Infrared reflectivity and lattice fundamentals in anatase TiO2
R. J. Gonzalez* and R. Zallen
Department of Physics, Virginia Tech, Blacksburg, Virginia 24061
H. Berger
Institut de Physique Appliquee, Ecole Polytechnique Federale Lausanne, Switzerland CH-1015
͑Received 9 July 1996; revised manuscript received 21 October 1996͒
Polarization-dependent far-infrared reflectivity measurements were carried out on single crystals of anatase
TiO2 . The results were analyzed to yield the dielectric dispersion properties of anatase in the lattice fundamentals
regime. The frequencies ͑in cmϪ1
͒ of the transverse optical ͑TO͒ and longitudinal optical ͑LO͒
zone-center phonons were determined to be 367͑755͒ for the TO ͑LO͒ of the A2u mode, 262͑366͒ and 435͑876͒
for the Eu modes. The large TO-LO splittings were used to estimate effective charges.
͓S0163-1829͑97͒02009-2͔
I. INTRODUCTION
The crystal forms of titanium dioxide ͑titania͒ are rutile,
anatase, and brookite. They are wide-bandgap semiconductors,
transparent in the visible, with high refractive indices.
Rutile is the stable phase at high temperature and is by far
the most-studied and best-understood phase. Because large
single crystals of rutile have long been long available, the
lattice dynamics of rutile TiO2 in the lattice-fundamentals
regime has been studied by polarization-dependent farinfrared
reflectivity measurements1,2
and by neutronscattering
measurements as well.3
To our knowledge, no
similar work has yet been reported for anatase TiO2, although
there have been studies of the Raman-active lattice
fundamentals.4
The near-bandgap optical absorption edge of
anatase has recently been measured.5
In this paper, we present the results of far-infrared reflectivity
measurements on anatase single crystals. The
polarization-dependent single-crystal results are analyzed to
yield the zone-center transverse-optical ͑TO͒ and
longitudinal-optical ͑LO͒ phonon frequencies, effective
charges, and polariton dispersion curves of anatase.
II. EXPERIMENT
Anatase crystals were grown by chemical transport
reactions.6
As-grown single crystals were up to 5 mm in size,
with an octahedral shape limited by ͑101͒ natural faces. Two
crystals were used in this study. There were cut and polished;
one to yield a surface containing the tetragonal c axis, the
other to yield a surface perpendicular to the c axis. Alumina
powder was used for polishing, the final grit size being 0.1
␮m. The area of the optical surface was about 8 mm2
for the
parallel-to-c surface and about 9 mm2
for the perpendicularto-c
surface.
The infrared measurements were performed with a
BOMEM DA-3 FTIR spectrometer. A pyroelectric detector
was used to cover the wave-number region from 50 to 700
cmϪ1
; a HgCdTe detector was used from 600 to 5000 cmϪ1
.
Spectra were collected with a 1-cmϪ1
resolution. At least 400
interferometer sweeps were added for each spectrum. For
measurements with light polarized parallel to the crystal c
axis, a wire-grid polarizer was used. ͑The low-frequency cutoff,
determined by the polarizer substrate, was 200 cmϪ1
.͒
Reflectivity was measured at an angle of incidence of 11°,
and was determined by comparison to an aluminum mirror
standard.
III. INFRARED-ACTIVE MODES OF ANATASE
Anatase is tetragonal, with two TiO2 formula units ͑six
atoms͒ per primitive cell. The space group is D4h
19
͑I4/amd),
number 141 in the standard listing. The structure is shown on
the left side of Fig. 1. The c-axis is vertical, small circles
denote Ti atoms, large circles denote O atoms. Oxygen atoms
labeled with the same number are equivalent. The octahedral
coordination of the titanium atoms is seen to be significantly
distorted.
The 18-dimensional reducible representation generated by
the atomic displacements contains the zone-center ͑kϭ0͒
modes: 3 acoustic modes and 15 optical modes. The irreducible
representations corresponding to the 15 optical modes
FIG. 1. The structure of the anatase primitive cell is shown at
the left. The c axis is vertical, small circles denote Ti atoms, large
circles denote O atoms. Oxygen atoms labeled with the same number
are equivalent. The center figure shows ͑ϩ͒ the position of the
inversion center and the vibrational eigenvector for the A2u mode.
