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Composition of AlGaAs
Z. R. Wasilewski,a) M. M. Dion, D. J. Lockwood, and P. Poole
Institute for Microstructural Sciences, National Research Council of Canada, Montreal Road, Ottawa,
Ontario K1A 0R6, Canada
R. W. Streater and A. J. SpringThorpe
Nortel Technology (formerly Bell-Northern Research), P.O. Box 3511, Station C, Ottawa,
Ontario K1Y 4H7, Canada
~Received 17 September 1996; accepted for publication 29 October 1996!
Although the AlxGa12xAs alloy system has been extensively investigated, there are still
considerable uncertainties in measuring the value of x. Here a new AlxGa12xAs calibration
structure, grown by molecular beam epitaxy, has been used to establish unambiguous alloy
compositions. Such ‘‘standard’’ AlxGa12xAs layers were measured by high-resolution x-ray
diffraction, photoluminescence, and Raman spectroscopy to determine the compositional variations
of the measured physical parameters. The phenomenological equations derived from these
measurements can now be used to establish the Al content of unknown alloys with confidence. In
addition, the results show that Vegard’s law does not hold for the variation of the AlxGa12xAs
lattice constant with x. The small quadratic term has very important implications for a correct
analysis of x-ray results. © 1997 American Institute of Physics. @S0021-8979~97!06303-2#
I. INTRODUCTION
Of all the semiconductor materials used in the electron-
ics industry, the AlGaAs–GaAs alloy system is arguably sec-
ond only to silicon in terms of its scientific and technological
importance. As the properties of AlGaAs strongly depend on
its aluminum content, it is not surprising that much effort has
been put into establishing reliable ways of measuring it. Of
all the characterization tools available, high-resolution x-ray
diffraction ~HRXRD!, photoluminescence ~PL!, and Raman
spectroscopy have proved to be the most popular due to their
nondestructive character. HRXRD and PL are particularly
useful when very high compositional resolution is needed.
However, among the multitude of equations proposed over
the last twenty years to relate observed peak positions to the
aluminum content of the layer, many are mutually inconsis-
tent. The increasing technological importance of AlGaAs has
spurred renewed activity in the area. In spite of this effort,
the divergence between different measurements has actually
widened. For the case of HRXRD characterization, the scat-
ter of proposed AlAs material parameters needed for the in-
terpretation of the rocking curves is very large.1–6 Further-
more, it is not uncommon to find that even within the same
laboratory there is no consensus as to how one should calcu-
late the Al fraction, with discrepancies approaching 0.05 at
the 0.5 Al fraction level. The responsibility of translating one
scale to the other is often left to the crystal growers and the
confusion is felt most acutely in the area of molecular beam
epitaxy ~MBE!. Indeed, for almost twenty years MBE grow-
ers have been calculating AlGaAs compositions and growth
rates using very simple algebra, which accurately reflects the
well established finding that for growth temperatures below
about 640 °C the re-evaporation of Al and Ga atoms from the
growing layer is negligible.7Since the crystalline quality and
stoichiometry of MBE-grown AlGaAs layers is excellent
~this holds true for most growth conditions where the growth
temperature is above 450 °C: Typical levels of vacancies,
antisites, and interstitials are ten orders of magnitude below
the host atoms concentration!, there is no room for error in
such a calculation if only the Ga and Al fluxes are suffi-
ciently well know. The advent of in situ growth rate moni-
toring techniques, in particular those based on optical
pyrometry,8made the inconsistency in ex situ characteriza-
tion of AlGaAs particularly apparent. Indeed, a 2% uncer-
tainty in both the Ga and Al fluxes ~considered average by
today’s MBE standards!will give an estimate of the
Al0.5Ga0.5As layer composition with an accuracy better than
0.007. This is a very respectable number in view of the ex-
isting discrepancies for the ex situ characterization. The abil-
ity of MBE technology to provide what is potentially the
most accurate standard for the AlGaAs layer composition is
well established, and has been used by some authors in the
past as a reference for calibrating HRXRD,5,6,9 PL5or
Raman.9However, in these studies, the authors were relying
on the reflection high-energy electron diffraction ~RHEED!
oscillation phenomenon as a primary tool to measure the
growth rates of GaAs and AlAs in situ. From these measure-
ments, the composition of the following thick AlGaAs layer
was calculated. Even though, with extreme care, such a tech-
nique can be used to reliably monitor growth rates, it is hard
to give a convincing estimate of its accuracy. Indeed, since
such a measurement has to be done on a stationary wafer, the
flux within the ‘‘footprint’’ of the RHEED streak varies
considerably,10 and an uncertainty in the beam position as
small as 1 mm can by itself introduce a 1% error in the
growth rate for the rotating substrate. This is typically the
smallest of the errors plaguing this technique, which can be
very strongly skewed by flux transients on shutter opening or
arsenic overpressure during the growth.5
In the present article, we return to the foundations of the
MBE technology of arsenides as a primary source of the
information about the AlGaAs composition. However, our
a!Electronic mail: zbig.wasilewski@nrc.ca
1683J. Appl. Phys. 81 (4), 15 February 1997 0021-8979/97/81(4)/1683/12/$10.00 © 1997 American Institute of Physics
approach differs considerably from those used previously. At
its very core, it refers only to the linearity of time and space
measurements, and does not rely on the accurate knowledge
of any material parameters, accuracy of instrument calibra-
tions, or other correction factors. We also provide an inde-
pendent measure of the growth rate stability, which is the
only source of possible error, as well as verify the validity of
the fundamental assumption that, with fixed Ga and Al
fluxes, the growth rate of AlGaAs is equal to the sum of the
growth rates of GaAs and AlAs. We use the derived compo-
sitions to revisit the calibrations of the three techniques dis-
cussed: HRXRD, low temperature PL, and Raman spectros-
copy. For each of the characterization techniques used, we
discuss the precautions to be taken and the factors that
should be addressed in order to avoid misinterpretation and
to fully utilize the accuracy of the formulas provided. We
also discuss the applicability of Vegard’s law, which postu-
lates a linear variation of the lattice constant with the alloy
composition, to the AlGaAs system. We show that it has
been the rigid commitment to Vegard’s law which has led in
recent years to the unacceptable discrepancies in AlGaAs
parameters obtained from different characterization schemes.
We reconcile these differences by introducing a small qua-
dratic term into the variation of the AlxGa12xAs lattice con-
stant with x.
II. EXPERIMENT
A. The concept
In its essence, the concept of the composition measure-
ment used in the present work is reduced to the measurement
of the thickness of GaAs and AlAs binary layers embedded
in the same structure as the thick AlGaAs layer, the target of
our study. This information, along with the growth times, is
used to calculate arrival rates of Ga and Al atoms, which in
turn are used to calculate the composition and thickness of
the AlGaAs. The main challenge in such an approach is to
find an accurate way of measuring the thickness of the binary
layers. Also, since the Al and Ga fluxes are not measured
here during the growth of the actual AlGaAs layer, it is very
important to ensure that their values accurately represent the
corresponding fluxes present during the growth of the
AlGaAs.
