Strain distribution and optical phonons in InAs/InP self-assembled quantum dots

Abstract

The strain distribution in self-assembled InAs/InP (001) quantum dots is calculated, using an atomistic valence force-field description. Two typical dot shapes are considered. Strain relaxation is found to depend much on the dot shape. From these modeling results we deduce the strain-induced phonon frequency shifts. Unlike confinement, strain induces large frequency shifts. The calculations agree well with experimental results obtained by Raman scattering. It is shown that alloying effects are small. Finally, we show that average strain values can be obtained experimentally if one combines longitudinal and transverse optical-phonon Raman scattering.

Full text

Strain distribution and optical phonons in InAs/InP self-assembled quantum dots
J. Groenen
Laboratoire de Physique des Solides, ESA 5477, Universite
´
P. Sabatier, F-31062 Toulouse Cedex 4, France
C. Priester
IEMN, De
´
partement ISEN, CNRS-UMR 8520, BP 69 F-59652 Villeneuve d’Ascq Cedex, France
R. Carles
Laboratoire de Physique des Solides, ESA 5477, Universite
´
P. Sabatier, F-31062 Toulouse Cedex 4, France
共Received 14 June 1999兲
The strain distribution in self-assembled InAs/InP 共001兲 quantum dots is calculated, using an atomistic
valence force-field description. Two typical dot shapes are considered. Strain relaxation is found to depend
much on the dot shape. From these modeling results we deduce the strain-induced phonon frequency shifts.
Unlike confinement, strain induces large frequency shifts. The calculations agree well with experimental results
obtained by Raman scattering. It is shown that alloying effects are small. Finally, we show that average strain
values can be obtained experimentally if one combines longitudinal and transverse optical-phonon Raman
scattering. 关S0163-1829共99兲10347-3兴
I. INTRODUCTION
Self-assembled quantum dot structures have been attract-
ing considerable attention in the past decade. Dots are ob-
tained during heteroepitaxy as a result from the elastic relax-
ation of misfit strain. Several studies were devoted to the
modeling of the strain inside the dots.
1–6
Calculations were
performed using either elastic continuum theory 共finite ele-
ment model兲
1–3
or an atomistic description 关valence force
field model 共VFF兲兴.
4–6
The strain field was shown to depend
strongly on dot shape. Thanks to the strain simulations, the
quantitative analysis of many experimental data has become
possible. In particular, they are very helpful to understand
the electronic properties.
2,3,5,6
Confinement effects were
shown 共see, for instance, Ref. 6兲 to depend strongly on dot
size and strain 共and thus on the dot shape兲.
Although phonons are efficient probes for investigating
low-dimensional structures, to date little work has been done
on phonons in self-assembled nanostructures.
2,7–16
Most of
the theoretical and experimental work on phonons in quan-
tum dots deals with unstrained systems. Only a few Raman
scattering investigations of self-assembled nanostructures
have been reported.
9–16
Valuable information 共about strain,
alloying, electronic properties, ...)hasbeen accessed using
resonant Raman scattering. In particular, Raman scattering
was shown to provide a means of determining independently
the residual strain and the alloy composition in SiGe/Si self-
assembled dots.
13
In this paper we shall address two issues. 共i兲 How is the
strain distributed in InAs/InP 共001兲 self-assembled dots? 共ii兲
How does it modify the optical-phonon frequencies? To an-
swer the first question, we use the VFF model.
17
A large
simulation cell is considered, in order to account for both the
strain distribution inside and around the dots. In contrast
with previous simulations of strain in capped dots, we shall
examine capped dots with truncated pyramidal shapes. The
strain-induced optical-phonon frequency shifts will be de-
rived from the calculated strain. We shall finally compare
these calculations to experimental results obtained by means
of resonant Raman scattering on InAs/InP 共001兲 self-
assembled dots. In contrast with InAs/GaAs or
InP/In
x
Ga
1⫺ x
P systems, the gap between the InAs and InP
optical-phonon frequencies is large. Consequently, the con-
finement of the InAs optical phonons inside the dots is very
efficient, providing us with local probes. Moreover, the dot-
related features can therefore be easily identified in the Ra-
man spectra.
