Raman scattering in HfxZr1-xO2 nanoparticles

Article abstract of DOI:10.1103/PhysRevB.71.115408

Raman spectroscopy demonstrates that ～5 nm dimension Hf _xZr_1-xO₂ nanocrystals prepared by a nonhydrolytic sol-gel synthesis method are solid solutions of hafnia and zirconia, with no discernable segregation within the individual nanoparticles. Zirconia-rich particles are tetragonal and ensembles of hafnia-rich particles show mixed tetragonal/monoclinic phases. Sintering at 1200°C produces larger particles (20-30 nm) that are monoclinic. A simple lattice dynamics model with composition-averaged cation mass and scaled force constants is used to understand how the Raman mode frequencies vary with composition in the tetragonal Hf_xZr_1-xO₂ nanoparticles. Background luminescence from these particles is minimized after oxygen treatment, suggesting possible oxygen defects in the as-prepared particles. Raman scattering is also used to estimate composition and the relative fractions of tetragonal and monoclinic phases. In some regimes there are mixed phases, and Raman analysis suggests that in these regimes the tetragonal phase particles are relatively rich in zirconium and the monoclinic phase particles are relatively rich in hafnium. 2005 The American Physical Society.

Full text of DOI:10.1103/PhysRevB.71.115408

PHYSICAL REVIEW B 71, 115408 ͑2005͒
Raman scattering in HfxZr1−xO2 nanoparticles
Richard D. Robinson, Jing Tang, Michael L. Steigerwald, Louis E. Brus, and Irving P. Herman
*
Materials Research Science and Engineering Center, Columbia University, New York, New York 10027, USA
͑Received 30 August 2004; published 10 March 2005͒
Raman spectroscopy demonstrates that ϳ5 nm dimension HfxZr1−xO2 nanocrystals prepared by a nonhydro-
lytic sol-gel synthesis method are solid solutions of hafnia and zirconia, with no discernable segregation within
the individual nanoparticles. Zirconia-rich particles are tetragonal and ensembles of hafnia-rich particles show
mixed tetragonal/monoclinic phases. Sintering at 1200 °C produces larger particles ͑20–30 nm͒ that are mono-
clinic. A simple lattice dynamics model with composition-averaged cation mass and scaled force constants is
used to understand how the Raman mode frequencies vary with composition in the tetragonal HfxZr1−xO2
nanoparticles. Background luminescence from these particles is minimized after oxygen treatment, suggesting
possible oxygen defects in the as-prepared particles. Raman scattering is also used to estimate composition and
the relative fractions of tetragonal and monoclinic phases. In some regimes there are mixed phases, and Raman
analysis suggests that in these regimes the tetragonal phase particles are relatively rich in zirconium and the
monoclinic phase particles are relatively rich in hafnium.
DOI: 10.1103/PhysRevB.71.115408
PACS number͑s͒: 61.46.ϩw, 78.30.Ϫj, 78.67.Bf, 64.70.Nd
I. INTRODUCTION
reported the first synthesis of hafnia-zirconia solid solution
nanoparticles.14 Solid solutions of hafnia-zirconia
͑HfxZr1−xO2͒ are of interest for several reasons, including the
need to understand the mode structure of such mixed nanoc-
rystals. This is still an issue in bulk mixed crystals. For in-
stance, the question of possible two-mode behavior in bulk
HfxZr1−xO2 has attracted much attention.15
Hafnia ͑HfO2͒ and zirconia ͑ZrO2͒ are called twin oxides
because of their similar chemical and physical properties.
They are isostructural in the bulk1 and this close correlation
in properties is due to the identical valence states and nearly
identical ionic radii for Hf and Zr. Still, there is slightly
stronger bonding in Hf compounds relative to the analogous
Zr compounds.1,2 The dielectric constants of hafnia and zir-
conia at low frequency are both very high, very roughly
20;3–5 this, along with their stability when in contact with Si,
make them interesting candidates for insulating barriers in
microelectronics.3 First principles calculations give
orientationally-averaged dielectric constants of 20 for zirco-
nia and 16–18 for hafnia in this monoclinic phase, and much
higher dielectric constants for the tetragonal phase, 47 and
70 respectively.4,5
Bulk hafnia and zirconia can each adopt three different
crystal structures at ambient pressures, i.e., monoclinic, te-
tragonal and cubic. In the bulk, each oxide is stable in the
monoclinic phase at room temperature and each transforms
to tetragonal at high temperatures, the former at 1720 °C and
the latter at 1170 °C.6 At even higher temperatures, 2600 °C
and 2370 °C, respectively, the tetragonal phases transform to
the cubic phase. At room temperature, zirconia is stable in
the monoclinic phase for dimensions տ100 nm, while
nanometer-sized zirconia is stable in the tetragonal phase.