As the tool for measuring layer thickness, we have cho-
sen the ex situ HRXRD technique. Indeed, of all the infor-
mation which can be extracted from the rocking curves, one
number is totally independent of any material parameters,
namely the layer thickness. The most reliable thickness mea-
surements that can be performed with this technique are on
periodic structures that give the rocking curves a character-
istic series of peaks well distinguished from each other and
the rest of the diffractogram. To make full use of this unique
feature, we embedded in the structures two periodic stacks
with different periods and different total thickness, one be-
low the AlGaAs layer, sampling the GaAs growth rate, and
one above, sampling the AlAs growth rate ~see Fig. 1 and
Sec. II B for details!. With this approach, we were able to
obtain with each rocking curve simultaneous information on
the local growth rates for both GaAs and AlAs materials, as
well as the AlGaAs Bragg peak. Even though we found the
Philips MRD system used for these measurements to be
highly accurate and reproducible, none of these qualities are
truly essential for the calculation of the local composition
using the measured stack periods. Indeed, such a calculation
relies only on the ratios of these numbers, and the informa-
tion about the periods themselves is distributed in the entire
diffractogram. Thus, the ratio we measured is insensitive to a
drift in the machine calibration over time or from point to
point.
Another important difference in our approach compared
to those followed previously is the selection of AlGaAs com-
positions used for the measurements. Rather than rely on a
large number of layers, each with a different composition,
we use primarily two layers, providing a continuous span of
Al compositions in two chosen ranges. This approach elimi-
nates variations between different substrates ~to be discussed
later!, while at the same time taking full advantage of the
high spatial resolutions of all the characterization methods
used.
As discussed in the introduction, in the absence of re-
evaporation of group III atoms, the growth rate of AlGaAs,
RAlGaAs , is given by the arrival rates FGa and FAl of Ga and
Al, respectively:
RAlGaAs5
k
GaAs3FGa1
k
AlAs3FAl ,~1!
while the layer composition is given by
FIG. 1. Schematic diagram of wafer MBE1671 showing ~a!the layer struc-
ture, and ~b!calculated Al-composition contour lines ~every 0.01 from 0.40
to 0.28, from left to right!, as well as the outline of the wafer, and the x-ray
measurement points.
1684 J. Appl. Phys., Vol. 81, No. 4, 15 February 1997 Wasilewski
et al.
x5FAl
FGa1FAl .~2!
Since all three compounds GaAs, AlAs, and AlGaAs
form perfect crystals in typical MBE growth, the coefficients
k
can be calculated from the lattice constants of GaAs
and AlAs and are, respectively:
k
GaAs54.525310215
and
k
AlAs54.540310215 cm3. With the chosen units, the
growth rate will be expressed in Å/s if the flux is given
in atoms/cm2s. The difference in
k
for GaAs and AlAs
comes from the small lattice constant difference for these
materials. For the case considered ~no relaxation!, this takes
into account the AlAs tetragonal distortion in the growth
direction. With the same assumption, the fluxes FGa and FAl
can be derived from the growth rates of GaAs and AlAs
using the following formulae: FGa5RAlGaAs/
k
GaAs and
FAl5RAlAs/
k
AlAs . Note that growth rates are often inserted
into Eq. ~2!in place of the fluxes. For the case of AlGaAs,
such a simplification is largely justified since the resulting
error in composition would be of the order of 0.001. The
error, however, may be significant for materials with a larger
lattice mismatch, such as InGaAs on GaAs.
B. The growth details
Layers were grown in a modified V80H VG-Semicon
MBE system equipped with Al, In, two Ga, Si, Be, and As
effusion cells, as well as an EPI high-capacity valved cracker
cell. The Ga and In cells are high-uniformity, dual-filament
cells with tilted crucible inserts from EPI,11,12 while the Al
cell is a 30 cc, EPI high-uniformity single-filament cell with
a dual crucible. All group III cells are retracted one inch
away from the shutters compared to the original position.
With this arrangement, we find that the flux transients on
shutter openings are less than 1.5% for all the cells, as moni-
tored with an ion gauge inserted in the wafer position. From
regular growth rate monitoring, we find the fluxes to be very
stable, with an average drift due to the charge depletion of
about 0.5% for the equivalent of 10
m
m of material evapo-
rated from the cell ~decrease of flux with time for Ga and In
cells and increase for Al!. The wafer temperature is stabi-
lized with a thermocouple, while the monitoring of the ap-
parent wafer temperature is done using an infrared pyrometer
calibrated against the melting point of InSb. During the same
time period, we were growing in the system modulation-
doped GaAs/AlGaAs heterostructures with ultrahigh-
mobility two-dimensional electron gases ~peak value of
6 400 000 cm2/V s!as well as very low threshold current
density 980 nm lasers ~44 A/cm2!. These results demonstrate
the very low levels of background contamination in the sys-
tem.A number of layers were grown for the purpose of the
present study, and additional details pertaining to the growth
conditions used will be given wherever relevant. The key
results presented come from two wafers grown without sub-
strate rotation in order to generate a desired range of Al
composition. Both wafers, MBE1671 and MBE1696, were
grown on 2 in. ~100!vertical gradient freeze ~VGF!GaAs
semi-insulating ~SI!substrates from American Xtal Tech-
nologies, indium-free mounted in molybdenum holders. Both
substrates were from the same GaAs ingot, cut approxi-
mately 20 mm apart from each other. The growths were done
at a 590 °C substrate temperature, using As2with a V/III flux
ratio a factor of three higher than the minimum needed to
ensure arsenic-stabilized reconstruction for the growth of Al-
GaAs at 3 Å/s. The Al and Ga cell temperatures were kept
constant throughout the growth of each structure. The nomi-
nal growth rates for the first wafer were chosen as 1 and 2
Å/s for GaAs and AlAs, respectively. A reversed proportion
of 2 and 1 Å/s was used for the second wafer. With this
approach, we obtained one wafer with an Al composition
changing from about 0.28 to 0.40 and the other with a com-
positional range from 0.60 to 0.72.
The schematic diagram of the first structure is shown in
Fig. 1~a!. Figure 1~b!shows contour plots of the Al compo-
sition across the wafer, calculated with modeled Al and Ga
flux profiles,10 along with the outline of the wafer used and
markers for the points sampled by the x-ray beam. The struc-
tures are composed of three distinct parts, with the main
AlGaAs layer Lin the central location. The bottom stack,
SGa , is grown by inserting 10 equally spaced 10-Å-thick
AlAs ‘‘x-ray markers’’ into a 1-
m
m-thick GaAs layer. ~All
thickness values quoted are the nominal values for a rotating
substrate.!Such thin periodic markers give distinct signa-
tures in the rocking curve from which their accurate spacing
can be extracted. It can be easily shown that with such a
design one can obtain very precisely the growth rate for the
stack’s prime component, which is GaAs in this case. The
measurement accuracy is not compromised even by consid-
erable uncertainty in the growth rate of the marker material.
