Presentation and experimental validation of a model for the effect of
thermal annealing on the photoluminescence of self-assembled InAs/GaAs
quantum dots
M. Srujan, K. Ghosh, S. Sengupta, and S. Chakrabarti
Citation: J. Appl. Phys. 107, 123107 (2010); doi: 10.1063/1.3431388
View online: http://dx.doi.org/10.1063/1.3431388
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v107/i12
Published by the American Institute of Physics.
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JOURNAL OF APPLIED PHYSICS 107, 123107 共2010兲
Presentation and experimental validation of a model for the effect of
thermal annealing on the photoluminescence of self-assembled InAs/GaAs
quantum dots
M. Srujan, K. Ghosh, S. Sengupta, and S. Chakrabartia兲
Center for Excellence in Nanoelectronics, Indian Institute of Technology Bombay, Mumbai, Maharashtra
400076, India
共Received 16 February 2010; accepted 25 April 2010; published online 18 June 2010兲
We present a model for the effect of thermal annealing on a single-layer InAs/GaAs quantum dot
共QD兲 heterostructure and study the corresponding variation in full photoluminescence 共PL兲
spectrum. In/Ga interdiffusion due to annealing is modeled by Fickian diffusion and the Schrödinger
equation is solved separately for electrons and holes to obtain ground state PL peaks of the
heterostructure at different annealing temperatures. We theoretically examine the decrease in strain
effects and carrier confinement potentials with annealing. PL spectra of the entire ensemble of QDs,
annealed at different temperatures, are calculated from a lognormal distribution of QD heights
derived from experimental atomic force microscopy 共AFM兲 data. Results from our calculations,
which illustrate the blueshift in emission wavelength and linewidth variation in PL with annealing,
are in excellent agreement with experimental PL observations on the same samples. This highlights
the potential of the model to assist in precisely engineering the optical properties of QD materials
for specific device applications. Moreover, the simplicity of the model and its multiple useful
features including computation of material interdiffusion, band profiles and full PL spectra make it
a valuable tool to study annealing effects on QD heterostructures. © 2010 American Institute of
Physics. 关doi:10.1063/1.3431388兴
I. INTRODUCTION
Self-assembled quantum dots 共QDs兲 have been gaining
increasing interest over the last years due to their discrete
energy levels and delta-like density of states, which render
them suitable for potential high-quality optoelectronic applications such as QD lasers and photodetectors.1–4 InAs/GaAs
QDs grown in the Stranski–Krastanow mode are particularly
well-studied in terms of their structural and optical
properties5–7 and are being considered for applications in the
IR region.8,9 A major difficulty that could arise in designing
such applications using QD heterostructures is the nonuniform size distribution of QDs, which arises during growth.
This results in significant broadening of optical response,10
an increase in laser threshold current and decrease in optical
gain.11 The impact of a QD size distribution is most evidently seen in the photoluminescence 共PL兲 spectrum of QD
materials, in which the envelope of emissions of all QDs in
the structure follows a distribution over emitted wavelengths,
as opposed to an ideally desired sharp peak. Rapid thermal
annealing 共RTA兲 of as-grown QD heterostructures at temperatures 600– 800 ° C is one technique commonly used to
reduce the linewidth and enhance the full intensity of the PL
emission.12 It is believed that RTA leads to a relatively homogenized size distribution in the QD ensemble, a relaxed
strain distribution and a reduction in structural defects,
thereby conferring desirable properties.13,14 The PL spectrum
of InAs/GaAs QDs with annealing has been extensively
a兲
Electronic mail: [email protected].
0021-8979/2010/107共12兲/123107/7/$30.00
studied in experiments on single as well as multilayer heterostructures with the interest of enhancing their potential for
device applications.15–19
Since QD materials are to be tailored for specific uses, it
is essential to develop a theoretical model to study and predict the experimental variation in their optical properties
with annealing. An approximate treatment of this problem is
presented by Kumar et al.20 in which as-grown and annealed
QDs are treated as triangular and rectangular quantum wells,
respectively, thereby qualitatively explaining trends in PL
peak energy with annealing. In another work,21 modeled PL
peaks are used to derive information about QD dimensions.
In this paper, we present modeling and simulations of the
In/Ga interdiffusion process in single-layer InAs/GaAs QDs
subjected to annealing, and for the first time, study the corresponding variation in full PL spectrum. The model is developed with a central emphasis on Fickian diffusion, derivation of potential profiles and solution of the Schrödinger
equation. We establish a quantitative correlation with our experimental PL spectra, and thereby attempt to provide a useful solution to engineer the optical emission of QD materials
subjected to thermal annealing.