Symmetry coordinates for the Eu modes are shown at the right.
PHYSICAL REVIEW B 15 MARCH 1997-IVOLUME 55, NUMBER 11
550163-1829/97/55͑11͒/7014͑4͒/$10.00 7014 © 1997 The American Physical Society
are4
1A1gϩ1A2uϩ2B1gϩ1B2uϩ3Egϩ2Eu . Three modes
are infrared active, the A2u mode and the two Eu modes.
͑The B2u mode is silent.͒ The A2u mode is active for light
polarized parallel to the c axis (Eʈc); the Eu modes are
active for light polarized perpendicular to the c axis
(EЌc). The vibrational eigenvector for the A2u is symmetrydetermined;
it is shown in Fig. 1. Symmetry coordinates for
the Eu modes are also shown. The actual Eu eigenvectors are
mutually orthogonal combinations of these.
IV. POLARIZATION-DEPENDENT REFLECTIVITY,
DIELECTRIC FUNCTION, AND POLARITON DISPERSION
Figure 2 presents our results for the far-infrared reflectivity
of single-crystal anatase, for Eʈc and EЌc. ͑The frequency
or photon-energy scale used throughout this paper is
in terms of wave number, ␯¯ϭ␭Ϫ1
.) The Eʈc results, obtained
for a surface containing the c axis, required the use of
a polarizer which cut off below 200 cmϪ1
. The incident light
was polarized perpendicular to the plane of incidence. The
EЌc results, obtained for a surface normal to the c axis,
required no polarizer and extended down to 50 cmϪ1
. ͑The
50–100 cmϪ1
range, not shown in Fig. 2, contained no discernible
structure.͒
The theoretical curves included in Fig. 2 are based on the
factorized form of the dielectric function:2,7–9
␧͑␯͒ϭ␧1͑␯͒Ϫi␧2͑␯͒ϭ␧ϱ͟n
␯LOn
2
Ϫ␯2
ϩi␥LOn␯
␯TOn
2
Ϫ␯2
ϩi␥TOn␯
. ͑1͒
The factorized form is more elegant than the classicaloscillator
form,7,8
and it is especially appropriate to highly
ionic crystals ͑such as anatase͒ which have large TO-LO
splittings. When ␯LO is much larger than ␯TO , a single
damping parameter ␥ ͑as is used in oscillator analysis͒ is
inadequate and the factorized-form analysis is more
successful.2,9
We carried out both oscillator and factorizedform
analyses of the data shown in Fig. 2. Both methods give
good fits ͑and essentially identical results for ␯TO and ␯LO)
for Eʈc. But for EЌc, only the factorized form gives a good
fit to the measured reflectivity spectrum.
Table I lists the TO and LO frequencies corresponding to
the fitted curves shown in Fig. 2. Also included in the table
are the corresponding damping parameters and two sets of
published calculations for the TO and LO frequencies using
rigid-ion10
and GF-matrix4
models. The calculations tend to
overestimate the vibrational frequencies and underestimate
the TO-LO splittings ͑oscillator strengths͒.
In the EЌc results of Fig. 2, it can be seen that the measured
reflectivity shows a small dip ͑relative to the theoretical
curve͒ near 750 cmϪ1
. It is not an accident that this dip
occurs near the LO frequency for the other (Eʈc) polarizaFIG.
2. The polarization-dependent far-infrared reflectivity of
single-crystal anatase. The Eʈc results, obtained for a surface containing
the c axis, required the use of a polarizer which cut off
below 200 cmϪ1
. The EЌc results, obtained for a surface normal to
the c axis, required no polarizer and extended down to 50 cmϪ1
.
͑The 50–100 cmϪ1
range, not shown, contained no discernible
structure.͒ The continuous curves included in this figure are fits
based on the factorized form of the dielectric function ͓Eq. ͑1͔͒.
TABLE I. TO and LO phonon frequencies of anatase TiO2 .