II. CALCULATIONS
A. Strain distribution
It should be noted that the accuracy of the VFF model
goes beyond classical elasticity theory as it decribes the elas-
tic properties and the relaxation on the atomic scale.
18
The
strain elastic energy
␦
E depends on the geometric deforma-
tions of bonds that each atom makes with its four nearest
neighbors. For each atom i of the zinc-blende structure, one
can write
␦
E
i
⫽
兺
j⫽ 1
4
3
8r
0
2
␣
关
r
ជ
ij
2
⫺ r
0
2
兴
2
⫹
兺
j⫽ 1
4
兺
k⫽ j⫹ 1
4
3
8r
0
2
␤
冋
r
ជ
ij
•r
ជ
ik
⫹
r
0
2
3
册
2
.
共1兲
r
ជ
ij
is the vector connecting the central atom i to one of its
four nearest neighbors j; r
0
is the unstrained bond length;
␣
and
␤
are, respectively, bond-bending and bond-stretching
elastic constants. They are related to the elastic constants
c
11
, c
12
, and c
44
of the continuum elasticity theory by the
following expressions:
c
11
⫽
␣
⫹ 3
␤
a
, c
12
⫽
␣
⫺
␤
a
, c
44
⫽
4
␣␤
a
共
␣
⫹
␤
兲
. 共2兲
PHYSICAL REVIEW B 15 DECEMBER 1999-IVOLUME 60, NUMBER 23
PRB 60
0163-1829/99/60共23兲/16013共5兲/$15.00 16 013 ©1999 The American Physical Society

a⫽ 4r
0
/
冑
3 is the lattice constant. It is not possible to per-
fectly fit all three c
ij
’s with only two elastic constants
␣
and
␤
; the less ionic is the material, the better is the fit. Thus, this
method basically works with covalent bonding, but Coulomb
corrections could be introduced.
19
In practice, for the sake of
simplicity one usually simply uses Eq. 共1兲.
20
This we do, and
choose to fit c
11
and c
12
and to drop c
44
, resulting in a
10–20 % error range on the VFF effective c
44
.
We examine two typical InAs/InP 共001兲 dot morpholo-
gies, corresponding to the ones already reported in Refs. 10
and 21. The misfit strain between InAs and InP equals
⫺ 3.1%. The first morphology, labeled A, corresponds to 3
nm high and 25 nm wide dots, whereas the second one,
labeled B, corresponds to 7 nm high and 45 nm wide dots.
The dots have truncated pyramidal shapes, with 共114兲 and
共113兲 side facets for A and B, respectively. InAs islands are
formed on the top of a 1.5 monolayer 共ML兲 wetting layer
共WL兲 and capped by a 25 nm InP layer.
10,21
For the sake of simplicity, we simulate pyramids with
square base „the InAs/InP islands are in fact slightly elon-
gated along
关
11
¯
0
兴
共Ref. 21兲…, and periodic boundary condi-
tions in the plane perpendicular to 关001兴 are used. At the
bottom of the modelized cell, atoms are kept fixed, in order
to simulate the thick substrate. We calculate the atomic po-
sitions that minimize the total elastic energy. Once the posi-
tions of all atoms are known, the local deformation distribu-
tion is derived straightforwardly. Calculations have been
performed with and without WL 共1 or 2 ML thick兲. Concern-
ing the strain field in the dots, no significant differences were
observed. The results we present here were obtained disre-
garding the WL. It has been shown that the subtrate and the
cladding layer are affected by the strain relaxation within the
dot. In particular, the strain field penetrates deeply into the
subtrate.
4
Consequently, to obtain a reliable and realistic
strain field, a rather thick nonfrozen substrate layer has to be
considered in the simulation. For that purpose, calculations
have been performed with up to 600 000 atoms and about
95% of the simulation cell corresponds to InP. A small scal-
ing factor 共1.6 for A and 5 for B) remains between the actual
pyramid size and the one used in the simulation. One can
avoid this scaling factor but, in counterpart, one has to re-
duce the substrate layer thickness. We have checked, on
smaller systems, the validity of using such a scaling factor,
which is bound to the fact that the strain field depends much
on shape and not on size.