This is useful to eliminate the possible influence of shutter
transients on the calculated growth rates. Indeed, the Ga
shutter stays open for most of the time, and 10 s shutter
closures ~5 s for the second wafer!are too short to generate
any measurable flux transient. On the other hand, the tran-
sient for the 10 Å AlAs layer can increase its average thick-
ness by not more than 0.15 Å ~1.5% transient in our case!,
which is negligible compared to the 1000 Å period of the
stack. A similar approach is taken for the top, mainly AlAs
stack, SAl . Here, 20 equally spaced 10 Å AlGaAs layers are
inserted into a 0.4-
m
m-thick AlAs layer. Note that in this
case the Al shutter stays open all the time and only the thick-
ness of its first layer can be skewed somewhat by the shutter
transient. The design for the second, high Al composition
structure, differed from the one shown in Fig. 1~a!only in
the thickness of the AlGaAs Llayer and the kind of markers
used in the top AlAs stack. We used a nominal thickness of
2
m
m for the Llayer, and markers of 7 Å GaAs rather than
10 Å AlGaAs. For both layers, we used a 2000-Å-thick
GaAs buffer and a 50 Å GaAs cap.
The final results were structures that provided a thick
AlGaAs layer of laterally varying composition along with
accurate local ‘‘HRXRD rulers’’ to measure the growth rates
of GaAs and AlAs.
C. Secondary ion mass spectroscopy (SIMS)
measurements
With all the precautions taken, and the excellent track
record for the stability of the fluxes in our system, basing the
1685J. Appl. Phys., Vol. 81, No. 4, 15 February 1997 Wasilewski
et al.
entire project on two wafers may leave it vulnerable to some
‘‘unforeseen’’ flux transients or drifts that may distort the
results, yet remain undetected. In order to eliminate such a
possibility, we performed high-precision SIMS measure-
ments on selected spots for both MBE1671 and MBE1696
wafers. SIMS was performed using a Cameca IMS 4f sys-
tem, utilizing a 10 keV Cs1primary beam and detecting
~M1Cs!1positive secondaries at 4.5 keV. The primary and
secondary ion optics used for the measurement have been
developed and optimized for high precision Al composition
and depth profiling.13 Crater depths were measured on a
Dektak IIA, which we have found over several years to be
reproducible to better than 1% or 100 Å, whichever is
greater, and which has had its calibration checked against a
National Bureau of Standards ~NBS!referenced SiO2thick-
ness standard. Successive measurements were done by step-
ping across the sample pieces on 200
m
m centers in a direc-
tion approximately parallel to the contours shown in Fig.
1~b!. Repeated measurements of the thickness of various lay-
ers, spaced apart by as much as 1 mm showed agreement to
within 1% ~generally better!. We took particular care not to
stop sputtering within AlAs layers, which have a tendency to
corrode quickly and introduce a noticeable drift in successive
Dektak profiles. Furthermore, an independent check was
made utilizing previously determined relative sputtering
rates of AlAs, GaAs, and AlGaAs layers on the full structure;
agreement to within 1% was achieved in all cases. The full
Ga and Al profiles for the MBE1671 structure are presented
in Fig. 2, with appropriate sputtering rate corrections. The
thicknesses were confirmed independently with Dektak
depths measurements on partial profiles. The same notation
as that for Fig. 1~a!is used to indicate different layer seg-
ments. The table in Fig. 2 summarizes the results of SIMS
profiles for both structures studied. A computer peak fitting
algorithm was used to obtain the values for the individual
periods in the stacks SGa and SAl . As seen from the table, the
standard deviations
s
from the average periods Lare of the
order of 1 monolayer ~ML!for both stacks. These periods are
used along with the original growth times to calculate the
growth rates RGaAs and RAlAs , also quoted in the table. The
thickness of the middle AlGaAs layer LSIMS was calculated
using very sharp signatures from the marker layers M1and
M2, correcting appropriately for their thickness. Both Ga
and Al profiles were used independently for this calculation
~they were two separate profiles!, and both values LGa1Cs and
LAl1Cs along with their average
^
L
&
are quoted in the table.
Note that LGa1Cs and LAl1Cs agree to better than 0.5% for
both structures measured.
Special attention should be given to the last two columns
in the table. The Lcalc quoted there was calculated using
the formula LAlGaAs5RAlGaAs3tAlGaAs , where, according
to Eq. ~1!,RAlGaAs was put as RGaAs1RAlAs ~values derived
from the SIMS profile!, and tAlGaAs is the time for the growth
of the AlGaAs Llayer, taken from the growth recipe.
The last column quotes the normalized difference between
the measured and calculated thickness of the AlGaAs
layer–~^L&–Lcalc!/^L&. The thicknesses agree to better than
0.7%, which from our experience is the instrumental accu-
racy of our profiling technique. Consistency of this algebra
illustrates in a rather elegant way the applicability of Eq. ~1!
for our growth conditions. From the stability of the Ga and
Al count levels ~confirmed additionally by the period stabil-
ity of both stacks SGa and SAl!, we conclude that the Ga and
Al fluxes were constant throughout the growth of both struc-
tures to better than 1%, giving 0.003 ~or 60.0015!as the
estimate of the uncertainty for the compositions derived from
Eq. ~2!in our experiment.
D. HRXRD measurements
The x-ray rocking curves were acquired with a Philips
Materials Research Diffractometer in the five-crystal con-
figuration using a four-crystal Ge ~220!monochromator and
Cu-K
a
1, radiation. V-2
u
scans were taken with a slit width
of 0.45 mm and step size of 0.00075°. All rocking curves
discussed in the present article were obtained at 293 K with
the ~400!reflection on as-grown ~i.e., not cleaved!wafers. A
number of rocking curves were collected along the wafers’
compositional gradient, with a typical spacing between
points of 5 mm @see Fig. 1~b!#. We have performed separate
calibrations of the sample stage translation mechanism and
found it accurate to within 0.1 mm, provided all scanned
points are approached from one x-scale direction to eliminate
backlash. The x-ray beam cross section was approximately
0.5 mm by 1 mm in the xand ydirections, respectively, thus
sampling an area of the layer with about 0.001 variation in
Al composition and 0.3% variation in SGa and SAl stack pe-
riods. Scans taken with azimuthal angle
f
50° and
f
5180°
gave the same peak splitting to better than 0.5 arc s, indicat-
FIG. 2. SIMS profiles of Al ~thin line!and Ga ~thick line!for wafer
MBE1671. The notation used to label the various layers in the structure is
the same as that in Fig. 1~a!. The period and L-layer thicknesses, confirmed
with Dektak measurements, are also indicated. The table summarizes the
SIMS results obtained for both wafers MBE1671 and MBE1696.
1686 J. Appl. Phys., Vol. 81, No. 4, 15 February 1997 Wasilewski
et al.
ing that the wafer off-cut angle from the @100#direction was
smaller than 0.025° for both layers @see Eq. ~3!below#.