Dielectric function fit to R͑␯¯͒ Published calculations
Rigid ion GF matrix
Frequency Damping ͑Ref. 12͒ ͑Ref. 4͒
Mode ␯¯ ͑cmϪ1
͒ ␥ ͑cmϪ1
͒ ␯¯ ͑cmϪ1
͒ ␯¯ ͑cmϪ1
͒
Eʈc axis TO 367 68 566 654
(A2u) LO 755 79 844
EЌc axis TO 262 36 329 169
(Eu) LO 366 4.1 428
TO 435 32 644 643
LO 876 33 855
␧ϱ(Eʈc)ϭ5.41 ␧0(Eʈc)ϭ22.7
␧ϱ(EЌc)ϭ5.82 ␧0(EЌc)ϭ45.1
55 7015INFRARED LATTICE FUNDAMENTALS IN TiO2 . . .
tion. The dip is a consequence of the experimental geometry,
the 11° deviation from normal incidence. The effects of offnormal
incidence, for optically uniaxial crystals ͑two independent
polarizations, Eʈc and EЌc), have been experimentally
established and theoretically analyzed by several
authors.11–13
With the c axis normal to the surface and the incident
beam polarized in the plane of incidence, an Eʈc LO mode
that is situated in frequency between a pair of EЌc TO and
LO modes will produce a dip in the EЌc high-reflectivity
plateau at the position of the Eʈc LO mode.11
Our measurements
for EЌc were obtained on a normal-to-c surface using
unpolarized light, which includes a component contributing
such ‘‘leakage’’ ͑structure arising from the unintended Eʈc
infrared response function͒. This off-normal-incidence leakage
effect accounts for the 750-cmϪ1
dip observed in the
measured EЌc spectrum of Fig. 2.
There is a similar off-normal-incidence leakage effect for
the Eʈc reflectivity when it is measured with the c axis and
the light polarization in the plane of incidence.12,13
Since our
measurements were carried out with the c axis and the light
polarization perpendicular to the plane of incidence, this effect
is absent here. The structure in the Eʈc spectrum near
640 cmϪ1
is not attributable to off-normal incidence. There is
a Raman-active mode in anatase at 639 cmϪ1
͑Ref. 4͒, but
anatase is centrosymmetric so that Raman-active modes are
infrared-forbidden and do not contribute to the infrared response.
͑Such modes are not turned on, in infrared reflectivity,
by oblique incidence.͒ The weak 640-cmϪ1
feature in
Fig. 2, if real, remains unexplained.
The ␧ϱ values in Table I, for frequencies well above the
lattice-fundamental regime, correspond to ␧’s at frequencies
below the electronic interband regime. Using ␧ϭn2
where n
is the refractive index, our ␧ϱ values correspond to refractive
indices of 2.33 (Eʈc) and 2.41 (EЌc). Reported values of n
in the visible region are14
2.49 (Eʈc) and 2.56 (EЌc), indicating
that our n values are about 6% low. This does not
affect our results for the TO and LO frequencies, which are
predominantly determined by the shape of the reflectivity
spectrum. We estimate the probable error in our frequency
values to be no more than Ϯ3 cmϪ1
for the TO and Eʈc LO
frequencies, and Ϯ5 cmϪ1
for the EЌc LO frequencies.
From the experimental values given in Table I and the
␧͑␯͒ expression of Eq. ͑1͒, it is straightforward to determine
the following spectroscopic quantities in the far infrared: the
real and imaginary parts of the dielectric function ␧, the real
and imaginary parts of the complex refractive index nc
, the
optical absorption coefficient, and the polariton dispersion
curves ␯(q1), where q1 is the real part of the complex propagation
vector describing the coupled photon-phonon wave in
the crystal.15
We limit our discussion of spectroscopic quantities
to the dielectric function and the polariton dispersion.
Figure 3 shows the results derived for the Eʈc and EЌc
dielectric functions for anatase, in the far infrared. The
shaded bars highlight the TO-LO splittings, which are large.
Polariton dispersion curves are shown in Fig. 4. The steps
from ␧͑␯¯͒ to ␯¯(q1) are15
nc
(␯¯)ϭͱ␧(␯¯), qc
(␯¯)
ϭ2␲␯¯nc
(␯¯), q1(␯¯)ϭRe͓qc
(␯¯)͔. The solid curves correspond
to the experimental factorized-form parameters of
Table I. The dashed curves results from setting the damping
parameters equal to zero; they display the classical coupledwave
from with qϭ0 intercepts at LO frequencies and q→ϱ
asymptotes at TO frequencies.16
The light lines show the
asymptotic slopes, which are inversely proportional to the
optical refractive index ͑long line͒ and static refractive index
͑short line͒.
