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Dispersive Phonon Linewidths: The E2Phonons of ZnO
J. Serrano,
1,
*F. J. Manjo
´n,
1,2
A. H. Romero,
3
F. Widulle,
1
R. Lauck,
1
and M. Cardona
1
1
Max-Planck-Institut fu
¨r Festko
¨rperforschung, Heisenbergstrasse 1, 70569 Stuttgart , Germany
2
Depar ta mento de Fı
´sica Aplicada , Universit at Polite
`cnica de Vale
`ncia, E.P.S.A. ES-03801 Alcoy, Spain
3
Advanced Materials Depar tment, IPICyT, 78231 San Luis Potosı
´, Mexico
(Received 14 October 2002; published 7 Febr uary 2003)
Phonon linewidths ca n exhibit a large variation when either pressure or isotopic masses are changed.
These effects yield deta iled information about the mechanisms responsible for linewidths and lifetimes,
e.g., a nhar monicit y or isotopic disorder. We report Raman measurements of the linewidth of the upper
E2phonons of ZnO cr ystals wit h several isotopic compositions and their dependence on pressure.
Changes by a factor of 12 are observed at a given temperature. Comparison with ca lculated densities of
one-phonon states, responsible for isotope scattering, and of two-phonon states, responsible for
anharmonic decay, yields a consistent picture of these phenomena. Isotopic disorder broadening by
7cm
1is found in samples with mixed 16 O–18Ocontent, whereas the anharmonic processes involve
decay into sums and differences of two phonons.
DOI: 10.1103/PhysRevLett.90.055510 PACS numbers: 63.20.Kr, 63.20.Dj, 78.30.Fs
Considerable progress has been made in recent yea rs in
the understanding of the mechanisms which determine
phonon linewidt hs [1– 3]. The relevant experi mental dat a-
base has been obtained mainly by Raman spectroscopy
on samples with diverse isotopic compositions and under
hydrostatic stress [4]. The theoretical underpinnings have
been extracted from ab initio electronic structure calcu-
lations, mostly for diamond- and zinc-blende-type semi-
conductors [5]. Unfortunately, these materials have only
one set of Ra man-active phonons because of the presence
of only two atoms per primitive cell (PC).
In this Letter we investigate the technologically im-
portant material ZnO whose wurtzite structure allows
four sets of Ra man-active phonons of symmetries A1,
E1,and2E2. We concentrate on the upper E2phonon,
Ehigh
2, located at 439 cm1at 300 K for the natural iso-
topic abundances, and show that its width can change by
as much as a factor of 5 upon application of pressure and
12 by changing the isotopic composition.
Natural ZnO has a mixed isotopic zinc composition but
nearly isotopically pure 16O. T his results in negligible
isotopic disorder effects for the Ehigh
2modes since they
involve mainly oxygen displacement s. In samples with
16O0:518 O0:5composition, a large broadening of the Ehigh
2
mode is observed and analyzed on the basis of standard
perturbation theor y using an ab initio calculated one-
phonon density of states (DOS). For samples containing
only natural oxygen or pure 18Othe phonon linewidth is
determined by anha rmonic decay into pairs of phonons.
Accidentally, the phonon frequency lies on a steep and
structured ridge of the two-phonon (sum) DOS. The
anharmonic width can thus be varied widely by applica -
tion of pressure, which displaces the Ehigh
2frequency along
the ridge, and also by isotopic mass substitution, since the
Ehigh
2frequency depends almost only on the oxygen mass,
whereas the two-phonon DOS depends only on the zinc
mass in that region. Measurements of these effects at 7 K
and at 300 K yield a con sistent picture of the mecha nisms
which determine the linewidth of the Ehigh
2phonons:
anharmonic decay into sums and differences of phonons
plus elastic scattering in the cr ystals with mixed isotopic
oxygen composition.
