Infrared response of vitreous titanium dioxide ﬁlms with anatase
short-range order
G. Scarel
Advanced Coatings Experimental Laboratory and Materials Department, University of WisconsinMilwaukee,
P.O. Box 784, Milwaukee, Wisconsin 53201
C. J. Hirschmugl
Department of Physics, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201
V. V. Yakovlev
Department of Physics, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201
R. S. Sorbello
Department of Physics, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201
C. R. Aitaa)
Advanced Coatings Experimental Laboratory and Materials Department, University of WisconsinMilwaukee,
P.O. Box 784, Milwaukee, Wisconsin 53201
H. Tanaka and K. Hisano
Department of Applied Physics, The National Defense Academy, Yokosuka 239, Japan
͑Received 12 July 2001; accepted for publication 20 October 2001͒
The polarization-dependent optical response of vitreous titanium dioxide ﬁlms was determined by
infrared reﬂection–absorption spectroscopy ͑IRAS͒. The predominant nearest-neighbor
coordination ͑short-range order͒ in the ﬁlms was characteristic of crystalline anatase. A dielectric
function ⑀͑␯͒ was constructed, based on random orientational disorder of anataselike, octahedral
TiO6 units relative to the substrate. This dielectric function, which is the directional average of the
dielectric function for an anatase crystal in the EЌc and Eʈc orientations, was used to calculate
energy loss functions from which theoretical IRAS spectra were obtained. Experimental absorption
band frequencies are in good agreement with peaks in the calculated transverse optic loss function,
Im͓⑀͑␯͔͒, at 261 and 436 cmϪ1
, and in the calculated longitudinal optic loss function, Im͓Ϫ1/⑀͑␯͔͒,
at 840 cmϪ1
. Agreement ͑i.e., polarization-dependent behavior, band frequency, and relative
intensity͒ between the experimental and theoretical IRAS spectra indicates that orientational
disorder of TiO6 units is the important factor governing infrared reﬂection–absorption behavior of
the ﬁlms. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1427430͔
I. INTRODUCTION
The behavior of titanium dioxide in response to optical
photons has made this material technologically important
and therefore the subject of continued study. Optical properties
of TiO2 include a wide electron energy band gap, transparency
throughout the visible spectrum, and a high refractive
index over a wide spectral range from the ultraviolet to
the far infrared. These properties led to the use of TiO2 for
diverse thin ﬁlm optical and electro-optic device
applications.1–6
Bulk TiO2 at atmospheric pressure and room temperature
can exist in three crystalline polymorphs.7–9
Rutile is the
thermodynamically stable phase at all temperatures, while
anatase and brookite form as metastable phases below
ϳ800 °C. Titanium is in octahedral coordination with O as
TiO6 units in all polymorphs. However, different Ti–O bond
length and angle, and consequently, different arrangements
of TiO6 octahedra, generate different lattice structures. The
rutile and anatase lattices are tetragonal with P42 /mnm and
I41 /amd space groups, respectively, and brookite is orthorhombic
with Pbca symmetry.10
In addition, a series of
TinO2nϪ1(4рnр10) daughter-structures form by shear operations
on a rutile mother-structure to accommodate
nonstoichiometry.11
Some bulk oxides have structural complexity, meaning
͑a͒ bond ﬂexibility giving rise to polymorphs with the same
chemistry and similar free energy of formation, and/or ͑b͒
the formation of mixed valence compounds with vernier,
block, or inﬁnitely adaptive physical structures to accommodate
changes in stoichiometry.12
A thin ﬁlm grown at room
temperature of such an oxide is likely to have an atomic
structure in which long-range crystallographic order ͑LRO͒
is lacking, i.e., the ﬁlm is noncrystalline. The precise atomic
arrangement within a noncrystalline oxide ﬁlm has long been
the subject of discussion, and depends somewhat upon thea͒
Electronic mail: aita@uwm.edu
JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 3 1 FEBRUARY 2002
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type of structural complexity of the bulk oxide. Following
Felner,12
we classify noncrystalline oxides in terms of increasing
nearest-neighbor atomic order as amorphous or vitreous.
An amorphous oxide ﬁlm has no short-range order
͑SRO͒, i.e., atomic coordination and chemistry at the
nearest-neighbor level differ throughout the ﬁlm. A vitreous
oxide ﬁlm has SRO, i.e., the same atomic coordination and
chemistry at the nearest-neighbor level throughout the ﬁlm,
and furthermore, may contain ordered domains extending beyond
nearest neighbors to the subnanometer or even nanometer
level. In other words, a vitreous oxide ﬁlm may contain
nanocrystallites, but has no LRO.
Titanium dioxide meets both criteria for structural complexity,
and it is not surprising that ﬁlms grown at low temperature
by diverse processes contain noncrystalline material,
either alone or coexisting with crystalline anatase and/or
rutile.13–29
Furthermore, Eastman30
reported a noncrystalline
structure coexistent with crystalline material in freestanding
nanostructured TiO2 , and astutely pointed out that this phase
is likely to be overlooked because its detection is extremely
difﬁcult.
We previously29
examined absorption of near ultraviolet
photons in noncrystalline TiO2 ﬁlms. A nonresonant Raman
spectroscopy study31
showed that these ﬁlms were vitreous
͑as distinguished from amorphous, in keeping with the aforementioned
deﬁnition͒ and had SRO characteristic of crystalline
anatase. The fundamental optical absorption edge was
effectively modeled within the framework of the coherent
potential approximation with Gaussian site disorder introduced
into the valence and conduction bands of a perfect
virtual anatase crystal. The ﬁlms were then annealed to produce
various amounts of crystalline anatase, hence LRO. The
results reafﬁrmed a general rule by Tauc:32
interband optical
transitions described by wave functions localized over distances
on the order of a lattice constant are relatively unchanged
by disorder. The transfer of electronic charge within
TiO6 octahedral units ͑i.e., between nearest-neighbor Ti and
O atoms͒ was found to be the key factor in determining the
material’s response to ultraviolet radiation, irrespective of
the degree of LRO.
However, a solid’s response to infrared photons is governed
by the relative motion of atoms, and indeed may be
different for crystalline and vitreous TiO2 .33
The goals of the
present paper are, therefore: ͑a͒ to determine the infrared
reﬂection–absorption behavior of vitreous TiO2 with anatase
SRO, ͑b͒ to model this behavior by choosing an appropriate
dielectric function, and ͑c͒ to identify the type of structural
disorder that is commensurate with the experimental and theoretical
modeling results.
Infrared reﬂection–absorption spectroscopy ͑IRAS͒ of
solids was historically33
used to measure vibrational modes.
In a crystal, long wavelength vibrational modes are associated
with atomic motion perpendicular and parallel to the
direction of wave propagation. The two types of modes are
denoted transverse optic ͑TO͒ and longitudinal optic ͑LO͒.