High-contrast Nomarski microscopy showed excellent sur-
face morphology for both layers, with no sign of layer relax-
ation ~from accumulated experience, we find that this tech-
nique is capable of detecting very early onsets of relaxation,
well before it can be detected with asymmetric reflections
typically used6to measure the difference in ‘‘in-plane’’ lat-
tice constants between the partially relaxed layer and the
substrate!and total point defect densities below 50 cm22.
For a sufficiently thick single pseudomorphic layer, the
angular separation D
u
~in radians!between the Bragg peaks
of the epilayer and the substrate is given in the second-order
approximation by14
D
u
5
$
12@112~
e
'2
e
'
2!cos2
a
tan2
u
B
1
e
'sin 2
a
tan
u
B#1/2
%
/tan
u
B,~3!
where
u
Bis the Bragg angle,
a
is the angle between the
surface and the diffracting planes, and
e
'is the perpendicular
strain caused by tetragonal distortion. For the case of an
AlGaAs layer on a thick GaAs substrate,
e
'is expressed by
e
'5aAlGaAs
'2aGaAs
'
aGaAs
'5aAlGaAs
'2aGaAs
aGaAs .~4!
Often, for the case of symmetrical reflections and
a
50~zero
off-cut angle!, a much simpler equation is used for D
u
:
D
u
52tan ~
u
B!3
e
',~5!
which approximates Eq. ~3!well for the case of very small
lattice mismatches. Indeed, for the case of a perpendicular
strain of 0.002 78 ~close to that of strained AlAs on a GaAs
substrate!, the difference between predictions of the tangen-
tial approximation @Eq. ~5!# and the more accurate second
order approximation @Eq. ~3!# is approximately 1.3 arc s,
with the second order approximation giving a smaller split-
ting. As the AlxGa12xAs composition xincreases from 0 to
1, its properties are smoothly changing from GaAs-like to
AlAs-like. In particular, this applies to the alloy lattice con-
stant aAlGaAs
o~the index ‘‘o’’ refers to the material lattice
constant in the absence of any strain!and its elastic proper-
ties, which for a tetragonal distortion in the @100#direction
can be conveniently folded into one parameter—the Poisson
ratio
n
AlGaAs . In this case, Eq. ~4!can be rewritten as
e
'511
n
12
n
3aAlGaAs
o2aGaAs
o
aGaAs
o.~6!
For the case of AlGaAs, a linear change in both
aAlGaAs
oand
n
AlGaAs with layer composition xhas been tradi-
tionally assumed ~Vegard’s law!:
aAlGaAs
o5aGaAs
o1x~aAlAs
o2aGaAs
o!
~7!
n
AlGaAs5
n
GaAs1x~
n
AlAs2
n
GaAs!.
Such an assumption is in fact embedded in the most
popular commercial rocking curve simulation software, such
as Philips’ HRS ~used in this work!and Bede’s RADS,
where the lattice parameter and the Poisson ratio of the ter-
nary compound are calculated using Eq. ~7!with appropriate
values of aoand
n
for its end binary compounds along with
the user-supplied composition x. Ideally, one knows accu-
rately the values of the needed material parameters from in-
dependent experiments done on bulk binary materials. Since
bulk AlAs is not available for accurate studies due to its
chemical instability, with the exception of relatively inaccu-
rate, early powder diffractometer data,15,16 published values
of aAlAs
oand
n
AlAs were derived indirectly from the properties
of pseudomorphic or partly relaxed AlAs epitaxial layers. In
recent years, proposed values for these parameters appear to
be binned in two widely spaced sets. One set is based on
Eqs. ~5!and ~7!, and the AlAs parameters needed for Eq. ~7!
are deduced from the experimentally measured D
u
~x!for
pseudomorphic AlGaAs layers. The Poisson ratios calculated
with this method in 1991 independently by Goorsky et al.4
~0.27560.015!, and Tanner et al.6~0.2860.01!appear to be
anomalously low, even when the quoted errors are taken into
account. On the other hand, their values of aAlAs
oare consid-
erably larger than those ever reported. This apparent abnor-
mality was challenged two years later by Bocchi et al.2who
measured
n
AlAs50.32260.005 in their x-ray study of
pseudomorphic and relaxed AlAs epitaxial layers. Their
value is somewhat higher than the 0.311 measured for GaAs
and agrees well with theoretical predictions17 and the general
trend for III–V compound constants. Also, the derived value
for aAlAs
ois close to that measured with powder diffracto-
metry, giving a relaxed lattice mismatch between GaAs and
AlAs of 1.4431023. A similar independent study by Leavitt
and Towner18 gave a slightly higher value of
n
AlAs50.32860.004, while
n
AlAs50.32460.004 was de-
duced in 1995 by Krieger et al.,1from a combination of
x-ray measurements and infrared Brillouin scattering. Unfor-
tunately, when this second set of parameters, clustering
n
AlAs
in the range between 0.322 and 0.328, is used in connection
with experimentally measured D
u
(x) for pseudomorphic
AlGaAs layers, the aluminum compositions derived are
higher than the MBE calibrations based on Eq. ~2!by as
much as 0.035 at the 0.5 aluminum fraction level, an equiva-
lent of a 14% error in the Al flux calibration. As mentioned
in the introduction, the outlined conflicts are resolved by
allowing a small quadratic term in the variation of the
AlxGa12xAs lattice constant with the aluminum composition
x. Before addressing such a correction, we revisit, using our
‘‘internal’’ composition calibration, the D
u
(x) dependence,
which is fundamental to this issue.
Unacceptable scatter exists presently in the published
experimental D
u
(x). It is natural to assume that in deriving
the experimental dependence of D
u
on the aluminum com-
position x, problems with the accurate determination of the
latter will be the main source of error. That this is not nec-
essarily the case is evident when we compare the values for
D
u
quoted by different authors for the case of pure AlAs,
where the measurement of the Al fraction is not an issue.
Indeed, even for more recent data, where authors eliminated
the possibility of layer relaxation, poor instrument calibra-
tion, or a finite
a
in Eq. ~3!,D
u
measured with Cu K
a
radia-
tion and the ~400!reflection span from 368 ~Ref. 4!to 379.5
arc s.6This range is far beyond the measurement error. The
most likely reason for such discrepancies is a difference in
1687J. Appl. Phys., Vol. 81, No. 4, 15 February 1997 Wasilewski
et al.
the lattice parameters of the GaAs substrates used in the
different experiments. Indeed, the values for aGaAs
oin the
literature span from 5.6528 ~Ref. 19!to 5.653 75 Å,20 while
the measurement errors quoted are smaller than 60.000 02
Å. Such a variation can be a result of lattice dilation due to
the presence of a large number of native defects ~vacancies,
interstitials, etc.!or dopant atoms, all of which are specific to
the technology used for the growth of the substrate material.
On the other hand, one expects that for high-quality epitaxy,
the lattice constant of the nominally undoped and stoichio-
metric GaAs and its alloys should be much more reproduc-
ible. In view of the above, it is important to establish for the
given epitaxy method how the lattice constant of the GaAs
layer grown relates to the lattice constant of the GaAs sub-
strate on which it is deposited before attempting to calculate
the composition of an AlGaAs layer using the measured D
u
.