We have performed Raman measurements on ZnO
grown by chemical vapor transport with NH4Cl as trans-
porting agent and a source temperature of 900 C[6]. As
starting constituents we have used 64 Zn,68 Zn,18O,and
natural oxygen (99.76% 16O), the first three elements
having an isotope purity of 99%. Cr ystals with all
four combinations of the ‘‘pure’’ isotopes, as well as
natural zinc, with an average mass of 65.39 amu, were
grown. We also grew cr ystals with 64 Zn0:568Zn0:5and
16O0:518 O0:5so as to investigate isotopic disorder effects.
The Ra man measurements were performed using the
676, 568, and 514 nm lines from a Kr-Armixed gas
laser, the resolution obtained being, of course, higher
with the 676 nm line, i.e., 0:6cm
1for measurements
at low temperature and 2cm
1at high temperature. The
pressure was applied at room temperature with a diamond
anvil cell, with a 4:1 methanol-ethanol mixture as pres-
sure transmitting fluid. Such a mixture gives hydrostatic
pressures within the resolution at hand and the pressure
range under consideration, 0 – 8 GPa [7]. The measure-
ments, involving different pressures, temperatures, and
isotopic composition, were restricted to the Ehigh
2Raman-
active phonons of ZnO for which we were aware that
anomalies took place [8–10]. Cursory checks for the other
Raman phonons did not show any effects as striking as
the ones discussed here. The Elow
2phonon was found to be
extremely na rrow, well below our spectral resolution
especially at low temperature (0:1cm
1)[11].
Figure 1 shows three typica l Raman spectra at 7 K,
two obtained for 64 Zn18Oand 68Zn16Owith a spectral
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resolution of full width at half ma ximum (FWHM) equal
to 0:6cm
1. The deconvoluted Lorentzian FWHM, L,
can be easily obtained from those in Fig. 1 using [12]
L2
G
;(1)
where is the measured FWHM and Gthat of the lines
obtained from a calibration neon lamp which yields
Gaussian line shapes. This procedure leads to L
4:5cm
1for 64Zn18 Oand L1cm
1for 68Zn16 O,a
rather remarkable change. The Ra man spectrum of
natZn16 O0:518O0:5of Fig. 1, natZn representing the natural
element, was measured with a resolution of 2cm
1.Its
large and asymmetric width reveals the effect of the mass
fluctuations of the oxygen atoms, that of the zinc fluctua-
tions being nea rly negligible since the Ehigh
2mode corre-
sponds (85% [8,13]) to a vibration of the oxygen atoms.
We have observed an effect of the zinc mass fluctuations
on the Elow
2phonon, but it is not discussed here in detail.
The dashed line through the experimental points for the
isotopically mixed sa mple of Fig. 1 represents a theoreti-
cal fit to be discussed below.
Figure 2 displays the frequency dependence of the
widths of the Ehigh
2phonons of all isotopically pure crys-
tals and those containing natural and 64 Zn0:568Zn0:5
mixed zinc compositions measured at low temperature.
The effect of the spectral resolution has been removed
with Eq. (1). The line through the points represents the
DOS for sums of two phonons with equal qvectors
calculated from the ab initio lattice dynamics [8,13].
Raman phonons, at q’0, are known to decay mainly
into such combinations [1–5]. The frequency range of
Fig. 2 corresponds to a sum of transverse and longitudinal
acoustic phonons (TA LA) in the vicinity of the Kpoint
of the Brillouin zone. The frequency of these phonons
varies like M1=2
Zn , where MZn is the average mass of the
zinc atoms, and is nearly independent of the oxygen mass,
a conjecture that was checked by cha nging the isotopic
masses in the ab initio calculation. The experimental
points of the 64 Zn samples were plotted at the measured
frequencies, whereas those with other zinc masses were
shifted to take into account the M1=2
Zn dependence of the
abscissa of t he two-phonon DOS, 2
!.We note in Fig. 2
that the experimental points follow rather well the calcu-
lated dependence of 2
!on frequency. The factor of
57 cm2, by which 2
!had to be multiplied to repro-
duce the measured FWHM, represents the corresponding
average anharmonic squared matrix element which was
found to have the rather simila r values of 56 cm2for
GaP [14] and 70 cm2for CuCl [15]. This matrix element
thus seems to increase with increasing ionic character of
the material. Notice that the calculated curve reproduces
the experimental trend and also reveals the large differ-
ence in widths between 64 Zn18Oand 68Zn16 Oalready
mentioned in connection with Fig. 1.