II. EXPERIMENTAL METHODS
To correlate with our theoretical interpretation, we have
grown single-layer InAs/GaAs QDs by solid source molecular beam epitaxy 共MBE兲 on a 具100典 oriented GaAs substrate.
An EPI MOD GEN II system, equipped with Ga, In, Al, Si,
and Be effusion cells and an As cracker was used. The
107, 123107-1
© 2010 American Institute of Physics
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123107-2
J. Appl. Phys. 107, 123107 共2010兲
Srujan et al.
FIG. 1. 共Color online兲 共a兲 Schematic of MBE grown InAs/GaAs QD heterostructure. 共b兲 Truncated pyramid type QD structure as obtained from the
HAADF-STEM image with the dimensions which are incorporated in simulations. 共c兲 Experimental PL spectra from samples annealed at various temperatures
between 650 and 850 ° C.
structure of the samples grown is depicted schematically in
Fig. 1共a兲. After initial oxide desorption, the GaAs buffer
layer was grown at about 590 ° C, and then the substrate
temperature was slowly reduced to 520 ° C to grow the InAs
QD layer. A slow growth rate of 0.032 ML s−1 was chosen
to deposit a 2.7 ML InAs layer. The QD layer was subsequently covered with relatively thick 共1000 Å兲 GaAs capping. The heterostructure was then terminated with an uncapped 2.7 ML InAs QD layer grown under identical
conditions as the previous ones for atomic force microscopy
共AFM兲 measurements. Samples cut from the central region
of the as-grown wafer were subjected to ex situ RTA in AnnealSys AS-One 150 RTP system for duration of 30 s in an
argon atmosphere at different annealing temperatures between 650 and 850 ° C. All the samples treated in this way
exhibited highly reflecting surfaces like that of the as-grown
samples. The annealing was done under GaAs proximity
capping in order to prevent degradation of sample quality
due to arsenic out-diffusion from the sample surface.14 PL
experiments are carried with a closed cycle He cryostat operating at 8 K using a 405 nm solid state blue laser at 40 mW
excitation power. Experimental PL spectra from samples annealed at various temperatures are shown in Fig. 1共c兲. We
observe a sharp decrease in PL full intensity in samples annealed at temperatures exceeding 800 ° C. Although annealing has the beneficial effect of full width at half maximum
共FWHM兲 reduction, it is to be borne in mind that annealing
at very high temperatures may lead to excessive interdiffusion, and thereby complete degradation of QD structure.
III. THEORETICAL MODEL AND SIMULATIONS
A. Annealing induced interdiffusion
High angle annular dark field-scanning tunneling electron microscopy 共HAADF-STEM兲 data from our as-grown
sample22 indicate a truncated pyramid type QD structure as
shown in Fig. 1共b兲. The exact QD dimensions, which we
incorporate in our simulations, are arrived at considering
both TEM results and a good correlation with PL data from
the as-grown sample. The dimensions predicted from asgrown PL are slightly larger than those from TEM data, and
this is likely due to presence of some Ga in the outer layers
of the QD.
We carry out interdiffusion simulations on this single
QD in the barrier. The system is taken in a cuboidal box of
dimensions 2wbot ⫻ 2wbot ⫻ 8hQD. These dimensions are chosen in such a way as to reduce spurious errors arising due to
abrupt boundary conditions, while at the same time, bearing
in mind time complexity of the computation. It is widely
established that RTA of InAs/GaAs QD heterostructures
leads to In/Ga interdiffusion between QD and barrier
material.12,13,23–25 We model this diffusion process for indium
using Fick’s second equation
⳵ x共r,t兲
= Dⵜ2x共r,t兲,
⳵t
共1兲
where x denotes the mole fraction of In in InxGa1−xAs, and D
the diffusion constant of InAs is assumed to be constant
throughout.
We assume the as-grown QD to be composed purely of
InAs, and use the following initial conditions for Eq. 共1兲:
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123107-3
J. Appl. Phys. 107, 123107 共2010兲
Srujan et al.
FIG. 2. 共Color online兲 共a兲 Indium concentration profile of the as-grown QD. 共b兲 QD annealed at 650 ° C, 共c兲 750 ° C, and 共d兲 850 ° C along the vertical
symmetry plane.
x = 1, in the QD and WL,
x = 0, in the barrier material.
