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Carrier thermal escape and retrapping in self-assembled quantum dots
S. Sanguinetti
Istituto Nazionale per la Fisica della Materia, Dipartimento di Scienza dei Materiali, Universita
`di Milano ‘‘Bicocca,’’
Via Cozzi 53, I-20125 Milano, Italy
M. Henini
School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
M. Grassi Alessi and M. Capizzi
Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica, Universita
`di Roma ‘‘La Sapienza,’’ Piazzale A. Moro 2,
I-00185 Roma, Italy
P. Frigeri and S. Franchi
CNR-MASPEC Institute, Pario delle Scienze 37a, I-43010 Fontanini, Parma, Italy
!Received 22 December 1998; revised manuscript received 7 May 1999"
The effects of carrier thermal escape and retrapping on the temperature dependence of the photolumines-
cence of InAs/GaAs self-assembled quantum dots are investigated. A systematic experimental study of the
temperature evolution of the photoluminescence spectra in two different sets of samples is reported. The
photoluminescence behavior is well reproduced in terms of a steady state model for the carrier dynamics which
takes into account the quantum-dot size distribution, random population effects, and carrier capture, relaxation,
and retrapping. The relative contributions of these processes to the photoluminescence thermal quenching is
discussed. #S0163-1829!99"07335-X$
I. INTRODUCTION
The quest for high performance optoelectronic devices
has promoted a growing interest for zero-dimensional semi-
conductor heterostructures #or quantum dots !QDs"$. In these
systems, indeed, the strong localization of the electronic
wave function leads to an atomiclike electronic density of
states and to the possible realization of novel and improved
photonic and electronic devices.1–3 Furthermore, the self ag-
gregation of defect-free QDs during the epitaxial deposition
of strained semiconductor layers4has stimulated a large
number of experimental works. QD injection-laser proto-
types, made from InAs/GaAs heterostructures, have now
characteristics as good as quantum well based devices.5
A crucial issue for the realization of room temperature
efficient photonic devices is the understanding of the tem-
perature dependence of the QD photoluminescence !PL". An
unusual temperature dependence of !i"the integrated inten-
sity, !ii"the peak energy position, and !iii"the width of the
PL bands in InAs/GaAs self-assembled QDs has been re-
ported by several groups.6– 11 In particular, the emission en-
ergy of the PL bands exhibits a sigmoidal behavior and the
PL band linewidth decreases with increasing temperature.
Such anomalies in the PL temperature dependence cannot be
explained on the basis of a direct coupling between QD
states,12 even if one introduces two activation energies
!mainly to account for the high-temperature quenching of the
PL bands". Moreover, the general description in terms of a
carrier thermal evaporation from the QD ground state is
complicated by the presence of a wetting layer !WL"con-
necting the QDs. Recently, the simple thermal escape model
has been criticized and the critical role of the carrier capture
and relaxation—as well as of their dependence on
temperature—in determining the QD radiative recombination
efficiency has been pointed out.13 However, the carrier relax-
ation in QDs is poorly understood, in particular, the phenom-
enology of the carrier interlevel scattering. In the PL spectra,
an emission from excited states has been observed before the
emission from the ground state was saturated. This feature
has been attributed to a slowdown of the interlevel scattering
since the energy difference of the electronic excited states
varies with QD shape and size and cannot always match the
optical phonon frequencies. However, no clear evidence of a
phonon bottleneck has been found, so far, in the time depen-
dence of the PL spectra.14– 16 Moreover, it has been demon-
strated that the observation of excited states in the PL spectra
is possible even in the case of very fast relaxation processes
like Auger-like recombination and impurity assisted
relaxation.17,18 In fact, because of the random nature of the
QD population mechanism,19 a number of QDs can show an
excited state emission before the ground state is saturated in
the whole QD ensemble.
In this paper, a steady-state thermal model which takes
into account all the relevant thermalization and quenching
processes active in QDs is proposed. This model well repro-
duces the temperature dependence of the PL spectrum in a
large number of samples. The model is based on a simple set
of rate equations connecting the QDs, the wetting layer, and
the GaAs barrier states, namely, the main components of the
system. It takes into account the random carrier capture by
QDs,19 the energy dispersion of the ground states, the lack of
direct coupling between different QDs, and the role of the
wetting layer which provides a channel for carrier exchange.