Typical local deformation distributions within the dots
共along lines defined in Fig. 1兲 are shown in Figs. 2 and 3 for
shape A. Except close to the pyramid boundaries, the strain
field is rather uniform and does not vary very rapidly. The
shear strain ␧
ij
(i⫽ j) turns out to be significant at the facet
edges and pyramid boundaries and very small inside the
pyramid 共Fig. 3兲. Let us point out that the previous simula-
tions of strain in capped islands all correspond to untruncated
pyramids. Unlike for capped untruncated pyramids, ␧
xx
and
␧
yy
never change sign inside the pyramid.
2,5
Figure 2 clearly shows that the strain field penetrates deep
into the InP barriers. At the surface of the InP capping layer,
some strain is still present just above the dot 共compare lines
a and b). As this tensile strain is rather localized within the
xOy plane, it is able to promote vertical order when several
layers with dots are grown.
22
From the numerical local deformation values of all the
InAs cells, we have computed the average strain within the
FIG. 1. Schematic plot of a type-A pyramid. The origin O is the
center of the pyramid base plane. x⫽
关
110
兴
, y⫽
关
11
¯
0
兴
, and z
⫽
关
001
兴
.
FIG. 2. Local deformations along the a, b, and c lines defined in
Fig. 1: ␧
xx
共solid line兲 and ␧
zz
共dashed line兲. z positions are normal-
ized with respect to the pyramid height (z⫽ 0 stands for the pyra-
mid bottom and z⫽1 for the pyramid top兲. S denotes the sample
surface.
FIG. 3. Local deformations along the lines defined in Fig. 1: 1
共solid line兲,2共dashed line兲, and 3 共dotted line兲. y positions are
normalized with respect to half of the pyramid base width (y⫽ 0
corresponds to the middle兲. ␧
xy
and ␧
xz
are not reported here as they
equal almost zero regardless of the y position. ␧
xx
are not reported
either; they display almost constant values, given by ␧
xx
⫽ ␧
yy
at
y⫽0.
16 014 PRB 60
J. GROENEN, C. PRIESTER, AND R. CARLES

InAs dot. The average values of the diagonal strain compo-
nents in the InAs dot are reported in Table I 关the values
corresponding to a pseudomorphic two-dimensional 共2D兲
layer are also given as a reference兴. According to symmetry
requirements, the average values of the shear strain compo-
nents vanish. The biaxial strain relationship, ␧
zz
/␧
xx
⫽⫺2C
12
/C
11
, has often been assumed to be valid for flat
islands. It is noteworthy that, even for the rather flat islands
examined here, this relationship does not hold 共Table I兲. Re-
sidual strain depends much on shape. The B-type dots are
more relaxed than the A-type ones 共their height/width ratio
equals 0.155 and 0.12, respectively兲.
B. Optical-phonon spectra
One could, in principle, obtain the vibrational eigenmodes
from the diagonalization of the dynamical matrix 共once the
relaxation procedure described above has been performed兲.
In polar materials, both short-range interaction 共covalent
bonding兲 and long-range Coulomb interaction have to be
taken into account. Such calculations have been performed
recently for free-standing GaP dots with up to 2000 atoms.
23
In our case, this procedure is, however, impractical: our sys-
tem, which includes necessarily a dot and a large part of the
matrix, contains too many atoms 共with regard to the simula-
tion capabilities兲.
Moreover, the dot vibrational eigenmodes are collective
excitations, involving all the atoms belonging to the dot. As
an approximation, we shall therefore consider that the
phonons experience the average strain field inside the dots.
Owing to the rather homogeneous strain field inside the dots
共Figs. 2 and 3兲, this should provide us with reasonable re-
sults. The frequencies of the optical phonon in presence of
strain can be derived from the secular equation given in Ref.
24. The frequency shifts depend on the strain tensor ␧
ij
and
the phonon deformation potentials K
˜
ij
. According to our
modeling, the average shear strain components can be disre-
garded. As ␧
xx
⫽ ␧
yy
, the strain splits the optical phonons
into a singlet and a doublet component.