FIG. 3. The dielectric functions of anatase TiO2 . These curves
correspond to the fits obtained with the factorized form of the dielectric
function. The shaded bars highlight the TO-LO splittings.
FIG. 4. Polariton dispersion curves for anatase TiO2 . The solid
curves correspond to the experimental parameters of Table I. The
dashed curves result from setting the damping parameters equal to
zero. The light lines show the asymptotic slopes, which are inversely
proportional to the optical refractive index ͑long line͒ and
static refractive index ͑short line͒.
7016 55R. J. GONZALEZ, R. ZALLEN, AND H. BERGER
V. EFFECTIVE CHARGES
The vibrational eigenvector of the lone A2u optical mode
in anatase is fully determined by symmetry: the atomic displacements
xTi and xO are parallel to the c axis and
(xTi /xO)ϭϪ0.67. ͓The mass ratio (mTi /mO) is 2.994.͔ This
presents us with an opportunity to estimate effective charges,
using the approach of Kurosawa.7
Since TiO2 is highly ionic,
we assign static charges (eTi* and eO*) to the ions and assume
these charges move with the ions. This is a rigid-ion model,
with dynamic charge17
neglected. The Eʈc dielectric function
can then be written7
␧͑␻͒ϭ␧ϱϩ4␲VϪ1
͚ͫi
ei*xiͬ2
͚ͫi
mixi
2
ͬϪ1
͑␻TO
2
Ϫ␻2
͒Ϫ1
,
͑2͒
where ␻ϭ2␲␯ is the angular frequency. V is the volume per
TiO2 unit, xi is the displacement of ion i, ei* is the ion’s
charge, and mi is its mass. From Eq. ͑2͒ and (xTi /xO)ϭ
Ϫ0.67 and (eTi*/eO*)ϭϪ2 ͑crystal neutrality͒ and
(mTi /mO)ϭ2.994, it follows that
␧0Ϫ␧ϱϭ4␲VϪ1
͑eO*͒2
͑0.30mO͒Ϫ1
␻TO
Ϫ2
. ͑3͒
Every quantity in Eq. ͑3͒ is known except for eO* , which is
thus determined: eO*ϭϪ2.8e, where e is the magnitude of
the electron charge and the negative sign is chosen for the
oxygen ion. The result is reasonable, but somewhat large; we
do not expect the oxygen ion to be more highly charged than
OϪϪ
, and any contribution from dynamic charge would be
expected to reduce the size of the observed effective
charge.18
Szigeti’s introduction of the local-field correction
in the case of a cubic crystal19
reduces the effective charge
by the factor 3/(␧ϱϩ2). TiO2 is not cubic, but the TO-LO
splittings of Fig. 3 are larger than the anisotropy shifts so
neglecting anisotropy may not be a terrible approximation.
Doing this yields a Szigeti effective charge, for the oxygen
ion, of Ϫ1.1e.
VI. SUMMARY
Single crystals of anatase, grown by transport reactions,
were studied by far-infrared reflectivity. Clean polarizationdependent
spectra were observed ͑Fig. 2͒, and the results
were analyzed to yield the dielectric dispersion properties in
the lattice fundamentals regime ͑Figs. 3 and 4͒. The TO ͑ and
LO͒ frequencies of the zone-center phonons were determined
to be ͑in units of cmϪ1
͒: 367͑755͒ for the A2u mode;
262͑366͒ and 435͑876͒ for the Eu modes. The large TO-LO
splitting was analyzed in terms of effective charges for the
A2u mode, whose vibrational eigenvector is symmetry deter-
mined.
ACKNOWLEDGMENTS
We are grateful to A. Gaynor and R. M. Davis of the
Chemical Engineering Department at Virginia Tech for
many discussions about titania, to B. Kutz of the Department
of Geological Sciences for her help in orienting, cutting, and
polishing the anatase crystals, and to G. Margaritondo of
EPFL for his support and encouragement. Work at EPFL
was supported in part by the Fonds Nationale Suisse de la
Recherche Scientifique.
*Present address: Sienna Biotech, Inc., Columbia, MD.
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55 7017INFRARED LATTICE FUNDAMENTALS IN TiO2 . . .