Figure 3 displays the pressure dependence of the
FWHM of the Ehigh
2phonon peaks of nat Zn16O,64Zn16 O,
and 64Zn18Omeasured at room temperature for pressures
up to 8 GPa. At this pressure a transition to the rocksalt
400 410 420 430 440 450 460
Raman shift [cm−1]
0.02
0.07
0.12
0.17
0.22
Normalized Raman intensity [a. u.]
64Zn18O
natZn16O0.5
18O0.5
68Zn16O
E2
high
7 K
FIG. 1. Raman spectra of the Ehigh
2phonons at 7 K for three
ZnO samples. Note the broad and asymmetric spectr um of the
16O0:518 O0:5sample. The dashed line is a fit with a Lorentzia n
including spectral resolution, the an ha rmon ic contribution and
the disorder effects (see text).
400 410 420 430 440 450 460 470
Frequency [cm−1]
0
2
4
6
8
10
12
ΓL (FWHM), ρ(2)
+(ω)x 57 [cm−1]
1
natZn16O0.5
18O0.5
68Zn16O
64Zn0.5
68Zn0.5
16O
natZn16O
64Zn16O
64Zn18O
natZn18O
68Zn18O
64Zn0.5
68Zn0.5
18O
0 200 400 600
Frequency [cm−1]
0.00
0.02
0.04
ρ(1)(ω) [states/cm−1]
7 K
FIG. 2. Resolution-corrected FWHM of the Ehigh
2phonons
obtained from 7 K Raman spectra for several isotopic compo-
sitions. The points corresponding to 64 ZnxOsamples a re plot-
ted at the measured frequencies, the others have been shifted as
explained in the text. The solid line displays the calculated
2
!scaled by a factor of 57 cm2. The la rge width observed
for natZn16 O0:518O0:5illustrates the effect of isotopic mass
fluctuations. Similar values of F WHM were found for ot her
samples containing the same mixture of O isotopes. The inset
displays the one-phonon DOS, 1!.
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phase occurs [9,10]. The widths in Fig. 3 have been
corrected for spectral resolution with Eq. (1) using G
2cm
1. The ma ximum width shown in Fig. 3, for
64Zn18 O,is13 cm1, whereas a minimum width of
3:5cm
1is seen for the natZn16Osample. The non-
monotonic changes with pressure seen in Fig. 3 corre-
spond rather well to the features in 2
!displayed in
Fig. 2. The difference between the curves in Fig. 3, drawn
as a guide to the eye th rough the measured points, arises
from the different zero pressure frequencies for the three
samples.
In order to compa re these curves with 2
!, one
has first to rescale the frequencies with the zinc mass
using one, i.e., 64Zn, as reference, as done for Fig. 2. T hen,
we must take into account the shift of 2
!with
pressure and subtract it from the mass-renormalized fre-
quencies. The pressure dependence of 2
!was ob-
tained from ab initio calculations performed at 0, 4,
and 8 GPa [13]. A fter this process, the experimental
points of Fig. 3 coalesce into a single curve, as displayed
in Fig. 4. The agreement among the three sets of mea-
surements and the calculated DOS is excellent provided
one adds a constant background to 2
!. Subtraction of
this background makes the ratio of ma ximum to mini-
mum width of about five seen in Fig. 2 equal to that in
Fig. 4, as expected for a mechanism involving decay into
the sum of two phonons of frequencies !1and !2, which
leads to the temperature dependent width
!B2
!1nBE!1nBE !2;(2)
where !!1!2,nBE!are Bose-Einstein factors,
and B57 cm2is the anharmonic squared matrix
element. The calculated dispersion relations of ZnO yield
!1250 cm1and !2190 cm1. According to Eq. (2)
the ratio of the widths for either two different samples or
pressures should be independent of temperat ure.