The reported “critical size” below which the tetragonal phase
is stable ranges from 9–30 nm.7–10 The reason for the forma-
tion and stability of tetragonal zirconia nanocrystals is still
uncertain, and has been attributed to one of several factors,
including the lower surface free energy for the high tempera-
ture phase7 and anionic vacancies that nucleate the tetragonal
phase.11,12 There have been relatively few studies on the
polymorphs of hafnia, and particularly nanosized
crystals.13,14 The first extensive synthesis of tetragonal hafnia
nanoparticles was reported recently by the authors, who also
For these binary oxides, six of the 18 normal modes are
allowed Raman modes in the tetragonal phase ͑space group
15
4h
D , Z=2͒ and 18 of the 36 normal modes are allowed Ra-
man modes in the lower symmetry monoclinic phase ͑space
group C52h, Z=4͒. Many of the studies of Raman scattering of
tetragonal zirconia present evidence for conflicting assign-
ments of the observed Raman modes.9,16–18 To our knowl-
edge, there have been no previous Raman scattering studies
on tetragonal hafnia nanoparticles or hafnia-zirconia nano-
particle solid solutions, other than a brief report by the
present authors in Ref. 14. The present authors presented
preliminary results regarding the Raman spectra of as-
synthesized HfxZr1−xO2 nanocrystals excited by the 325-nm
line from a He-Cd laser; Raman analysis was not possible
with excitation at 488 and 514.5 nm by an argon-ion laser,
because the background luminescence was too intense.
This paper reports on the Raman spectra of HfxZr1−xO2
nanocrystals over a range of composition, particularly for
those small particles in the tetragonal phase. These spectra
demonstrate that the particles are solid solutions of hafnia
and zirconia, with no discernable segregation within the in-
dividual nanoparticles. A simple lattice dynamics model is
used to explore how changes in the average cation mass and
interatomic force constants with alloy composition affect the
Raman-allowed mode frequencies. The minimization of
background luminescence from these particles, which in
many cases is needed to be able to observe the Raman spec-
trum, is also addressed. The control of luminescence in these
and other nanocrystals can provide insight into the location
and control of defects.
1098-0121/2005/71͑11͒/115408͑8͒/$23.00
115408-1
©2005 The American Physical Society

ROBINSON et al.
II. EXPERIMENTAL PROCEDURE
PHYSICAL REVIEW B 71, 115408 ͑2005͒
TABLE I. Sizes of HfxZr1−xO2 nanoparticles analyzed after an-
nealing at 600 °C or sintering at 1200 °C, both for 1 h in air. x is
from ICP measurements. Phases are also shown from XRD, with t
for tetragonal and m for monoclinic. Particles were also sintered for
1 h in air at lower temperature: 900 °C ͑x=0, 8.4 nm; x=0.45, 5.4
nm; x=0.46, 7.2 nm͒ and 1000 °C ͑x=0.35,15,3 nm͒.
HfxZr1−xO2 nanoparticles were synthesized through a non-
hydrolytic sol-gel synthesis. The appropriate amounts ͑dic-
tated by the value of x͒ of Hf͑iso-propoxide͒4, HfCl4,
Zr͑iso-propoxide͒ and/or ZrCl4 were added, under argon, to
4
degassed trioctylphosphine oxide ͑TOPO, which acts as both
solvent for the reaction and surface ligand for the product
nanocrystals͒. The reaction mixture was heated quickly to
350 °C and held at this temperature for 2 h with vigorous
stirring. The reaction mixture was then cooled to ϳ60 °C
and acetone was added to precipitate the hafnia nanopar-
ticles. The precipitate was retrieved by centrifugation and
washed several times with acetone to remove excess TOPO.
The particles are capped with TOPO ligands and can be
readily redispersed in hexane producing a colorless solution.
Further details of these preparations can be found in Ref. 14.
Unless otherwise stated, before Raman analysis all as-
synthesized particles were annealed at 600 °C for 1 h in air.
Larger particles were obtained by sintering in air at tempera-
tures up to 1200 °C for 1 h. The average nanoparticle size
was determined by Debye Scherrer analysis of x-ray powder
diffraction ͑XRD͒ scans, which were recorded on a Scintag
X2 diffractometer using Cu Ka radiation ͑␭=1.54056 Å͒ op-
erating at 35 mA, 45 kV. The elemental composition of these
nanoparticles was determined by inductively coupled plasma
͑ICP͒ analysis.
Raman scattering was performed in a backscattering con-
figuration using the 488 nm and 514.5 nm lines of a continu-
ous wave argon ion laser ͑Coherent Innova 100͒, and the 325
nm line of a helium-cadmium laser ͑Omnichrome͒, at room
temperature unless otherwise stated. The beam was focused
to a spot size of ϳ2 ␮m and all incident power was less than
1 mW, to minimize heating in these powder samples. A 0.6 m
triple spectrometer ͑SPEX 1877, Triplemate͒ in subtractive
configuration was used to collect and disperse the spectra,
and a UV coated Si CCD array detector ͑SPEX Spectrum
One͒ collected the spectra. Plasma lines were used to cali-
brate the 70 to 800 cm−1 frequency range ͑resolution
ϳ2 cm−1͒. All peak intensities and positions are the result of
Lorentzian fitting. The luminescence from as-synthesized
nanoparticles was monitored as a function of temperature in
oxidizing ͑air or oxygen͒, reducing ͑forming gas: 95% N2
+5% H2͒, and presumably nonreactive ͑N2͒ gas environ-
ments. The heat-treated particles were then reexamined at
room temperature in the same gas environment. Any back-
ground due to luminescence was subtracted before determin-
ing peak shifts.
