VOLUME 88, NUMBER 21

PHYSICAL REVIEW LETTERS

27 MAY 2002

Anomalous LO Phonon Lifetime in AlAs
M. Canonico, C. Poweleit, and J. Menéndez
Department of Physics and Astronomy, Arizona State University, Box 871504, Tempe, Arizona 85287-1504

A. Debernardi
Istituto Nazionale per la Fisica della Materia (INFM), Unitá di ricerca de Trieste Universitá e SISSA,
Dipartimento Fisica Teorica, Strada Costiera 11, 34014 Trieste, Italy

S. R. Johnson and Y.-H. Zhang
Department of Electrical Engineering, Arizona State University, Box 875706, Tempe, Arizona 85287-5706
(Received 20 November 2001; published 14 May 2002)
The temperature dependence of the frequencies and linewidths of the Raman-active longitudinal optical
(LO) phonons in GaAs and AlAs have been measured. The low-temperature lifetime of the LO phonon
in AlAs is found to be 9.7 ps, very close to the corresponding GaAs value of 9.5 ps. This contradicts
early theoretical predictions. The agreement between theory and experiment can be restored when the
accidental degeneracy between the AlAs LO phonon frequency and a feature in the two-phonon density
of states is taken into account.
DOI: 10.1103/PhysRevLett.88.215502

PACS numbers: 63.20.Kr, 78.30.Fs

The lifetime of the Raman-active longitudinal optical
(LO) phonon in III-V semiconductor compounds is of critical importance for the coupled dynamics of carriers and
phonons in these materials. Accordingly, an intense effort has been devoted to the study of this quantity for the
past 20 years [1]. The recent availability of ab initio predictions of phonon lifetimes [2–5] represents a significant
breakthrough, since they pinpoint the relevant LO phonon
decay channels that may open the door for the engineering
of phonon lifetimes.
In spite of the technological relevance of the Al-Ga-As
system, the lifetime t of LO phonons in AlAs has not been
reported thus far. From time-domain measurements on
an Al0.3 Ga0.7 As alloy sample, Tsen and Morkoç find t 苷
7 6 1 ps (at 10 K) for both the GaAs-like and AlAs-like
LO vibrations [6]. These results are quite puzzling since,
as pointed out by Kash and Tsang [1], the phonon dispersion curves for GaAs and AlAs (which determine their
possible LO decay channels) are quite different. A comparison of these curves for several semiconductors suggests
that t AlAs should be significantly longer than t GaAs . The
first ab initio estimates of phonon lifetimes added to the
puzzle [3], since they predict t AlAs 苷 12.7 ps . t GaAs 苷
9.2 ps, in agreement with Kash and Tsang’s qualitative argument. The reliability of the theoretical prediction was
underscored by the good agreement with the existing experimental data for GaAs [7–11].
The above discrepancies between theory and experiment could be dismissed by arguing that the GaAs-like and
AlAs-like LO vibrations in AlGaAs alloys are not equivalent to the LO phonons in binary GaAs and AlAs crystals, so that a direct comparison is not legitimate. In this
Letter, however, we report Raman measurements of the
LO phonon frequencies and lifetimes in GaAs and AlAs.
Our results show that the low-temperature lifetimes of LO

phonons in GaAs and AlAs are indeed very similar. But
we also find that these experimental results do not imply a
failure of the ab initio method. Instead, they arise from a
breakdown — in the case of AlAs — of the standard formalism used to compare theory with experiment. This
breakdown is caused by the accidental degeneracy between
the frequency of the AlAs LO phonon and a Van Hove singularity in the two-phonon density of states. Such degeneracies have been proposed in the past — most notably in
the case of the transverse-optic phonon in GaP— to explain
highly distorted Raman line shapes [12,13]. The remarkable feature of our measurements is that, within experimental error, the Raman line shape of the LO phonon in
AlAs remains Lorentzian, so that the anomalous aspects of
the phonon decay can be unraveled only by detailed theoretical calculations.
We measured the frequency and the linewidth of the LO
phonon Raman peaks as a function of temperature. Good
quality AlAs samples are available only as thin epitaxial
layers on GaAs substrates. This makes it very difficult
to perform reliable Raman linewidth measurements. Our
studies were made possible by the availability of a 2 m
double monochromator (SOPRA). When combined with
charge-coupled device detection, this instrument makes it
possible to measure Raman spectra from thin layers with
a resolution as good as 0.01 cm21 . Our AlAs sample was
a 0.5 micron layer grown by molecular beam epitaxy on a
GaAs substrate and capped with a 50 Å GaAs layer for
protection. For the Raman measurements, we used the
488 nm line from an Ar1 laser in the backscattering geometry. The samples were placed in a continuous flow
cryostat and the temperature was varied between 10 and
300 K. To minimize laser heating, the laser power was
lowered and the monochromator’s slits were opened until
the instrument broadening reached a value of 0.15 cm21 ,