There is no indication that such a screening was performed
in any of the previously published work, with the exception
of Bocchi et al.,2whose study was limited to pure AlAs lay-
ers. For the purpose of the present experiment, we have per-
formed numerous MBE growths of undoped, Si- and Be-
doped GaAs layers on SI and N1two inch ~100!GaAs VGF
substrates. All of these layers were grown at a substrate tem-
perature of 600 °C with As2using a growth rate of 2 Å/s. We
find that undoped MBE GaAs is lattice matched to VGF SI
substrates to better than 0.000 022 Å ~D
u
less than 1 arc s!.
On the other hand, we systematically found a mismatch of
0.000 11–0.000 13 Å ~D
u
'5–6 arc s!for undoped MBE
GaAs grown on N1GaAs substrates, with the N1substrate
having the smaller lattice constant. For undoped AlGaAs lay-
ers grown simultaneously on SI and N1wafer pieces, we
find the difference between the measured D
u
to be in the
same range ~5–6 arc s!, even though low-temperature PL
lines overlap to better than 0.5 Å on the lscale. Interest-
ingly, results from MBE layers doped to 231018 cm23with
Si indicate an increase of 0.0001 Å in the GaAs lattice con-
stant compared to undoped MBE layers. No change in lattice
constant was detected on Be-doped layers up to the level of
631018 cm23. Similar findings were recently reported for Si-
and Sn-doped InP layers grown on InP substrates by meta-
lorganic vapor phase epitaxy ~MOVPE!and liquid phase ep-
itaxy ~LPE!.21 As stated earlier, both structures ~MBE1671
and MBE1696!used in this work were grown on SI VGF
wafers, thus eliminating the complication of the lattice mis-
match between the substrate GaAs and MBE GaAs. It is
worthwhile to stress at this point that when comparing or
quoting the measured D
u
, one more parameter ~seldom in-
cluded in the earlier articles!should be considered, namely
the measurement temperature. Indeed, due to the different
temperature coefficients for the GaAs and AlAs lattice con-
stants, the strain between the two decreases as the tempera-
ture increases.15 This effect causes the measured splitting
between the GaAs and AlAs ~400!Bragg reflections to de-
crease at a rate of about 0.5 arc s for every degree Celsius
increase in the sample temperature.
Figure 3~a!shows a typical rocking curve obtained in
one of the scans on the MBE1671 wafer along with an opti-
mized rocking curve simulation displaced for clarity. The
three central peaks, shown expanded in Fig. 3~b!, are, from
left to right, the ~400!zero-order reflection from the top SAl
stack, the ~400!reflection from the thick AlGaAs Llayer,
and the ~400!reflection from the GaAs substrate overlaid
with the zero-order reflection from the bottom SGa stack. The
higher-order superlattice peaks from the SAl and SGa stacks,
used to measure their periods accurately, are indicated in Fig.
3~c!. As mentioned earlier, the Philips HRS simulation pro-
gram used here has embedded in it Vegard’s law. Even
though such an assumption may be largely unfounded, it has
a practical advantage. Indeed, for the simulation, apart from
the layer structure itself, the program needs only two param-
eters for each of the binary compounds involved: the relaxed
lattice constant and the Poisson ratio. Since, for all simula-
tions, we use the same values of
n
GaAs50.3114,6,22 and
aGaAs
o55.653 3860.000 02 Å,23 only two parameters are
unknown—
n
AlAs and aAlAs
o. They can be obtained indepen-
dently for every rocking curve by fitting the relative posi-
tions of the three main Bragg reflections shown in Fig. 3~b!.
All of the parameters of the layer structure fed into the simu-
lation ~i.e., thickness and composition of every layer!scale
according to the local GaAs and AlAs growth rates. These
growth rates are easily extracted from the relative spacing of
the superlattice reflections for the SAl and SGa stacks as
shown in Fig. 3~c!. Typically, two to four iterations of such a
FIG. 3. ~a!Typical HRXRD ~400!rocking curve for wafer MBE1671,
showing both the experimental data and the dynamical simulation, displaced
vertically with respect to each other for clarity; ~b!Close-up view of the
zero-order diffraction peaks, along with their appropriate labeling; ~c!
Close-up view of the higher-order superlattice diffraction peaks and pendel-
losung fringes, also properly labeled.
1688 J. Appl. Phys., Vol. 81, No. 4, 15 February 1997 Wasilewski
et al.
procedure are sufficient to obtain a nearly perfect match be-
tween the simulated and the experimental rocking curves,
along with very accurate local growth rates ~which determine
the local x!and a set of the ‘‘best’’
n
AlAs and aAlAs
oparam-
eters. From all the rocking curves collected on layers
MBE1671 and MBE1696, we obtain with this method
n
AlAs50.255 with a standard deviation
s
n
50.003 and
aAlAs
o55.662 73 Å with a standard deviation
s
a50.000 06 Å.
The value of
n
AlAs obtained here is lower than that derived
by Goorsky et al.4~0.27560.015!or Tanner et al.6~0.28
60.01, based on only 4 points!. The difference between their
results and ours may be related to the uncertainty in the
substrate lattice constants in those previous studies and the
different methods used for compositional calibration.
Figure 4 shows another ‘‘cross section’’ of our experi-
mental results, namely the compositional dependence of D
u
for the AlGaAs L-layer Bragg peak. The point at x50.154
comes from another layer, MBE1880, also grown on a SI
VGF substrate, but with substrate rotation. As with the other
layers, this one also had two ‘‘flux-transient-free’’ periodic
stacks for the independent measurement of the GaAs and
AlAs growth rates from which the layer composition was
established. Unlike MBE1671 and MBE1696, the substrate
for MBE1880 came from a different batch and no SIMS
profiling was done to verify the stability of the fluxes for this
structure. The continuous line drawn through the experimen-
tal points represents the prediction of Eqs. ~5!–~7!with the
parameters
n
AlAs and aAlAs
ooptimized according to the least-
squares algorithm ~Rfactor of 0.999 98!. Obtained in such a
way, the values are the same as those calculated from the set
of rocking curve simulations. If the second order approxima-
tion @Eq. ~3!# for D
u
is used instead of the tangential approxi-
mation @Eq. ~5!# the same value for aAlAs
ois obtained with a
somewhat higher value of 0.256 for
n
AlAs . The D
u
values
measured here can also be described very well with a phe-
nomenological second-order polynomial
D
u
~x!5420x246.5x2arc s.
This dependence, within the scale used in Fig. 4, gives a line
identical to the one calculated using Eqs. ~5!–~7!. Very little
scatter of the experimental data from the calculated solid line
is seen. To emphasize the bowing measured, we show with
the dashed line the linear dependence D
u
(x)5373.5x, while
the inset in Fig. 4 shows the difference between the two. It
should be stressed at this point that the parameters
n
AlAs and
aAlAs
oderived above have only a ‘‘virtual’’ character, serving
as a convenient intermediate stage connecting the layer com-
position xwith the actually measured physical quantity,
e
'.