We then solve Eq. 共1兲 in three dimensions by discretization in both time and space, with Dirichlet boundary conditions and annealing time t = 30 s to obtain the composition of
the annealed heterostructure. To account for the variation in
properties with annealing temperature Ta, we consider an
Arrhenius-type temperature dependence of the diffusion
coefficient.21
冉 冊
D = D0 exp −
Ea
,
kTa
共2兲
where the parameters D0 = 8.5⫻ 10−14 m2 s−1 and Ea
= 1.23 eV, respectively.
Results from our simulations 共Fig. 2兲 show appreciable
change in indium concentration profile of the QD structure
with increase in annealing temperature. Gradual desorption
of indium from the QD with gallium in-diffusion is clearly
seen commencing at an annealing temperature of 650 ° C
and total dissolution of the QD in the wetting layer 共WL兲 is
observable at 850 ° C.
B. Variation in carrier confinement potentials
A single-band assumption is used to calculate the carrier
confinement potentials and effective masses, which are necessary to solve the Schrödinger equation for carrier energy
states. We use the modified concentration profile of the annealed heterostructure and bulk band parameters of InAs and
GaAs to obtain the electron and hole confinement potentials.
The position-dependent bulk band gap after interdiffusion is
obtained from the following empirical relation 关Eq. 共3兲兴 共Ref.
26兲 involving indium content x in InxGa1−xAs.
Eg共r兲 = 1.519 − 1.584x共r兲 + 0.475x2共r兲.
共3兲
To calculate the electron and hole confinement potentials, it
is assumed that bulk CB and VB offsets, ⌬Ee and ⌬Eh bear
the constant ratio 76/34.27 However, crystal strain arising due
to lattice mismatch between InAs and GaAs considerably
affects the band profiles.28,29 A simplified model, which enables us to recalculate the confinement potentials, is adopted
to account for the effect of strain.21 In the case of the asgrown QD, it is assumed that the energy shifts in the calculated CB and VB due to strain in the dot material are constant. This is justified since the variation in strain inside the
dot material is small. We take these shifts to be ⌬Ee =
−450 meV and ⌬Eh = 90 meV.30,31 In case of an annealed
QD, the strain is assumed to vary in a linear fashion with the
mole fraction of indium, thereby resulting in ⌬Ee and ⌬Eh
varying in space. Effectively, these strain corrections cause
the portion of the calculated CB inside the QD to move
closer to the bulk GaAs CB, and the calculated VB inside to
move farther above the bulk GaAs VB. Variation in effective
mass is taken care of by a linear interpolation of electron and
hole effective masses in InAs and GaAs using Eqs. 共4兲 and
共5兲
mⴱe 共r兲 = 0.0665关1 − x共r兲兴 + 0.04x共r兲,
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共4兲
123107-4
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Srujan et al.
FIG. 3. 共Color online兲 Conduction band profiles of the 共a兲 as-grown QD and the QD annealed at 共b兲 650 ° C, 共c兲 750 ° C, and 共d兲 850 ° C, along a horizontal
cross-sectional plane.
mⴱh共r兲 = 0.2774关1 − x共r兲兴 + 0.241x共r兲.
共5兲
It can be seen from the results of our band profile calculations 共Fig. 3兲 that annealing leads to a decrease in carrier
confinement potentials inside the QD. The reason for the
same may be understood by considering two effects. First,
interdiffusion causes an overall increase in QD Ga content,
which leads to a greater difference between both the calculated bands inside the QD, and hence reduced band offsets.
Second, the strain effect, which is directly related to x in our
simulations, also becomes more uniform and decreases in
magnitude with annealing. As the correction due to the strain
energy distribution decreases, it is expected that the band
profiles deviate lesser from the originally calculated profiles,
thereby maintaining the confinement potential for electrons
at a greater value but for holes at a lower value. Both these
effects, that is interdiffusion induced decrease in band offsets
and variation in strain appear to favor a decrease in hole
confinement potential. In case of electrons, the former effect
dominates, ultimately causing their confinement potential to
reduce too. Thus, both potential profiles become observably
smoother and shallower with an increase in annealing temperature. Such a variation in confinement potential is likely
to lead to an increase in ground-state carrier energy.
C. Schrödinger solver and calculation of ground state
PL peaks
The three-dimensional Schrödinger equation 关Eq. 共6兲兴 is
solved to obtain the energy states of an electron and a hole in
the QD
−
冉
冊
1
ប2
ⵜ ·
ⵜ ␺ + V␺ = E␺ .
