SM ARTICLE 669 Revise 1st proof 15.2.96
Superlattices and Microstructures, Vol. 19, No. 2, 1996
Electronic structure and energy relaxation in strained InAs/GaAs
quantum pyramids
M. G*, R. H, N. L†, O. S, D. B
Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstr. 36, D-10623 Berlin,
Germany
V. M. U, P. S. K’, Z. I. A
A.F. Ioffe Physical-Technical Institute, 194021, Politekhnicheskaya 26, St Petersburg, Russia
S. S. R†‡, P. W, U. G̈, J. H
Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany
(Received 21 August 1995)
We perform experimental and theoretical studies of the electronic structure and relaxation
processes in pyramid shaped InAs/GaAs quantum dots (QDs), grown by molecular beam
epitaxy in the Stranski–Krastanow growth mode. Structural properties are characterized with
plan view and cross section transmission electron microscopy.
Finite difference calculations of the strain and the 3D Schrödinger equation, taking
into account piezoelectric and excitonic effects, agree with experimental results on transition
energies of ground and excited states, revealed in luminescence and absorption spectra. We
find as relative standard deviation of the size fluctuation n\0.04; the pyramid shape fluctuates
between R101S and R203S side facets.
Carrier capture into the QD ground state after carrier excitation above barrier is a
very efficient process. No luminescence from excited states is observed at low excitation density. Energy relaxation processes in the zero-dimensional energy states are found to be
dominated by phonon energy selection rules. However, multi-phonon emission (involving
GaAs barrier, InAs wetting layer, InAs QD and interface modes) allows for a large variety
of relaxation channels and thus a phonon bottleneck effect does not exist here.
( 1996 Academic Press Limited
1. Introduction
The fabrication of quantum dot arrays with heteroepitaxy in the coherent islands Stranski–Krastanow growth regime [1–11] seems to be one of the most promising approaches to reach one of the
ultimate goals of nanostructure research: the quantum dot (QD) laser with its predicted temperature
independent low threshold current density [12,13]. This unique property is due to the d-function like
density of electronic states in each QD which was demonstrated experimentally [9,14]. Recently, the
first lasers based on such nm-scale, self-organized quantum dots were reported [15,16].
*E-mail: mariusgrww422zrz.physik.tu-berlin.de, WWW: http://sol.physik.tu-berlin.de/.
†On leave from A. F. Ioffe Institute, St Petersburg, Russia.
‡Present address: Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA.
0749–6036/96/020081]15 $18.00/0
( 1996 Academic Press Limited
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Fig. 1. (A) Cross-section HTEM and (B) plan view TEM image of laterally ordered array of InAs/GaAs
quantum pyramids.
Detailed understanding and tailoring of properties and performance of quantum dot based
optical devices requires theoretical modelling of their electronic states. In the case of strained islands,
simulation of the three-dimensional strain distribution and subsequent solution of the 3D Schrödinger equation in the modified confinement potential is necessary and has been first performed in
Grundmann et al. [17]. In this paper we will investigate the influence of size and shape on the
electronic states in InAs/GaAs quantum pyramids. The theoretical results will be compared with
experimental luminescence spectra.
Equally important as the electronic structure of the quantum dot are the carrier capture and
energy relaxation processes. Generally, capture and relaxation into the QD ground state from above
barrier excitation seem to be very efficient since the quantum efficiency of QD luminescence is comparable to best quantum well samples and at low excitation density no other than QD ground state
luminescence is present. Only at higher excitation densities, when the ground state saturates, excited
levels appear. At first surprisingly, however, the relevation of excited electronic states with photoluminescence excitation spectroscopy fails in the sense that the spectrum is dominated by peaks at
multiple LO phonon energies rather than by excited electronic states [6,18]. Such experiments thus
allow to study the energy relaxation in the QDs.
The paper is organized as follows: after a brief description of our experimental methods
(Section 2), we discuss the strain distribution in and around the InAs pyramids (Section 3). In addition
the driving mechanism for vertical ordering of QDs is discussed. In Section 4 we present the electronic
structure of the quantum pyramids as a function of size and shape. In the light of the theoretical
simulation we interpret luminescence data obtained from arrays of such QDs (Section 5). In Section 6
we describe PL excitation experiments and discuss the applicability of the ‘phonon bottleneck’ effect.