Particles annealed at
600 °C
Particles sintered at
1200 °C
Phase
from
Phase
from
Particle type
Size ͑nm͒
XRD
Size ͑nm͒
XRD
ZrO2
3.8
4.8
5.1
4.6
4.3
4.1
5.1
5.5
t
20.8
26.8
26.2
23.1
26.1
23.6
19.9
29.5
m
m
m
m
m
m
m
m
Hf0.12Zr0.88O2
Hf0.19Zr0.81O2
Hf0.35Zr0.65O2
Hf0.45Zr0.55O2
Hf0.46Zr0.54O2
Hf0.75Zr0.25O2
HfO2
t
t
t
t
t+m
t+m
t+m
increase noticeably around 900 °C. In the earlier work by the
authors,14 TEM showed that these particles are single crys-
tals. Table I gives the size and Hf fraction x ͑obtained from
ICP analysis͒ of particles examined by Raman scattering af-
ter annealing at 600 °C ͑3.8–5.5 nm dimension͒ and sintering
in air at 1200 °C for 1 h ͑20–30 nm͒. Raman scattering was
also examined for several particles sintered in air for 1 h at
lower temperatures: 900 °C ͑x=0, 8.4 nm; x=0.45, 5.4 nm;
x=0.46, 7.2 nm͒ and 1000 °C ͑x=0.35, 15.3 nm͒.
There was strong background luminescence when all as-
synthesized nanoparticles were excited with sub-bandgap ra-
diation at 488 and 514.5 nm. The emission was ϳ200 nm
broad, starting at the excitation wavelength, and prevented
obtaining a Raman spectrum. This luminescence was absent
or weak enough with 325-nm excitation in some ͑3.7 nm
HfO2, 5.5 nm HfO2, 4.3 nm Hf0.45Zr0.55O2, and 4.6 nm
Hf0.35Zr0.65O2͒ but not all particles.14
Heating as-synthesized particles to 150 °C for 10 min in
air decreased room-temperature luminescence by 40–70%
͑␭=514.5 nm͒, but no change was seen after heating under
these conditions in either N2 or forming gas. Heating at
higher temperature and longer times in air or O2 decreased
emission even more. All particles heated to 600 °C for 1 h in
air produced Raman spectra with no interfering background
emission at any of these three excitation wavelengths. The
luminescence intensity of as-synthesized particles measured
in air decreased by 20–70% after treatment by an oxygen
plasma for 30 min.
III. EXPERIMENTAL RESULTS
The Raman spectra of HfO2, ZrO2, and several
HfxZr1−xO2 nanoparticles taken at room temperature ͑␭
=514.5 nm͒ after annealing at 600 °C in air ͑Table I͒ are
shown in Fig. 1. The peak frequencies of the six modes that
are prominent for the xϽ0.46 particles—the six tetragonal
modes—are labeled T1–T6 in Fig. 1 and are plotted in Fig. 2
vs x. At each x examined, the mode structure in the spectra of
particles sintered at or below 1000 °C ͑Table I caption͒ was
The synthesized particles were typically quasispherical
nanocrystals with Ӎ3.8–5.5 nm dimensions, typically for x
Ͻ0.46, while those with xϾ0.46 were typically nanorods
with dimensions ϳ3 nm by ϳ8 nm, as shown by earlier
transmission electron microscope ͑TEM͒ analysis.14 Only the
quasispherical nanocrystals were examined here.
The average particle size, as determined by XRD, did not
change with heating up to 600 °C. Particle size began to
115408-2

RAMAN SCATTERING IN HfxZr1−xO2…
PHYSICAL REVIEW B 71, 115408 ͑2005͒
FIG. 1. Room temperature Raman spectra of HfxZr1−xO2 nano-
particles after heating at 600 °C in air for 1 h, taken with at ␭
=514.5 nm: ͑a͒ ZrO2, ͑b͒ Hf0.12Zr0.88O2, ͑c͒ Hf0.45Zr0.55O2, ͑d͒
Hf0.46Zr0.54O2, ͑e͒ Hf0.75Zr0.25O2, and ͑f͒ HfO2. The Lorentzian fits
of the peaks are shown only in ͑a͒. Peaks labeled T1–T6 are the six
tetragonal phase first order Raman modes. The two arrows point
from monoclinic phase peaks in spectra of lower x to the same
peaks at higher x. Plasma lines from the laser are denoted by an
asterisk.
similar to the corresponding structure in Fig. 1 ͑but is not
shown͒. Figure 3 shows the Raman spectra for the
ϳ20–30 nm dimension particles after sintering at 1200 °C
͑Table I͒.