For thin crystalline ﬁlms ͑thicknessӶinfrared wavelength͒,
TO modes are detected as absorption bands in transmission
and reﬂection spectra in both s and p polarization, whereas
LO modes are detected only in p polarization.34
Polarizationdependent
bands are also detected in thin noncrystalline
ﬁlms, and have been attributed to TO-like or LO-like
modes.35–41
In reality, however, the vibrational modes in a noncrystalline
solid do not cleanly separate into atomic motion perpendicular
and parallel to the propagation direction. Rather,
the atomic displacements generally form a complicated pattern
varying markedly in magnitude and direction throughout
the medium. Nonetheless, a noncrystalline solid can be described
by a dielectric function, ⑀͑␯͒, that contains all the
relevant microscopic information as far as the response to the
incident electromagnetic wave is concerned. Such information
includes the resonant frequencies of the individual vibrating
units, their associated lifetimes, and their oscillator
strengths ͑or effective charges͒. The dielectric function is
also affected by coupling between units through the longrange
Coulomb interaction as well as through short-range
chemical bonding.
Thus, by modeling the dielectric function and comparing
the results of theoretical IRAS spectra with experimental
IRAS spectra, one can, in principle, deduce the value of the
microscopic parameters of a noncrystalline solid. Furthermore,
structural information concerning the shape, orientation,
and distribution of the individual vibrating units can
also be obtained.42
Alternatively, if one is interested only in
the electromagnetic response of macroscopic samples, one
could forgo the microscopic information entirely and use the
experimental IRAS spectra to deduce ⑀͑␯͒. Once ⑀͑␯͒ is determined,
the electromagnetic response of any macroscopic
sample of the noncrystalline solid can be found.
We report the optical response of sputter-deposited vitreous
TiO2 ﬁlms with anatase SRO. Experimental IRAS
spectra were obtained in the 100–1000 cmϪ1
wave number
range. Our data were found to differ from those for single
crystal anatase.43
We therefore constructed a dielectric function
to account for random orientational disorder within the
ﬁlms. Calculated energy loss functions and theoretical
reﬂection–absorption spectra using this dielectric function
were compared with experimental results. Good qualitative
agreement between experimental and theoretical IRAS spectra
indicates that orientational disorder is the important factor
governing the reﬂection–absorption behavior of our vitreous
ﬁlms.
II. EXPERIMENTAL PROCEDURE
Titanium dioxide ﬁlms ͑0.25-␮m-thick ﬁlm A and 0.70␮m-thick-ﬁlm
B͒, were grown by sputter deposition from a
99.995% Ti target in a radio-frequency-excited O2 plasma as
described in Ref. 29. Films were grown on substrates cut
from a 99.0% pure rolled Al sheet.
Double-angle x-ray diffraction ͑XRD͒ was used to identify
the small amount of nanocrystalline material in each
ﬁlm. Diffraction patterns were obtained using unresolved
0.15418-nm-wavelength Cu K␣ radiation. Peak position
(2␪hkl), maximum intensity (Imax), and full width at one-half
maximum intensity ͑FWHM͒ were measured from high resolution
scans of individual peaks. Phase identiﬁcation, preferred
orientation, and phase composition were determined
1119J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
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from these data. The Debye–Scherrer relationship44
was
used to estimate the crystallite dimension parallel to the
growth direction, assuming the absence of peak broadening
due to random lattice strain.
Infrared reﬂection–absorption spectroscopy data in the
far-infrared spectra range were obtained with a Fourier transform
infrared spectrometer by Bruker ͑Model IFS 113v͒
equipped with a polyethylene-supported polarizer. For each
spectrum, 100 scans were taken at 4 cmϪ1
resolution and at
an angle of incidence ␾ϭ45°. ͑An oblique incident beam
was required to observe LO-like features.͒ Measurements
were made at a pressure of 1ϫ10Ϫ3
Torr and room temperature.
Middle-IRAS data was obtained with a Bruker V22
Fourier transform infrared spectrometer equipped with a
ZnSe-supported polarizer. For each spectrum, 300 scans
were taken at a resolution of 4 cmϪ1
and at an angle of
incidence ␾ϭ45°. Data were taken at ambient conditions. A
bare Al substrate was used as reference for both far- and
mid-IRAS measurements. Theoretical IRAS spectra were
calculated from the Fresnel equations34,45
using MATH-
EMATICA.
III. FILM STRUCTURE
The XRD spectra ͑not shown͒ of 0.25-␮m-thick ﬁlm A
and 0.70-␮m-thick ﬁlm B had the following common features:
͑a͒ Reﬂections from both rutile and anatase phases
were present. ͑b͒ The dominant anatase reﬂection was from
͑101͒ planes at 2␪ϭ25.4°, with additional minor reﬂections
from ͑004͒ planes at 2␪Ӎ38° and ͑200͒ planes at 2␪Ӎ48°.46
A comparison of peak intensities with that for a randomly
oriented powder sample46
showed that anatase preferably
crystallized with ͑101͒ planes parallel to the substrate. ͑c͒
Solely rutile-͑110͒ reﬂections were present, at 2␪ϭ27.5°,47
i.e., rutile crystallized with ͑110͒ planes parallel to the substrate.
͑d͒ All peaks were broad, suggesting nanocrystallinity.
͑e͒ All peaks had low intensity, indicating that a TiO2 structure
not detectable by XRD coexists along with the nanocrystalline
phases.
The maximum intensity and FWHM for anatase-͑101͒
and rutile-͑110͒ peaks are recorded in Table I. For reference,
the maximum intensities in Table I should be compared to a
fully crystallized, single ͑110͒ orientation, 0.25-␮m-thick
rutile ﬁlm with a ﬁrst order diffraction peak intensity of 2300
cps and a FWHM equal to that for instrumental broadening
effects. The average minimum crystallite dimension ͑D͒ parallel
to the growth direction anatase-͑101͒ and rutile-͑110͒
are included in Table I.
The detectable rutile volume fraction in each ﬁlm was
determined using the external standard method.44
The formalism
developed for the application of this method to disordered
TiO2 ﬁlms is given in Ref. 29. In summary, the
volume fraction of the rutile component of a uniform multiphase
mixture, fr , is related to the integrated intensity of a
speciﬁc rutile reﬂection, Ihkl
(r)
, as
Ihkl
͑r͒
Ihkl
͑s͒ ϭ
fr
fs
, ͑1͒
where Ihkl
(s)
is the XRD intensity of a rutile standard that has
the same preferred orientation as the ﬁlm to be analyzed, and
fs ͑standard͒ is the volume fraction of rutile in the standard
and is equal to unity by deﬁnition. In order to apply Eq. ͑1͒
to our ﬁlms, we used a previously fabricated, fully crystallized
rutile standard ﬁlm:29
an annealed 0.25-␮m-thick TiO2
ﬁlm on fused silica whose XRD pattern showed two orders
of a single rutile-͑110͒ peak. For ﬁlm A, Eq. ͑1͒ becomes:
fr(ﬁlm A)ϭIr110(ﬁlm A)/Ir110(standard). A similar procedure
was followed to calculate the rutile volume fraction in
ﬁlm B, with the added stipulation that the thickness difference
between ﬁlm B and ﬁlm A must be taken into account.
A previous calculation29
lead to the relationship: fr(ﬁlm B)
ϭ0.43͓Ir110(ﬁlm B)/Ir110(standard)͔. The volume fraction
of rutile in each ﬁlm is recorded in Table II.