2. Experimental
Our samples are grown by molecular beam epitaxy on GaAs (001) substrates. Details about the
growth procedure can be found in Grundmann et al. [9]. For an average deposited InAs thickness
t above 1.6 monolayers (ML), the InAs layer transforms into coherent islands. For 4 ML deposition
!7
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83
z [001]
A
B
y [010]
x [100]
InAs
WL
GaAs
Fig. 2. Schematic drawing of the dot geometry. Lines A and B denote linescans in [001] direction used below.
The origin of the z-axis is at the lower interface of the wetting layer quantum dot.
we obtain dense arrays (about 1011 cm~2) of equilibrium size quantum pyramids with a typical base
length of L \12 nm. The base sides are oriented along [100] and [010] interface directions, the side
"
facets are close to R101S and R203S [9,11] (Fig. 1). In dense arrays, when island–island interaction via
the substrate is significant, the dots arrange into a square lattice, distorted by statistical fluctuations
[9,11]. Our numerical calculations show that this arrangement is energetically more favorable than
rectangular, checkerboard or hexagonal arrays for the same area coverage. More dilute arrays, but
with the same pyramid size, are obtained for only 2 ML deposition and growth interruption [19].
Transmission electron microscopy (TEM) studies are carried out in a high voltage JEOL
JEM1000 microscope operated at 1 MV. Photoluminescence (PL) is excited using the 632.8 nm line
of a He–Ne laser and detected using a cooled Ge photodetector. PL excitation spectroscopy is
performed with tungsten halogen lamp dispersed by a 0.35 m double grating monochromator as
tunable light source and detected with a 0.75 m double grating monochromator and a cooled Ge
photodiode. Calorimetric absorption spectroscopy (CAS) is carried out at T\500 mK. Minimal
detectable ad products are as low as 10~5 [20]. Our numerical calculations (typically solving for 106
variables) are performed on a DEC Alpha AXP 3000/600 workstation with 128 MByte RAM.
3. Strain distribution
Figure 2 depicts the QD geometry we will use subsequently. First we discuss pyramids with R101S
facets, while later on also R203S facets are considered. Before we start our discussion, we like to
emphasize that the following does only apply to QDs surrounded by barrier material. In the case of
uncovered QDs just after deposition, surface contributions to the total energy are essential in particular to explain self-ordering of size and shape [16,21].
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10
5
A
B
Strain (%)
5
0
0
–5
–5
–10
–10
0
–10
10
Strain (%)
5
C
0
–5
–10
0
10
z (nm)
Fig. 3. Strain distribution in and around a pyramidal QD for linescans in [001] direction: (A) through the
wetting layer far away from the dot; (B) along line A; and (C), along line B of Fig. 2. The solid line denotes e ,
;;
the dashed line e and the dotted-dashed line e .
99
::
The strain distribution around a 3D confined object depends only on its shape but not on its
size [17,22]. The strain distribution e scales linearly with the lattice mismatch e , e©e . The strain
0
0
generally decays with the third power of the distance. Analytical solutions can be obtained for a
sphere of isotropic material [17]. For realistic QD geometries we use a numerical approach, employing a finite difference method [17]. An important point is that for a strained layer quantum well the
barrier is unstrained and the strain distribution [23] does not depend on the elastic properties of the
barrier. The amount of strain relaxation of a quantum dot depends on the stiffness of the barrier,
thus making it dependent on both the QD and barrier elastic constants. The isotropic (hydrostatic)
part of the strain is mainly inside the QD, while the biaxial and shear components are shared between
inner and outer material, as expected from a simple analytical model [17].
Figure 3 shows the strain distribution in the wetting layer and along the lines A and B of Fig.