We now explain the background of 2cm
1used in
Fig. 4, but not in Fig. 2, as a decay into differences of two
phonons represented by 2
!. This contribution is
!B2
!nBE!2nBE !1;(3)
where !!1!2,and!1;2are the average fre-
quencies which contribute to the difference mode. Bis
an adjustable parameter corresponding to the an harmonic
squared matrix element. 2
!extends from !0
to !Max, where !Max is the highest phonon frequency
550 cm1[8,13]. A calculation [13] reveals that
2
!is almost flat in the 400–460 cm1region of
interest in Figs. 2 and 4, 2
!0:25 states=cm1per
PC. It also shows that !1is a longitudinal optic mode of
average frequency 550 cm1, whereas !2110 cm1
has TA character. According to Eq. (3), !can be
neglected at 7 K, whereas nBE!2nBE !11:36 at
300 K. The dashed line of Fig. 4 can therefore be assigned
to 2
!and the cor responding matrix element squared
is B6:3cm
2, considerably smaller than that which
corresponds to 2
!,B57 cm2.
For completeness, we display in Fig. 5 all data col-
lected at 300 K for cr ystals containing either 16 Oor 18O
with t he frequencies shifted as done in Fig. 2 in order to
compare them with a single 2
!curve. For the sake of
clarity the pressure data of Fig. 4 have been replaced by a
400 410 420 430 440 450 460 470
Frequency [cm−1]
2
4
6
8
10
12
14
ΓL (FWHM), Γ
++Γ
− (ω) [cm−1]
theory
64Zn18O
64Zn16O
natZn16O
300 K
Γ
+(ω) + Γ
−(ω)
Γ
− (ω)
FIG. 4. Experimental points of Fig. 3 replotted vs pea k fre-
quency, with the frequency shifts and scaling described in the
text. The solid line was obtained by adding to the !of
Eq. (2) a constant at tributed to !(dashed line).
02468
Pressure [GPa]
2
4
6
8
10
12
14
ΓL (FWHM) [cm−1]
64Zn18O
64Zn16O
natZn16O
300 K
T
T
FIG. 3. Effect of hydrostatic pressure on the linewidths of
three ZnO samples after correcting for spectra l resolution. The
measurement s were performed at 300 K. T he lines are g uides to
the eye.
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smooth dot-dashed curve drawn through the experimen-
tal points. All isotopes measured at zero pressure agree
with this curve within error ba rs.
Finally, we discuss the effect of the oxygen mass
fluctuations on the Ehigh
2linewidths as determined at 7 K
and at 300 K for sa mples containing 16O0:518 O0:5.Four
different sa mples were measured containing 64Zn,68Zn,
natZn,and64Zn0:568Zn0:5, respectively. The 7 K Raman
spectrum for natZn16O0:518 O0:5, as displayed in Fig. 1,
shows a FWHM equal to 12 cm1,7cm
1larger than
the expected value of 5cm
1which can be read off Fig. 2
(see ar row). A similar increase in FWHM is found to be
induced by the isotopic disorder at 300 K, as expected.