2
mⴱ
共6兲
Here simple finite difference approximation is used, thereby
resulting in a discretization of the QD in barrier system described earlier. We impose ␺ = 0 on the boundaries of the
cube, under the assumption that QDs in the structure are
spaced sufficiently enough to neglect mutual interaction. The
discretized Schrödinger equation is a large-scale matrix eigenvalue problem, in which the energies of the system are
given by the eigenvalues of a large 共106 ⫻ 106 order兲 sparse
matrix. The open-source MATLAB program EIGIFP.M 共Ref. 32兲
is used to compute the first few eigenvalues of H, in which
lies our interest. It is noticed that this program, which uses a
preconditioned Krylov subspace method,33 is especially efficient in our context when compared to conventional largescale eigenvalue estimators based on Arnoldi iteration methods.
Electrostatic interaction between electron and hole
charge densities in the QD is accounted for by addition of the
interaction energy ECoul. Here a full self-consistent
Schrödinger–Poisson method is not adopted, since it is observed that ECoul is itself small compared to the initially calculated confinement potentials.
To determine the Coulomb potential due to the electron
共hole兲 charge density, we solve the Poisson equation, given
by
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123107-5
J. Appl. Phys. 107, 123107 共2010兲
Srujan et al.
FIG. 4. 共Color online兲 Comparison between model and experimental variation in PL peak energy with annealing temperature.
ⵜ2Ve,h = −
␳e,h
.
苸
共7兲
E. Calculation of the full PL spectrum
The electrostatic interaction energy is given by
ECoul =
冕
␳h,eVe,hd␶ .
共8兲
In the above equations, ␳e = −e␺2e and ␳h = e␺2h, ␺e and ␺h
being the ground state electron and hole wave functions
which are obtained from the Schrödinger solver. Note that in
case of an annealed QD, the permittivity 苸 is itself varying
in space, which we approximate by a linear interpolation
between 苸InAs and 苸GaAs.
PL peak energy EPL is calculated by
EPL = Eeconf + Ehconf + ECoul
FIG. 5. 共Color online兲 Variation in electron and hole confinement energies
EConf,e and EConf,h and corresponding variation in the PL peak, EPL with
reference to the bulk GaAs conduction and valence bands.
So far, we have described the calculation of PL energy
from a single QD and studied its variation with annealing
temperature. To calculate the PL emission from the ensemble
of QDs of varying size in the heterostructure, we use the QD
size distribution to calculate the intensity distribution as a
function of PL energy, which is essentially the PL spectrum.
AFM data from the surface layer of QDs in the sample reveal
a distribution of QD heights as shown in Fig. 6共a兲. We observe that the data is best fitted with a lognormal distribution
given by
共9兲
Eeconf and Ehconf are the electron and hole energy states, calculated with a common reference, in our case, the bottom of
the conduction band in the barrier material.
D. Correlation with experimental PL peaks
Figure 4 shows the comparison between calculated
ground state PL peak energies and our experimental results.
The theoretically calculated values are in very good agreement with the experimental data, with a standard deviation of
8 nm, and a maximum deviation of 20 nm at 700 ° C. This
variation in PL peak energy may be explained in terms of the
change in confinement energies of the carriers, as shown in
Fig. 5. The decrease in carrier confinement potentials, which
occurs as elucidated earlier, causes the respective confinement energies to increase with annealing, thereby blueshifting the PL peak. Further, our earlier observation that variation in strain due to annealing aids the decrease in hole
confinement potential but opposes the decrease in electron
confinement potential is reflected in the rate of change in
electron and hole energy states with annealing temperature.
It can be observed that the ground state energy of holes increases at a faster rate than that of electrons.
FIG. 6. 共Color online兲 共a兲 Distribution of QD heights in the uncapped layer
of the heterostructure as extracted from AFM observations. A lognormal
distribution fits the data more closely than a Gaussian. 共b兲 Shell-like formation of an InAs/GaAs QD. An ␧ variation in QD base width causes an equal
variation in QD height, and corresponding variations along other
dimensions.
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123107-6
J. Appl. Phys. 107, 123107 共2010兲
Srujan et al.
FIG. 7. 共Color online兲 Comparison between model and experimental variation in FWHM of the PL spectrum.
