Influence of Poisson's ratio uncertainty on calculations of the bowing parameter for strained InGaN layers
Thin
InGaN layers form the active region of most nitride-based light-emitting
devices. The thickness of InGaN layers in these devices ranges from 3-5 nm in
conventional quantum-well structures to 10-50 nm in charge asymmetric resonance
tunneling structures [1]. The dependence of the emission wavelength on the In
content in InGaN layers is important for device design. This dependence has
been measured by many groups [2] [3] [4] [5] [6] [7] [8], and significantly
"different" results have been obtained. The bowing parameter has
been measured to be from 1 eV to 3.2 eV (see Table 1). The possible
reasons for "scatter" in bowing parameter values have been widely
discussed. It has been pointed out that determination of the In content by
x-ray diffraction (XRD), using the linear interpolation of the c lattice
parameter between GaN and InN (Vegard's law), may result in a systematic
overestimation of the In content, and low values of bowing parameter. This
determination implicitly assumes that InGaN layers are relaxed. It was
observed, however, that due to difficulty of dislocation formation at typical
InGaN growth temperatures, InGaN layers with thickness much larger
than the critical thickness remained pseudomorphic to GaN [2] [5] [9] [8] [10],
and no misfit dislocations were formed at InGaN/GaN interface. Thus, the In
content needs to be corrected with respect to elastic properties of the layers,
resulting in higher values of the bowing parameter. The "scatter"
in bowing parameter exists, however, even in this case. Direct measurement of
the In content by Rutherford back-scattering spectrometry (RBS) [5] and
secondary ion mass spectroscopy (SIMS) with calibrated standards [8] confirms
some of XRD results but differs significantly from others. Besides, it has
been proven that the bowing parameter also depends on the method of the optical
transition measurement. Photoluminescence (PL) [2] [3] [11] gives typically
lower values of energy than optical transmission (OT) [4] [5], photothermal
deflection spectroscopy (PDS) [7], photorefraction (PR) [6] [11] [8] and
spectroscopic ellipsometry (SE) [12]. This difference, however, in addition to
uncertainty in energy measurement due to possible In content fluctuations, is
not large enough to explain "scatter" in the published values of
bowing parameter.
In this paper, we report the InxGa1-xN emission
wavelength dependence on In content up to x=0.2. We discuss the main reason
for the "scatter" in measured values of bowing parameter for InGaN
alloys. We estimate also the uncertainty of bowing parameter measurement. The
main result is that no significant difference exists in bowing parameter
values measured by various groups.
Sets
of samples have been grown by low pressure MOCVD on (0001)-oriented 2"
sapphire rotated substrates. We used an AIX 200/4HT-S reactor equipped with
in-situ optical reflectance monitoring for optimization of the growth
parameters and control of the epilayer thickness. The growth pressure was 200
mbar. TMGa/TEGa, TMIn, NH3 were employed as precursors and
H2/N2 as gas carriers. Structures consisted of a GaN
nucleation layer grown at low temperature, a 1.2 µm-thick GaN buffer
layer and upper InxGa1-xN layers with thickness
15-70 nm. Ternary alloys with x in the range 0.05-0.20 were grown under
temperatures in the range 700-840°C.
The In content in the layers was determined using a high-resolution Bede D1 XRD
system with Cu K
1 radiation. The (00.2) reflections
from GaN and (00.6) reflections from sapphire were measured to calculate the
c-lattice constant of GaN. The (00.6) reflections from GaN and InGaN
were measured to calculate the c-lattice constant of InGaN. Despite
lower x-ray intensity for these reflections compared to more typically measured
(00.2) reflections, the accuracy of lattice constant measurements in this case
is significantly higher. The in-plane a-lattice constant was extracted
from measurement of the asymmetrical (10.5), (20.5) and (11.4) reflections.
The resulting a-lattice constants for the layer were calculated after
averaging the values calculated from each of these three reflections.
Room-temperature (RT) PL spectra were measured to obtain the emission
wavelength of the InGaN layers. A 0.1-1 mW He-Cd laser was used as the
excitation light source for PL.
For
a strained InGaN layer on GaN, biaxial compressive strain
|| and the corresponding perpendicular strain
are induced in the layer, and
where 
is the Poisson's ratio. For hexagonal layers grown
along (00.1) axis the Poisson's ratio is expressed through elastic
constants C13 and C33 as
Measuring the Poisson's ratio is difficult for InGaN films. For the
first-order estimation of the In content in a strained or partially relaxed
InGaN layer, the fixed Poisson's ratio can be assumed. Taking into
account the dependence of the Poisson's ratio on the In content leads to
more sophisticated estimation of the In content [10]. However, this estimation
yields essentially the same results considering experimental uncertainty.
For calculation of the In content, it was assumed that Vegard's law is
valid for both lattice constants of relaxed InGaN. The fixed relationship
c0/a0 = 1.629 was assumed for relaxed
InGaN and GaN, according to the measured values of cGaN =
5.187 Å and aGaN = 3.185 Å for GaN. For relaxed
InN, the value of cInN = 5.705 Å was assumed. The In
content in the InGaN layers can then be calculated as
For InGaN layers pseudomorphic to GaN, the second component of the sum can
be neglected.
