t) is the deconvoluted wave form 共free of artifacts
due to multiple reflections of the sampling beam兲, and a
⫽0.025 is the ratio of the intensities of the first two sampling
pulses calculated using Fresnel equations. The denominator
of Eq. 共4兲accounts for normalization of the signal by the
voltage of the reference photodiode. In view of the small
value of a, all higher order terms in Eq. 共4兲can be neglected.
FIG. 6. Complex reflectance and dielectric permittivity of SBT ceramics
from TDTRS measurements. 共a兲Complex reflectance for
⫽12.5°, ppolar-
ization; 共䊉兲amplitude; 共䊊兲phase. 共b兲Dielectric function; 共c兲dielectric loss;
共⫹兲
⫽12.5°, ppolarization; 共䊏兲
⫽45°, ppolarization, 共䊐兲
⫽45°, spo-
larization; solid lines: fit of the FTIR reflectance data based on the sum of
damped harmonic oscillators.
4715Rev. Sci. Instrum., Vol. 74, No. 11, November 2003 Terahertz reflection spectroscopy
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Transforming Eq. 共4兲into the frequency domain and dividing
it by the spectrum of the reference pulse we obtain for the
complex reflectance
r0共
兲⫽r共
兲1⫹a
1⫹ae⫺i
⌬t,共5兲
where r(
) is the measured reflectance and r0(
) the cor-
rected one which should be used for the evaluation of dielec-
tric properties. This correction mainly leads to changes in the
imaginary part of the permittivity. In the case of SBT film it
increases the value of the dielectric loss peak by about 7%.
The complex permittivity was calculated by numerically
solving a system of two equations derived by Berreman14
which relates the complex reflectance of a thin film on a
substrate at an arbitrary angle of incidence to the dielectric
constants of the thin film and substrate.
The resulting complex permittivity obtained from both
transmission and reflection measurements and a fit of the
transmission data by two damped harmonic oscillators are
shown in Fig. 7. The fit yields the following parameters of
the soft mode:
0⫽28 cm⫺1,
␥
0⫽26 cm⫺1, and ⌬
⑀
0⫽54.
For comparison we also show the complex permittivity cal-
culated from the amplitudes of the reflectance and transmit-
tance 共disregarding the respective phases兲. We would like to
stress three points.
共1兲The transmission data can comprise large errors in the
static value of the permittivity and in the strength of
modes due to the uncertainty in substrate thickness.3In
contrast, the substrate thickness does not play any role in
the reflection experiment. Hence, one can use, e.g., the
static value of the dielectric function determined by the
reflection experiment for small corrections 共within 1 or 2
m兲of the substrate thickness: trial substrate thicknesses
can be used during the transmission data evaluation in
order to match the resulting permittivity to that obtained
from reflectance. Such an approach has been used to
evaluate the transmission data shown in Fig. 7.
共2兲Evaluation of the complex permittivity using reflectance
and transmittance amplitudes is indeed possible; more-
over it does not require the value of the substrate thick-
ness for transparent substrates. However, our experience
shows that the results obtained by this method are not as
accurate as the results of phase sensitive methods 共note
the appreciable error in the imaginary part of the permit-
tivity in Fig. 7兲.
共3兲The data obtained from the transmission measurement
using the above-described procedure fulfill slightly bet-
ter the Kramers–Kronig relations than those obtained
from the reflectance only. In this respect, if the substrate
thickness is very precisely known, the transmission ex-
periment seems to provide slightly more accurate data
for this film. The transmission and reflection experiments
are thus complementary in this sense and their combina-
tion allows unambiguous determination of the dielectric
strength of the polar modes detected.
In summary, we have introduced a new approach for
TDTRS which allows precise measurement of the dielectric
function from reflectivity measurements. We have solved the
key phase problem; as a result, the phase is independent of
the relative position of the sample and reference mirror. This
feature makes the method attractive also in situations where
it is not possible to precisely align the sample holder, which
is the case, e.g., in temperature-dependent measurements. We
have shown that this method can be successfully applied to
the characterization of a variety of materials.
ACKNOWLEDGMENTS
This work was supported by the Ministry of Education
of the Czech Republic 共Project No. LN00A032兲,bythe
Grant Agency of the Czech Republic 共Project No. 202/01/
0612兲, and by the Volkswagen Stiftung 共Grant No. I/75908兲.
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FIG. 7. Real and imaginary parts of the permittivity of the thin SBT film
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4716 Rev. Sci. Instrum., Vol. 74, No. 11, November 2003 Pashkin
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