In the model, the low-temperature PL spectrum provides the
PHYSICAL REVIEW B 15 SEPTEMBER 1999-IVOLUME 60, NUMBER 11
PRB 60
0163-1829/99/60!11"/8276!8"/$15.00 8276 ©1999 The American Physical Society
QD density of states !DOS", while the ground state recom-
bination time and the WL-to-QD excitation transfer time are
set equal to values reported in the literature. A limited num-
ber of fitting parameters are introduced in the model. A criti-
cal discussion of the relative weight of the different phenom-
ena involved in the recombination process is presented on
the grounds of a quantitative comparison between experi-
mental observations and model simulations.
This paper is organized in the following way. Section II is
dedicated to the description of the sample preparation proce-
dure and of the experimental setup. In Sec. III, the experi-
mental findings and models proposed in the literature for the
PL temperature dependence are briefly reviewed. Our rate
equation model is presented in Sec. IV. Finally, the experi-
mental results and the model simulations are reported and
discussed in Secs. V and VI, respectively.
II. SAMPLE GROWTH AND EXPERIMENTAL DETAILS
Two different sets of samples obtained from different
sources were studied. The growth procedures are summa-
rized here. The first set of samples was grown by molecular
beam epitaxy !MBE"!Varian Gen-II system"on liquid encap-
sulated Czochralski semi-insulating GaAs substrates with
!100"!0.5° !sample NU1468"and !311"A/B !0.5°
!samples NU1422 and NU1478, respectively"surfaces. The
structures consisted of the substrate, a 0.3
%
m thick undoped
GaAs buffer followed by a 15"(3.8 nm Al0.33Ga0.67As
#3.4 nm GaAs"superlattice, a 0.2
%
m undoped GaAs, 1.8
monolayers !ML"of InAs !the QD region", and, finally, a 30
nm undoped GaAs capping layer. The growth temperature
(Tg$630 °C) was monitored by a pyrometer, except during
the growth of InAs and of the capping layer (Tg$500 °C) .
The growth rates were 1 ML/s for GaAs, 0.5 ML/s for AlAs,
and 0.066 ML/s for InAs !average thickness 1.8 ML". At the
growth temperature of 630 °C, a very low As/Ga ratio
(&13) was chosen to achieve a (2"4 ) reflection high-
energy-electron diffraction !RHEED"pattern during the
growth on !100"surfaces. The samples were rotated during
the growth to improve their uniformity. The change from a
streaky to a spotty RHEED pattern has been taken as a fin-
gerprint of the onset of the three-dimensional growth mode.
After growth, the epitaxial surfaces were examined by using
a Nomarski phase-contrast optical microscope and were
found to be mirror smooth and nearly defect-free. The sec-
ond set of samples !MA882, MA884, and MA886"have
been prepared in a MBE Varian Gen-II modular system
equipped with a valved cracker cell for As. The cracker stage
of the cell was kept at 500 °C so that As4was used during
epitaxial growth, with a beam equivalent pressure ratio
As/Ga &17. The substrates were radiatively heated and the
growth temperature was monitored by an optical pyrometer
suitable for GaAs. The growth rates were 0.87 and 0.133
ML/s for GaAs and InAs, respectively. The structures consist
of a semi-insulating (100)!0.25° GaAs substrate, a 100 nm
thick GaAs buffer grown at 600 °C, a InAs deposit grown at
500 °C with a coverage ranging between 1.6 !MA882"and
3.0 ML !MA886", and a 21.5 nm thick GaAs cap layer. At
the end of the buffer a growth interruption of 210 s was
performed to change and stabilize the substrate temperature
for the growth of the InAs dots. A very sharp (2"4)
RHEED pattern was observed after this stage, which indi-
cates a very smooth surface. During the deposition of InAs
the onset from 2D to 3D growth mechanism was observed
after 12.1 s, corresponding to a deposition of 1.6 ML of
InAs. A second 210 s growth interruption was performed
before growing the GaAs cap in order to lower Tgdown to
350 °C. At this temperature 5 ML of GaAs were grown, in
order to minimize the interactions between InAs and GaAs,
then the temperature was raised up to 600 °C and 20 nm of
GaAs were grown to complete the cap. This procedure has
been shown to give high quality structures.