2. The solid line in Fig. 3 denotes e , the dashed line e and the dotted-dashed line e . Figure 3A
;;
99
::
shows the linescan intersecting the wetting layer far from the dot is. In Fig. 3B the intersection goes
through the top of the pyramid (line A in Fig. 2). In both cases symmetry imposes e \e . In Fig.
99 ::
3C the linescan intersects the dot at one half of the base length in [100] direction from the center
(line B in Fig. 2). In the wetting layer (Fig. 3A) the strain is biaxial and entirely confined to InAs.
The wetting layer is affected by the QD only in its vicinity, within a distance of about half of the
pyramid’s base length.
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Change of unit cell area (%)
101
A
100
85
B
1.0
0.5
0.0
10–1
0
5
10
15
Surface distance (nm)
–0.5
0
5
–15 –10 –5
10 15
Distance from center (nm)
Fig. 4. Decay of strain above a 12 nm InAs/GaAs pyramid. Upper part: G2e ]e (relative change of in-plane
99 Lower
:: part: Lateral distribution
unit cell area) as a function of distance of the top surface from the pyramid top.
of e ]e for a distance of 5 nm between the top surface and the pyramid top.
99 ::
The linescan through the center of the QD (Fig. 3B) reveals a very different situation. Close
to the lower interfaces, e is still positive but much smaller ([3%) than in the wetting layer because
;;
the substrate can no longer force the interface lattice constant to be that of GaAs. With increasing
height within the dot e changes its sign and becomes negative at the top of the pyramid. This
;;
happens because at the very top only little forces act on the QD in the xy-plane, but the GaAs barrier
compresses the pyramid mainly from the sides along the z-direction, imposing tensile strain components in the xy-plane (e \e at the top becomes positive). We emphasize that generally, however,
99 ::
the strain is still compressive, even at the top of the pyramid (Trace e\0). Around the pyramid also
the barrier becomes significantly strained ([3% close to the interfaces).
Directly above the InAs pyramid the GaAs is under tensile in-plane strain, while in between
dots the strain is compressive. This is driving the vertical ordering of multiple QD layers [1,24]. Figure
4A shows the decay of the tensile in-plane strain e ]e (the relative change of surface unit cell area)
99 ::
at the surface above a 12 nm base length InAs pyramid with R101S facets for different distances of
the surface from the top of the pyramid. Even for quite large barrier thickness of 10 nm a strain in
the 10~3 range is still present. Figure 4B depicts the (almost radially symmetric) strain distribution
e ]e on a surface 5 nm above the pyramid. InAs deposited on top of such an inhomogeneously
99 ::
strained surface [1,24] finds directly on top of the pyramid the largest substrate lattice constant and
thus the most favorable energetic conditions.
4. Electronic states
The strain distribution modifies the confinement potential which enters the effective mass Schrödinger
equation. From the strain we calculate the local modification of bulk bandgaps as described in
Grundmann et al. [17]. The electron experiences the hydrostatic bandgap shift. For the holes we
diagonalize the 6]6 Hamiltonian [25,26], thus taking into account corrections due to the split-off
band. Only heavy holes play a role for the localized states since the biaxial strain strongly pushes
light holes away from the valence band edge. The heterostructure band offsets are taken from modelsolid theory [27]. The influence of the superlattices underneath and above the pyramid has been
modeled. It can though be neglected since it affects the electronic levels by less than 1 meV for the
given geometry.
An additional effect has to be included at this stage. The shear strains give rise to a piezoelectric polarization which results in non-vanishing local charges at the R112S-like edges of the R101S
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A
000
B
000
{101}
000
{203}
100
001
000
010
100
010
001
011
101
002
6 nm
6 nm
C
D
6 nm
Fig. 5. Wavefunctions for electrons and holes in a 12 nm InAs/GaAs quantum pyramid with (A) R101S and (B)
R203S side facets. Orbitals are given for 70% probability. Hole wavefunctions are indexed according to nodes in
the different directions. In (C) the electron wavefunctions for a 7 nm and a 12 nm base side R101S facet pyramid
are compared. In (D) the same comparison is given for the hole ground state.