再
冎
关ln共h兲 − ␮兴2
N0
N共h兲 = exp −
,
h
2␴2
共10兲
with ␮ = 1.1429, ␴ = 0.3199= 0.28␮ and N0 an appropriate
normalizing constant. We expect that the distribution of QDs
in the internal layer follows a similar height distribution
共nearly equal ␴ / ␮ value兲 but with a lower modal height due
to the effect of the capping layer. Modal QD dimensions are
those which are used to calculate the PL peak, as discussed
earlier. Further we assume that QD growth occurs in a shelllike manner,34 thereby allowing us to calculate the horizontal
dimensions of any QD in the ensemble from its height and
the modal dimensions 关Fig. 6共b兲兴.
PL ground state energies for QDs of heights varying in a
narrow regime are calculated, and we observe a near linear
relation between height of the unannealed QD and PL emission energy. Using such a linear fit for every annealing temperature, we obtain the PL emission energies from several
QDs in the ensemble. It is assumed that the intensity of emission at a particular energy is directly proportional to the
number of emitting QDs, which we infer from the QD height
distribution. We, therefore, obtain intensity data points at
various emission energies. The Gaussian fit of these points is
considered the ground state PL spectrum of the QD ensemble.
FIG. 8. 共Color online兲 Predicted variation in PL emission energy with annealing of hemispherical QDs of varying diameter.
allow us to make simple qualitative observations. From the
results in Fig. 8, we observe that smaller QDs undergo a
more rapid blueshift at lower annealing temperatures than
larger QDs. When an as-grown QD is subjected to annealing,
indium present in the outer layer of the QD diffuses into the
barrier due to a greater concentration gradient at the boundary. In the case of small QDs, the QD core is thereby exposed to the barrier at low annealing temperatures leading to
rapid indium out-diffusion. Indeed, with very small QDs
共around 5 nm兲, we observe from our results that such an
effect causes the PL emission energy to increase rapidly and
saturate to the bulk GaAs band gap. This QD size-dependent
blueshift causes different portions of the PL spectrum from
the as-grown heterostructure to respond differently to annealing. The tail at the lower wavelength 共smaller QD size兲 end
undergoes a larger blueshift than the higher wavelength
共larger QDs兲 end. This results in an effective broadening of
the PL spectrum with an increase in annealing temperature.
However, at around 750 ° C, the blueshifts from large and
small QDs become more comparable, reducing the significance of this effect. Beyond this temperature, it is likely that
the effect of an overall size homogenization in the ensemble
causes a reduction in FWHM as the QDs begin to merge into
the WL.
IV. CONCLUSION
F. Correlation with experimental variation in FWHM
From the PL spectra obtained from fitting, we observe
that the FWHM increases up to 750 ° C and then decreases
共Fig. 7兲, which captures the experimental trend in FWHM
variation. A slight difference between model and experimental values is likely to arise due to neglecting higher order
effects, for instance, the effect of QD volume on intensity of
emission. This trend in FWHM variation may be explained
by considering two competing effects viz. dependence of the
annealing-induced blueshift on dot size and homogenization
of the dot size distribution. We first examine the difference in
the effect of annealing on the emission energies of different
sized QDs in the ensemble by carrying out extended simulations. Hemispherical dots are used in the study, since they
We have presented a model for the effect of thermal
annealing on the PL properties of single-layer InAs/GaAs
QDs, and followed it up with an experimental correlation.
We initiated our study with computation of chemical composition of the annealed heterostructure. Subsequently the corresponding band profiles were calculated and their variation
with annealing was examined. The band profiles were used
to solve for carrier energy states from the Schrödinger equation, and we observed a well correlated blueshift in PL peak
energy. Operating within a similar framework, PL spectra
from the QD ensemble in the heterostructure, annealed at
different temperatures were calculated. In addition to quantitatively reproducing the variation in PL spectra, our studies
shed light on changes in strain effects and potential profiles
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123107-7
in QD materials, which may form the basis for investigations
into other phenomena induced by annealing. The close experimental correlation achieved by the model represents an
advance in predicting annealing effects on QD heterostructures and specifically addresses the necessity for a design
tool for optical properties of QD materials. A detailed calculation of strain effects on confinement potentials may help
offer a richer picture of the effect of interdiffusion on band
profiles and also perhaps obtain a better correlation with experimental results.
ACKNOWLEDGMENTS
The authors acknowledge the financial support provided
by the Department of Science and Technology, India. The
authors also acknowledge the partial financial support provided by the Ministry of Communication and Information
Technology, Government of India, through the Centre of Excellence in Nanoelectronics and by the European Commission under Contract No. SES6-CT-2003-502620 共FULLSPECTRUM兲.
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