FIG. 2. Measured frequencies of the tetragonal modes of the
HfxZr1−xO2 nanoparticles from Fig. 1 ͑circles fit to a solid line͒,
compared to model predictions. Model predictions include assum-
ing linear mass averaging only ͑– – – lines͒, and mass scaling with
force constant scaling of all six force constants of zirconia by a
factor of 1+0.09x ͑· · ·͒, similar scaling of only the four Zr-O
interplane force constants ͑– · – · –͒, and scaling of only these four
force constants by a factor of 1+0.2x ͑– · · – · · –͒. The results for
1+0.09x scaling of the four or all six force constants nearly overlap,
except for the T4 mode. The modes are T1–T6 from bottom to top.
͑Note the measured shifts of T3 exceed those of T2 for all x.͒
IV. DISCUSSION
A. Background emission
With 514.5 nm excitation at room temperature, lumines-
cence decreased greatly after either heating to only 150 °C in
oxygen-containing environments or exposure to oxygen plas-
mas, but not after heating in forming gas. This suggests that
oxygen is removing bulk or surface oxygen deficiencies or
defects, or is reacting with the surface ligands or other de-
fects that luminesce. After treatment at 600 °C, the back-
ground luminescence is weaker than the Raman scattering.
When the as-synthesized particles were excited at 325
nm, relatively low levels of luminescence were observed;
this luminscence did not interfere with Raman measure-
ments. ͑See also Ref. 14.͒ Note the band gap in bulk hafnia-
zirconia alloys is ϳ5–6 eV ͑ϳ200–250 nm͒ and is expected
to be no smaller than this in hafnia-zirconia nanoparticles.
Therefore, the 325, 488, and 514.5 nm excitations are all
below the band gap.
also the only ones seen for Hf0.12Zr0.88O2 nanoparticles. For
xϾ0.45, additional peaks are seen due to monoclinic-phase
modes. A very weak monoclinic peak is seen for
Hf0.45Zr0.55O2 nanoparticles at 513 cm−1 ͑see the tail of the
right arrow͒, which is likely due to a mode in the monoclinic
phase.1 More dominant monoclinic peaks are seen for larger
x. This suggests that only the tetragonal phase is present for
the smaller x͑Ͻ0.45͒, while there is evidence of both tetrag-
onal and monoclinic phases for larger x͑Ͼ0.45͒ and for
hafnium nanoparticles. This likely means that there is a mix-
ture of tetragonal and monoclinic particles and not mixed
phases within a particle, because TEM has shown that each
particle is a single crystal. Raman spectra were taken at vari-
ous points in the probed powder, because the employed Ra-
man diagnostic probes a lateral dimension of about 2 ␮m.
These spectra of different spots showed no essential differ-
ences.
B. Raman spectra
The room temperature Raman spectrum of the small zir-
conia nanoparticles in Fig. 1 ͑3.8 nm, annealed at 600 °C͒
has six peaks, labeled T1–T6 for increasing mode frequency
͑with peaks T2 and T3 overlapping͒; these six peaks are a
signature of the tetragonal phase. These tetragonal peaks are
The Raman mode frequencies vary with hafnium content.
Also, no pure zirconia or pure hafnia Raman modes are ob-
served for these alloy nanoparticles. Both observations dem-
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ROBINSON et al.
PHYSICAL REVIEW B 71, 115408 ͑2005͒
perature for bulk monoclinic HfxZr1−xO2.15 Given the large
width of the peaks for these very small particles, it is un-
likely that any two-mode behavior could be observed.
Zirconia-rich particles ͑xϽ0.45͒ sintered at 900 °C ͑5–8
nm dimension͒ have only tetragonal phase Raman peaks,
while the spectra of the even larger nanoparticles ͑xഛ0.75͒
sintered at 1200 °C ͑20–30 nm͒ have only monoclinic peaks
͑Fig. 3͒. This suggests the critical diameter for the transition
from the tetragonal to monoclinic phases is roughly between
6–8 and 20–30 nm in this composition range. In the one
sintering run at 1000 °C, Hf0.35Zr0.65O2 nanoparticles grew to
15.3 nm. The room temperature Raman spectrum of these
nanoparticles has only tetragonal peaks, so the critical diam-
eter for x=0.35 is between 15 and 23 nm.
The Raman identifications of tetragonal ͑Fig. 1͒, mono-
clinic ͑Fig. 3͒, and mixed phases ͑Fig. 1͒ were confirmed by
XRD analysis ͑Table I͒.
FIG. 3. Room temperature Raman spectra of HfxZr1−xO2 nano-
particles after sintering at 1200 °C in air for 1 h, taken with at ␭
=514.5 nm: ͑a͒ ZrO2, ͑b͒ Hf0.12Zr0.88O2, ͑c͒ Hf0.45Zr0.55O2, ͑d͒
Hf0.46Zr0.54O2, ͑e͒ Hf0.75Zr0.25O2, and ͑f͒ HfO2. The Lorentzian fits
Using Raman scattering for quantitative analysis
Peak T2 seems relatively unaffected by monoclinic peaks
and peak M12 seems to be relatively unaffected by tetrago-
nal peaks, and they are both strong, making them good can-
didates for use in composition and phase analysis.
of the M12 ͑475 cm−1͒, M16 ͑616 cm−1͒, and M17 ͑637 cm−1
͒
peaks that are used for composition analysis are shown only in ͑a͒.