Since both ﬁlms contain a large component that cannot
be detected by XRD, a straightforward calculation using fa
ϭ1Ϫfr , where fa is the anatase volume fraction, is not
valid. Furthermore, precise calculation of fa using the external
standard method, analogous to the above calculation for
fr , requires a completely crystallized anatase external standard
with the same preferred orientation as the ﬁlms. Preparation
of an anatase ﬁlm standard is not possible by annealing
because the transformation of crystalline anatase to rutile
occurs in these ﬁlms before complete crystallization of the
noncrystalline component.21,22
However, the standardless direct
comparison method of Averbach and Cohen48,49
will
yield a reasonable estimate of fa . In this case, the anatase͑101͒
peak is compared to the rutile-͑110͒ peak in both ﬁlms:
Ia(101) /Ir(110)ϷXa fa /Xr fr . X for each component is the
product of ﬁve factors: ͑a͒ the inverse of the square of the
unit cell volume, ͑b͒ the square of the absolute value of the
structure factor, ͑c͒ the multiplicity of a particular set of diffracting
planes, ͑d͒ the Lorentz polarization factor, and ͑e͒
the Debye–Waller temperature factor. Calculation44
of these
factors leads to the relationship Ia(101) /Ir(110)Ϸ0.65fa /fr .
Substitution of the relative anatase-͑101͒ and rutile-͑110͒ integrated
intensities into the above expression yields an estimate
of the volume fraction of detectable nanocrystalline
anatase, recorded in Table II, neglecting the small contribution
of nanocrystalline anatase with other than a ͑101͒ orienTABLE
I. XRD parameters and crystallite size in titanium dioxide ﬁlms on
aluminum.
Film/Peak Imax (cps) FWHM ͑deg͒ D ͑nm͒
A/r ͑110͒ 60 0.65 24
A/a ͑101͒ 107 0.65 24
B/r ͑110͒ 287 0.50 17
B/a ͑101͒ 135 0.50 17
TABLE II. Phase composition of titanium dioxide ﬁlms on aluminum.
Film fr(ϫ102
) fa(ϫ102
) f␣(ϫ102
)
A 3.7 3.2 93.1
B 2.0 6.5 91.5
1120 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
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tation. Also recorded is the total volume fraction of material
that cannot be detected by XRD, calculated from f␣ϭ1
Ϫ(faϩfr).
Nonresonant Raman spectra ͑see Ref. 31͒ of ﬁlms on Si
grown under the same conditions as ﬁlms A and B showed
four absorption peaks attributable to Raman-active vibrational
modes of anatase.50
The assignment and wave numbers
of these vibrations are: ͑1͒ unresolved B1g at 144 cmϪ1
and Eg at 147 cmϪ1
, ͑2͒ B1g and A1g at 198 cmϪ1
, ͑3͒ B1g at
398 cmϪ1
, and ͑4͒ Eg at 640 cmϪ1
. These peaks account for
ﬁve of the six Raman-active vibrational modes of anatase. A
sixth mode, Eg at 515 cmϪ1
, was masked in the ﬁlms by the
strong LO phonon mode of the Si substrate at 520 cmϪ1
.
Markedly, the three characteristic rutile modes, Eg at 448
cmϪ1
, A1g at 612 cmϪ1
, and B1g at 827 cmϪ1
, were absent
from spectra of ﬁlms A and B.
The XRD data given in Table II show that two nanocrystalline
phases, anatase and rutile, are present in both ﬁlms A
and B, but these phases account for a small amount of the
ﬁlms’ volume. Of the detectable nanocrystalline material,
less than 4% in ﬁlm A and 2% in ﬁlm B is crystalline rutile.
Raman spectroscopy of companion ﬁlms on Si detects only
anatase Ti–O short-range order. From these results, we conclude
that greater than 96% of ﬁlm A and 98% of ﬁlm B is
comprised of material that has anatase short-range order.
IV. EXPERIMENTAL IRAS SPECTRA
The experimental IRAS spectra for ﬁlms A and B are
shown in Figs. 1 and 2, respectively. In these ﬁgures, the
quantity RﬁlmϩAl /RAl is graphed as a function of wave number,
␯/c, from 102
to 103
cmϪ1
, where RﬁlmϩAl is the reﬂectivity
of the sample obtained using p or s polarization, and
RAl is the reﬂectivity of the bare Al substrate. The insert
shows an expanded region from 225–475 cmϪ1
to highlight
small features in that spectral range. ͓For the sake of comparison,
theoretical spectra are included in Figs. 1 and 2
͑broken curves͒, but discussion of these curves is postponed
until Sec. V.͔ The spectral position of all features is recorded
in Table III.
With respect to ﬁlm A, Fig. 1 shows that absorption
bands are present at 260, 355, and 442 cmϪ1
when an
s-polarized beam is used. Absorption bands at 260, 443, and
853 cmϪ1
͑with a shoulder at 962 cmϪ1
͒ occur for p polarization.
In addition, weak features occur in Rp between 300–
400 cmϪ1
.
With respect to ﬁlm B, Fig. 2 shows absorption bands at
259 and 439 cmϪ1
when an s-polarized beam is used. In
addition, extremely weak features centered at ϳ310 and
ϳ345 cmϪ1
are present in Rs . Absorption bands at 258, 438,
and 849 cmϪ1
͑with shoulders at 785 and 962 cmϪ1
͒ are
observed for p polarization. In addition, the features at ϳ310
and ϳ345 cmϪ1
in Rs broaden into a single, extremely weak,
featureless band extending from 300–350 cmϪ1
in Rp .
Comparing the spectra of ﬁlms A and B, three absorption
bands can be identiﬁed as having the same frequency within
experimental error. Two absorption bands common to both
polarizations of both ﬁlms have average frequencies of 259
cmϪ1
and 441 cmϪ1
. Furthermore, a band with an average
frequency of 851 cmϪ1
occurs in the p-polarization spectra
of both ﬁlms and is absent from their s-polarization spectra.
The above results should be compared to those of Trasferetti
et al.,51
who used IRAS over a spectral range of 400–
1400 cmϪ1
to study plasma-enhanced chemical vapor deposited
noncrystalline TiO2 on Al. The SRO was not measured
by an independent method and was unspeciﬁed. Only one
peak was reported, a single broad band that appeared in Rp ,
but not in Rs , centered at ϳ850 cmϪ1
for an incident beam
angle of 50°. This frequency is in excellent agreement with
the highest frequency absorption band in our ﬁlms. Trasferetti
et al. discussed this band’s frequency as being in disagreement
with the prominent LO mode of rutile at 806
cmϪ1
,52
but we suggest, on the basis of agreement with our
spectra, that the SRO order of their ﬁlms was most likely
anatase.
V. THEORETICAL IRAS SPECTRA
Gonzalez et al.43
recently published experimentally determined
TO and LO phonon frequencies of single-crystal
anatase in which the applied electric ﬁeld, E, was either parallel
or perpendicular to the c crystallographic vector. Those
researchers found that Eʈc gave rise to one pair of A2u
modes: a transverse optical phonon mode at 367 cmϪ1
and a
longitudinal optical phonon mode at 755 cmϪ1
and EЌc gave
FIG. 1. Solid curves show experimental IRAS spectra and broken curves
show theoretical IRAS spectra for 0.25-␮m-thick ﬁlm A. RﬁlmϩAl /RAl is
graphed as a function of wave number, ␯/c. The insert shows an expanded
region from 225–475 cmϪ1
to highlight small features in that spectral range.