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© 2002 The American Physical Society

0031-9007兾02兾 88(21)兾215502(4)$20.00

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VOLUME 88, NUMBER 21

PHYSICAL REVIEW LETTERS

27 MAY 2002

about 1兾4 of the intrinsic linewidth. This allowed us to perform all measurements with power levels below 10 mW.
Figures 1 and 2 (insets) show typical Raman spectra.
For both materials, the Raman line shape can be fit with a
standard Voigt profile. The Lorentzian component of the
profile is written as
I共V兲 苷 GR 共T 兲兾兵关VR 共T兲 2 V兴2 1 GR2 共T 兲其 ,

(1)

and it is assumed to represent the intrinsic line shape of the
Raman line. The Gaussian component represents the instrumental broadening. The width of the Gaussian is set
equal to the width of the laser line, and it is kept fixed
for the fit of the Raman line [14]. Hence, aside from an
overall amplitude, our fits include two adjustable parameters: the peak position, VR 共T兲, and the Lorentzian width,
GR 共T兲 苷 1兾共2t兲.
The full width at half maximum [FWHM 苷 2GR 共T兲]
and line shifts obtained from our line shape fits are plotted
in Figs. 1 and 2 as a function of temperature. In GaAs,
the LO Raman peak is broadened by isotopic disorder.
This broadening was found to be 0.12 cm21 at low temperature from measurements of isotopically pure samples
[15]. The isotopic disorder contribution has been subtracted from our experimental GaAs linewidth. No isotopic
disorder is present in AlAs. By averaging data below 50 K
(GaAs), or 70 K (AlAs), we estimate 2GR 共0兲 苷 0.56 6
0.01 cm21 for GaAs and 2GR 共0兲 苷 0.55 6 0.02 cm21 for
AlAs. These values correspond to lifetimes of 9.5 and
9.7 ps, respectively, confirming the apparent disagreement
between experiment and the ab initio results of Ref. [3]
for the case of AlAs.
The existence of a discrepancy between the theoretical
and the experimental low-temperature Raman linewidth for
AlAs is troubling because the predicted relative AlAs兾
GaAs value appears to be in error. This is surprising when
one considers the similarity between the interatomic force
constants for the two materials [16] and the excellent
agreement between theoretical and experimental harmonic
properties [17–20]. Thus, we are led to a critical review
of the standard approximations used to compute phonon
lifetimes in tetrahedral semiconductors. For these materials, it is assumed that the phonon self-energy (i.e., the
physical quantity that describes its interaction with other
phonons) can be obtained from a perturbative expansion
[21] based on the quasiharmonic model, which consists in
expanding the interatomic potential in powers of the displacements from the equilibrium configuration at temperature T. When the width of the peak is small compared
with the frequency of its maximum, the Raman line shape
is given, to an excellent approximation, by [22]
T
I共V兲 ~ GLO 共T, V兲兾兵关vLO
1 DLO 共T , V兲 2 V兴2

1 GLO 共T, V兲2 其 .
T
vLO

(2)

Here
is the quasiharmonic frequency of the
Raman mode at temperature T, whereas DLO 共T , V兲 and
GLO 共T, V兲 are, respectively, the real and imaginary part of
the phonon self-energy. To lowest order in the expansion
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FIG. 1. The inset shows a typical GaAs LO phonon Raman
spectrum obtained at 95 K and its Voigt profile fit. Circles (triangles) represent the Raman linewidth (line shift). The broadening due to isotopic disorder has been subtracted from the data.
The errors in the Voigt fit parameters are typically smaller than
the symbol size. The dashed line is a calculation of the linewidth
based on Eq. (5b), which ignores the anharmonic shift of the LO
phonon. The solid lines are predictions based on Eqs. (4a) and
(4b), as discussed in the text. The theoretical frequency curve
was rigidly upshifted by 9.3 cm21 to match the lowest temperature experimental frequency.

of the potential, the imaginary part of the self-energy is
given by [23]
18p 2 X
GLO 共T, V兲 苷
jV T 共LO, qj1 , 2qj2 兲j2
h̄2 qj1 j2 3
3 共n1 1 n2 1 1兲d共v1T 1 v2T 2 V兲 ,
(3)