The PL spectra were measured using a grating mono-
chomator. The excitation source was a multiline Ar#laser.
The laser spot diameter ranged from 100 to 500
%
m, with an
excitation power density varying from a few Wcm%2to
some hundredths Wcm%2. Measurements between 10 and
300 K were performed using a cold-finger, closed cycle He
refrigerator. The low-temperature spectra of the six samples
are displayed in Fig. 1.
III. QD TEMPERATURE BEHAVIOR
The experimental findings and the models developed so
far to explain the complex phenomenology shown by the
temperature dependence of the InAs/GaAs QD
photoluminescence6– 11 are summarized in the following.
!1"The PL line of a single QD shifts with temperature as
the bandgap does.10
!2"The PL bands of an ensemble of QDs undergo a red
shift with temperature much faster than that of the InAs band
gap and, sometimes, their peak energies exhibit a sigmoidal
dependence on temperature.6
!3"In the latter case, the full width at half maximum
!FWHM"is constant up to &30 K and undergoes a strong
reduction !up to &50%) at higher temperatures. After hav-
ing reached a minimum value at T&100 K, the FWHM
slowly recovers the low-Tvalue.
!4"The spectrum of those bands which exhibit a complex
structure at low-Tundergoes strong changes with increasing
temperature, in particular, the emission on the low-energy
side of the QD band gains in intensity with respect to that on
the high-energy side, at least for T&30 K.6
FIG. 1. PL spectra of the samples measured at T$10 K. !a"
NU1468 !continuous line", NU1422 !short– dotted line", NU1478
!dashed line";!b"MA882 !continuous line", MA886 !short– dotted
line"and MA884 !dashed line". Each spectrum is normalized to its
maximum.
PRB 60 8277
CARRIER THERMAL ESCAPE AND RETRAPPING IN . . .
After the works of Fafard et al.10 and Brusaferri et al.6the
PL anomalous temperature dependence has been attributed
mainly to excitation-distribution effects in the DOS of QDs
with random size. In fact, the single QD emission follows
well that of the InAs band gap and a carrier redistribution
over DOS favors the emission on the low-energy side of the
band.
In spite of this complex phenomenology, the temperature
dependence of the integrated intensity of QDs is usually
fitted by assuming that the carrier dynamics is
quantum-well-like.6,7,9,12 Therefore, a single activation en-
ergy for the carrier escape from coupled, not saturated QD
ground states to a quenching channel is introduced. Although
such a simplified model roughly describes the temperature
dependence of the PL integrated intensity, it does not ac-
count for the other features mentioned above. In particular,
the model predicts a change in slope between the low- and
high-temperature ranges much sharper than the round
‘‘knee’’ observed and reported in the literature. In order to
take into account the enhanced red shift #item !2"$ and the
‘‘knee’’ feature, several authors introduced an inhomoge-
neous broadening in their models, thus representing the QD
ensemble as a multiple, decoupled system of QWs. The
quenching state has been identified6,9,10 with the wetting
layer, through which excited carriers can reach non radiative
recombination centers at the heterostructure interface or in
the barrier !alternatively, they can be recaptured by QDs".A
rate equation model developed for the multiple QW case20
has been used, then, with the WL acting the role of the QW
barrier and QD parameters being used in place of their QW
counterparts. On the contrary, the effect of QD ground-state
random population and saturation were not taken into ac-
count. The emission intensity of each single dot in the en-
semble is given by
Ii$PRi
!
'
i#Ri/Ui"
!
R!#(
j$1
nd
Rj/!