1121J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
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rise to two pairs of Eu modes: a low frequency pair at 262
cmϪ1
͑TO͒–366 cmϪ1
͑LO͒ and a high frequency pair at 435
cmϪ1
͑TO͒–876 cmϪ1
͑LO͒. However, a comparison of
single-crystal data with the experimental IRAS data ͑Table
III͒ shows that only the two Eu ͑TO͒ modes correlate with
bands in both ﬁlms A and B.
We suggest that the primary cause of the disparity between
our experimental results and those for single-crystal
anatase is that the Ti–O units in the vitreous ﬁlm are not
solely in either the EЌc or Eʈc orientation. Consequently, the
electromagnetic response of the ﬁlm is described not by either
of the dielectric functions ⑀Ќ(␯) or ⑀ʈ(␯) alone. In the
following text, we construct a suitable dielectric function,
⑀͑␯͒, that is the directional average of ⑀Ќ(␯) and ⑀ʈ(␯). We
then show that the theoretical spectra calculated using ⑀͑␯͒
do, in fact, differ from the reported spectra for single-crystal
anatase.
In order to calculate theoretical IRAS spectra, we must
determine the reﬂection coefﬁcient for infrared radiation incident
from air onto a dielectric ﬁlm that lies on a metal
substrate, i.e., the standard electromagnetic-wave-reﬂection
problem for a ‘‘three-layer model.’’ Its explicit solution, embodied
in the Fresnel equations, can be found, for example,
in the work of Berreman34
and Hansen.45
The dielectric function
of the ﬁlm and the metal, denoted ⑀͑␯͒ and ⑀m(␯), respectively,
are required to apply this solution. ͓The corresponding
indices of refraction are ⑀(␯)1/2
and ⑀m(␯)1/2
.͔
In constructing our model, we assume that each individual
Ti–O unit, which may be as small as a single TiO6
molecule, in the vitreous structure has the same polarizability
and the same dielectric function that the unit would have in
an anatase crystal environment. We are thus considering orientational
disorder of Ti–O units, but ignoring chemical effects,
local strains, and local defects that might cause the
polarizability parameters of an individual Ti–O unit to be
different in the vitreous solid compared to a perfect crystal.
͑Such parameters include the frequencies, oscillator
strengths, and lifetimes of the local vibrational modes.͒ Evidence
that this type of orientational disorder is the dominant
contribution to line broadening in the spectra of noncrystalline
ﬁlms has been given by Ovchinnikov and Wight in their
studies of N2O, CO2 , and O3 ﬁlms.42
In keeping with the assumption of randomly oriented
Ti–O units described by the dielectric functions appropriate
to the crystal, we construct a dielectric function that is the
directional average of the dielectric functions for a crystal in
the EЌc and Eʈc orientations:
⑀͑␯͒ϭ͓2⑀Ќ͑␯͒ϩ⑀ʈ͑␯͔͒/3, ͑4͒
where ⑀Ќ(␯) and ⑀ʈ(␯) are obtained from the factorized form
of the dielectric function for the EЌc and Eʈc orientations,
respectively, using the resonant frequencies and damping
factors for phonon modes given in Ref. 43. The same prescription
for constructing an effective dielectric function was
used by Aoki et al. to model polycrystalline MgF2 IRAS
spectra.53
The real and imaginary parts of the dielectric functions
⑀ʈ(␯), ⑀Ќ(␯), and ⑀͑␯͒ are shown in Fig. 3.
For a single crystal, the TO loss function, deﬁned as the
imaginary part of the dielectric function, has a peak at a TO
mode resonance. These resonances can be seen in the graphs
of Im͓⑀ʈ(␯)͔ ͓Fig. 3͑a͔͒ and Im͓⑀Ќ(␯)͔ ͓Fig. 3͑b͔͒ for singlecrystal
anatase. Consistent with established nomenclature,
we refer to Im͓⑀͑␯͔͒ as the TO loss function for the vitreous
TABLE III. Wave number of absorption bands of titanium dioxide ﬁlms on
aluminum.
Film Polarization ␯/c (cmϪ1
)
Experimental Theoretical
A s 260 259
355 374
442 434
A p 260 261
a b
443 435
853, c 841
B s 259 247
d 367
439 423
B p 258 247
e 368
438 423
849, ͑c,f͒ 846
a
Weak features between 300–400 cmϪ1
.
b
Weak bands at 330 and 380 cmϪ1
.
c
Shoulder at 962 cmϪ1
.
d
Extremely weak bands at ϳ310 and ϳ345 cmϪ1
.
e
Broad, extremely weak band from 300–350 cmϪ1
.
f
Shoulder at 785 cmϪ1
.
FIG. 2. Solid curves show experimental IRAS spectra and broken curves
show theoretical IRAS spectra obtained using s and p polarization for 0.70␮m-thick
ﬁlm B. RﬁlmϩAl /RAl is graphed as a function of wave number,
␯/c.
1122 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
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ﬁlms. The peaks in Im͓⑀͑␯͔͒ resulting from Eq. ͑4͒ are recorded
in Table IV. These peaks are essentially at the same
frequencies as the TO loss function peaks of the anatase
single crystal.
Likewise, for a single crystal, the LO loss function, deﬁned
as the imaginary part of the negative inverse dielectric
function, has a peak at an LO mode resonance. These resonances
can be seen in the graphs of Im͓Ϫ1/⑀ʈ(␯)͔ ͓Fig. 4͑a͔͒
and Im͓Ϫ1/⑀Ќ(␯)͔ ͓Fig. 4͑b͔͒ for an anatase single crystal.
Again, using established nomenclature, Im͓Ϫ1/⑀͑␯͔͒ is referred
to as the LO loss function for the vitreous ﬁlms. The
peaks in Im͓Ϫ1/⑀͑␯͔͒ resulting from Eq. ͑4͒ are recorded in
Table IV. Unlike the case of the TO loss function peaks, the
LO loss function peaks are signiﬁcantly different from those
for single-crystal anatase. Notably, the strong high-frequency
peak in Im͓Ϫ1/⑀͑␯͔͒ is at 840 cmϪ1
and lies between the
anatase single crystal LO loss peaks at 755 cmϪ1
͓Fig. 4͑a͔͒
and 876 cmϪ1
͑Fig. 4b͒ for Eʈc and EЌc beam conﬁgurations,
respectively.