24,25
Their relative
frequency shifts are given by
冉
⌬
␻
␻
0
冊
S
⫽
1
2
K
˜
12
共
␧
xx
⫹ ␧
yy
兲
⫹
1
2
K
˜
11
␧
zz
, 共3兲
冉
⌬
␻
␻
0
冊
D
⫽
1
2
K
˜
11
␧
xx
⫹
1
2
K
˜
12
共
␧
yy
⫹ ␧
zz
兲
. 共4兲
The vibrations are along 关001兴 for the singlet mode and in
the plane normal to 关001兴 for the doublet modes.
25
Notice
that, depending on whether the longitudinal optical 共LO兲 or
transverse optical 共TO兲 deformation potentials are used, the
resolution of the secular equation provides us either with the
LO singlet and the LO doublet or with the TO singlet and the
TO doublet. Concerning Fig. 6 in Ref. 2, let us indicate that
the peaks assigned to LO and TO correspond in fact to the
histogram of the LO singlet and the LO doublet relative
changes in phonon energy. One can easily identify three
peaks for the dot: two having a similar location around 7%
共i.e., the doublet兲 and another one around 10% 共i.e., the sin-
glet兲.
According to Aoki et al.,
27
(K
˜
11
⫹ 2K
˜
12
)
LO
InAs
⫽⫺6.4 and
(K
˜
11
⫹ 2K
˜
12
)
TO
InAs
⫽⫺7.3. According to Yang et al.,
28
(K
˜
11
⫺ K
˜
12
)
TO
InAs
⫽ 0.51. Unfortunately, (K
˜
11
⫺ K
˜
12
)
LO
InAs
has not
been measured. The corresponding TO value has been used
in most of the previous calculations. One has, however, to
note that in III-V compounds, (K
˜
11
⫺ K
˜
12
)
LO
is usually larger
than (K
˜
11
⫺ K
˜
12
)
TO
.
26
On the other hand, Tran et al. did in-
vestigate strained InAs/InP superlattices combining Raman
scattering and x-ray diffraction.
29
Their data can therefore be
used to estimate (K
˜
11
⫺ K
˜
12
)
LO
InAs
. One obtains (K
˜
11
⫺ K
˜
12
)
LO
InAs
⫽ 0.92, which is about twice as large as the corre-
sponding TO value 关thus, one observes the same trend as for
GaAs and InP 共Ref. 26兲兴. One finally obtains K
˜
11
⫽⫺1.50
and K
˜
12
⫽⫺2.43 for the InAs LO and K
˜
11
⫽⫺2.09 and K
˜
12
⫽⫺2.60 for the InAs TO.
30
According to measurements per-
formed at room temperature on InAs 共111兲,
␻
0
equals
239.8 cm
⫺ 1
and 218.8 cm
⫺ 1
for the InAs LO and TO, re-
spectively.
In backscattering geometry from the 共001兲 surface, only
the LO singlet is Raman-active. The calculated strain-
induced frequency shifts of the LO singlet are reported in
Table II. The shift expected for a pseudomorphic 2D layer is
also given. Although the strain relaxation in these systems is
quite different, the calculated strain-induced frequency shifts
关Eq. 共3兲兴 are rather similar.
The island heights are small and confinement may modify
the phonon frequencies. As the island widths are much
larger, the effects of lateral confinement on the phonon fre-
quencies can be neglected. From the island heights and the
optical-phonon dispersion relation
29
共i.e., applying the usual
linear chain model
25
兲, we have calculated the confinement
induced frequency shifts ⌬
␻
conf
. The values of ⌬
␻
conf
de-
duced in this way for the first-order confined mode are re-
ported in Table II; ⌬
␻
conf
is found to be very small for the 3
and 7 nm high islands. In our case, the frequency changes
related to dot size fluctuations are thus also small.
III. COMPARISON WITH EXPERIMENT
AND DISCUSSION
We present Raman spectra ofa2MLInAs single quan-
tum well sample 共SQW兲 and a sample with A-type dots. De-
TABLE I. Average strain components for A and B dot shapes.
2D stands for a pseudomorphic InAs layer on InP.