'
j#Rj/Uj"
"
,!1"
where the subscripts iand jstand for different QDs, ndis the
total number of QDs, Riis the radiative rate of each QD, R!
the nonradiative recombination rate in the WL, Uithe trap-
ping rate constants,
'
i$exp(%Ei/kT)—Eibeing the depths
of the confined QD states below the WL state, and Pis the
excitation rate into the WL. Finally, the above equation has
been fitted to the PL integrated intensity.11
IV. THEORETICAL MODEL
Equation !1"is the basis for the development of an ex-
tended set of rate equations modeling the excitation and re-
combination processes in QDs. In these models, where car-
riers injected in the barrier at a constant rate by an external
excitation source fall in the QDs or recombine nonradiatively
directly in the barrier, the role of carrier reservoir, played by
the barrier in QWs, is taken by the barrier itself and by the
wetting layer. This doubles the number of activation energies
and gives rise to a carrier redistribution among the QD levels
which is observed much before a substantial quenching of
PL takes place6and is more important at high temperatures.
We model the GaAs barrier, wetting layer, and QD system
under optical excitation by the rate-equation scheme shown
in Fig. 2.
The following points determine the set of rate equations.
!1"The external excitation fills a reservoir, the GaAs bar-
rier, from which the excitation is transferred to the WL at
rate g.
!2"The direct QD-to-barrier capture/emission channels
are not taken into account.
!3"The nrexcitons in the WL are either captured by dots
with probability
)
c, or are thermally excited to the barrier at
a rate given by
)
eexp(%*EWL barrier /kBT), or are captured by
nonradiative recombination centers with probability
)
t.
!4"Only the ground state of each of the nddifferent dots
enters the rate equations since only PL from the ground state
has been observed in the present experiment, consistently
with the low-excitation density used in our experiment.
Moreover, QD excited states are usually 50– 70 meV higher
in energy than the ground state and, therefore, do not con-
tribute to the QD density of occupied states. Their role as an
excitation buffer for the QD has been also neglected, mainly
because the magnitude of the transition times to the QD
ground state is expected to be similar to that of the WL– QD
capture process.
!5"The QDs are filled randomly with only one e-hpair
and
nd$nf#ne,!2"
where nfis the number of filled QDs and nethat of empty
QDs. This assumption is not conflicting with recent reports
of biexciton emission from QDs.21 In fact, the exciton-
exciton binding energy is low and the model for the tempera-
ture dependence of a doubly degenerate state is almost
equivalent to that of two singly degenerate, decoupled states.
!6"Since the process of QD population is intrinsically
random,19 the QD DOS is assumed to be proportional to the
low-TPL emission band. This allows us to treat on an equal
footing both monomodal and multimodal PL spectra, with no
a priori assumption on the nature of the QD DOS.
FIG. 2. Schematic representation of the rate equation model
described in the text.
8278 PRB 60
S. SANGUINETTI et al.
!7"A general consensus has not been reached yet on the
correlated or uncorrelated nature of carriers in QDs. In our
model carriers are assumed to behave as correlated e-hpairs
and all dots are neutral, mainly for sake of simplicity. How-
ever, in the case of QWs it has been shown that the tempera-
ture dependence of the QW photoluminescence is not af-
fected by the band-offset ratio.20,22 In that case, there is no
need of a separate dynamics for electrons and holes for de-
scribing thermal quenching and retrapping of carriers. The
assumption of a carrier dynamics dominated by coupled e-h
pairs is quite sound in QDs, where the zero-dimensional con-
finement gives rise to a strong Coulomb interaction between
carriers.23
!8"The radiative recombination frequency
)
r+1/
,
ris the
same in all QDs, which are assumed to be defect free, in
particular free of nonradiative channels in the bulk and at the
interfaces.
!9"The frequency of thermal escape of carriers from the
QDs to the WL is given by exp(%*EQD-WL /kBT)*
)
b
)
c,
where
)
bis the effective DOS of the WL. This assumption is
consistent with the observation that the density of the WL
states is much larger than that of the QD states.
We neglect several expected temperature dependencies of
the model parameters. This choice is motivated by our sake
of simplicity and by our effort of presenting a model with the
lowest number of parameters that still leads to a significant
progress in reproducing the experimental data. Therefore, we
did not introduce any new parameter, or guessed temperature
dependences, unless they sizeably improved our fits.
The rate equations of the QD system are the following:
nd!E"$nf!E"#ne!E",
nf
˙
!E"$%nf!E"
)
r%nf!E"
-)
c
)
b#nrne!E"
)
c$0, !3"
nr
˙
$g%nr
)
t%nr
')
e#
#
nf!E"
-)
c
)
bdE
%
#
nrne!E"
)
cdE$0,
where
.
nd(E)dE$nd,
-
$exp(%*EQD-WL /kBT) and
'
$exp(%*EWL%barrier /kBT).