The ﬁnal ingredient needed to calculate theoretical IRAS
spectra is the dielectric function of the metal. Here we assume
the free-electron model, and take ͑in Gaussian units͒
⑀m͑␯͒ϭ1ϩ2i␴͑␯͒/␯, ͑5͒
where ␴͑␯͒ is the frequency-dependent conductivity, which
has the Drude form,
␴͑␯͒ϭ␴͑0͒/͑1Ϫ2␲i␯␶͒, ͑6͒
where ␶ is the relaxation time. In our calculations, we used
the values ␴(0)ϭ3.67ϫ1017
sϪ1
and ␶ϭ0.80ϫ10Ϫ14
s for
Al at room temperature.54
The theoretical reﬂection coefﬁcients for ﬁlms A and B
are shown for s polarization (Rs) in Figs. 5 and 6, and for p
polarization (Rp) in Figs. 7 and 8. Each ﬁgure is comprised
of three panels calculated using a dielectric function: ͑a͒
⑀ʈ(␯), ͑b͒ ⑀Ќ(␯), and ͑c͒ ⑀͑␯͒. The wave numbers at the
peaks of the absorption bands ͑reﬂectivity minima for case
͑c͒͒ are recorded in Table III.
With respect to ﬁlm A, Fig. 5͑c͒ shows that absorption
bands are present at 259, 374, and 434 cmϪ1
when an
s-polarized beam is used. Figure 6͑c͒ shows absorption
bands at 261, 435, and 841 cmϪ1
occur for p polarization. In
addition, weak bands occur in Rp at 330 and 380 cmϪ1
.
With respect to ﬁlm B, Fig. 7͑c͒ shows absorption bands
at 247, 367, and 423 cmϪ1
when an s-polarized beam is used.
Figure 8͑c͒ shows absorption bands at 247, 368, 423, and
846 cmϪ1
for p polarization.
A comparison of Figs. 5͑c͒ and 7͑c͒ ͑ﬁlm A͒ with Figs.
6͑c͒ and 8͑c͒ ͑ﬁlm B͒ shows that for the same polarization
FIG. 3. The real ͑broken curve͒ and imaginary ͑solid curve͒ parts of the
dielectric function vs wave number, showing ͑a͒ ⑀ʈ(␯), ͑b͒ ⑀Ќ(␯), and ͑c͒
⑀(␯)ϭ͓2⑀Ќ(␯)ϩ⑀ʈ(␯)͔/3.
TABLE IV. Wave numbers of theoretical loss function peaks ͓see Figs. 3͑c͒
and 4͑c͔͒.
Loss function peak ␯/c (cmϪ1
) Maximum intensity
TO: Im͓⑀͑␯͔͒ 262 158
369 32.6
436 80.3
LO: Im͓Ϫ1/⑀͑␯͔͒ 328 0.06
396 0.04
840 2.62
FIG. 4. LO loss function vs wave number showing ͑a͒ Im͓Ϫ1/⑀ʈ(␯)͔, ͑b͒
Im͓Ϫ1/⑀Ќ(␯)͔, and ͑c͒ Im͓Ϫ1/⑀͑␯͔͒.
1123J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
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state, the shapes of the theoretical spectra are similar, but
there are signiﬁcant changes in peak intensity, as might be
expected in going from a thinner to thicker ﬁlm, and small
shifts in peak position.
VI. DISCUSSION
A. Comparison of experimental and theoretical IRAS
spectra
The quantity RﬁlmϩAl /RAl , calculated using ⑀͑␯͒, is included
in Figs. 1 and 2 ͑broken curves͒, along with the experimental
curves. A comparison of the experimental and
theoretical spectra for both ﬁlms shows similar peak position,
but the theoretical bands are more intense. In addition, a
band at 367–368 cmϪ1
in the theoretical spectra of ﬁlm B is
absent from the corresponding experimental spectra.
The experimental and theoretical absorption band frequencies
for ﬁlm A ͑Table III͒ can be compared with TO and
LO loss function peaks ͑Table IV͒. Both the experimental
and theoretical band frequencies correlate well with TO loss
function peaks at 262 and 435 cmϪ1
. Weaker bands in experimental
Rs at 355 cmϪ1
and in theoretical Rs at 374 cmϪ1
can be associated with an TO loss function peak at 367
cmϪ1
. A strong band in experimental Rp at 853 cmϪ1
and
theoretical Rp at 841 cmϪ1
can be associated with an LO loss
function peak at 840 cmϪ1
. Note that this band is absent in
Rs . We therefore associate this band with an LO-like resoFIG.
5. Theoretical reﬂection coefﬁcients for ﬁlm A on aluminum for
s-polarized beam conditions as a function of wave number using ͑a͒ ⑀ʈ(␯),
͑b͒ ⑀Ќ(␯), and ͑c͒ ⑀(␯)ϭ͓2⑀Ќ(␯)ϩ⑀ʈ(␯)͔/3.
FIG. 6. Theoretical reﬂection coefﬁcients for ﬁlm B on aluminum for
s-polarized beam conditions as a function of wave number using ͑a͒ ⑀ʈ(␯),
͑b͒ ⑀Ќ(␯), and ͑c͒ ⑀(␯)ϭ͓2⑀Ќ(␯)ϩ⑀ʈ(␯)͔/3.
FIG. 7. Theoretical reﬂection coefﬁcients for ﬁlm A on aluminum for
p-polarized beam conditions as a function of wave number using ͑a͒ ⑀ʈ(␯),
͑b͒ ⑀Ќ(␯), and ͑c͒ ⑀(␯)ϭ͓2⑀Ќ(␯)ϩ⑀ʈ(␯)͔/3.
1124 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
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nance in ﬁlm A, a manifestation of the Berreman effect,34
which ﬁnds LO resonances in p-polarized spectra, but not in
s-polarized spectra. Similarly, weak structures in the Rp
spectrum can be traced to minor peaks of the LO loss function
at 328 and 396 cmϪ1
.
The experimental and theoretical absorption band frequencies
for ﬁlm B ͑Table III͒ can be compared with TO and
LO loss function peaks ͑Table IV͒. Both the experimental
and theoretical band frequencies can be correlated with TO
loss function peaks at 262 and 435 cmϪ1
. A strong band in
experimental Rp at 849 cmϪ1
and in theoretical Rp at 846
cmϪ1
is in good agreement with an LO loss function peak at
840 cmϪ1
. As in the case of ﬁlm A, these peaks are absent
from Rs , again, the Berreman effect.
The experimental splitting, ⌬LO–TO , of the high frequency
‘‘LO–TO’’ pair for both ﬁlms is equal to 410–411
cmϪ1
, in excellent agreement with that predicted from the
difference in corresponding theoretical LO–TO loss function
peaks in the vitreous ﬁlms, ⌬LO–TOϭ404 cmϪ1
. For a crystalline
solid, this correspondence yields the width of the high
reﬂectivity Reststrahlen band, and ⌬LO–TOϭ441 cmϪ1
, for
the analogous high frequency EЌc LO–TO mode pair in
single-crystal anatase.43
These results show that introducing
random disorder into an anatase crystal has the effect of narrowing
the Reststrahlen region. Furthermore, ⌬LO–TO in a
crystalline solid is related to its ionicity,33
and this concept
has been extended to glasses as well.40
The fact that ⌬LO–TO
in the vitreous ﬁlms is reduced from the single crystal value
indicates that random orientational disorder reduces the effective
dipole moment per unit ﬁlm volume.