a. Determining composition. Raman scattering can deter-
mine composition, and as such is a nondestructive alternative
to ICP analysis. Raman scattering has been used to determine
the composition in bulk, monoclinic HfxZr1−xO2 by tracking
the Raman shifts of two of the higher frequency peaks that
have been shown to vary smoothly and linearly with
composition.20,21 Although, XRD analysis can be used to de-
termine the composition in bulk HfxZr1−xO2 by measuring
the lattice spacing, this requires high-resolution XRD be-
cause the lattice constants of HfO2 and ZrO2 are nearly
equal.20 Such analysis is much more difficult for the alloy
nanoparticles because of peak broadening in the
nanoparticles.
onstrate that the particles are solid solutions with at least
fairly good mixing of Hf and Zr, with no discernable segre-
gation of hafnia and zirconia within the individual nanopar-
ticles. ͑Such mixing cannot be determined by usual powder
XRD because of the nearly identical lattice constants of
hafnia and zirconia.͒ This also indicates there are no distinct,
pure hafnia and pure zirconia particles for x not equal to 0 or
1. The dependence of the tetragonal mode frequencies are
modeled below on the basis of this mixing. Also, particle
compositions are determined below on the basis of this mix-
ing and the consequent dependence of the mode frequency
on x.
Not all 18 modes of the monoclinic phase can be identi-
fied in the Raman spectra of the hafnia-rich particles in Fig.
1, in part because of the peak broadening for these very small
particles.19 The assignments of the stronger monoclinic
modes M1-M18 are now presented. ͑They are as in Ref. 1,
but with M7 and M8 reversed and M16 and M17 reversed,
so the frequencies monotonically increase.͒ The strong and
sharp modes M2 near 135 cm−1 and M4 near 150 cm−1 in
pure monoclinic hafnia overlap the T1 peak. The observed
shoulder at 240 cm−1 in the hafnia particles here is due to the
medium strong hafnia M6-M8 peaks at 242, 256, and
270 cm−1. For mixed tetragonal/monoclinic phases, spectral
analysis of T2 and T3 could be weakly affected by these
peaks and by the hafnia M9 mode near 336 cm−1. Strong
hafnia monoclinic peaks near 384 and 398 cm−1 ͑M10 and
M11͒ are seen as a broad peak near 390 cm−1 in the mixed
phases. The hafnia M12 peak at 503 cm−1 is very strong, and
it is seen here as a peak for xജ0.46 nanoparticles, which has
a low frequency shoulder due to T4. The strong and broad
hafnia modes M16 ͑642 cm−1͒ and M17 ͑672 cm−1͒ are seen
in Figs. 1͑d͒–1͑f͒ to overlap T6.
Figure 2 shows that the T2 peak frequency varies fairly
linearly with x, and so composition could be determined by
linearly interpolating between the measured mode 2 frequen-
cies for the zirconia ͑x=0, 274 cm−1͒ and hafnia ͑x
=1, 293 cm−1͒ particles. Table II shows the agreement be-
tween the ICP and Raman scattering is quite good and within
Ϯ 0.1. In principle, variations in mode shifts with diameter
could be significant in using such methods to determine com-
position in alloy particles. ͑All of the tetragonal particles
have about the same diameter here.͒
The M12 peak frequencies for samples annealed at
600 °C show
a
clear variation with composition:
Hf0.46Zr0.54O2: 487 cm−1, Hf0.75Zr0.25O2: 502 cm−1, and
HfO2: 505 cm−1. This monoclinic-phase peak M12 was used
to analyze composition in the larger ͑20–30 nm͒,
monoclinic-phase particles that had been heated at 1200 °C
͑Fig. 3͒. Linearly interpolating between the frequencies mea-
sured for these zirconia and hafnia nanoparticles ͑475 and
503 cm−1͒, assuming single-mode behavior,15 gives x in
Table II. This was also done using peaks M16 and M17 ͑616
and 637 cm−1 in zirconia, and 641 and 675 cm−1 in hafnia͒,
No evidence of two-mode behavior is seen for any modes;
evidence of some two-mode behavior was seen at low tem-
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RAMAN SCATTERING IN HfxZr1−xO2…
PHYSICAL REVIEW B 71, 115408 ͑2005͒
TABLE II. Comparison of composition determined from Raman
scattering to that from ICP ͑given as x in HfxZr1−xO2͒ for tetragonal
particles ͑annealed at 600 °C; using peak T2͒ and monoclinic par-
ticles ͑sintered at 1200 °C; using either peak M12 or peaks M16 and
M17͒. For Hf0.46Zr0.54O2 particles annealed at 600 °C ͑mixed
phase͒, xm=0.43 from the monoclinic peak 12. See Table I for par-
ticle dimensions.