The effect of the introduction of a random population of
QDs in the model may be made more clear by considering
the case of a
/
-like DOS for QDs. In the case of stationary
conditions, one ends up with
g$!
)
t#
)
e
'
"nr#
)
rnf,!4"
)
cndnr$!
)
r#
)
cnr#
-)
c
)
b"nf.!5"
Equation !4"is the steady-state equation for excitons in
the QD#WL system. Equation !5"is similar, but pertains
only to the QD system. It contains in the right hand side an
additional term (
)
cnrnf) which is induced by the presence in
the dot of a single level to accommodate the excitons; see
Eq. !2". This last term accounts for an increase of the capture
time as the dots are filled and plays a major role in the carrier
dynamics only when the number of full dots is a sizeable
fraction of the total.
In summary, the present model takes into account three
major physical effects neglected by previous models,8,11
namely, the random nature of the QD population process, the
saturation of the QD ground state, and the thermal escape of
carriers from QDs into the barrier. The former two processes
play an important role in determining the PL spectral depen-
dence on Tfor different excitation power densities by giving
rise to a temperature and size dependent capture probability.
This favors the capture of carriers by small, high-energy QDs
which, otherwise, should loose their carriers for increasing T
before large QDs do. This effect slows down the shift with
temperature of the carrier population from small to large
QDs. The carrier thermal escape into the barrier gives rise,
instead, to an increase in the PL thermal quenching and to a
second activation energy, observed at high temperature.
V. EXPERIMENTAL RESULTS
The set of equations !3"has been solved numerically. The
QD PL spectrum has been calculated then at different tem-
peratures via the equation I(E)$A
)
rnf(E), where Ais a
normalization factor.
The model contains eight parameters, namely,
g,nd,
)
r,
)
c,
)
b,
)
t,
)
e, and EWL , some of which fixed to
experimentally determined values.
!1"Since the QD density just after the transition is a
steeply growing function of the coverage,4the QD densities
reported in the literature cover a wide range of values, with
an upper value of &1011cm%2. In the present work, ndhas
been determined for the first set of samples by contact mi-
croscopy on uncapped samples grown under the same con-
ditions than those of the capped samples investigated by PL.
The ndvalue (2"1010 cm%2) well agrees with previous
findings.24 Because of the lack of direct measurements, we
assumed nd$2"1010 cm%2also in the case of the second
set of samples. This assumption does not reduce the validity
of the model, nevertheless, some care must be paid when
comparing the values of the fitting parameters of the two sets
of samples. In fact, the dot density enters as a multiplicative
factor in various terms of the model rate equations.
!2"It has been shown that, in CW excitation at low tem-
perature, the first excited state can be seen in PL when the
excitation g
,
r/nd&0.2; see Ref. 19. By looking at the onset
with excitation power density of the excited state emission
and by rescaling this power threshold to the effective power,
one gets g
,
r/nd$0.001, a value which reproduces well the
temperature dependence of the QD PL band. This is not af-
fected by small changes of g, thus confirming that QDs are
far from saturation, at least in our experimental conditions.
In fact, gdetermines the relative number of filled dots at T
$0 and sizeably affects the QD dynamics only when nf
&nd, namely, at excitation much higher than those used
here; see Eq. !5".
!3"
)
rand
)
care related, respectively, to the times for
carrier radiative recombination and capture in QDs. Values
of 1 ns for
)
r,and of 30 ps for the WL-to-QD capture time
)
chave been used !i.e., the values reported in Refs. 14,16".
In summary, only four parameters,
)
b,
)
t,
)
e, and EWL
are let free to vary when fitting the temperature dependence
of the PL integrated intensity. In the fitting process, the tem-
perature evolution of the PL spectral line shape provides
PRB 60 8279
CARRIER THERMAL ESCAPE AND RETRAPPING IN . . .
only a criterion for the chi-square minima that prevents the
fit from being trapped in an unrealistic local minimum in the
parameter space. This criterion is the level of reproduction of
the PL spectrum taken at the highest temperature. Model
simulations of PL spectra are then calculated by solving Eq.