There is also a difference to note in the experimental
versus theoretical spectra when comparing the frequency of
equivalent bands between ﬁlms. Referring to Table III, the
difference in experimental frequencies of equivalent bands is
within experimental error when comparing ﬁlms A and B.
However, this is not in general the case for the theoretical
band frequencies. For example, experimental Rp bands at
260, 443, and 853 cmϪ1
in ﬁlm A shift by Ϫ2, Ϫ5, and Ϫ4
cmϪ1
, respectively, in ﬁlm B ͑within experimental error͒,
whereas the theoretical shifts in these bands in going from
ﬁlm A to B are Ϫ14, Ϫ12, and ϩ5 cmϪ1
, respectively.
B. Connection between absorption bands and loss
functions
To understand the relationship of the theoretical absorption
bands to the loss functions for the vitreous ﬁlms,
Im͓⑀͑␯͔͒ and Im͓Ϫ1/⑀͑␯͔͒, and in particular to trace the
source of the shifts in the theoretical spectra with increasing
ﬁlm thickness, we next consider the case in which the substrate
is a perfect metal ͓␴͑0͒→ϱ͔. Figure 9 compares Rs and
Rp curves for ﬁlm A on perfect and real metal substrates and
shows the validity of the ␴͑0͒→ϱ assumption. The effect of
ﬁnite conductivity is to increase the overall absorption ͑or
decrease the reﬂectivity͒, but there is very little change in the
shape of the curves or in the positions of the absorption
bands. The same conclusion applies to ﬁlm B, where the
perfect-metal and real-metal results are actually much closer
together than they are for ﬁlm A.
The expressions for Rs and Rp for a ﬁlm of thickness d
on a perfect metal substrate can be obtained from the
Berreman34
or Hansen45
expressions by taking the limit
␴͑0͒→ϱ ͑also see Ref. 55͒. The results for both polarizations
can be written in the following compact form,
FIG. 8. Theoretical reﬂection coefﬁcients for ﬁlm B on aluminum for
s-polarized beam conditions as a function of wave number using ͑a͒ ⑀ʈ(␯),
͑b͒ ⑀Ќ(␯), and ͑c͒ ⑀(␯)ϭ͓2⑀Ќ(␯)ϩ⑀ʈ(␯)͔/3.
FIG. 9. Theoretical reﬂection coefﬁcients ͑a͒ Rs and ͑b͒ Rp as a function of
wave number for ﬁlm A on perfect ͑solid curve͒ and real metal ͑broken
curve͒ substrates.
1125J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
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Rϭ͉͑1ϩi␣ tan ␤d͒/͑1Ϫi␣ tan ␤d͉͒2
, ͑7͒
where ␣ϭ(cos ␾)͓⑀(␯)Ϫsin2
␾͔Ϫ1/2
for s polarization, ␣
ϭ͓⑀(␯)Ϫsin2
␾͔1/2
/͓⑀(␯)cos ␾͔ for p polarization, and ␤
ϭk0͓⑀(␯)Ϫsin2
␾͔1/2
with k0ϭ2␲␯/c.
Expanding Eq. ͑7͒ for a very thin ﬁlm, i.e., for ͉␤d͉
Ӷ1 and k0dӶ1, we ﬁnd,
Rsϭ1Ϫ͑4/3͒͑cos ␾͒͑k0d͒3
Im͓⑀͑␯͔͒, ͑8͒
and
Rpϭ1Ϫ͑4/3͒͑1/cos ␾͒͑k0d͒3
Im͓⑀͑␯͔͒
ϩ4͑sin2
␾/cos ␾͒͑k0d͒Im͓1/⑀͑␯͔͒, ͑9͒
where higher-order terms in k0d in both Im͓⑀͑␯͔͒ and Im͓1/
⑀͑␯͔͒ have been omitted, and ␾ is assumed to be not too close
to 90°. The condition ͉␤d͉Ӷ1 can be relaxed to ͉␤d͉Ͻ0.5
since this condition is only used to justify the series expansion
tan ␤dϭ␤dϩ(␤d)3
/3 in Eq. ͑7͒. We have numerically
veriﬁed that Eqs. ͑8͒ and ͑9͒ are good approximations to the
‘‘exact’’ theoretical results, from Eq. ͑7͒, for ﬁlm A on an
ideal-metal substrate, except that Eq. ͑9͒ overestimates the
strength of the absorption band at 840 cmϪ1
in the Rp spec-
trum.
Equations ͑8͒ and ͑9͒ show that for very thin ﬁlms, the
absorption bands for s polarization are located at the peaks of
Im͓⑀͑␯͔͒, and for p polarization the absorption bands are located
at the peaks of Im͓⑀͑␯͔͒ and Im͓Ϫ1/⑀͑␯͔͒. This phenomenon
is the essence of the Berreman effect: absorption
bands for thin ﬁlms occur in p polarization, but not in s
polarization at the peaks of Im͓Ϫ1/⑀͑␯͔͒, which is the LO
loss function for our ﬁlm. Further, note that for very thin
ﬁlms, the p-polarization absorption is dominated by Im͓Ϫ1/
⑀͑␯͔͒ peaks rather than by Im͓⑀͑␯͔͒ peaks because absorption
bands associated with Im͓⑀͑␯͔͒ peaks are of order (k0d)3
,
while the absorption bands associated with Im͓Ϫ1/⑀͑␯͔͒
peaks are of order k0d.
To understand the physical origin of these absorption
band intensities for very thin ﬁlms, note that the electric
ﬁeld, E in the ﬁlm is parallel to the ﬁlm surface in
s-polarization, whereas in p polarization E is dominated by a
component EЌ that is perpendicular to the surface. The absorption
due to the parallel component of E is proportional to
the TO loss function Im͓⑀͑␯͔͒, and the absorption due to EЌ
is proportional to the LO loss function Im͓Ϫ1/⑀͑␯͔͒. This is a
very general statement about ⑀͑␯͒ and absorption bands in
very thin ﬁlms, regardless of the physical processes occurring
in the ﬁlm. Thus, despite our use of the usual terminology
of TO and LO loss functions for Im͓⑀͑␯͔͒ and Im͓Ϫ1/
⑀͑␯͔͒, respectively, this does not imply that vibrational modes
with atomic motion perpendicular or parallel to the direction
of propagation exist in our ﬁlms. Indeed, Eqs. ͑8͒ and ͑9͒
would hold in the hypothetical case that a ﬁlm did not support
any vibrational modes at all, provided of course that all
relevant dynamical processes were included in ⑀͑␯͒.
The relative strength of s-polarization and p-polarization
absorption bands can be understood from the fact that the
electric ﬁeld E in the ﬁlm is much weaker for an s-polarized
beam than for a p-polarized beam.56
This is a consequence of
the boundary condition requiring the tangential component
of E to be continuous across the ﬁlm/metal interface. Since E
vanishes in a perfect metal, and E is parallel to the surface
only for the s-polarized beam, there must be a node in E at
the ﬁlm/metal interface for an s-polarized beam, but not for a
p-polarized beam. More quantitatively, note that for s polarization,
the sine-wave standing wave pattern for E in a very
thin ﬁlm results in an essentially linear variation for E, rising
from zero at the ﬁlm/metal interface to a constant at the
air/ﬁlm interface. Since the absorbed power is proportional
to integral of ͉E͉2
over the ﬁlm volume, we deduce that the
band intensity is proportional to (k0d)3
for s polarization.