is more difficult to address.͒ For the x=0.46 mixed phase
particles previously heated to 600 °C, xt ͑using the T2 peak͒
is 0.32 and xm ͑using the M12 peak͒ is 0.43 ͑see Table II
caption͒. For the x=0.75 mixed phase particles previously
heated to 600 °C, xt ͑using the T2 peak͒ is 0.63 and xm ͑using
the M12 peak͒ is 0.96. In both cases the tetragonal phase
seems to be relatively rich in zirconium and the monoclinic
phase relatively rich in hafnium. This is reasonable because
the monoclinic phase appears in these smaller particles when
the hafnium fraction increases. As the particles grow during
heating, one might expect xt and xm to both approach x ͑and
xm to decrease here͒. For the x=0.46 and 0.75 particles pre-
viously sintered at 1200 °C ͑all monoclinic͒, x=xm ͑using the
M12 peak͒ is 0.36 and 0.71 respectively, and so both cases
show this trend. ͑Note that if the different phases have dif-
ferent compositions, perhaps these nanoparticles should not
be termed HfxZr1−xO2 particles, although these mixed phase
particle ensembles can still be characterized by an average
hafnia fraction x.͒
Particles
annealed
at 600 °C
Particles sintered at
1200 °C
xt from
tetragonal
peak 2
xm from
monoclinic
peak 12
xm from
monoclinic
peaks 16 and 17
Particle type
ZrO2
0.00
0
0
Hf0.11Zr0.89O2
Hf0.12Zr0.88O2
Hf0.19Zr0.81O2
Hf0.21Zr0.79O2
Hf0.35Zr0.65O2
Hf0.45Zr0.55O2
Hf0.46Zr0.54O2
Hf0.75Zr0.25O2
HfO2
0.07
0.04
0.07
0.00
0.25
0.39
0.36
0.71
1
0.02
0.01
0.12
0.19
0.32
0.42
0.05
0.16
0.10
0.42
0.42
0.32
0.63
1.00
C. Lattice dynamics model
The simple linear chain model developed by Bouvier and
Lucazeau in Ref. 18 to characterize and assign the six al-
lowed tetragonal Raman modes in zirconia nanoparticles is
slightly modified and used here to examine the variations of
the mode frequencies in tetragonal HfxZr1−xO2 nanoparticles
with composition ͑3.8–5.5 nm, annealed at 600 °C͒. For the
solid solution nanoparticles, the model uses masses and force
constants that are averaged over composition. In the original
linear lattice chain model for zirconia, each atom in the
primitive cell is assigned to one of six planes successively
containing Zr, O, O, Zr, O, O, and these planes are coupled
through interactions with differerent force constants, as out-
lined in the Appendix here. Reference 18 used this model to
deduce that the ͑T1,T2,T3,T4,T5,T6͒ modes have
͑Eg,Eg,B1g,Eg,A1g,B1g͒ symmetry. The force constants Cd1
and Cd2 represent the two different effective stretching inter-
actions of the Zr and O atom planes and can be thought of as
projections of the real force constants on the axis normal to
the planes, and Cw describes non-nearest neighbor contribu-
tions and angular interactions that are effectively described
as stretching interactions of two O planes. The frequency of
the A1g mode is determinied by Cd1+Cd2, while the two B1g
modes are determined by Cd1, Cd2, and Cw. Analogous shear
interactions are characterized by the Csd1, Cds2, and Cws that
describe the three Eg modes. The six force constants for te-
tragonal zirconia were determined from the Raman shifts of
the T1-T6 modes of the 3.8 nm zirconia particles, and are
given in the Appendix.
0.96
1
as in Ref. 21 ͑by averaging x determined using either M16 or
M17͒. There is poor agreement with the ICP results for x
Ͻ0.2 and good agreement for higher x using the M12 peak
and for x between 0.2 and 0.45 when using the M16 and
M17 peaks.
b. Determining the fractions of tetragonal and monoclinic
phases. Peaks T2 and M12 can be used to determine the
fractions of tetragonal and monoclinic phases, ft and fm ͑with
ft+ fm=1͒, for any x, and serve as an alternative to XRD. For
xജ0.46 ͓Figs. 1͑d͒–1͑f͔͒, the ratios of the peak intensities
and integrated areas of M12 and T2 are 1.3 and 2.2 for the
x=0.46 particles and 1.7 and 2.0 for the x=0.75 particles.
Analysis of the XRD spectrum gives the ratio of monoclinic
to tetragonal phase of 1.9 for the x=0.46 particles and 1.0 for
the x=0.75 particles, so the M12 and T2 peaks are very
roughly the same strength for the same amount of mono-
clinic and tetragonal material probed at 514.5 nm. In con-
trast, Kim et al.22 claim that monoclinic zironica has a stron-
ger Raman spectra than tetragonal zirconia, on the basis of
the T1 peak near 148 cm−1 and the monoclinic peaks from
180–192 cm−1 ͑M2–M4͒ peaks, which are not resolvable for
the very small particles here.
c. Evaluating possible different compositons in different
phases. In the mixed phase solid solution samples, x deter-
mined by ICP is an average of both phases. It is possible that
the hafnium fractions in the tetragonal and monoclinic phase
particles, xt and xm, are different; either xt and xm could be
Ͻx and the other Ͼx—yet still with xtft+xmfm=x. Raman
scattering is well suited to see if this occurs ͑Table II͒. ͑Also,
xt and xm, could be averaged values for each phase and there
could be dispersion about these averages for each—but this
In the alloy model, the average cation mass ͗mcation,1͑x͒͘
=͑1−x͒mZr+xmHf replaces mZr, where mZr and mHf are the
masses of zirconium and hafnium ͑91.22 and 178.49 amu,
respectively͒, to model the changes in mass with composi-
tion. Such mass scaling is not unreasonble with one-mode
alloy behavior. The force constant for the Hf-O stretch is
about 9% higher than the Zr-O stretch in diatomic molecules.