!3"with the parameter values determined by the fit of the
integrated intensity.
In Fig. 3, the experimental !symbols"and theoretical
simulations !full lines"for the temperature dependence of the
integrated intensity, the FWHM, and the peak position are
reported for the case of sample NU1422. This sample is
characterized by a structured low-TPL band, see Fig. 1, and
exhibits, as well as the other two samples in the set, all the
anomalous temperature dependencies reported in Sec. III.
For increasing temperature, the QD-band shift towards lower
energy is faster than that of the InAs band gap, then it slows
down thus giving rise to a sigmoidal shape. At the same
time, the FWHM of the QD band reduces by &30– 40 % and
reaches a minimum at about 110 K. At higher temperatures
the FWHM slowly increases and recovers the low-
temperature value at about 200 K. The agreement between
experimental data and model simulations goes from good to
excellent, despite the model simplicity. A similar spectral
behavior and good degree of data simulation has been
achieved also for the other two samples, NU1468 and
NU1478, which exhibit a simpler PL-band line shape; see
Fig. 1. The values of the parameters entering the PL fits are
given in Table I for all samples.
Also the fine-grain characteristics of the PL spectra are
accurately reproduced by the present model, as shown in Fig.
4. Therein, the changes of the FWHM and line shape of the
PL band produced by the redistribution of carriers among the
QD states is quantitatively reproduced. The QD disordered
system, not thermalized at low temperature, reaches condi-
tion of partial !full"thermalization at intermediate !high"
temperatures via the WL. This recovery of thermalization
conditions produces the observed shift of weight from the
high- to the low-energy side of the PL band.
The present model is validated by a comparison of the
WL energy estimated directly, from PL excitation measure-
ments or PL spectra at high power density, with that obtained
from the fit of the theoretical model to PL data. In fact, the
energy of the WL state critically determines the evolution
with temperature of the PL spectra. As a matter of fact, the
value of EWL obtained from the fit of sample NU1478
(1.420!0.015 eV, see Table I"perfectly matches the energy
of the WL emission !1.420 eV"which appears for exciting
power densities on the order of 1 kW cm%2. This supports
our assumption of a carrier dynamics dominated by the es-
cape of e-hpairs. In the case of an independent escape of
carriers, indeed, the activation energy for carrier escape
would be equal to that of the less bound carrier.
In the second set of samples, the WL energy position
determined by PLE measurements changes by small amounts
!from 1.40 to 1.41 eV, see Table I". Nevertheless, the QD
spectrum shows major changes for increasing InAs cover-
FIG. 3. Integrated PL intensity, FWHM and peak position vs
temperature of sample NU1422. The full lines are calculated by
solving the set of rate equations 3 with the parameters reported in
Table I. The dotted line in the lowest panel shows the InAs gap
temperature dependence.
TABLE I. Parameters used to fit PL temperature dependencies. Bold numbers indicate fixed parameters. The last column reports the
parameter set of sample MA886 when the constraint on
)
eis relaxed.
NU1468 NU1422 NU1478 MA882 MA884 MA886 MA886 !rel."
EWL !eV"1.390!0.015 1.410!0.015 1.420!0.015 1.410 1.406 1.400 1.400
)
t!GHz"772!230 92!30 138!48 1677!500 3800!1000 3060!750 3300!900
)
e(105GHz"5.10!1.60 0.60!0.25 3.80!1.15 130!30 130 130 450!200
)
b(1017 cm%2) 2.7!0.8 4.7!1.4 8.7!2.6 3.8!1.2 10!3 600!180 350!100
FIG. 4. Experimental !full line"and calculated !dotted line"PL
spectra for sample NU1422 at four different temperatures.
8280 PRB 60
S. SANGUINETTI et al.
ages L. Samples MA882 and MA884 (L$1.6 and 2.0 ML"
show a PL spectrum characterized by a single, Gaussian QD
emission band; see Fig. 1. In sample MA886, with a higher
In coverage (L$3.0 ML", the PL spectrum shows a main
peak and a low-energy shoulder whose relative strength in-
creases for increasing temperature. We have fitted the inte-
grated intensity of sample MA882 in terms of three param-
eters
)
b,
)
t, and
)
e, the WL energy having been fixed to
the experimentally determined value. Then, the integrated
intensity of the other two samples has been reproduced by
using
)
tand
)
bas free parameters, the other two parameters
being either given by the PLE results (EWL), or being taken
equal to the value derived from the fit of the sample MA882
(
)
e).