For p polarization, on the other hand, E contains a dominant
component EЌ , that is perpendicular to the ﬁlm surface, and
this component has no node at the ﬁlm/metal boundary. The
band intensity is now proportional to k0d, since ͉E͉2
is essentially
constant throughout the ﬁlm and the d dependence
of the absorbed power enters only through the volume inte-
gral.
For ﬁlm A, the conditions k0dӶ1 and ͉␤d͉Ͻ0.5 are
both satisﬁed. Thus, ﬁlm A is in the very-thin-ﬁlm limit, so
Eqs. ͑8͒ and ͑9͒ are applicable, and therefore absorption
bands do occur at the peaks of Im͓⑀͑␯͔͒ and Im͓Ϫ1/⑀͑␯͔͒. For
ﬁlm B, however, ͉␤d͉Ͼ1 in the vicinity of the peaks in
Im͓⑀͑␯͔͒. This difference accounts for the shifts in the theoretical
absorption band frequencies associated with the peaks
of Im͓⑀͑␯͔͒ in going from ﬁlm A to ﬁlm B. If ﬁlm B was still
in the very-thin-ﬁlm limit, the absorption bands in the theoretical
spectra for this ﬁlm would remain where they are for
ﬁlm A, consistent with our experimental observation. The
reason for the frequency shifts in the theoretical spectra appears
to be that our ⑀͑␯͒ expression, determined from Eq. ͑4͒,
is too resonant, i.e., the values of ⑀͑␯͒ at the TO loss function
peaks are too large. A sharp resonance would also account
for the theoretical absorption bands being more intense than
the experimental absorption bands.
C. Choice of dielectric function
The problem of constructing an appropriate dielectric
function for the vitreous ﬁlms starting from orientationally
disordered anatase Ti–O units is a difﬁcult one. Within a
mean-ﬁeld-theory approach, this requires consideration of local
effective ﬁelds in inhomogeneous media.57
The result depends
on the shape of the unit and its orientation with respect
to the macroscopic electric ﬁeld E. The choice of Eq. ͑4͒ for
⑀͑␯͒ implies that ͑on average͒ the effective ﬁeld acting on a
unit is the macroscopic electric ﬁeld in the medium, or, in
other words, there are no local-ﬁeld corrections.57,58
Strictly
speaking, such a result is valid only if the individual units
have rodlike, needlelike, or platelike shapes with the long
axis along the direction of E.57,58
If, on the other hand, the
units are platelike and are oriented perpendicular to E, the
correct choice would be given by the ‘‘parallel capacitor formula,’’
⑀(␯)Ϫ1
ϭ͓2⑀Ќ(␯)Ϫ1
ϩ⑀ʈ(␯)Ϫ1
͔/3.58
Clearly, in the
latter case, one would ﬁnd strong absorption bands in
p-polarized spectra at the LO mode resonances in both ⑀Ќ(␯)
and ⑀ʈ(␯). This is not the case in our experiments. Furthermore,
dielectric functions based on a preferred geometrical
1126 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
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arrangement of units that are either parallel or perpendicular
to E can be excluded because the direction of E is different
in s and p polarizations.
We have also considered alternative prescriptions for
constructing a dielectric function within a mean-ﬁeld-theory
framework. IRAS spectra were calculated using dielectric
functions obtained from effective-medium theory,59
and from
Maxwell–Garnett and Clausius–Mossotti expressions.60
None of these dielectric functions resulted in reﬂectivity
curves that ﬁt the experimental data as well as those obtained
from our original choice of ⑀͑␯͒ given by Eq. ͑4͒.
Evidently, there are signiﬁcant contributions to inhomogeneous
broadening that are left out of our model, which is a
mean-ﬁeld model consisting of component parts, ⑀ʈ(␯) and
⑀Ќ(␯) that are those for a perfect anatase crystal. Part of the
difﬁculty may be in the mean-ﬁeld aspect of the model. By
using an average ͑or effective-medium͒ prescription to construct
a dielectric function, we are not faithfully including
short-range correlations between dipoles in neighboring
Ti–O units except in some mean-ﬁeld manner. Perhaps even
more signiﬁcantly, we have neglected additional life-time
broadening and frequency-shift effects that are associated
with chemical bonding between Ti–O units that may differ
from the chemical bonding in the crystalline material. This
includes local strain ﬁelds and local defects in the vitreous
ﬁlms that are not present in crystalline material.
We have been able to infer from these studies that orientationally
disordered Ti–O units with a local structure
characteristic of anatase play a dominant role in the dielectric
response of our vitreous ﬁlms. Further reﬁnement of our
model will include consideration of local ͑dis͒order, traceable
to short-range correlations in the coupling and/or chemical
bonding between neighboring Ti–O units.
VII. SUMMARY
We reported the optical response of sputter-deposited
vitreous TiO2 ﬁlms with anatase short-range order. Experimental
IRAS spectra were obtained in the 100–1000 cmϪ1
wave number range. Our data were found to differ from
those for single crystal anatase. We constructed a dielectric
function to account for random orientational disorder within
the ﬁlms. Calculated energy loss functions and theoretical
reﬂection–absorption spectra using this dielectric function
were compared with experimental results. Good qualitative
agreement between experimental and theoretical IRAS spectra
indicates that orientational disorder is the important factor
governing the reﬂection–absorption behavior of our vitreous
ﬁlms.
ACKNOWLEDGMENTS
We thank Dr. J. D. DeLoach, M. A. MacLaurin, and E.
E. Hoppe for technical assistance, Dr. A. V. Sklyarov for
useful discussions, and U. W.-Milwaukee Chancellor N. L.
Zimpher for creating AceLab ͑Advanced Coatings Experimental
Laboratory͒, through which G. S., R. S. S., and C. R.
A. were partially supported. The following grants are also
acknowledged: NSF-DMR-9806055 and NSF-CHE-9984931
͑CJH͒; NSF-ECS 9984225, ACS-PRF 34396-G5, and a Wisconsin
Space Grant Consortium Fellowship ͑VVY͒.
1
T. Nakayama, K. Onisawa, M. Fuyama, and M. Hanasono, J. Electrochem.
Soc. 139, 1204 ͑1992͒.
2
K. Jurek, M. Guglielni, G. Kuncova, O. Renner, F. Lukes, M. Navratil, E.
Rousky, V. Vorlicek, and K. Kokesova, J. Mater. Sci. 27, 2549 ͑1992͒.
3
K. S. Yeung and Y. W. Lam, Thin Solid Films 109, 169 ͑1983͒.
4
B.-H. Jun et al., Appl. Opt. 36, 1482 ͑1997͒.
5
E. Sirbegovic, M. Cevro, and I. Tucak, Vacuum 43, 723 ͑1992͒.
6
J. D. DeLoach and C. R. Aita, J. Mater. Sci. Lett. 19, 1123 ͑2000͒.
7
W. L. Bragg, Atomic Structure of Minerals ͑Cornell University Press,
Ithaca, 1937͒, pp. 102–106.