FIG. 2. The inset shows a typical AlAs LO phonon Raman
spectrum obtained at 95 K and its Voigt profile fit. Circles (triangles) represent the Raman linewidth (line shift). The errors
in the Voigt fit parameters are typically smaller than the symbol
size. The dashed line is a calculation of the linewidth based on
Eq. (5b), which ignores the anharmonic shift of the LO phonon.
The solid lines are predictions based on Eqs. (4a) and (4b),
as discussed in the text. The theoretical frequency curve was
rigidly upshifted by 15.8 cm21 to match the lowest temperature
experimental frequency.

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VOLUME 88, NUMBER 21

PHYSICAL REVIEW LETTERS

where the sum is over phonon wave vectors and branches,
ni 苷 关exp共h̄viT 兾kB T 兲 2 1兴21 is the phonon occupation
number, and the matrix element V3T is essentially a Fourier transform of the third spatial derivative of the interatomic potential evaluated at the equilibrium position at
temperature T [21,23]. The lowest-order expansion of
DLO 共T, V兲 includes third and fourth order anharmonic
terms [5,21,23]. The real and imaginary parts of the selfenergy are Hilbert transforms of each other, and this relationship is used for the calculation of the real part [5].
Equation (3) corresponds to the decay of a zone center
phonon of frequency V into pairs of phonons v1 共q兲 and
v2 共2q兲, with the decay probability being proportional
to the two-phonon density of states at the LO frequency
times a (squared) anharmonic matrix element and a thermal factor.
In practical evaluations of Eq. (3), including those to be
discussed here, the quantities V3T , v1T , and v2T are replaced
by V3M , v1M , and v2M , which are computed from potential derivatives at the minimum of the interatomic potential
(frequencies calculated in this manner are often referred to
as “bare harmonic frequencies” in the literature). Within
this approximation, the T dependence of the self-energy
is entirely due to the explicit T dependence of the occupation numbers for the decay products. For the AlAs LO
phonon, the ab initio calculations indicate that the main decay channel consists of combinations of LA phonons near
the L point of the Brillouin zone [3].
The line shape functions in Eqs. (1) and (2) are not
equivalent due to the frequency dependence of the phonon
self-energy. Hence, extreme care must be exercised when
comparing theory with experiment. If the self-energy can
be modeled as a linear function of the frequency over
the width of the Raman peak, a fit of Eq. (2) with an
expression proportional to Eq. (1) gives, to an excellent
approximation,
T
VR 共T 兲 苷 vLO
1 DLO 关T, VR 共T兲兴 ,

(4a)

GLO 关T, VR 共T兲兴
GR 共T 兲 苷
,
共1 1 G12 兲 共1 2 D1 兲

(4b)

27 MAY 2002

convenient because it does not require a calculation of the
real part of the self-energy. These facts explain the widespread use of Eq. (5b) in the experimental and theoretical
literature [3]. We will now show, however, that in the case
of AlAs Eq. (5b) is a poor approximation, even though
the Raman line shape remains Lorentzian to within experimental error. The failure of Eq. (5b) is not due to the neglect of the denominators in Eq. (4b) (which for the AlAs
LO phonon tend to cancel each other and represent a correction of the order of 1%) but to the different values of
M
GLO 共T, V兲 at V 苷 VR 共T兲 and V 苷 vLO
.
Our calculations are based on the ab initio approach developed by Debernardi and co-workers [2–5]. The interatomic force constants are computed within the zero
field anharmonic approximation [3] up to fourth order in
the perturbative expansion, following the prescriptions of
Refs. [3,5]. Other computational details are the same as in
Ref. [5]. Figures 3 and 4 show the LO phonon self-energy
for GaAs and AlAs, respectively, and the corresponding
Raman line shape based on Eq. (2). The predicted temperature-dependent line shifts and linewidths can be extracted directly from these computed line shapes, and we
find that for both materials these values agree with those
predicted using Eqs. (4a) and (4b), the latter with the denominator set equal to unity. These predictions are also
shown in Figs. 1 and 2 as solid lines, and the agreement
with experiment is found to be excellent. However, the
dashed lines in Figs. 1 and 2, based on Eq. (5b), represent
a good approximation only for GaAs, in spite of the fact
that the predicted line shape in both materials is approximately Lorentzian.
The failure of Eq. (5b) in the case of AlAs is due to
the stronger frequency dependence of its LO phonon selfenergy. In the GaAs case, the LO phonon self-energy
is weakly dependent on the frequency over the range of
interest, as seen in Fig. 3. In contrast, the frequency

where D1 (G1 ) is the first derivative of the real (imaginary)
part of the self-energy with respect to the frequency V,
evaluated at V 苷 VR 共T兲.
If the self-energy can be assumed to be essentially constant over a frequency range comparable to the anharmonic
shift of the LO phonon (about 6 cm21 for AlAs at room
temperature), Eqs. (4a)(4b) can be simplified to
T
M
VR 共T兲 苷 vLO
1 DLO 关T , vLO
兴,