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87
pyramid facets [17]. Neighboring edges have opposite charges. The resulting piezoelectric potential
V [28], which has to be added to the confinement potential, has thus no 90° rotational symmetry
P
around the [001] axis through the pyramid’s center and reduces the symmetry of the pyramid from
C to C . This will subsequently lift some degeneracies of hole levels as outlined further below. We
47
27
note that the piezoelectric potential depends linearly on the QD size and scales like V ©L e .
P
"0
In the confinement potential we solve the 3D effective mass (anisotropic masses) single-particle
Schrödinger equation using a modified Davidson algorithm [17]. In Fig. 5A the wavefunctions for
localized electron and hole levels in a 12 nm quantum pyramid with R101S facets are shown. In Fig.
5B the wavefunctions for a pyramid with the same volume but R203S side facets are given. The orbitals
are for 70% probability to find the particle inside. In (C) and (D), ground state electron and hole
wavefunctions, respectively, are compared for a 7 nm and 12 nm pyramid with R101S facets.
For R101S facets and a base length between 6 and 18 nm only one electron level exists (which can be
populated with up to two electrons of opposite spin). For smaller pyramids no localized electron states
exist. The hole levels are indexed according to their nodes in the different directions. The P100[ and
P010[ state would be degenerate if the piezoelectric potential had been neglected. Inclusion of V leads to
P
a size dependent splitting (about 20 meV for L \12 nm) [17]. Non-degenerate levels are, however,
"
affected only in second order perturbation. Optical transitions are possible between the electron ground
state and the hole P000[ level; the hole P001[ level has about 50% of the ground state wavefunction
overlap and contributes to another allowed transition, which is found in luminescence as described in
Section 5. This is particular to the pyramidal geometry and would not occur for a box-like quantum dot.
The comparison of 7 nm and 12 nm quantum pyramids (Figs 5C and D) shows that the
extension of the electron wavefunction stays almost constant due to strong tunneling into the barrier
in the case of small dots. The hole ground state wavefunction follows more closely the QD shape
due to the stronger localization caused by the larger mass.
In the case of R203S side facets, the hole wavefunctions extend considerably more into the
wetting layer. With respect to R101S facets the sequence of the electronic states is changed.
Additionally Coulomb forces between electrons and holes can be treated as a perturbation [17].
For L \12 nm we find an exciton binding energy of 20 meV which is a 20-fold enhancement with
"
respect to the InAs bulk exciton energy of 1 meV. The exciton binding energy is small compared to
the kinetic energy in the order of 500 meV for electrons and holes together, thus a posteriori justifying
the perturbational approach.
In Fig. 6A we compare the ground state transition energy as a function a pyramid base length
for pyramids with R101S and R203S facets. For the same base length the flatter pyramid exhibits the
higher energy due to reduced volume. The electron level actually depends in a good approximation
on the volume only and not on the facet shape (even an isovolumetric sphere was a good approximation [17]). Hole levels are sensitive to the QD shape. The energy separation between the P000[
and P001[ hole states is shown in Fig. 6B for the two different facet types as a function of ground
state energy (solid circles are experimental data for various samples and will be discussed below in
Section 5). The variation of energy of the excited state with a variation of ground state energy is
larger for the R203S-facet case. It has a positive slope in both cases, i.e. for increasing ground state
energy, the energy of the excited state increases even further. Thus in general we expect the excited
transitions to be broadened more strongly than the ground state transition [29].
5. Luminescence experiments
We have shown in [9,14] that single QDs luminescence lines are ultrasharp, similar to atoms, even at
elevated temperatures, proving the true zero-dimensional nature of the QDs. In the present paper we
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1.2
A
Ground state transition (eV)
{203}
InAs/GaAs QD
1.1
{101}
1.0
120
10
8
12
14
Pyramid base length (nm)
16
18
B
110
h 001 – h 000 (meV)
100
{203}
G
E
90
F
A
D
C
80
{101}
B
70
60
50
InAs/GaAs QD
1.05
1.10
Ground state transition energy (eV)
Fig. 6. (A) Ground state transition energy as a function of pyramid base length for R101S and R203S facets. (B)
Separation of P001[ hole level from P000[ ground state as a function of ground state transition energy for a
variety of quantum pyramid samples. Solid lines are theoretical results for R101S and R203S side facets. Samples
have been grown under similar conditions, in particular A: T\460°C, t \1.2 nm, B,C,D: T\480°C,
!7 s annealing).