8
J. G. Clark, The Chemistry of Titanium and Vanadium ͑Elsevier, New
York, 1968͒, pp. 66–277.
9
F. Dachille, P. Y. Simons, and R. Roy, Am. Mineral. 53, 1929 ͑1968͒.
10
S.-D. Mo and W. Y. Ching, Phys. Rev. B 51, 13023 ͑1995͒.
11
K. Kosuge, Chemistry of Non-stoichiometric Compounds ͑Oxford University
Press, Oxford, UK, 1994͒, pp. 121–128.
12
F. P. Felner, Low Temperature Oxidation ͑Wiley-Interscience, New York,
1981͒, pp. 31–49 and references therein.
13
M. Shiojiri, J. Phys. Soc. Jpn. 21, 335 ͑1966͒.
14
T. M. Salama, T. Tanaka, T. Yamaguchi, and K. Tanabe, Surf. Sci. Lett.
227, L100 ͑1990͒.
15
W. G. Lee, S. I. Woo, J. C. Kim, S. H. Choi, and K. H. Oh, Thin Solid
Films 237, 105 ͑1994͒.
16
D. G. Howitt and A. B. Harker, J. Mater. Sci. 2, 201 ͑1987͒.
17
L. S. Hsu, R. Rujkorakarn, J. R. Sites, and C. Y. She, J. Appl. Phys. 59,
3475 ͑1986͒.
18
T. Nakayama, J. Electrochem. Soc. 141, 237 ͑1994͒.
19
R.-Y. Tasi and M.-Y. Hua, Appl. Opt. 35, 5073 ͑1996͒.
20
H. Nozoye and N. Nishimiya, Appl. Phys. Lett. 54, 231 ͑1989͒.
21
J. D. DeLoach and C. R. Aita, J. Vac. Sci. Technol. A 16, 1963 ͑1998͒.
22
M. D. Wiggins, M. C. Nelson, and C. R. Aita, J. Vac. Sci. Technol. A 14,
772 ͑1996͒; Proc. Mat. Res. Soc. 398, 381 ͑1996͒.
23
P. Alexandrov, J. Koprinarova, and D. Todorov, Vacuum 47, 1333 ͑1996͒.
24
D. Wicaksana, A. Kobayashi, and A. Kinbara, J. Vac. Sci. Technol. A 10,
1479 ͑1992͒.
25
D. Guerin and S. Ismat Shah, J. Vac. Sci. Technol. A 15, 712 ͑1997͒.
26
M. G. Krishna, N. Rao, and S. Mohan, J. Appl. Phys. 73, 434 ͑1993͒.
27
L. M. Williams and D. W. Hess, J. Vac. Sci. Technol. A 1, 1810 ͑1983͒.
28
N. Martin, C. Rousselot, D. Rondot, F. Palmiro, and R. Mercier, Thin
Solid Films 3000, 113 ͑1997͒.
29
J. D. DeLoach, G. Scarel, and C. R. Aita, J. Appl. Phys. 85, 2377 ͑1999͒.
30
J. A. Eastman, J. Appl. Phys. 75, 770 ͑1994͒.
31
V. V. Yakovlev, G. Scarel, C. R. Aita, and S. Mochizuki, Appl. Phys. Lett.
76, 1107 ͑2000͒.
32
J. Tauc, in Amorphous and Liquid Semiconductors, edited by J. Tauc ͑Plenum,
London, 1974͒, Chap. 4.
33
P. M. A. Sherwood, Vibrational Spectroscopy Of Solids ͑Cambridge University
Press, Cambridge, UK, 1972͒.
34
D. W. Berreman, Phys. Rev. 130, 2193 ͑1963͒.
35
F. L. Galeener and G. Lucovsky, Phys. Rev. Lett. 37, 1474 ͑1976͒.
36
F. L. Galeener, A. J. Leadbetter, and M. W. Stringfellow, Phys. Rev. B 27,
1052 ͑1983͒.
37
S. E. de Leeuw and M. F. Thorpe, Phys. Rev. Lett. 55, 2879 ͑1985͒.
38
M. F. Thorpe and S. E. de Leeuw, Phys. Rev. B 33, 8490 ͑1986͒.
39
J. M. Gales, B. Boltjes, M. F. Thorpe, and S. E. de Leeuw, Phys. Rev. B
40, 7214 ͑1989͒.
40
R. M. Almeida, Phys. Rev. B 45, 161 ͑1992͒.
41
E. I. Kamitsos, A. P. Patsis, and G. Kordas, Phys. Rev. B 48, 12499
͑1993͒.
42
M. A. Ovchinnikov and C. A. Wight, J. Chem. Phys. 99, 3374 ͑1993͒; 100,
972 ͑1994͒; 102, 67 ͑1995͒.
43
R. J. Gonzalez, R. Zallen, and H. Berger, Phys. Rev. B 55, 7014 ͑1997͒.
44
See for example: H. P. Klug and L. E. Alexander, X-Ray Diffraction Processes
͑Wiley, New York, 1974͒.
45
W. N. Hansen, J. Opt. Soc. Am. 58, 380 ͑1968͒.
46
Anatase: ASTM JCPDS File No. 21-1272 ͑1991͒.
47
Rutile: ASTM JCPDS File No. 21-1276 ͑1991͒.
48
B. L. Averbach and M. Cohen, Trans. AIME 176, 401 ͑1948͒.
49
L. S. Zevin and G. Kimmel, Quantitative X-ray Diffractometry ͑Springer,
New York, 1995͒, pp. 188–204.
1127J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
Downloaded 13 Feb 2009 to 193.52.108.46. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp
50
M. Ocana, J. V. Garcia-Ramos, and C. J. Serna, J. Am. Ceram. Soc. 75,
2010 ͑1992͒.
51
B. C. Trasferetti, C. U. Davanzo, N. C. da Cruz, and M. A. B. de Moraes,
Appl. Spectrosc. 54, 687 ͑2000͒.
52
D. M. Eagles, J. Phys. Chem. Solids 25, 1243 ͑1964͒.
53
A. Aoki, H. Tanaka, and K. Hisano, Proceedings of the Twelfth Japan
Symposium on Thermophysical Properties, A103 ͑1991͒, p. 9.
54
N. W. Ashcroft and N. D. Mermin, Solid State Physics ͑Saunders College
Press, Philadelphia, 1976͒.
55
R. Fuchs, K. L. Kliewer, and W. J. Pardee, Phys. Rev. 150, 589 ͑1966͒.
56
R. G. Greenler, J. Chem. Phys. 44, 310 ͑1966͒.
57
R. Landauer, in Electrical Transport and Optical Properties of Inhomogeous
Media edited by J. C. Garland and D. B. Tanner ͑American Institute of
Physics, New York, 1978͒, p. 2.
58
R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, Phys. Rev. B 8,
3689 ͑1973͒.
59
D. Stroud, Phys. Rev. B 12, 3368 ͑1975͒.
60
R. Landauer, op. cit., p. 10 ͑see Eqs. 3.5–3.7͒.
1128 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Scarel et al.
Downloaded 13 Feb 2009 to 193.52.108.46. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp