Far-infrared transmittance and reflectance of YBa2Cu3O7

Far-infrared transmittance and reflectance of YBa2Cu3O7
A. R. Kumar
Z. M. Zhang1
Department of Mechanical Engineering
V. A. Boychev
D. B. Tanner
Department of Physics
University of Florida,
Gainesville, FL 32611
L. R. Vale
D. A. Rudman
Electromagnetic Technology Division,
National Institute of Standards and Technology,
Boulder, CO 80303
Far-Infrared Transmittance and
Reflectance of YBa2Cu307-s
Films on Si Substrates
The transmittance and reflectance of superconductive YBa2Cu307.s (YBCO) thin films
deposited on Si substrates have been measured in the far-infrared frequency region from
10 to 100 cm'1 (wavelength from 1000 to 100 pm) at temperatures between 10 and 300
K. The effects of interference, optical resonance, and antireflection on the radiative
properties of high-temperature superconducting (HTSC) films are observed and quantitatively analyzed. Furthermore, we have measured the reflectance of the HTSC filmsubstrate composites for radiation incident on the substrate side (backside reflectance) for
the first time. The backside reflectance increases significantly from the normal state to the
superconducting state at certain frequencies; this experimentally demonstrates that HTSC
films can be used to build far-infrared intensity modulators. The complex refractive index
of the YBCO films is determined from the measured transmittance using the Drude model
in the normal state and a two-fluid model in the superconducting state. The complex
refractive index obtained from this study is useful for various applications of YBCO films,
including radiation modulators, detectors, and Fabry-Perot resonators.
1 Introduction
The radiative properties of high-temperature superconducting
(HTSC) films change rapidly from the normal state to the superconducting state in the far-infrared region. This distinguishing
characteristic of HTSC films may be used in designing thermooptoelectronic devices such as infrared detectors, intensity and
phase modulators, and radiation shields (Zhang and Frenkel, 1994;
Zhang, 1998). Additionally, the reflectance of HTSC thin films
differs significantly for radiation incident on the substrate side of
the film-substrate composite (also called backside illumination) as
compared to radiation incident on the film side. Recently, Zhang
(1998) presented a design analysis of far-infrared intensity modulators using YBa2Cu307.6 (YBCO) films by evaluating the reflectance for various design structures in both the superconducting and
normal states. This work predicted large differences in the backside reflectance between the superconducting and normal states.
To date, the reflectance of HTSC films has been measured only for
radiation incident on the film side (Renk, 1992; Tanner and Timusk, 1992; Zhang et al., 1994). In order to confirm the features
associated with the backside illumination, there is a need for
measurements of the radiative properties of HTSC films for radiation incident on the substrate.
For the development of HTSC applications, the choice of the
substrate material is an important issue. Substrates commonly used
for growing HTSC films include MgO, SrTi0 3 , LaA103, yttriastabilized zirconia (YSZ), and sapphire (Chen et al, 1995; Phillips,
1996). Several promising far-infrared applications of HTSC films,
such as bolometers, modulators, and resonators, demand that the
substrate be transparent in the spectral region of interest. Fenner et
al. (1993) presented transmittance measurements for various substrates, NdGaOj, LaA103, MgO, YSZ, and Si, in the mid and
far-infrared regions. This study showed that the transmittance of Si
' To whom correspondence should be addressed.
Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT
TRANSFER and presented at 1998 ASME IMECE, Anaheim. Manuscript received by
the Heat Transfer Division, Sept. 8, 1998; revision received, May 5,1999. Keywords:
Experimental, Heat Transfer, Interferometry, Radiation, Thin Films. Associate Technical Editor: P. Menguc.
844 / Vol. 121, NOVEMBER 1999
is generally higher than that of the other substrates in the measured
spectral region. (The transmittance of pure Si is slightly greater
than 0.5 at wavelengths longer than 20 |U.m.) The higher transmittance of Si in a broad spectral region allows new optical designs
such as backside-illuminated HTSC microbolometers (Rice et al.,
1994). At present, Si substrates have received significant attention
from the electronics industry due to the feasibility of lithographically patterning HTSC films and the potential integration of semiconductor and superconducting electronics (Phillips, 1996). Highquality YBCO films have been successfully grown on Si substrates
using pulsed laser ablation (Fork et al., 1991; Mechin et al., 1996)
but very few measurements have been done to determine the
radiative properties of such films. Berberich et al. (1993) measured
the transmittance and reflectance of imperfect YBCO films deposited on Si substrates. These films were of poor quality because they
were deposited on Si substrates without buffer layers, and the
reflectance was measured only at room temperature. Karrai et al.
(1992) studied the transmittance of YBCO thin films on Si substrates with and without a magnetic field. However, the substrates
were intentionally wedged to avoid interference effects. Most of
the reported radiative properties of HTSC superconductors were
for opaque samples or thin films on thick substrates (Tanner and
Timusk, 1992; Zhang et al., 1994). For transmittance measurements, the interference effects associated with the substrate were
often neglected by either averaging over a free spectral range or
using wedged substrates (Gao et al., 1991; Karrai et al, 1992;
Zhang et al., 1992; Cunsolo et al., 1993). Knowledge of the
radiative properties of thin YBCO films deposited on thin substrates is essential for designing optoelectronic devices including
radiation modulators and Fabry-Perot resonators (Renk et al.,
1990; Genzel et al., 1992; Malone et al., 1993; Zhang, 1998).
The transmittance and reflectance of thin films on transparent
substrates typically oscillate periodically as the optical frequency
changes, as a result of multiple reflections inside the substrate.
This oscillation is particularly prevalent in the far infrared because
the wavelength is comparable with the substrate thickness. Hadni
et al. (1995) measured the far-infrared transmittance of several
YBCO films deposited on MgO substrates. The interference features associated with the substrate were clearly seen in their study
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at frequencies from 10 to 40 cm"1. In the present study, several
infrared spectrometers have been used to measure both the transmittance and reflectance of YBCO films (^35-nm thick) deposited
on Si substrates (=200-/xm thick), in the frequency region from 10
cm -1 to 100 cm"' (wavelength from 1000 to 100 /im) from room
temperature down to 10 K. The transmittance was measured with
radiation incident on the film side and the reflectance was measured for radiation incident both on the film and on the substrate.
The transmittance of the film-substrate composite is the same for
radiation incident on the film side and the substrate side. The
spectral resolution is chosen high enough to measure the effects of
interference associated with the substrate.
Accurate assessments of the potential of YBCO films in optoelectronic applications require the determination of the frequencydependent radiative properties. Knowledge of the complex dielectric function of the YBCO film facilitates the computation of
radiative properties in desired spectral regions (Phelan et al., 1991;
1992). Due to their complicated crystalline structures, the refractive index of the HTSC materials may vary significantly depending
on the method of preparation, oxygen content, thickness, and
microstructure (Choi et al, 1992; Renk, 1992; Tanner and Timusk,
1992; Flik et al., 1992). In the present study, the measured transmittance spectra are used to determine the complex dielectric
function e((o) of YBCO films deposited on Si substrates. In the
normal state, a two-component model, consisting of a temperaturedependent free-carrier absorption term (the Drude term) and a
temperature-independent mid-infrared term (the Lorentz term), is
used for computing e(w). In the superconducting state, the conventional two-fluid model is used for computing e(<w) (Tanner and
Timusk, 1992). The spectral transmittance and reflectance are
calculated with the transfer-matrix method using the refractive
indices and thicknesses of the film and the substrate. The dielectric
functions are determined at different temperatures by comparing
model predictions with measured results. Our measurements demonstrate the feasibility of using thin HTSC films deposited on
transparent substrates as radiation modulators. The frequency and
temperature-dependent refractive index of the HTSC films obtained from the present study will facilitate future design of potential far-infrared devices based on YBCO films on Si substrates.
2 Experiments
2.1 YBCO Film Preparation. The YBCO films were deposited by pulsed laser ablation using an excimer laser operated at
a wavelength of 248 nm. A single-crystal (100) Si wafer, polished
on both sides with a thickness of approximately 200 /i,m and a
diameter of 76 mm, was used as the substrate. The wafer is slightly
boron-doped and has an electric resistivity of ^1000 ft-cm. The Si
wafer, was cut into approximately 12 X 12 mm2 pieces for the
deposition of the YBCO. Ag paste was used to mount the substrate
onto a substrate holder in the deposition chamber. At high temperatures, Ag may diffuse into the Si substrate. Therefore, a
320-nm thick Si0 2 layer was deposited by chemical vapor depo-
^^^^sxss^^^NSN^^^s^^s>c<^^^
- YBCO (35 nm)
-CeO,(10nm)
' YSZ (20 nm)
Si SUBSTRATE (»200 nm)
Fig. 1 Structure of the film-substrate composite (the lateral dimensions
are ~12 x 12 mm2)
sition (CVD) on the backside of the Si substrate before the application of the Ag paste. The substrate was heated to 1063 K for the
deposition of a 20-nm thick YSZ layer and then a 10-nm thick
Ce0 2 layer on the Si substrate. These buffer layers are required for
growing high-quality superconducting films (Mechin et al., 1996).
A YBCO film of 35 nm was deposited on the top of the buffer
layers at 1043 K in an optimized 0 2 environment (Rice et al,
1994). The thicknesses of the thin films were determined by
calibrations of the rate of deposition. The YBCO films formed this
way are a-b plane oriented (c-axis is parallel to the surface
normal) with a critical temperature (Tc) between 86 and 88 K.
Several steps were followed to remove the Ag paste and the
Si0 2 layer. First, the YBCO film was covered by a photoresist.
Second, nitric acid (HN0 3 ) was used to remove Ag paste and then
hydrofluoric acid (HF) was used to etch off the Si0 2 layer. Third,
the photoresist was removed using acetone. Finally the film was
rinsed with isopropanol. The structure of the film is shown in Fig.
1. Some damage to the film may have taken place during this
stripping process since the critical temperature of several films
dropped to 80 to 82 K after the removal of the photoresist.
However, scanning electron microscopic (SEM) images showed
no evidence of microcracks or other surface damage.
2.2 Transmittance and Reflection Measurements. The
transmittance spectra of the films were measured using a slow-scan
Michelson interferometer with a Hg-arc lamp as the source (Gao,
1992). The entire interferometer chamber is evacuated to eliminate
the absorption by atmospheric gases. The source radiation is modulated by a rotating chopper inside the chamber to allow lock-in
detection. This procedure allows only in-phase modulated radiation to be detected and thus minimizes the effect of external noise.
The radiation is guided by mirrors and light pipes to the sample
and thence to the detector. Polyethylene windows, which are
transparent in the far-infrared region, are used to seal the cryostat.
The detection system consists of a 4.2 K He-cooled Si bolometer.
A preamplifier and lock-in electronics are used to measure the
detector output signal.
The sample holder consists of two identical copper plates with
equally sized apertures in the middle. The copper plates are
mounted at a right angle. One aperture is covered by the specimen
and the other is left blank for the reference measurement. The
sample holder is mounted on the cold finger of the cryostat. The
cold finger is kept in high vacuum and cooled by flowing liquid He
Nomenclature
c0 = speed of light in vacuum, 2.9979 X
108 m/s
d = thickness, m
fs = fraction of superconducting electrons
•• = (-1)" 2
k = imaginary part of refractive index
N = complex refractive index
n = real part of the refractive index
ne= electron number density, m"'
r — reflection coefficient
T = transmittance
Tc= critical temperature, K
Journal of Heat Transfer
y
8
Aw
7«
£
dimensionless admittance of the
film
phase change
free spectral range, m"1
damping constant in the Lorentz
term, rad/s; (1 rad/s = 5.3089 X
10"12 cm"1)
dielectric function
electric permittivity of free space,
8.8542 X 10~12C -V" 1 • m"1
high-frequency dielectric constant
wavelength in vacuum, m
1/T = electron scattering rate, rad/s
crDC= DC conductivity, (ft • m)"1
w = angular frequency, rad/s
ioc = center frequency, rad/s
a),, = plasma frequency, rad/s
(iipi. = plasma frequency in the Lorentz
term, rad/s
Subscripts
/=film
Si = silicon
NOVEMBER 1999, Vol. 121 / 845
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at 4.2 K, which results in conductive cooling of the specimen. A
diffusion pump is used to achieve a vacuum of 1CT2 Pa inside the
cryostat before cooling; the actual pressure should be much lower
due to cryopumping. The liquid He is transferred from a liquidhelium tank via an evacuated transfer line. The specimen temperature is measured using a Si diode. An automatic temperature
controller was used to adjust the temperature of the specimen using
a proportional-integral-derivative (PID) control scheme and an
electric heater wrapped around the cold finger. The temperature
variation was within 0.5 K of the set temperature for measurements
above 100 K. During the measurements at 10 K and 50 K, the
variation of temperature was about 1 K from the set temperature.
The cryostat, along with the sample holder, is introduced vertically into the interferometer chamber. The cryostat can be rotated
so that either the blank aperture or the specimen is in the beam path
of the far-infrared radiation. For the transmittance measurement,
the spectrum obtained with the blank is taken as the background
spectrum. The cryostat is then rotated by 90 degrees to introduce
the specimen in the beam path. The exact position of the cryostat
for the blank aperture or for the specimen is determined by gently
rotating the cryostat until the signal is maximum. The transmitted
spectrum of the specimen is divided by the background spectrum
to obtain the transmittance. The spectral resolution is approximately 0.5 cnT 1 .
The reflectance was measured using a commercial fast-scan
Fourier transform spectrometer. The spectrum is averaged over
128 scans and the resolution is «°1 cm"1. Here, the far-infrared
beam is directed to either the specimen or a Au mirror by a
combination of plane and elliptical mirrors. The angle of incidence
in the reflectance measurement is *»7.5 deg with a beam divergence of 7.5 deg. The Au mirror is used as the reference for the
reflectance measurements at all temperatures. In the far-infrared
region, the reflectance of the gold is greater than 0.995 at room
temperature and increases at lower temperatures. Hence, the reflectivity of the Au mirror is taken to be 1.
The transmittance and reflectance of bare Si substrates were
measured at different temperatures to evaluate the measurement
uncertainty. The refractive index of single-crystal Si has been well
documented (Loewenstein et al., 1973). The use of the highresistivity single-crystal Si has essentially eliminated absorption in
the substrate. The measured transmittance shows interference patterns with TmM about 1, Tmin about 0.3, and a free spectral range Aw
(the wave number interval between two interference maxima)
about 7 cm"1. Even a small variation in the substrate thickness
(dsi) or its refractive index (»si) can affect Aw, since Aw =
(2nsirfsi)"1. The Si thickness is determined from the free spectral
range by assuming that n si = 3.42 at frequencies from 10 to 100
cm -1 (Loewenstein et al., 1973). The actual angles of incidence in
the transmittance measurement have a spread up to 18 degrees as
a result of the beam divergence in the light pipe. The effect of
inclined incidence on Aw (and hence ds{) is less than 0.5 percent
since the angle of refraction inside the silicon is small. The
transmittance extrema shift toward higher frequencies as the temperature decreases, indicating a decrease of the refractive index
since the thickness change can be neglected (Loewenstein et al,
1973). The measured transmittance is compared with the calculated transmittance to determine the refractive index at different
temperatures. The results are nsi = 3.405 at 200 K, 3.395 at 100 K,
and 3.39 at 50 K and 10 K. For the same Si wafer, the transmittance maxima become the reflectance minima at all temperatures.
Although the interference patterns in the measured transmittance and reflectance of the Si substrate match theoretical predictions, the radiometric accuracy is not as high. The maximum
transmittance often varies between 0.95 and 1.05 or more near the
spectral cutoffs. The reflectance minimum is always greater than
zero, which may be caused by the insufficient spectral resolution.
The root-mean-square difference between the measured and the
calculated values shows a standard uncertainty of =0.05 in both
the transmittance and reflectance. Hence, the expanded uncertainty
(95 percent confidence) is estimated to be 0.1 for all measure846 / Vol. 121, NOVEMBER 1999
(a)
0.5
10
(b)
20
30
40
50
60
Frequency (cm'1)
70
80
90
0.5
Sample B
0.4 ;
|
10 K
50 K
100 K
200 K
300 K
Tc = 80.2 K
0.3
0.2
0.1
10
20
30
40
50
60
Frequency (cm'1)
70
80
90
Fig. 2 Measured transmittance of YBCO films on Si substrates at various temperatures
ments. This uncertainty is large compared with the measurements
without sharp interference fringes, which could be caused by many
parameters, such as detector nonlinearity, misalignment, phase
error, and multiple reflections between the sample, windows, and
the beam splitter (Griffiths and de Haseth, 1986; Zhang et al.,
1996).
Figure 2 shows the measured transmittance of two specimens,
identified as Sample A and Sample B, at various temperatures.
The transmittance spectra oscillate periodically due to interference effects inside the substrate with Aw slightly higher than 7
cm"1. In the normal state, the fringe-averaged transmittance,
defined as f(w) = (1/Aw) ST-^Jil T(co')dco', is nearly uniform
for all the measured frequencies at any given temperature but
decreases gradually as the temperature is lowered. This decrease is expected because the electrical conductivity of the
YBCO film increases as temperature decreases. The interference pattern is quite periodic over the studied wave number
range at any given temperature, but varies significantly as the
temperature is changed from 300 K to 100 K. Not only do the
peak locations vary with temperature but also the fringe contrast (the relative amplitude of oscillation) changes significantly. There is a phase shift of IT rad between the 300 K and
100 K data; that is, the transmittance maxima at one temperature correspond to the transmittance minima at the other. Furthermore, the spectrum at 200 K for Sample A has no discernible interference fringes, which can be attributed to an
antireflection effect of the YBCO film. In the superconducting
state, the fringe-averaged transmittance is lower at smaller
frequencies and increases toward higher frequencies. The peak
transmittance is higher at lower temperatures. This is caused by
the optical resonance effect in the film-substrate composite and
by decreasing losses in the film. The antireflection effect on the
interference pattern at 200 K and the effect of optical resonance
on the peak transmittance in the superconducting state are
discussed in Section 5.
The measured reflectance spectra of Sample A are shown in Fig.
3 for radiation incident on the substrate. The fringe-averaged
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From Eqs. (2) and (3), only crDC and 1/T are needed to calculate
€
0
10
20
30
40
50
60
70
80
90
100
Frequency (cm'1)
Fig. 3 Measured reflectance for radiation incident on the substrate
(backside)
reflectance is nearly independent of wave number in the normal
state, but increases toward smaller frequencies in the superconducting state. The reflectance increases sharply as the temperature
is reduced to below the critical temperature and increases slightly
as the temperature is further reduced. The fringe contrast is much
higher in the superconducting state than in the normal state. In a
way similar to the transmittance, the fringe contrast is much
smaller at 200 K than at 300 K and 100 K. At frequencies where
the transmittance is maximum, the reflectance for radiation incident on the substrate decreases to minimum values. The large
change in the reflectance of YBCO films at particular frequencies,
from the superconducting to the normal state, experimentally demonstrate that HTSC films can be used to construct far-infrared
intensity modulators as proposed by Zhang (1998).
Using suitable dielectric function models, the complex refractive index of the YBCO film is determined by fitting the measured
transmittance data because the transmittance measurement has a
better spectral resolution, which is especially important in the
region where sharp interference extrema occur. Comparison is also
made between the calculated and measured reflectance whenever
the data are available. Detailed discussions are given below.
Dmdc'
The Lorentz term can be derived by assuming that the electrons
are bound to their nuclei by harmonic forces and are subjected to
damping forces. Lorentz terms are also commonly used to model
infrared active phonons. The phonon contributions, however, can
be neglected compared to the electronic contributions (Choi et al.,
1992). This neglect is particularly valid in the far infrared because
the resonant frequencies of most phonons are in the mid-infrared
region. For HTSC materials, however, there is a broadband midinfrared electronic absorption, which is typically modeled with a
Lorentz term (Tanner and Timusk, 1992). This contribution is
therefore expressed as
(4)
2
(£07,
to„
where <*>,„, a>e, and ye are, respectively, the plasma frequency,
center frequency, and damping constant of the mid-infrared band.
Due to the relatively weak effect of the Lorentz term in the far
infrared, a single oscillator is used with the parameters fixed to
those recommended by Zhang et al. (1994), i.e., a>e = 1800 cm"',
<A = 18000 cm' 1 , and ye = 5400 cm"'. The effect of the
mid-infrared band on the radiative properties of the film is very
weak at frequencies between 10 and 100 cm"', because eLore„te is
essentially a real constant, ^(w^/to,,) 2 in the far infrared. Calculations using different Lorentz parameters, such as we = 1800
cm"1, co„, = 24150 cm"1, ye = 7500 cm"1 (Flik et al., 1992), do
not modify significantly the transmittance and reflectance of the
film-substrate composite.
Below the critical temperature Tc, the free-carrier part of the
dielectric function is described by a two-fluid model. In this model,
only a fraction of electrons (/,) are assumed to be in the condensed
phase (or superconducting state) and the remaining electrons are in
the normal state. The superconducting electrons move without any
scattering, and the value of fs is assumed temperature-dependent.
The contribution of the superconducting electrons to the dielectric
function is
^Lonmlz
(5)
3 Analysis
3.1 Dielectric Function Models for the Superconductor
YBCO. The optical properties of superconducting YBCO
change abruptly from the normal state to the superconducting state.
Hence, different dielectric function models are used in the normal
and superconducting states (Tanner and Timusk, 1992). In the
normal state, the frequency-dependent complex dielectric function
can be modeled as a sum of the Drude term (eDrodc), the Lorentz
term (eUorcntz), and a high-frequency constant (e„ ra 5):
The Drude term remains due to the presence of normal electrons
with a number density of (1 — fs)ne. The dielectric function in the
superconducting state is therefore modeled as
* M =/ I e su „ + (1 ~/,)eDradC + eLorentz + e«
(6)
3.2 Fitting Procedure. The transfer-matrix method, described by Zhang and Flik (1993), is used to compute the transmittance and reflectance of the film-substrate composite shown in
Fig. 1. The Si substrate is assumed to be nonabsorbing and to have
the temperature-dependent refractive index described in Section
e(0>) — e D r a < | e + fiLorenlz "•" £*
0) 2.2. The effective thickness of the Si substrate is determined by
The Drude term describes the electronic behavior in the infrared matching the interference patterns between the calculated and
region by assuming that free electrons (or holes) are accelerated in measured spectra. The angle of incidence has little effect except
the presence of an electric field and that collisions result in a for a slight shift in the frequencies of the interference fringes,
which has already been accounted for using the effective thickness
damping force. Thus, the Drude term can be expressed as
of the Si substrate. Hence, the angle of incidence is assumed to be
normal in all calculations.
(2)
The absorption of dielectric materials, such as YSZ and Ce0 2 ,
(o(a> + i/r)
is very weak in the far-infrared region, especially for thickness less
where w is the angular frequency, a>P is the plasma frequency, and than 20 nm. The dielectric constant of YSZ is 25 and that of Ce0 2
1/T is the electron scattering rate. The plasma frequency is defined is 17 in the far-infrared and microwave regions (Grischkowsky and
as top = nee2/me0, where ne, e, and m are the electron number Keiding, 1990; Phillips, 1996). In all cases, the addition of the
density, charge, and effective mass, and e0 is the electric permit- YSZ and Ce0 2 layers has essentially no effect on the calculated
tivity of free space. The plasma frequency, the scattering rate, and transmittance and reflectance. Therefore, the YSZ and Ce0 2 layers
the DC electrical conductivity crDC are related by
are omitted in the calculations and not discussed further.
The real part nf((o) and imaginary part kf(,a>) of the complex
refractive index of the YBCO film are related to the dielectric
(3) function by
eoi"'
Journal of Heat Transfer
NOVEMBER 1999, Vol. 121 / 847
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Table 1 Fitting parameters, where the expanded uncertainties are estimated to be ten percent in 1/o-DC, 20 percent in 1/T, and 20 percent in fs
Sampl e A (L = 81.5)
Temp.
1/CJDC
ft
Samp! e B {Tc =80.2)
1/T
1/CTDC
(cm"')
(uQ-cm)
ft
Vx
(cm"1)
(K)
(|iQ-cm)
300
800
-
600
700
-
600
200
538
-
400
460
-
440
100
306
-
230
290
-
275
50
-
0.25
150
-
0.30
180
10
-
0.35
190
-
0.40
220
ikt)2 = e(ay)
(J)
where e(w) is calculated from Eq. (1) and Eq. (6) for the normal
and superconducting states, respectively. The refractive index of
YBCO is used to calculate the transmittance and reflectance of the
film-substrate composite.
In the normal state, <xnc and 1/T are taken as adjustable parameters. Their values are determined by fitting the calculated transmittance to the measured transmittance. The best fit is obtained
when the root-mean-square difference is the smallest. In some
cases, the differences in the fringe-averaged transmittance and the
peak transmittance are also used to determine the best fit when the
root-mean-square difference is not so sensitive to the parameters.
Our calculations show that the radiative properties depend strongly
on crDC but weakly on 1/T, suggesting a larger uncertainty in 1/T.
The fitted values of o-DC and 1/T at room temperature are used to
compute the plasma frequency from Eq. (3). The plasma frequency
is proportional to the total electron density and is typically constant
over temperature (Kamaras et al., 1990; Tanner and Timusk,
1992). Hence, at other temperatures in the normal state, the scattering rate is the only adjustable parameter. Both the
superconducting-electron fraction/, and the scattering rate 1/T are
considered to be adjustable parameters in computing the dielectric
function of YBCO in the superconducting state.
for a single-crystal YBa 2 Cu 3 0 7 of extremely high quality with
fs <= 1 at very low temperatures.
Even at very low temperatures, there is a large fraction (1 — /,)
of normal-state electrons in the thin YBCO films. For conventional
superconductors, / , should be zero at temperatures much lower
than Tc. The definition of/, for HTSC materials is rather ambiguous. Some researchers obtained higher values of/, by assuming
that a portion of the YBCO film is not superconductive, also
known as "dead layer" (Renk, 1992; Hadni et al, 1995). In the
present study, the superconducting fraction is calculated without
assuming any dead layer, and this may account for the small values
of/, obtained in this study. Another reason may be the higher DC
resistivity (l/(rDC) of our YBCO films in the normal state. The DC
resistivity is strongly affected by the choice of the substrate and
buffer layers, by their thicknesses, as well as by the thickness of
the YBCO film (Mechin et al, 1996). The damage caused by the
stripping process as mentioned in Section 2.1 may have reduced/,
as well. Although the critical temperature of Sample A is slightly
higher than that of sample B, the fraction of superconducting
electrons of Sample A is less than that of sample B. This may be
explained by the fact that Sample A has a higher electrical resistivity and scattering rate.
The real and imaginary parts of the refractive index of YBCO
calculated from Eq. (7) using the fitting parameters for Sample A
are shown in Fig. 5. Both nf and ks increase as the wave number
decreases at all temperatures. In the normal state, nf «* kf, as
(a)
100 K
10
40
50
60
Frequency (cm-1)
Fitted
70
80
50 K
r
4 Results
848 / Vol. 121, NOVEMBER 1999
30
"-300K
0.4
(b)
.
The fitting parameters for both specimens are shown in Table 1
(Kumar, 1999). The difference in the fitted values may be caused
by the slight variation in the film deposition conditions. Figure 4
compares the fitted and measured transmittance of Sample A. The
fitted and measured values agree well with a root-mean-square
difference of less than 0.03 in most cases. At temperatures greater
than Tc, both the scattering rate and the resistivity drop almost
linearly with decreasing temperature. At these temperatures, the
scattering rate is dominated by the temperature-dependent electronic scattering, which governs the linear dependence of resistivity on the temperature. At low temperatures, below Tc, the
temperature-dependent scattering ceases to exist, and the remaining scattering is mainly due to impurities and lattice defects, which
should be temperature-independent (Flik et al., 1992). This scattering rate should be constant at low temperatures. However, the
best agreement between the measured and calculated transmittance
at 10 K was achieved with a scattering rate higher than that for 50
K. This unexpected higher scattering rate at lower temperatures
needs further investigation, although it could simply be caused by
the experimental uncertainty. Bonn et al. (1992) showed that the
scattering rate of YBCO decreases drastically as the temperature is
reduced below Tc, due to the rapid reduction of the density of
thermally activated quasi-particles. Their results, however, were
20
*-200K
i-
•
:
0.1
10
(C)
Measured
— Fitted
\ J\J\J \J VI \J \J \J\J
20
30
40
50
60
Frequency (cm'')
70
80
V"7
90
0.4
10 K
0.3 •
Measured
— Fitted
40
50
60
Frequency (cm'1)
Fig. 4 Comparison between the measured and fitted transmittance for
Sample A: (a) normal state; (b) 50 K; (c) 10 K
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phase change in the film. Note that iV, = 1, N2 = Nf, and N3 =
nsi correspond to the refractive indices of the air, film, and the
substrate, respectively, and c 0 is speed of light in vacuum. If 12/8/1
<^ 1, which is satisfied for the thin YBCO film in the normal state,
then exp(2('8/) «* 1 + 2i8f. The requirement of 12(8/1 <S 1 is
equivalent to (1) df < A/4TT&/ (the film thickness is much smaller
than the radiation penetration depth), where A is the wavelength in
vacuum, and (2) df <^ A/4irn/. After some manipulation, Eq. (8)
becomes
100
0
10
20
30
40
50 60
Frequency (cm'1)
70
80
90
N3 - iV, + iSfN2{l + N3/N2)(l
"N3 + Ni - i8fN2(\ - N3/N2){\
100
-
Nl/Nl)
N^/Nij
(9)
Assume |Arr1/A^2| *^ 1 and \N3/N2\ < 1, which are also valid since
the refractive index and extinction coefficient of metallic films
(such as YBCO in the normal state) are generally much greater
than the refractive index of dielectric materials (such as the Si
substrate used here). Then,
N3- AT, + iSfN2
' N3 + N] - idfN2
0
10
20
30
40
50
60
70
80
90
100
Frequency (cm'1)
Fig. 5 Complex refractive index of the YBCO film (Sample A)
expected for a metallic film, and both ns and kf increase with
decreasing temperature. As the film becomes superconducting, nf
drops suddenly but k{ exhibits an abrupt increase, especially at low
frequencies. As the temperature is further reduced from 50 K to 10
K, nf continues to decrease whereas kf continues to increase.
Below Tc, kf > nf; because the film is largely inductive in its
optical response. The refractive index of Sample B has a similar
trend (Kumar, 1999), but now in the superconducting state the
imaginary part is even greater and the real part is even smaller due
to a slightly larger fraction of superconducting electrons (see Table
1). The expanded uncertainty in the real and imaginary parts of the
refractive index is estimated to be ten percent.
Figure 6 compares the measured with the calculated reflectance
at 50 K and 10 K for radiation incident on the substrate. The
measured and calculated reflectance is in reasonable agreement. At
the reflectance minima, the calculated values are much lower than
those measured, which may be caused by insufficient resolution in
the measured reflectance or by the beam divergence. Further
improvements in radiometric accuracy and spectral resolution are
required in order to resolve the discrepancy between the measured
and calculated reflectance.
5
(10)
Equation (10) is the same as that derived by McKnight et al.
(1987), but here it is obtained directly from Maxwell's equations
without assuming that the film is infinitely thin. It is convenient to
define a dimensionless admittance, v = —i8f(Nf), which in
general is a complex quantity. As discussed above, the dielectric
function of YBCO in the normal state is dominated by the Drude
term, which becomes purely imaginary at frequencies much
smaller than the scattering rate (WT < 1); hence, e(a>) «* i<jocla>e0.
From the definitions of y and 8/, y «* crDCdf/coe0 is a real positive
constant in the very far infrared. When y = nsi — 1, the reflection
coefficient given by Eq. (10) becomes zero for incidence on the
substrate. For N3 = n si °= 3.4, the required DC resistivity (1/CTDC)
is «*550 jail-cm for df = 35 nm. The DC resistivity obtained by
fitting the transmittance at 200 K of 538 ^Cl-cm is very close to
0.8
0.6
| 0.4
0.2
0
10
20
30
40
50
60
70
80
90 100
Frequency (cm"1)
1
Discussion
5.1 The Antireflection Effect. For Sample A, measurements reveal that the minima in the transmittance at 300 K approximately correspond to the maxima at 100 K, 50 K, and 10 K
(see Fig. 2(a)). The measured transmittance at 200 K has no
discernible interference fringes. This interesting change in the
interference pattern is analyzed below by considering the radiation
incident from the substrate and reflected by the film. The reflection
coefficient at the substrate-film interface is the ratio of the reflected
to the incident electric waves (Heavens, 1965):
-r23 + r 2 ,e
1 -
0.6
0.4
0.2
128/
r23r2le'
(8)
where, for normal incidence, r2i = (N2 — NI)/(N2 + N,) and
fn = (N2 — N3)/(N2 + N3), and Sf = wdfN2/c0 is the complex
Journal of Heat Transfer
Measured
Calculated
0.8
0
10
20
30
40
50
60
70
80
90
100
Frequency (cm' 1 )
Fig. 6 Measured and calculated backside reflectance for Sample A at 50
K and 10 K
NOVEMBER 1999, Vol. 121 / 849
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this value. However, because the scattering rate is only 400 cm -1
at 200 K (see Table 1), there are residual interference fringes.
Because of all the assumptions made earlier, interference fringes
persist in the calculated transmittance even with l/ffDC ^ 550
IxCl-cm. McKnight et al. (1987) analyzed and experimentally
demonstrated the effect of antireflection coating (also called
impedance-matched coating) on the transmittance of a Si wafer
coated with a Ni-Cr film. In that work, the scattering rate was
much greater than 100 cm"1, and the fringe contrast in the measured transmittance was much smaller.
The change from transmittance maxima at 300 K to transmittance minima at 100 K at fixed wave number can be explained by
the sign change of the reflection coefficient given by Eq. (10) as
the DC resistivity passes ^550 ;uXl-cm. The change in the sign of
this coefficient, combined with the interference effect inside the Si
substrate, will result in a phase shift of IT rad in the interference
pattern and thus causing the transmittance to change from maxima
to minima and vise versa. The calculated transmittance and reflectance agree extremely well in terms of the interference patterns at
all temperatures except for Sample A at 200 K. The actual phase
at 200 K is very complicated because the assumptions used in
deriving Eq. (10) are not perfectly met. The discrepancy in the
phase between the calculated and the measured transmittance at
200 K suggests that the scattering rate is complex and may depend
on the wave number. Modified Drude models with a complex and
frequency-dependent scattering rate have been used for the study
of HTSC materials (Varma et al., 1989; Virosztek and Ruvalds
1990; Renk, 1992; Quinlan et al., 1996). Further studies are needed
to investigate the applicability of these models to thin YBCO films
on Si substrates.