t \1.0 nm, E,F: T\480°C, t \1.2 nm, G: T\480°C, t \6](0.1 nm with 100
!7
!7
!7
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89
002 and hWL
002 and hWL
001
001
10
4
000
–2
500 W cm
InAs/GaAs
tav = 1.0 nm
T=8K
125
103
50
5
0.5
2
10
101
0.9
1.0
1.1
1.2
1.3
1.4
1.5
PL-intensity (arbitrary units)
PL-intensity (arbitrary units)
000
A
200
E000 = 1.098 eV
FWHM =
50 meV
1.0 nm InAs/GaAs
T=8K
–2
D = 500 W cm
150
E001 = 1.175 eV
FWHM = 61 meV
100
E002 = 1.248 eV
FWHM = 108 meV
50
B
0
0.9 1.0
1.1
1.2
1.3
1.4
1.5
Peak PL intensity (arbitrary units)
Energy (eV)
6
10
C
γ=1
5
10
104
103
101
101 –2
10
InAs/GaAs QD
3.3 ML
4.0 ML
10–1
100
101
102
Excitation density (W cm–2)
103
Fig. 7. (A) PL spectra (T\8 K, unscaled, semilogarithmic) of t \1.0 nm QDs for different excitation density
D given in W cm~2. Predicted transition energies of excited hole!7levels are indicated by arrows. (B) Lineshape
fit (three Gaussians) to the 500 W cm~2 spectrum of part (A), shown in linear scale together with fit parameters.
(C) Peak intensity of QD ground state emission at 8 K as a function of excitation density for two samples with
t \1.0 nm and t \1.2 nm, respectively.
!7
!7
focus on QD ensemble spectra. Figure 7A depicts a series of luminescence spectra taken at different
excitation density on a QD array with well developed pyramids. The intensity of the ground state
transition (1.1 eV) increases linearly with a slope close to 1 for increasing excitation density (see Fig.
7C) and starts to saturate at 50 W cm~2, when a second peak appears at 1.17 eV, which we attribute
to the transition involving the P001[ hole state [30]. Finally a third peak at about 1.24 eV evolves
due to the P002[ hole state and the hole wetting layer continuum. Figure 7B shows a nearly perfect
lineshape fit (linear scale) using three Gaussians to the experimental spectrum. The energy positions
agree very well with the hole level scheme we have predicted theoretically. The transitions of ground
state electrons with excited holes can also be observed in absorption spectra, as discussed in detail
in [30]. The fitted inhomogeneous broadenings of the excited P001[ state is larger than that of the
ground state luminescence as expected from theory, since oEP [/oEP [[0.
001
000
By attributing the inhomogeneous broadening of the ground state transition to size fluctuations only we obtain an upper limit for the size broadening. Using the theoretical dependence of
transition energy on pyramid base length L we find a standard variation of r \0.5 nm for the fit
"
L"
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PL-intensity (arbitrary units)
10
Ea = 170 meV
1
InAs/GaAs QD's
tav = 1.2 nm
–2
1 W cm
λex = 632.8 nm
0.1
0.0
0.5
1.0
1.5
100/T (K–1)
2.0
2.5
3.0
Fig. 8. QD emission intensity as a function of temperature for t \1.2 nm with optimal fit to the curve I/I \1/
!7
0
R1]C exp([E /kT)S with E \170 meV.
!
!
value of FWHM\50 meV. Thus the relative standard deviation, as introduced in [29], is n\0.04.
For smaller quantum dots, with ground state energy at 1.35 eV (L [7 nm), the variation of quantum
"
dot size by a single InAs molecule amounts to 0.06 meV and could be resolved experimentally by us
[14].