͓This is determined using the HfO and ZrO vibrational ener-
115408-5

ROBINSON et al.
PHYSICAL REVIEW B 71, 115408 ͑2005͒
gies, which are almost exactly the same, ϳ969 cm−1,23–25
and their ͑different͒ reduced masses.͔ The force constants for
zirconia were either used directly ͑as a base case to examine
mass effects only͒ or the zirconia constants were multiplied
by 1+0.09x to provide scaling to hafnia.
modes are well described by the model. This again suggests
consistency with ͑though not proof of͒ the assignment that
the two lowest modes have this same symmetry.
The measured frequencies are higher than predictions for
modes 3 and 6, the two B1g modes, with either model using
1+0.09x force constant scaling. This could indicate different
scalings of several of the force constants ͑some should scale
much faster than assumed here on the basis of the different
IR frequencies in zirconia and hafnia4,5͒, an incorrect mode
assignment, or an inherent weakness of this simple lattice
plane model with averages to account for the composition
changes. Figure 2 also shows the faster 1+0.2x scaling of the
four Zr-O interplane force constants, which explains mode 6
better. There is no indication of a decrease of mode fre-
quency in tetragonal particles as x approaches 0.46 ͑and the
transition to the monoclinic phase at higher x͒ beyond that of
the model expectations ͑for every model that includes the
scaling of the force constants͒, and as such there is no evi-
dence of a soft mode.
Use of such a simple lattice model presupposes bulklike
material. For successively smaller particles, the Raman spec-
tra change for several reasons and each Raman mode can
change differently. A lattice model can account for some, but
not all, of these effects. Strain can change with particle size
due to surface tension, defects, and so on, and this can be
incorporated as force constants that change with size. Pho-
non dispersion, and consequently phonon confinement, is
different in smaller particles and this and other factors, in-
cluding particle shape, are less easily incorporated into a
simple lattice model. For very small particles, even small
dispersion in particle size can strongly affect the Raman
spectrum.19 For example, in ceria nanoparticles, the cubic
Raman peak becomes more redshifted and broader for
smaller particles sizes because of strain, phonon confine-
ment, and size dispersion.19 The decrease in the Raman peak
linewidth of zirconia nanoparticles from room temperature to
96 K, measured in Ref. 10, suggests that phonon confine-
ment may not make the dominant contribution to the line-
widths observed here, at least for zirconia nanoparticles. In
any case, such analysis of the effect of size is beyond the
scope of the present treatment. ͑All of the tetragonal particles
described in Figs. 1 and 2 have about the same dimension, so
no scaling with diameter is needed for the model presented
in Fig. 2.͒
The model was tested using the mode assignments for the
T1-T6 modes zirconia from Bouvier and Lucazeau, and al-
ternatively, earlier proposed and different symmetry assign-
ments of the six modes: ͑B1g,Eg,B1g,Eg,A1g,Eg͒,16
͑Eg,B1g,A1g,Eg,B1g,Eg͒,26 ͑B1g,Eg,A1g,Eg,B1g,Eg͒,27 and
͑Eg,A1g,B1g,Eg,B1g,Eg͒.28 The assignments from Ref. 18
were the only ones to yield reasonable results for the ZrO2
and HfxZr1−xO2 nanoparticle mode models. Assumption of
the other mode assignments led to negative or imaginary
force constants. This is noted, although it is not proof that the
Ref. 18 assignments are correct for zirconia or the alloys.
This assignment has been confirmed for zirconia by the lat-
tice dynamics study of Ref. 29. Another strength of the Bou-
vier and Lucazeau assignment is that it is based in part on
their observation of anticrossing of the two lowest frequency
modes ͑T1 and T2͒ at elevated pressure, which indicates they
have the same symmetry. The frequency of the T2 mode
decreased with pressure, and was identified as the soft mode
leading to the transition to an intermediate tetragonal struc-
ture preceding the transformation to the cubic structure.30
Figure 2 compares the experimental Raman shifts with the
lattice model predictions, with the model using
compositionally-averaged cation masses, either with or with-
out the scaled force constants described above. This com-
parison should be evaluated seriously only for x up to 0.45—
although some data are provided for larger x—because of
uncertainties with overlapping monoclinic modes for larger
x. With such scaling of the cation mass and of all six force
constants, the model accounts for modes 1, 2, and 5 fairly
well. Model improvements could come from different scal-
ing of the cation masses or force constants.