)
tand
)
bwere let free to vary, instead, in order to take
into account the changes from sample to sample of the ra-
diative efficiency and WL DOS. The values of the fitting
parameters for this second set of samples are reported in
Table I. The fits !full lines"of the PL integrated intensity,
FWHM, and peak position for sample MA886 are reported
in Fig. 5, together with the experimental data !symbols". The
reproduction of the spectral features is fully satisfactory, as
found also in the case of sample MA882 !the case of sample
MA884 is slightly different, as will be discussed in the fol-
lowing Sec. VI". The rapid temperature shift of the QD-band
peak position is followed accurately by the model simula-
tions, as well as that of FWHM !which varies with Tless
than it was found in the first set of samples". Moreover, the
PL spectra as a function of temperature are well reproduced
by the model, as shown in Fig. 6 in the case of sample
MA886.
VI. DISCUSSION
Despite its simplicity, the model introduced in the previ-
ous Sec. V has demonstrated a high degree of prediction and
transferability. Besides the six samples reported in this pa-
per, the model has been verified on a wide set of samples
!about fifteen"coming from three different sources and
grown on substrates with various orientations. In almost all
those samples, the agreement between the model simulations
and the experimental data is as good as for the samples re-
ported here. The model fails only for some samples
(&10% of the total"which show temperature characteristics
different from the general behavior presented in Sec. III. A
typical example of such subset is sample MA884. It belongs
to the same series of samples MA882 and MA886, but with
an InAs coverage of 2.0 ML. The low-TPL of this sample is
characterized by a monomodal broad band, see Fig. 1, whose
dependence on temperature is that typical of the other
samples for what concerns the integrated intensity and peak
position, but not for the FWHM. The PL band width does not
have a minimum at intermediate temperatures: it remains
roughly constant up to 90 K and broadens at higher tempera-
tures, reaching values 30% higher than the low-temperature
ones. For what concerns the theoretical model, it accurately
fits the dependence of the integrated intensity on temperature
and reproduces well the redshift of the peak position. It com-
pletely fails, instead, to describe the FWHM temperature de-
pendence. Moreover, the simulated PL spectra only roughly
follow the experimental ones. Finally, while the experimen-
tal spectrum becomes bimodal at high temperature, the simu-
lated one remains monomodal.
These shortcomings point out a delicate point in the
model assumptions, i.e., the determination of the QD DOS
on the grounds of low-temperature PL measurements. In
fact, the disagreement between the simulated and experimen-
tal data might be due to a determinination of the low-energy
tail of the QD DOS which is not accurate enough. Indeed,
states which are hardly measurable in a PL spectrum at 10 K
may become important at higher temperature because of an
increased carrier thermalization. As a matter of fact, the
agreement between experimental data and model simulations
in most !90%"of the samples suggests a description of the
carrier quenching process in QDs which is consistent with
carrier thermal escape from the ground states of an ensemble
FIG. 5. Integrated PL intensity, FWHM and peak position vs
temperature of sample MA886. The full lines are calculated by
solving the set of rate equations !3"with the parameters reported in
Table I. The dotted line in the lowest panel shows the InAs gap
temperature dependence.
FIG. 6. Experimental !full line"and calculated !dotted line"PL
spectra for sample MA886 at four different temperatures.
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CARRIER THERMAL ESCAPE AND RETRAPPING IN . . .
of QDs with different sizes and shapes whose population is
not in thermal equilibrium with the lattice. The FWHM re-
duction and the peak energy rapid redshift with temperature
occurring at low-Tresult from an increase of the carrier ther-
mal emission from small QDs which is not fully compen-
sated by an enhancement of the capture probability induced
by random population and ground state saturation effects.
The increase in the QD band linewidth observed at higher
T’s !and the concomitant reduction in the rate of decrease of
the QD peak energy vs temperature"is, instead, a conse-
quence of the broadening of the carrier distribution function
with increasing temperature. This is confirmed by the obser-
vation that the high-Tlimit of the FWHM is approximately
equal to the low-Tvalue. In this context, the temperature
dependence of carrier capture and relaxation processes seems
to play, instead, a minor role. If one assumes
)
c$
)
c
0/T,
namely, a multiphonon transition capture of carriers in QD,25
no major improvement in the overall quality of the model
simulations is obtained.
It is worth to discuss here the values of the four param-
eters entering the model EWL ,
)
t,
)
e, and
)
b, which al-
lowed us to reproduce the PL temperature behavior in a large
number of samples, since this may provide a further insight
into the photoexcited carrier relaxation and PL quenching
processes. In order to compare the numerical values of the
whole set of samples on an equal footing, we relaxed the
constraint on
)
ewhile fitting sample MA886. The new pa-
rameter values are reported in Table I.
The changes in shape and FWHM of the QD PL band are
related to transfer of carriers via the WL. In particular, the
value of the energy difference EWL between the WL and the
QD ground state is crucial for obtaining the temperature de-
pendence of the PL spectral shape. A change of a few tens of
meV in EWL much worsen the agreement between simulated
and experimental spectra, thus giving an indirect support to
the assumption of a carrier dynamics dominated by corre-
lated e-hpairs.
For what concerns
)
t, this parameter widely varies from
sample to sample and tends to be lower for samples which
exhibit PL emission at higher temperature, since a low
quenching probability through the WL increases the high-
temperature efficiency. The presence of a superlattice be-
tween the substrate and the QD layer helps to prevent the
diffusion of dislocations from the substrate and may account
for the lower quenching probability through the WL found in
the first set of samples with respect to the second set; see
Table I.
Another parameter affected by the sample source is the
WL to barrier coupling,
)
e, whose value changes by roughly
two orders of magnitude on going from the first to the second
set of samples. Such differences may be related to the den-
sity of defect states accessible from the GaAs barrier.
In the second set of samples, the effective DOS of the
WL,
)
b, exhibits a coverage dependence and drops by two
orders of magnitude on going from the 3.0 to the 1.6 ML
sample. At least part of this drop is due to changes in QD
density, whose dependence on InAs coverage4is not in-
cluded in the model.
Finally, it’s worth noticing that the the WL-to-QD capture
time and the radiative QD recombination lifetime were fixed
to the experimental values and that slower decay rates did
not give good results. This indirectly supports the observa-
tion of a suppression of the theoretically predicted phonon
bottleneck reported in time resolved PL experiment.
VII. CONCLUSION
We have presented a system of rate equations for the tem-
perature dependence of the PL in InAs/GaAs QDs which
requires a maximum of four free parameters. An excellent
agreement between the model simulations of the PL spectra
and the experimental results has been obtained for tempera-
tures ranging from 10 to &150 K in a large number of
samples coming from different sources. The theoretical
model underlying the rate equations takes into account all the
most important mechanisms invoked for explaining the pe-
culiar evolution of the InAs QD-PL band with temperature,
namely, carrier thermal escape and retrapping, QD inhomo-
geneous broadening, random population of the QD ground
states, saturation effects, and increasing carrier thermaliza-
tion for increasing temperature. The mechanism governing
the carrier emission is the thermal excitation of correlated e-
hpairs to the WL state, which acts as transit channel to other
QDs or to quenching states inside and/or outside the GaAs
barrier. The non monothonic dependence on Tof the line-
width and peak energy is accounted for in terms of an in-
creasing carrier thermalization within the QD disordered sys-
tem and in the WL. Finally, although the microscopic carrier
relaxation mechanism cannot be established via the present
model, a fast relaxation channel seems to be active in the
system.
ACKNOWLEDGMENTS
We acknowledge helpful discussions with M. Colocci, A.
Polimeni, and A. Patane
´. This work has been supported by
the Engineering and Physical Sciences Research Council
!EPSRC"in the U.K. and partially by the Consiglio Nazion-
ale delle Ricerche !CNR", MADESS project, in Italy. We
acknowledge NATO for funds enabling the Milano-
Nottingham Collaboration.
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