The fringe patterns of Sample B differs from those of Sample A,
because their characteristics are different. From Fig. 2(b) and
Table 1, we can estimate that the interference contrast for Sample
B would be the smallest at a temperature somewhere between 300
K and 200 K, where its DC electric resistivity is near 550 /ull-cm.
The condition for impedance matching with the Si substrate is that
(<TDCdf)~l «* 157 CI. Hence, it is possible to construct antireflection coatings with a YBCO film on Si substrate at room temperature by increasing the film thickness. The practical applications of
the antireflection effect using HTSC films need further exploration.
5.2 The Effect of Optical Resonance. The measured spectra for both specimens at 10 K and 50 K have sharper transmittance
maxima and reflectance minima than the spectra obtained above
the critical temperature (see Figs. 2 and 3). In addition, the peak
transmittance at 10 K is even higher than that at 50 K. The
transmittance is expected to decrease as the temperature is lowered
since the YBCO material becomes more and more conductive. The
unexpected higher transmittance at lower temperatures in certain
spectral bands is the result of optical resonance in the filmsubstrate composite. A simple optical resonator is a dielectric layer
coated with highly reflecting films on both sides. By varying the
reflection coefficient of one or both coatings, the transmittance of
the resonator can be altered significantly. This resonance effect
will result in very high transmittance values at particular frequencies (Klein and Furtak, 1986). In the present study, the specimen
is analogous to an optical resonator made of a thin YBCO film
coated on only one side of the Si substrate. As the temperature is
reduced to below the critical temperature, the reflection coefficient
of the YBCO film increases sharply. The large reflection coefficient, coupled with the interference effects in the substrate, can
significantly increase the measured transmittance at certain frequencies. The observed resonance effects demonstrate that YBCO
films deposited on Si substrates have the potential for construction
of Fabry-Perot resonators (Renk et al., 1990; Genzel et al, 1992;
Malone et al., 1993). Further studies are needed to construct
resonance structures using two identical films facing each other to
optimize the performance of such resonators. The calculated minimum transmittance is slightly lower at 10 K than at 50 K. This is
consistent with the measured values for Sample B. The measured
850 / Vol. 121, NOVEMBER 1999
0
10
20
30
40
50
60
70
80
90 100
1
Frequency (cm" )
Fig. 7
Calculated film side reflectance for Sample A
minimum transmittance for Sample A is nearly the same at 10 K
and 50 K at large frequencies. However, the expected difference is
smaller than the experimental uncertainty.
5.3 The Film Side Reflectance. The film side reflectance
for Sample A is calculated using the fitted parameters and is shown
in Fig. 7. In the normal state, the peak locations of the interference
fringes remain at the same frequency. The film side reflectance
shows a higher fringe contrast in the normal state and a lower
fringe contrast in the superconducting state when compared to the
substrate side reflectance. The reflectance measurements on several sample films for the film side incidence agree well with the
above-mentioned pattern (Kumar, 1999).
6 Conclusions
We have measured the transmittance and reflectance of thin
YBCO films deposited on transparent Si substrates in the farinfrared region at temperatures from 10 K to 300 K. The measurements show fringes caused by interference effects within the Si
substrate. The change in the DC conductivity of the YBCO film
with temperature causes a change in the reflection coefficient at the
substrate-film interface, which results in a variation of the interference pattern. Matching of the admittance of the YBCO film with
the refractive index of the Si substrate will yield a zero reflection
coefficient at the substrate-film interface and thus eliminate the
fringes in the measured transmittance. A change in the sign of the
reflection coefficient results in a phase change of TT rad in the
interference pattern.
Sharp increases in the transmittance maxima in the superconducting state indicate that optical resonance has occurred within
the film-substrate composite. The reflectance for radiation incident
on the substrate increases significantly upon switching from the
normal state to the superconducting state. These phenomena demonstrate that the HTSC thin films on transparent substrates can be
used to build far-infrared devices, such as optical resonators and
radiation modulators.
The transmittance and reflectance are calculated using the Drude
model in the normal state and the two-fluid model in the superconducting state. The predicted values agree closely with the
measured values using only a few adjustable parameters. The
complex refractive index of the YBCO films is thus obtained at
different temperatures. The imaginary part increases steeply as the
superconducting state is reached but the real part shows a sudden
decrease. The refractive index of YBCO films obtained from this
study is important for the design of promising devices using HTSC
thin films. Future research is needed to improve the quality of the
YBCO films on Si substrates as well as the reproducibility of the
deposition process. Methods for improving the radiometric accuracy in the spectrometric measurements with sharp interference
fringes need to be further investigated.
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Acknowledgments
This research was partially supported by the University of
Florida through an Interdisciplinary Research Initiative (IRI)
award, and the National Science Foundation through grants DMR9705108 and CTS-9812027.
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