The isotopic disorder effect for the FWHM of the Ehigh
2
phonons of ZnO ca n be calculated with [3]
iso g2
6!21!jeOj4;(4)
where g2is the oxygen mass variance (g23:5103
for 16O0:518 O0:5), 1!is the one-phonon DOS, and eO
is the eigenvector component of the oxygen for the
mode under consideration. 1!, as calculated for the
ab initio lattice dynamics, has a sha rp peak with a
maximum slightly above the Ehigh
2frequency (see the inset
in Fig. 2). At this frequency, 1!0:021 states=cm1
per PC and jeOj20:85 [8]. Using Eq. (4) we thus find
iso 5:05 cm1, in reasonable agreement with the value
determined above. It is also possible to calculate
‘‘ab initio’’ the asymmetric profile of Ehigh
2given in
Fig. 1 for natZn16O0:518 O0:5. For this purpose we approxi-
mate the calculated 1!by a quadratic polynomia l fit
and replace the corresponding iso of Fig. 4 into a Lorent-
zia n expression. We obt ain in th is ma nner t he da shed line
through the experimental points given in Fig. 1.
In summar y, we have performed Raman measurements
of the Ehigh
2phonons of ZnO with several isotopic compo-
sitions at 7 K and at 300 K, and pressure dependence
measurements at 300 K. Striking differences a re found in
the linewidth at low and high temperat ures between the
different sa mples, together with a nonmonotonic depen-
dence of the linewidth on pressure. This unusual behavior
can be explained by the existence of a steep ridge in the
two-phonon DOS and its mass and pressure dependence,
obtained with the aid of first principles calculations. The
role played in the linewidth by isotopic oxygen mass
fluctuations is also reproduced by the enhanced one-
phonon DOS using perturbation theor y. This DOS is
responsible for the asymmetric line shape observed in
the spectra of samples containing mixed oxygen isotopes.
We are indebted to I. Loa for a critical reading of the
manuscript. We have benefited from discussions with
A. Rubio and A. Polian, and we thank K. Syassen for
allowing us to use his pressure equipment. F. J. M. and
A. H. R. acknowledge support by the ‘‘P rograma Incen-
tivo a la investigacio
´ndelaU.P.V.’’andbyMillen-
nium Conacyt, Initiative Mexico, Grant No. W-8001,
respect ively. M. C. a lso t hanks the Fonds der
Chemischen Industrie for support.
*Corresponding aut hor.
Electronic address: J.Serrano@f kf.mpg.de
[1] M. Canonico et al., Phys. Rev. Lett. 88, 215502 (2002).
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075206 (2002).
[4] C. Ulrich et al., Phys. Rev. Lett. 78, 1283 (1997); F. J.
Manjo
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[6] R. Lauck (unpublished).
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[8] J. Serrano et al., Phys. Status Solidi (b) 235, 260 (2003).
[9] F. Decremps et a l., Phys. Rev. B 65, 092101 (2002). A
change in the width of Ehigh
2with pressure can be seen in
Fig. 1 of t his work.
[10] F. J. Manjo
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[12] J. F. Kielkopf, J. Opt. Soc. Am. 63, 987 (1973).
[13] A. H. Romero (unpublished). The calculated pressure
dependence for the ridge is 1:56 cm1=GPa.
[14] F. Widulle et al., Phys. Rev. Lett. 82, 5281 (1999).
[15] A. Go
¨bel et al., Phys. Rev. B 56, 210 (1997).
400 410 420 430 440 450 460 470
Frequency [cm−1]
2
4
6
8
10
12
14
ΓL (FWHM), Γ
++Γ
− (ω) [cm−1]
300 K
Γ
+(ω)+Γ
− (ω)
Γ
−(ω)
64Zn18O
natZn16O
64Zn0.5
68Zn0.5
16O
68Zn16O
64Zn16O
68Zn18O
64Zn0.5
68Zn0.5
18O
natZn18O
FIG. 5. Linewidths of ZnO with several isotopic composi-
tions at 300 K, after cor rection for resolution. The solid line
represents !!, from Eqs. (2) and (3). The points
have been shifted as done in Fig. 2. The dot-dashed line
indicates the pressure dependence shown in Fig. 4.
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