Shape ␧
xx
␧
yy
␧
zz
␧
zz
/␧
xx
2D ⫺ 3.1 ⫺ 3.1 ⫹ 3.4 ⫺ 1.1
A ⫺ 2.81 ⫺ 2.80 ⫹ 2.67 ⫺ 0.95
B ⫺ 2.62 ⫺ 2.60 ⫹ 2.18 ⫺ 0.83
TABLE II. Strain (S⫽ singlet) and confinement 共conf兲 induced
LO frequency shifts 共in cm
⫺ 1
) for A and B dot shapes. ‘‘2D’’
stands for a pseudomorphic 2 ML InAs layer in InP. ‘‘exp’’ denotes
experimental data.
Shape ⌬
␻
S
⌬
␻
conf
⌬
␻
S⫹conf
⌬
␻
exp
2D 12 ⫺ 48 9
A 11.7 ⫺ 0.2 11.5 12.1
B 11.3 ⬇0 11.3 11.8
PRB 60
16 015STRAIN DISTRIBUTION AND OPTICAL PHONONS IN . . .

tails about the sample growth can be found in Refs. 10 and
21. The spectra were recorded at room temperature with an
XY Dilor spectrometer equipped with a cooled charge
coupled device detector. Depending on the excitation energy
used 共incoming or outgoing resonance兲, the Raman spectra
display InAs confined phonon or/and interface 共IF兲 mode
peaks. As we intend to discuss the effects of strain on con-
fined optical phonons, we only reported here 共Fig. 4兲 spectra
displaying confined LO-related peaks 关scattering by TO
phonons is forbidden in backscattering geometry from 共001兲
surfaces兴. The krypton laser lines we used are in incoming
resonance with either the SQW or the dot InAs E
1
-like tran-
sition 关520.8 nm and 482.5 nm for Figs. 4共a兲 and 4共b兲,
respectively兴.
10
We have been able to discriminate between
the island-related signal and the WL-related one using dif-
ferent polarization configurations and the 2 ML SQW sample
as a reference.
10
The InAs LO peaks related to the WL and
the islands are observed in the crossed-polarization and
parallel-polarization configurations, respectively. The InP
substrate LO peak is observed in the crossed-polarization
configuration. The peak observed in the parallel-polarization
configuration in the InP frequency range is attributed to a
symmetric InP-like interface mode 共IF兲.
10,29
Despite the dot
size fluctuations and the inhomogeneous strain fields, one
observes rather sharp Raman lines 共Fig. 4兲. This is likely due
to 共i兲 the weak dependence on the dot size of both ⌬
␻
S
and
⌬
␻
conf
and 共ii兲 the fact that the phonons are collective vibra-
tional modes 共the inhomogeneous strain field inside the dot
therefore does not induce significant line broadening兲.
The LO frequency shifts are reported in Table II. It is
noteworthy that the WL and the 2 ML SQW LO frequencies
are similar 共Fig. 4兲. The calculations ⌬
␻
S⫹ conf
⫽ ⌬
␻
S
⫹ ⌬
␻
conf
account rather well for the experimental data.
Moreover, using grazing incidence, we were able to observe
the InAs TO doublet peak 共for dots with shape A). Its fre-
quency is shifted up by ⫹ 7.2 cm
⫺ 1
with respect to the bulk
InAs TO frequency. From the average strain components 共for
shape A) and the TO deformation potentials, we obtain 关Eq.
共4兲兴 ⌬
␻
⫽ 6.8 cm
⫺ 1
, which is in good agreement with ex-
periment 共due to the very weak TO dispersion, confinement
effects are negligible兲.
Notice that we did not take into account in our simula-
tions that the dots are sometimes slightly elongated 共one ex-
pects a little less strain relaxation and higher phonon fre-
quencies兲. However, the main actual limitation we are
concerned with 共for either the comparison between the cal-
culations and the experimental results or the experimental
determination of the average strain values兲 is due to our poor
knowledge of the phonon deformation potentials. The latter
are indeed difficult to measure accurately.
It is noteworthy that if one is able to measure both the dot
LO and TO frequencies, one can deduce the ␧
xx
and ␧
zz
average values by considering simultaneously Eqs. 共3兲 and
共4兲共without any numerical strain simulation兲. Considering
the experimental LO and TO frequency shifts reported here
for A-type dots 共Table II兲, one obtains ␧
xx
⫽⫺2.92%, ␧
zz
⫽ 2.82%, and ␧
zz
/␧
xx
⫽⫺0.93; these values are very close
to the ones predicted by the simulation 共Table I兲. The relative
differences between the experimental and calculated values
do not exceed 4%. This good agreement supports the as-
sumptions we made 共in particular the one concerning the
phonons probing the average strain field兲. Combining LO
and TO Raman scattering provides thus a means of measur-
ing average strain values in self-assembled dots.
The rather good agreement between the calculations and
the experimental data suggests that the assumptions we made
are reasonable. It also suggests that alloying inside the is-
lands is not important. Optical phonons in InAsP alloys dis-
play the usual two-mode behavior: the InAs-like 共InP-like兲
LO frequencies decrease with decreasing In 共P兲 content.
31
If
one considers InAsP dots 共instead of InAs dots兲 in InP, the
calculations yield lower phonon frequencies 共due to the al-
loying induced frequency shift and the lower mismatch with
respect to InP兲 which obviously do not account for the ex-
perimental data 共Table II兲. Moreover, it has been shown that
the formation of an intermediate InAsP alloy layer during the
growth of InAs/InP structures gives rise to additional inter-
face modes.
32
We do not observe the corresponding features
in the Raman spectra.
One may wonder whether one can find some evidence in
the Raman spectra for the strain inside the InP barriers. As
most of the InP LO Raman signal originates from unstrained
regions, we are not able to identify the contributions of the
strained InP. The shift one expects from the tensile strain
underneath the InAs dot 共Fig. 2兲 and the InP LO deformation
potentials
33
is very small (⫺ 0.5 cm
⫺ 1
). On the other hand,
one expects the InP-like IF to be sensitive to the strain
around the islands.
14
However, as the IF frequencies depend
also on both the island size and shape,
23,34,35
it is not obvious
to obtain from the Raman spectra some reliable and quanti-
tative information concerning the strained InP.
IV. CONCLUSION
In summary, we calculated the strain distribution in typi-
cal self-assembled InAs/InP 共001兲 dots using the valence-
FIG. 4. Raman spectra: 共a兲 2 ML InAs/InP SQW, 共b兲 type A
dots, corresponding to 2.5 ML InAs deposited on InP and capped
by 25 nm InP. Spectra were recorded with the z(X,Y)z
¯
crossed-
polarization configuration 共solid line兲 and with the z(X,X)z
¯
parallel-polarization configuration 共dashed line兲, with X⫽
关
100
兴
and
Y⫽
关
010
兴
. The dot-dashed line indicates the bulk InAs LO fre-
quency.
16 016 PRB 60
J. GROENEN, C. PRIESTER, AND R. CARLES

force-field method. The residual strain was shown to depend
much on the dot shape. Even for rather flat dots, the average
strain field is quite different from the 2D case. As the strain
field penetrates deeply into the subtrate, a large part of the
simulation cell has to be devoted to the subtrate.
We calculated the strain-induced frequency shifts, assum-
ing the InAs phonons experience the average strain field in-
side the dot. It is shown that confinement does not modify
much the dot phonon frequencies. Our calculations show
good agreement with experimental results obtained by Ra-
man scattering. Both the Raman spectra and the comparison
between the calculated frequencies and the experimental val-
ues indicate that alloying effects are small.
Unlike electronic spectra, the dot phonon frequencies do
not depend much on dot size. One can therefore analyze the
dot phonon frequencies by 共solely兲 considering the residual
strain, and vice versa.
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⫺ K
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/(K
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Notice that if one calculates the strain-induced LO frequency
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⫹ 2K
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)
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and (K
˜
11
⫺ K
˜
12
)
TO
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˜
11
⫺ K
˜
12
)
LO
value we deduced from
Ref. 29兴, one would obtain shifts which are typically 25% lower
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PRB 60
16 017STRAIN DISTRIBUTION AND OPTICAL PHONONS IN . . .