In Fig. 6B we have compiled results from various samples grown under similar conditions
(details are given in the figure caption). The separation of the excited hole state from the ground state
shows a slight trend (opposite to the theoretical result for fixed dot shape) to increase for decreasing
ground state energy. This change is interpreted by us as a change in average dot shape towards a
smaller facet angle for larger dots. The observed variation is, however, rather subtle because the
changes in energy levels are smaller than the inhomogeneous broadening.
The carrier capture (for above barrier generation) into the QD ground state is observed to be
surprisingly efficient since at low excitation density no other luminescence channels are present and
the intensity of the QD line is comparable to the emission intensity from very good quantum wells.
We like to emphasize that we do not observe luminescence from excited states at low excitation
density, thus we do not conclude for population of excited levels due to inhibited or reduced energy
relaxation as had been reported in Bockelmann et al. [31]. In Bockelmann et al. [31] 125 nW laser
power (at 1.96 eV) were focused with a spot size of 1.5 lm diameter on to a single QD of 450 nm
base width. This corresponds to an excitation density of D7 W cm~2. Assuming, as in Bockelmann
et al. [31], an exciton lifetime of 100 ps, the steady state carrier density, however, is 40 electron hole
pairs. Since the QD area is 10% of the excitation spot, the spectra in Bockelmann et al. [31] thus may
not represent the limit of low excitation density.
The luminescence intensity stays almost constant up to 150 K and then decreases with increas-
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InAsQD
Superlattices and Microstructures, Vol. 19, No. 2, 1996
Eexc. = 1.165 eV
Eexc. = 1.959 eV
IF
Nd:YAG
InAsQD
InAsWL
GaAs
IF
InAsQD
InAsWL
IF
Intensity (arbitrary units)
InAsWL
InAs/GaAs
T = 1.8 K
3LO
1.05
2LO
1.10
Energy (meV)
1.15
Fig. 9. Luminescence spectra of QD array (t \1.2 nm) excited selectively at 1.165 eV (solid line) and via the
!7
GaAs barrier (dotted line).
ing temperature with an activation energy of about 170 meV (Fig. 8). Evaporation of carriers from
the QD cause the intensity drop [14]. The activation energy is similar to the localization energy
(150 meV) of electrons or holes in the QDs. At high excitation density (500 W cm~2) the intensity
drop between low and room temperature is only a factor of three.
6. Energy relaxation
So far we have considered only luminescence response upon excitation of carriers in the barrier with
energy far above the QD or WL. In this section we discuss a number of experiments involving charge
carriers excited with a defined, small (\200 meV) excess energy above the ground state transition.
Detailed predictions have been published about the reduced energy relaxation by phonon emission
in nanostructures (‘phonon bottleneck’ effect), especially quantum box systems [32–34]. The basic
idea is that phonon relaxation between fully quantized electronic states can occur only if the level
separation matches a phonon energy due to the law of energy conservation. LO and LO]LA
emission have been considered. In the following we will show that the energy is conserved in our
structures via multi-phonon relaxation which allows for a variety of phonon energies to contribute
to energy relaxation [18].
The first key experiment is the selectively excited luminescence (SEL) spectrum (fixed excitation wavelength 1064 nm) (Fig. 9) of a typical quantum pyramid ensemble. The spectrum exhibits
several peaks grouped around 1, 2 and 3 LO phonon energies; the envelope of the spectrum follows
the inhomogeneous broadening of the regular PL peak. Based on known bulk phonon energies and
model calculations for strained pyramids [17] we have identified various phonons (GaAs barrier
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Intensity (arbitrary units)
InAs/GaAs
T = 1.8 K
A
B
C
D
+
+
+
+
+
+
+
+
+
+
PL
1.1
+
+
1.2
1.3
Excitation energy (eV)
WL
1.4
Fig. 10. PLE spectra of the QD luminescence (scaled to same height of peak C). The detection energies are
indicated by crosses. Dotted line is PL spectrum upon above barrier excitation.