Their values are an artifact of Vegard’s law constraint used
in both the simulation program and the D
u
modeling shown
in Fig. 4. As such, they should be used only to calculate
e
'
using Eqs. ~6!and ~7!. Upon close inspection of the equa-
tions used, it is revealed that, within the scatter of the lattice
parameters of GaAs available in the literature, neither D
u
(x)
nor the simulated profiles change as long as the Poisson ra-
tios and aAlAs
o2aGaAs
oare kept constant. We believe that the
values of
n
AlAs and aAlAs
oderived here should work well for
all epitaxial techniques, provided that the AlGaAs and GaAs
layers are highly stoichiometric with a total concentration of
native defects and residual doping well below 1018 cm23.As
discussed earlier, precautions must be taken when interpret-
ing results on doped layers or when growing on GaAs sub-
strates that have a different lattice constant from that of the
epitaxial GaAs.
In contrast to the rather limited applicability of the pa-
rameters derived above, our experimental data, when com-
bined with the reported AlAs Poisson ratio, can be used to
derive an accurate compositional dependence of the AlGaAs
lattice constant. As three independent recent evaluations of
the AlAs Poisson ratio are available, we choose the average
value of 0.325 ~0.32260.005;20.32460.004;1
0.32860.00418!which lies within the error bars quoted in all
three reports. Figure 5 shows three sets of
n
AlGaAs(x) and
aAlGaAs
o(x) dependencies. Each of these sets, when inserted
into Eq. ~6!, will, in connection with Eq. ~5!, give the same
experimentally measured dependence of D
u
on x. The depen-
dencies marked with the dot-dash lines have already been
discussed and represent the Vegard’s law related parameters.
The dotted lines, on the other hand, show the compositional
variation of the Poisson ratio required to explain the experi-
mental D
u
(x), assuming a linear dependence for
aAlGaAs
o(x). In this case, we fix the Poisson ratios of GaAs
and AlAs at their experimentally measured values of 0.311
and 0.325, respectively. The bowing obtained here is much
too large and illustrates the poor sensitivity of D
u
(x) to the
Poisson ratio. It is clear that any physically reasonable bow-
ing of
n
AlGaAs(x) will have a very small impact on the actual
D
u
(x), and the main effect must come from the quadratic
term in the aAlGaAs
o(x). The continuous lines show the depen-
dencies assuming a linear change in Poisson ratio between
the GaAs value of 0.311 and the AlAs value of 0.325. This
constraint, along with the measured D
u
(x), gives a nonlinear
FIG. 4. AlAs–GaAs peak splitting, D
u
, vs aluminum fraction x. The per-
pendicular strain on the right-hand vertical scale, was calculated using Eq.
~3!. The open symbols are the experimentally measured x-ray points, while
the solid line is a least-squares fit using Eqs. ~5!–~7!with
n
AlAs and aAlAs
oas
fitting parameters. The dashed line is the linear dependence, D
u
(x)5373.5x.
The inset shows the difference in peak splitting between the least-squares fit
and the linear relation.
1689J. Appl. Phys., Vol. 81, No. 4, 15 February 1997 Wasilewski
et al.
variation of the AlGaAs lattice constant with Al composi-
tion. It is interesting to see that only a very small departure
from Vegard’s law is required to explain the observed
D
u
(x). As stated earlier, some nonlinearity in
n
AlGaAs(x) will
not change significantly the coefficients in the quadratic
equation for aAlGaAs
o(x). To summarize, we propose here a
set of equations for the compositional dependence of the
AlGaAs lattice parameter and its Poisson ratio:
aAlGaAs
o~x!55.653 3819.2931023~x20.134x2!Å,
n
AlGaAs~x!50.31110.014x.
The above equations are consistent with all the recent experi-
mental findings. The coefficients in the expression for the
lattice parameter were obtained using the more accurate sec-
ond order approximation @Eq. ~3!#. For x51, we obtain
aAlAs
o55.661 43 Å, or a value of 1.42431023for the free
lattice parameter mismatch between semi-insulating GaAs
and AlAs, in good agreement with that derived by Bocchi
et al.2and early powder diffractometry measurements.15,16
E. Optical measurements
For the optical measurement, we relied on the composi-
tional maps established with HRXRD and described in the
previous section. The points on which the PL and Raman
were measured lie on the same path as the one used for the
x-ray scans @see Fig. 1~b!#. Because of the much faster ac-
quisition rate for the optical techniques, and their better spa-
tial resolution ~sampled areas of less than 80
m
m width!,
measurements were made along the compositional gradient
every 1 mm. To obtain the AlGaAs composition for the in-
termediate points, we used the least-squares polynomial fit
through the HRXRD measured points. For both wafers, the
standard deviation of the measured HRXRD aluminum com-
positions from such a continuous phenomenological curve
was less than 0.0005. The technique used for relating the
spatial alignment of the probing laser beams to the compo-
sitional map obtained with x-ray diffraction was accurate to
better than 500
m
m. Consequently, the systematic shift of the
Al composition which may have been introduced to the x
scale for each structure was less than 0.001.
1. Low-temperature PL
Photoluminescence measurements were done using a
grating spectrometer with a 0.2 meV energy resolution
equipped with a liquid-N2-cooled CCD camera. The samples
were mounted stress-free on the cold finger of the cryostat
and kept at T5861 K. Such a temperature uncertainty does
not cause any measurable shifts in the peak positions, as the
changes in the band gap are less than 0.03 meV/K24 at this
temperature. Very low excitation power densities, from 0.01
to 10 W/cm2, were used for all measurements. For the lower
composition structure ~MBE1671!, we used a HeNe laser
with a maximum power of 0.5 mW at l5632.8 nm, while
for the higher composition structure ~MBE1696!we used an
Ar1-ion laser with a maximum power of 0.5 mW at l5457.9
nm.In Fig. 6, we show the PL spectra taken for a pure GaAs
layer and three different compositions of AlGaAs. The PL
for x50.154 was taken on the MBE1880 structure, while the
PLs for x50.361 and x50.721 were taken on MBE1671 and
MBE1696 structures, respectively. The top SAl stack was
etched off from the MBE1696 sample for this measurement
to avoid overlapping with the AlAs PL lines. In the top left-
FIG. 5. Dependence of the AlGaAs lattice constant and Poisson ratio on the
Al composition considering three different scenarios to fit our data: the
dot-dash lines assume Vegard’s law to be correct ~as in the simulation
programs!; the dotted lines assume a linear dependence for aAlGaAs(x); and,
the continuous lines assume a linear relation for
n
AlGaAs(x). In the latter two
cases, the experimentally measured Poisson ratios of GaAs and AlAs, 0.311
and 0.325 ~mean of three recent measurements!, respectively, are used.
FIG. 6. Photoluminescence spectra of GaAs, and three representative spec-
tra for AlxGa12xAs: x50.154 ~MBE1880!,x50.361 ~MBE1671!, and
x50.721 ~MBE1696!.