The
emission energy of an InxGa1-xN alloy system is usually
described by
where b is a bowing parameter. The accuracy of bowing parameter
measurements is limited by two main factors. The first is InGaN composition
fluctuations over the sample surface. The second is the energy-position
fluctuations in room-temperature PL measurements. We assume that both these
factors have approximately the same influence on experimental accuracy, so that
the uncertainty in emission-energy measurements is 
E 
kT. It gives the bowing parameter measurement uncertainty
b 
4E 0.1 eV. The uncertainty depends on
maximal In content measured. The lower the In content, the higher the
uncertainty. For example, for InGaN layers with maximal In content of only
0.10-0.12, the uncertainty should be about 0.25 eV. For layers with maximal In
content of 0.20, the uncertainty should be about 0.15 eV.
As
it was mentioned above, the deformation of the unit cell in InGaN affects the
interpretation of the In content from XRD measurements. The calculated In
content depends on the Poisson's ratio. Accuracy required for the
Poisson's ratio determination depends on the bowing parameter
uncertainty. The admissible uncertainty of the In content calculation
x can be estimated from (4) as follows:
Thus, the admissible Poisson's ratio uncertainty for measuring the
bowing parameter of strained InGaN alloys with In content up to 0.2 can be
estimated from (3) and (5) as
The values of the Poisson's ratio used by various groups in their
calculations are listed in Table 1. It is evident that existing uncertainty in
these values is higher than the admissible Poisson's ratio uncertainty.
In several works, InGaN layers were assumed relaxed, formally corresponding to
= 0 in (3). It results in low values of the bowing parameter. In
other works, the strain in the layers is considered. In these works, however,
there exists a scatter in values of the Poisson's ratio. The uncertainty
results from the differences between theoretical and various experimental data
on the elastic constants of GaN. Theoretical calculations [13] [14] predict
the value of 2C13/C33 to be about 0.51 for
GaN, leading to a Poisson's ratio of about 0.20. Calculation with this
value yields, however, a lower In content compared to direct measurement by
SIMS [8] or RBS [5]. The measured Poisson's ratio in the case of RBS
In-content determination should be 0.18. Several experimental works were
performed for measuring the Poisson's ratio. Detchprohm et al.
[15] reported 2C13/C33 = 0.38 (
= 0.16) after fitting their data on c- and a- lattice constants
for thick relaxed GaN layers on sapphire. Deger et al. [16] reported
2C13/C33 = 0.56 ( = 0.22) of GaN
from the surface acoustic-wave measurements on epitaxial AlGaN films.
Yamaguchi et al. [17] reported
2C13/C33 = 0.60 ( = 0.23) from
the Brillouin scattering study of epitaxial GaN films. This value was widely
used by many authors for determination of the bowing parameter. However, as it
was mentioned by Yamaguchi et al. [18], the measured elastic constants
of thin GaN epitaxial films differ significantly from actual values due to the
presence of residual stains in epitaxial layers. In addition, epitaxial films
have high dislocation densities, invariability affecting properties of GaN.
The Brillouin scattering study for bulk GaN [18] [19] gave
2C13/C33 = 0.42 ( = 0.17).
As it is seen, the most reliable data on the Poisson's ratio measurements
[5] [15] [18] [19] give the same results as compared to the admissible
Poisson's ratio uncertainty. In our calculations, we use =
0.17 as an average of these data.
Figure 4 presents the measured emission energy dependence on the In content
in the InGaN layers. Error bars represent the possible In content range (as
measured from c-lattice constant only) depending on the relaxation
degree of the InGaN layers. The right end of the bar corresponds to fully
relaxed material and the left end corresponds to fully strained material. This
curve also illustrates possible errors in In content determination when
calculations do not assume the strain in the layers (model of relaxed
material). The same data in comparison with previously published data are
presented on Figure 5. On Figure 6, the published data are recalculated with
respect to the same Poisson's ratio = 0.17. All curves are
corrected also to the same GaN emission energy of 3.42 eV. The curves show
that all the data on InGaN emission energy (band-gap) vs. In content agree well
in the framework of the admissible bowing parameter uncertainty discussed
above. Additional changes in measured dependence can be made after correcting
to the same lattice parameters of GaN and InN.
The
RT emission energy dependence on In content (c-lattice constant) of
pseudomorphically strained InGaN layers grown on GaN is measured. It is shown
that the existing "scatter" in various published data on this
dependence originates mostly from strong uncertainty in the values of the
(In)GaN Poisson's ratio used. After correcting to the same
Poisson's ratio value, all published data on the bowing parameter in
strained InGaN layers yield essentially the same results considering the
experimental uncertainty in emission energy and lattice constant measurements.
The work was supported by Arima Optoelectronic Co. (UK). This is gratefully acknowledged.
last updated Friday, May 11, 2001 2:18:53 PM.© 2001 The Materials Research Society