(36.6 meV), InAs wetting layer (29.5 meV [35]), InAs QD (31.9 meV) and interface mode (35.0 meV))
as the cause of the peaks. We emphasize that the different peaks are all due to different QDs (of
different ground state energy). Two conclusions can be drawn: (i) luminescence response is enhanced
when the condition for relaxation with a particular possible phonon energy is fulfilled for a particular
dot; (ii) a number of different phonons contribute and due to the contribution of various optical and
additionally acoustic phonons [18] the peaks merge to a continuum, thus allowing for efficient relaxation over an energy band.
A slightly different access to the relaxation of carriers is obtained with photoluminescence
excitation (PLE) spectroscopy. In this experiment the detection energy is fixed and the excitation light
energy is tuned across the high energy side. This means that the PLE spectrum belongs to QDs with
exactly the same ground state energy. However, for a given ground state energy, shape fluctuations
vary the level energies for excited states. Thus the PLE spectrum at different energies is in general
still due to response of different dots (with the same ground state energy). The PLE technique has
been widely-used to reveal excited electronic states in quantum wells and wires, e.g. [36]. At first
surprisingly, the PLE spectra of our quantum dots (Fig. 10) show a multi-peak structure which
cannot be interpreted to origin from excited electronic states. We previously reported the observation
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93
160
002
WL
110
120
∆E (meV)
010
001
80
100
40
InAs/GaAs
T = 1.8 K
000
1.05
1.10
1.15
Detection energy (eV)
1.20
Fig. 11. Energy differences between PLE peaks and detection energy for various detection energies. Solid lines
are theoretical hole level energies (for R101S side facets).
of a series of three almost equidistant peaks whose separation does not change with the detection
energy (Fig. 11) at the high energy side of the detection energy [6]. Since any higher electronic level
has a characteristic dependence on the ground state energy (i.e. basically the QD volume), as shown
as lines in Fig. 11, these peaks do not simply relate to excited electronic levels in the QD as claimed
in Ref. [37]. Their constant energy separation leads us to the conclusion that they are due to multiphonon emission processes which involve 1, 2 or 3 LO phonons. This also corresponds to the SEL
spectra discussed before. The 3 LO resonance is strongest due to the fact that it is nearly resonant
with the allowed transition into the P001[ excited hole state. Averaging of the PLE spectra over the
detection wavelength reveals a peak at the position of the P001[ hole level. The PLE peak at 1.39 eV
does not depend on the detection wavelength and is assigned, in agreement with absorption spectra
[9,30], to the wetting layer ground state.
We note that the PLE spectrum becomes broad and rather structureless for excess energies
larger than 150 meV. According to our calculations this is the typical localization energy for electrons
and also for holes in the QDs [17]. Thus the continuum onset in PLE is due to excitation into the
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wetting layer. We note that for small QDs (prepared with kinetically limited growth [9,14,19]) with
a ground state energy around 1.35 eV (size L [7 nm) no LO resonances are observed in PLE because
"
the electron level exhibits so weak localization that the continuum onset starts already close to (about
30–40 meV above) the detection energy.
7. Conclusion
We have prepared dense, laterally ordered arrays of quantum pyramids with narrow size distribution
(relative standard deviation of size broadening n[0.04). Strain distribution and electronic states have
been modelled numerically. Excited states are revealed only when the ground state luminescence
strongly saturates. The excited levels in the luminescence spectra agree with our theoretical calculations for pyramid facets between R101S and R203S and are due to allowed transitions between excited
hole states with the only electron (ground) state in the QD. The observed enhanced broadening of
the higher transition is expected from theory.
The energy relaxation for excitation into the zero-dimensional density of QD states is
dominated by the phonon energy selection rule. Energy relaxation is possible with multi-phonon or
multiple phonon processes, thus increasing the number of possible relaxation channels. A phonon
bottleneck effect does not exist. Relaxation into the QD ground state occurs fast upon excitation in
the wetting layer or in the barrier.
Acknowledgements—Parts of this work are supported by Deutsche Forschungsgemeinschaft in the
framework of Sfb 296, INTAS (grant 94-1028) and Volkswagen-Stiftung. One of us (N.N.L.) is
grateful to Alexander-von-Humboldt Stiftung.
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