1690 J. Appl. Phys., Vol. 81, No. 4, 15 February 1997 Wasilewski
et al.
hand corner of the AlGaAs PL panels, we show in the circle
the shift of the PL energy that would be induced by a change
in the AlGaAs composition of 0.01. The very high composi-
tional resolution of the PL technique for x,0.38 is evident.
Note that the GaAs PL is plotted on an energy scale ex-
panded by a factor of 10 compared to that for the other
samples. A more detailed discussion of the features shown
may be found in the literature.25–29 Their relative intensities
may change with the incident laser power, sample tempera-
ture, and sample quality. For that reason, great care must be
taken to positively identify the PL line used for the compo-
sition measurement as being associated with donor-bound
excitons (D0,X). For high-purity AlGaAs, and PL measure-
ments done at T,15 K, this is typically the narrowest feature
at the high-energy end of the spectrum. The presence of pho-
non replicas ~\
v
p!AlAs and ~\
v
p!GaAs , for the case of the
indirect AlGaAs ~x.0.4!, aids identification of this line for
high Al composition materials.
In Fig. 7, we plot the experimental dependencies of the
PL energy for the (D0,X) transition as a function of Al com-
position. The points marked with open diamonds and open
triangles were all measured on the samples with an ‘‘inter-
nal’’ composition standard as described in the previous sec-
tion. The dark continuous line represents the linear least-
squares fit ~with Rfactor of 0.9998!to the points marked
with open diamonds related to the direct-gap AlGaAs ~x
,0.38!. We obtain for this dependence
E~D0,X!
G~x!51.51511.403xeV.
The light continuous line represents the linear least-
squares fit ~with Rfactor of 0.9995!to the points marked
with open diamonds related to the indirect-gap AlGaAs
~x.0.4!. Here we obtain
E~D0,X!
X~x!51.94910.250xeV.
The data points marked with the open triangles are not used
in deriving the formulas, as they are in the region of Al
compositions where the excitonic transitions change their
character from Glike to Xlike. Plotted in Fig. 7 are also the
most recently published data30,31 on the compositional de-
pendence of the (D0,X) transition energy. For the case of
direct-gap AlGaAs, we find that our data agrees well with
previously published results30,31 for the lower Al composi-
tions, practically overlapping with the dependence proposed
by Bosio et al.31 for x,0.25. However, unlike the case of
these earlier experiments, there is no evidence of bowing in
our data and a purely linear dependence gives an excellent fit
to the experimental points ~see the Rfactor quoted above!.
The quadratic relations in the previous studies cause a deg-
radation of the agreement with our data for x.0.25. Never-
theless, even for the highest direct-gap compositions, the dis-
crepancy would amount to a deviation of less than 0.01 in
aluminum composition for a given transition energy. Such a
deviation is well within the compositional error brackets
quoted in previous articles.
For the case of the indirect-gap AlGaAs, our data falls
between the recently published results by Guzzi et al.32
~1992!and Olegart et al.28 ~1993!. Similarly to Guzzi et al.,
we find no evidence for the higher-order terms in the com-
positional dependence, and our linear coefficient of 0.25 eV
is identical to the one proposed in their article. A more com-
plicated dependence is postulated by Olegart et al., as seen
from the figure. The discrepancies observed for this indirect-
gap material are unlikely to be caused by inaccurate compo-
sitional calibrations. Indeed, with the transition energy fixed
at 2.124 eV, the predictions for the layer composition would
have to span from 0.668 to 0.753, exceeding by far any pos-
sible calibration errors. However, a close inspection of the
fourth panel in Fig. 6 shows that, for a given composition,
the different curves are spaced energetically by approxi-
mately the same amounts as the different components ~D1–
D4!of the (D0,X) transition. These components are, most
likely, signatures of excitons bound to different neutral point
defects, and the energy spacings between them are related to
their individual short-range potentials ~chemical shifts!.
Since all of our measurements come from the same structure,
we are ‘‘tracking’’ an exciton bound to the same point defect
D2. The same most likely holds true for the data presented
by Guzzi et al., except that, for their case, the defect D1 is
most likely responsible for the exciton trapping. The samples
studied by Olegart et al. were grown by variety of growth
techniques, and some of the layers were intentionally doped
with Si or Mg. We believe that the complicated composi-
tional dependence postulated by them is a result of the PL
spectra being biased by different defects over different com-
positional ranges.
It is evident from the above discussion that PL would not
be a very good choice for measuring the Al composition for
layers with x.0.4. Nevertheless, with our measured value of
0.25 eV for the coefficient of the linear term, the PL in high-
purity layers can still be used as an accurate and sensitive
tool for compositional mapping, provided that care is taken
FIG. 7. Dependence of the PL energy on the Al fraction. The open dia-
monds and triangles are experimental points. The continuous lines are least-
squares fits to the points: the dark line is for x,0.38, and the light one is for
x.0.40. The various dotted/dashed curves are previous experimentally de-
termined relationships from other groups ~as indicated in the figure!.
1691J. Appl. Phys., Vol. 81, No. 4, 15 February 1997 Wasilewski
et al.
to track reliably the (D0,X) components present in the spec-
trum.
2. Raman spectroscopy
The Raman scattering experiments were carried out at a
temperature of 295 K in a quasibackscattering geometry33
with the incident light at an angle of 77.7° from the normal
to the sample surface. The spectra were excited with 300
mW of 530.9 nm krypton laser light, as it was found that this
wavelength gave the best compromise concerning sample
penetration depth ~not too deep to see the bottom stack, but
deep enough to penetrate well into the alloy layer!. The scat-
tered light was analyzed with a Spex 14018 double mono-
chromator under computer control and detected with a
cooled RCA 31034A photomultiplier. The spectral resolution
was 1.2 cm21in all of the measurements. The incident light
was polarized in the scattering plane while the scattered light
was recorded without polarization analysis. The frequencies
of the Raman peaks were obtained with high accuracy using
a computer search algorithm34 and each spectrum was fre-
quency calibrated with respect to three reference points that
covered the complete scan @the laser ‘‘zero’’ position, the
GaAs longitudinal optic ~LO!phonon peak, and the AlAs
LO phonon peak#.
Representative Raman spectra from the graded concen-
tration samples are shown in Fig. 8. Sharp features at 290
~269!and 402 ~360!cm21are the LO ~TO!modes of the
GaAs and AlAs layers in the samples, respectively. The
peaks in between the LO–TO pairs are LO modes of the
alloy due to GaAs-like ~lower frequency!and AlAs-like ~up-
per frequency!modes of vibration.35 The results for the alu-
minum concentration dependence of the LO phonon Raman
frequencies in the alloy are given in Fig. 9. Each phonon
peak frequency could be located to better than 0.1 cm21
accuracy with the computer analysis. The main uncertainty
comes from the frequency calibration of each spectrum: The
absolute error of the Spex spectrometer is 60.4 cm21and
this would be the error involved when comparing spectra
recorded using different laser lines. However, for a given
laser line, the measurement reproducibility of our spectrom-
eter was excellent. With the 530.9 nm line used here, 77
scans over the GaAs and AlAs peaks gave average frequen-
cies ~with maximum deviation!of 290.2 ~60.2!, 360.3
~60.1!, and 402.2 ~60.1!cm21for the GaAs LO, AlAs TO,
and AlAs LO modes, respectively. Thus, making use of these
nearby peaks as internal calibration markers improved the
absolute accuracy for the alloy modes to 60.2 cm21. Sepa-
rate experiments on other samples showed that the reference
AlAs LO and TO phonon frequencies were not measurably
affected by the strain or the confinement of the AlAs layers
by GaAs layers.