From Eqs. ͑A1͒–͑A3͒, one Eg mode ͑mode 2͒ and the A1g
mode ͑mode 5͒ do not depend on cation mass, because the
cation does not move in the vibration. The increase of force
constants with x accounts for the observed slow increase of
their frequencies with x. For the other four modes, this mass
variation decreases the frequency by a factor between 1 and
1/2
͓͑mZr+2mO͒/͑Ͻmcation͑x͒Ͼ +2mO͔͒ ͑which is Ͻ1͒, while
bond stiffening tends to increase it with x. This mass factor is
nearly 1 for modes 4 and 6, and nearer the lower limit for
modes 1 and 3. With the stated force scaling, the predicted
decrease of frequency with x is faster than the observed de-
crease for modes 1 and 3, and, the predicted increase with x
is slower than the observed increase for mode 6, which sug-
gest the need for a slower mass variation and/or a faster
stiffening with x. Indeed, averaging the reciprocals of the
cation masses ͗mcation,2͑x͒͘=1/͓͑1−x͒/mZr+x/mHf͔ gives a
slower mass variation, but this change has a relatively minor
effect.
V. CONCLUDING REMARKS
Raman scattering demonstrates that the HfxZr1−xO2 par-
ticles are solid solutions of hafnia and zirconia, with no dis-
cernable segregation within the nanoparticles and there are
no distinct hafnia and zirconia particles. A simple lattice dy-
namics model with composition-averaged cation mass and
scaled force constants is used to understand how the Raman
mode frequencies vary for these alloys. Background lumi-
nescence from these particles is minimized after oxygen
treatment, suggesting possible oxygen defects in the as-
prepared particles. Raman scattering can also provide semi-
quantitative, nondestructive analysis of composition and the
Only mode 4 near 460 cm−1 is significantly affected when
instead of scaling all six force constants, only the four Zr-O
interplane force constants ͑Cd1,Cd2,Csd1,Cds2͒ are scaled and
the two effective interactions between the O-O planes
͑Cw,Csw͒ are not. Then this mode and therefore all three Eg
115408-6

RAMAN SCATTERING IN HfxZr1−xO2…
PHYSICAL REVIEW B 71, 115408 ͑2005͒
The effective shearing interactions by the Csd1, Cds2, and Cws
are force constants between the analogous pairs of planes.
The eigenfrequency ␻ of the A1g mode is given by
fraction of each phase. In some regimes there are mixed
phases, and Raman analysis suggests that in these regimes
the tetragonal phase particles are relatively rich in zirconium
and the monoclinic phase particles are relatively rich in
hafnium.
2
MO␻ ͑A1g͒ − ␣ = 0,
͑A1͒
ACKNOWLEDGMENTS
where ␣=Cd1+Cd2 and MO is the mass of an oxygen atom.
The eigenfrequencies of the two B1g modes are given by
This work was supported primarily by the MRSEC Pro-
gram of the National Science Foundation under Grant Num-
ber DMR-0213574 and by the New York State Office of
Science, Technology and Academic Research ͑NYSTAR͒,
and also by the Ford Foundation and the NSF IGERT Pro-
gram Grant Number DGE-9972892. The authors would like
to thank Siu-Wai Chan for discussions and Jason Fabbri for
help in synthesizing the particles and analyzing them by
XRD.
2
2
͓MZr␻ ͑B1g͒ − 2␣͔͓MO␻ ͑B1g͒ − ͑␣ + 2Cw͔͒ − 2␤2 = 0,
͑A2͒
where ␤=Cd1−Cd2 and MZr is the mass of a zirconium atom.
The eigenfrequencies of the three Eg modes are given by
2
2
2
͓MO␻ ͑Eg͒ − ␣s͔͕͓MZr␻ ͑Eg͒ − 2␣s͔͓MO␻ ͑Eg͒
− ͑␣s + 2Csw͔͒ − 2͑␤s͒2͖ = 0,
͑A3͒
APPENDIX
In the Bouvier and Lucazeau model of tetragonal
zirconia18 the cell is modeled as six planes successively con-
taining Zr, O, O, Zr, O, O ͑planes 1 to 6͒. The force constant
Cd1 represents the effective stretching interactions of Zr
planes with the nearest neighbor O atom planes ͑1-2, 3-4, 4-5
and those with adjacent cells͒ and Cd2 represents the effec-
tive stretching interactions of Zr planes with the next-nearest
neighbor O atom planes ͑1-3, 2-4, 4-6 and those with adja-
cent cells͒. Cw describes non-nearest neighbor contributions
and angular interactions and are effectively stretching inter-
actions of pairs of next-nearest neighbor O planes ͑2-5, 3-6͒.
where ␣s=Csd1+Cds2 and ␤s=Csd1−Cds2.
In the current model the cation mass ͑above MZr͒ is re-
placed by the cation-averaged mass, and either the zirconia
or cation-scaled force constants are used. The zirconia inter-
plane force constants used here were fit to the measured Ra-
man shifts of the T1-T6 modes of the 3.8 nm zirconia par-
ticles, 145, 274, 312, 457, 592, and 635, cm−1, and were
Cd1=2.33, Cd2=0.99, Cw=0.13, Csd1=0.60, Cds2=0.11, and
Csw=0.61 N cm−1, which are a bit different from those deter-
mined in Ref. 18.
Corresponding author. Email address: iph1@columbia.edu
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