GR 共T兲 苷
M
vLO

M
GLO 关T, vLO
兴,

(5a)
(5b)

is the bare harmonic frequency of the LO
where
phonon. When Eqs. (5) are inserted into Eq. (2), one recovers a Lorentzian line shape. Therefore, it is not surprising that the observation of Lorentzian line shapes is
implicitly interpreted in the literature as justifying the validity of Eqs. (5). On the theoretical side, Eq. (5b) is very
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FIG. 3. The real (dotted) and imaginary (solid) parts of the LO
phonon self-energy in GaAs at zero temperature. The thin solid
line represents the predicted line shape using Eq. (2). The arrow
marks the theoretical bare harmonic frequency of the LO mode
at zero temperature. The value of 2GLO at this frequency is the
Raman linewidth according to Eq. (5b).

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27 MAY 2002

FIG. 4. The real (dotted) and imaginary (solid) parts of the
phonon self-energy of the LO mode in AlAs at zero temperature.
The thin solid line represents the predicted line shape using
Eq. (2). The arrow marks the theoretical bare harmonic frequency of the LO mode at zero temperature. The value of 2GLO
at this frequency is the Raman linewidth according to Eq. (5b).

tal proximity of the LO phonon frequency to a maximum
in the two-phonon density of states formed by acoustic
phonons that represent the main anharmonic decay channel for the LO phonon. Were it not for this accidental
degeneracy, the original linewidth calculations for AlAs
[3] as well as the earlier qualitative predictions by Kash
and Tsang [1] would be correct. The agreement between
Tsen and Morkoç’s measurements in AlGaAs alloys [6]
and our results for the binary compounds is likely due to
the main decay products consisting of acoustic phonons
with small amplitudes on the cation sites, thereby causing
these phonons to show little effect from alloy disorder.
The strong dependence on small details of the phonon
dispersion relations suggests that in many semiconductors
it should be possible to alter the LO phonon lifetimes in
appropriately engineered nanostructures.
This work was partially supported by NSF Grant No.
DMR 9503904. We acknowledge the allocation of computer resources from INFM Progetto Calcolo Parallelo.

dependence of the LO phonon self-energy in AlAs, shown
in Fig. 4, is much stronger. This is because the LO phonon
frequency in this material is accidentally close to a local
maximum in the two-phonon density of states. This local
maximum— produced by decay phonons near the L point
of the Brillouin zone — can be seen as a cusplike feature
in the calculated one-phonon density of states for AlAs
[16]. Since the imaginary part of the self-energy changes
significantly over a frequency range of the order of the
M
, the use of Eq. (5b)
difference between VR 共T兲 and vLO
introduces sizable errors. By properly accounting for the
anharmonic shift of the LO frequency, as done in Eq. (4b),
the LO phonon is shifted even closer to the local maximum
in the two-phonon density of states, and this causes a 30%
increase in the linewidth value predicted from Eq. (4b)
relative to the prediction from Eq. (5b).
As the temperature changes, the LO phonon frequencies move across the fine structure in the self-energy, as
seen in Fig. 4, and this causes discontinuities in the first
derivative of the linewidth with respect to T . This is the
fingerprint of an anomalous decay process. Although the
overall T dependence of the solid line in Fig. 2 agrees with
experiment, the superimposed “oscillations” are not observed experimentally (or their amplitude is smaller than
the experimental error). This particular aspect of the theoretical predictions, however, should be treated with caution. The contribution from higher order terms in the
perturbative expansion of the self-energy, neglected here,
might be of the same order as the amplitude of the oscillations. Moreover, our calculations of the LO phonon
self-energies neglect the anharmonic shifts and lifetimes
of the decay products, which might affect the details of
the predicted oscillations.
In summary, we have shown that the unexpectedly large
LO phonon Raman linewidth in AlAs is due to the acciden-

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