The Raman data versus Al fraction are well represented
by the polynomial expressions shown in Fig. 9 with continu-
ous lines. For the GaAs-like mode, a best fit ~Rfactor of
0.999 78!was obtained with
v
LO
GaAs~x!5290.2236.7xcm21
~a second-order polynomial gave the same Rfactor with a
very small x2coefficient!. For the AlAs-like mode, the best
least-squares fit ~Rfactor of 0.999 52!was obtained with
v
LO
AlAs~x!5364.7146.7x29.4x2cm21.
In the same figure, we show with dotted lines the most
recent results obtained by Solomon et al.9For the case of the
GaAs-like LO phonon, our results are practically the same
for x.0.3. The discrepancy for x,0.3 is caused mainly by
the difference in the GaAs LO mode frequency for pure
GaAs, which we measure to be 290.2 cm21. It is interesting
to note that if in the least-squares fit to our data this point is
FIG. 8. Raman spectra representative of wafers MBE1671 and MBE1696.
Note that the intensity scales for the two compositions differ by a factor of
5. The apparent change in the spectra is mainly due to a very significant
increase in Raman intensities for the AlGaAs related peaks for the higher
compositions. FIG. 9. Aluminum concentration dependence of the AlGaAs LO-phonon
Raman frequencies. The open diamonds are experimental points. The light
continuous line is a linear least-squares fit to the GaAs-like LO-phonon
frequencies, while the dark continuous line is a second-order polynomial fit
to the AlAs-like LO-phonon frequencies. The dotted lines represent the
results obtained by Solomon et al. ~Ref. 9!.
1692 J. Appl. Phys., Vol. 81, No. 4, 15 February 1997 Wasilewski
et al.
omitted, the resulting equation remains the same, still ex-
trapolating to 290.2 cm21for x50. The origin of the discrep-
ancy between our data and that of Solomon for the case of
the AlAs-like LO mode in the lower composition range is
not clear. However, it should be noted that in this range the
AlAs-like LO peak is much weaker and the accuracy of de-
riving its position is decreased. Since in our case we have in
this region almost 50 points ~with very little scatter!versus 4
points in Solomon’s data, we believe our equation is more
accurate. Furthermore, if only the AlGaAs AlAs-like LO
mode frequencies are used for the least-squares fitting pro-
cedure, the extrapolated AlAs LO mode frequency would
give 401.7 cm21, which agrees very well with the measured
value of 402.2 cm21.
It is clear from the small slopes of the measured depen-
dencies, and the fact that it is difficult to normally maintain
Raman spectrometer calibrations to better than 60.4 cm21,
that the compositional accuracy which can be obtained with
this method is 60.01 at best. Although this accuracy is
worse than that expected from properly carried out x-ray
measurements or low-temperature PL, the technique is more
robust as there is less possibility for misinterpretation of the
data. Also, this technique, unlike x ray or PL, can be used to
derive compositions of very thin layers ~less than 3 nm in
thickness!, without the need for significant corrections.36
III. CONCLUSIONS
We have performed compositional studies of
AlxGa12xAs using HRXRD, low temperature PL, and Ra-
man spectroscopy. The method of compositional calibration
chosen relies only on the linearity of time and space mea-
surements. In this way, possible errors in empirical or theo-
retical parameters and correction factors, and uncertainties in
the instrument calibration, are eliminated. This reliable mea-
surement of the layer composition, along with the very low
scatter in the experimental points for all of the techniques
used, gives the most accurate compositional dependencies
available to date.
For the correct interpretation of HRXRD measurements,
we point out the importance of the possible lattice mismatch
between the epitaxial GaAs and the substrate GaAs. This
may introduce a considerable error in the measured angular
splitting between the Bragg reflections used to derive the
layer composition. We provide a set of AlAs lattice param-
eters to be used in the existing simulation software which
assumes the applicability of Vegard’s law to the AlGaAs
material system. We show that the considerable deviation of
such parameters from those recently measured for pure AlAs
epitaxial layers is due to a small bowing in the compositional
dependence of the AlGaAs lattice constant. The incorpora-
tion of this quadratic compositional dependence into the cal-
culation of strain used in the rocking curve analysis shows
excellent agreement with the experimentally measured
D
u
(x), using a mean value of 0.325 for the AlAs Poisson
ratio.
We used the same set of samples to measure the depen-
dence of the donor-bound exciton ~D0,X!transition energy
on the aluminum composition with low-temperature photo-
luminescence. For aluminum compositions smaller than 0.38
~direct band gap!, our measurements show a purely linear
variation of (D0,X!energy with composition. This contrasts
with previously postulated quadratic dependencies. How-
ever, for the two most recently published reports, we find
that our analytical expression fits the experimental results
within the reported compositional errors. For aluminum
compositions larger than 0.4 ~indirect band gap!, our mea-
surements also show a purely linear variation of ~D0,X!en-
ergy with composition. We compare this result with those
published in the literature and point out the existence of
chemical shifts for excitons bound to different neutral de-
fects in this compositional range. Thus the discrepancies in
transition energies reported earlier are most likely attribut-
able to the different predominant defects responsible for the
measured PL in the variety of samples studied.
For the case of Raman scattering experiments, we find a
linear decrease of the GaAs-like LO phonon frequency with
aluminum composition and a nonlinear increase of the AlAs-
like LO phonon frequency. With the exception of the lower
compositions ~less than about 0.4!, our data is in agreement
with that published by Solomon et al.9
The formulae provided here should give a reliable cali-
bration base for other important characterization tools, such
as SIMS or ellipsometry. They should also allow many ex-
isting discrepancies in AlGaAs characterization, processing,
and modeling, all of which rely on knowing the correct
AlGaAs compositions, to be reconciled.
Note added in proof. Recent reports37 confirm the ob-
served increase in the lattice constant of MBE GaAs layers
doped with Si ~see Sec. II D!. This effect is attributed prima-
rily to the influence of the free electrons on the lattice con-
stant via the deformation potential of the conduction band
minimum. The contribution due to free electrons is stronger
than and of opposite sign to that coming from the atomic size
effect, which is proportional to the difference in ionic radii of
dopant and host atoms. The electronic contribution is equally
important for the AlGaAs lattice constant, although in this
case a redistribution of electrons between the different con-
duction band minima has to be considered in order to explain
the experimental results.37
ACKNOWLEDGMENTS
The expert technical assistance of H. J. Labbe
´in the
Raman measurements is gratefully acknowledged.
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