Tuning the Exciton Binding Energies in Single Self-Assembled InGaAs=GaAs Quantum Dots
by Piezoelectric-Induced Biaxial Stress
F. Ding,
1,2,3,
*
R. Singh,
3
J. D. Plumhof,
1
T. Zander,
3
V. K r
ˇ
a
´
pek,
1
Y. H. Chen,
2
M. Benyoucef,
1
V. Zwiller,
4
K. Do
¨
rr,
5
G. Bester,
3,†
A. Rastelli,
1,‡
and O. G. Schmidt
1
1
Institute for Integrative Nanosciences, IFW Dresden, Helmholtzstrasse 20, D-01069 Dresden, Germany
2
Key Laboratory of Semiconductor Materials Science, Institute of Semiconductors, Chinese Academy of Sciences,
Beijing 100083, China
3
Max-Planck-Institut fu
¨
r Festko
¨
rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
4
Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands
5
Institute for Metallic Materials, IFW Dresden, Helmholtzstrasse 20, D-01069 Dresden, Germany
(Received 3 November 2009; published 12 February 2010)
We study the effect of an external biaxial stress on the light emission of single InGaAs=GaAsð001Þ
quantum dots placed onto piezoelectric actuators. With increasing compression, the emission blueshifts
and the binding energies of the positive trion (X
þ
) and biexciton (XX) relative to the neutral exciton (X)
show a monotonic increase. This phenomenon is mainly ascribed to changes in electron and hole
localization and it provides a robust method to achieve color coincidence in the emission of X and
XX, which is a prerequisite for the possible generation of entangled photon pairs via the recently proposed
‘‘time reordering’’ scheme.
DOI: 10.1103/PhysRevLett.104.067405 PACS numbers: 78.67.Hc, 78.20.hb, 81.05.Ea, 81.40.Tv
Sources of entangled photon pairs on demand are a
major building block for quantum computation and com-
munication [ 1]. Recently, the generation of entangled pho-
ton pairs from semiconductor quantum dots (QDs) has
attracted great interest [2–5]. The polarization-entangled
photons are produced in an idealized QD with degenerate
intermediate exciton states in the cascade: biexciton
ðXXÞ!excitonðXÞ!ground state (G), where the polar-
ization of a photon pair is determined by the spin of the
intermediate exciton state [6]. However, real self-
assembled QDs exhibit intermediate exciton ground states
split into two states by an energy called fine structure
splitting (FSS) [7]. This is the consequence of shape and
atomistic crystal anisotropy and the electron-hole ex-
change interactions [8]. The FSS in self-assembled
InðGaÞAs=GaAs QDs grown along the [001] crystal direc-
tion is typically quite large as compared to the radiative
linewidth (1:0 eV). The nonvanishing FSS encodes the
which-path information and destroys the polarization en-
tanglement. The generation of entangled photon pairs by
simple preselection of rare dots with close to zero [2,4]or
by spectral filtering [3] has been demonstrated.
Furthermore, a number of postgrowth techniques have
been used to reduce , such as in-plane magnetic fields
[9], lateral electric fields [10,11], uniaxial stress [12], and
rapid thermal annealing [13].
An alternative proposal to generate entangled photon
pairs from QDs, without any fundamental requirements
on the FSS to be smaller than the radiative linewidth, is
the so-called time reordering scheme [14]. However, this
scheme requires the emission energies of X (E
X
) and XX
(E
XX
) to be the same. In this scheme, one entangles the red
photons H1ðV2Þ and the blue photons V1ðH2Þ [see the
central panel of Fig. 1(a)] across generations in a QD. It is
accomplished by performing a unitary operation (time
reordering) U on the two-photon state such that
jh
H
jU
y
H
U
V
j
V
ij > jh
H
j
V
ij [14,15], where j
HðVÞ
i is
the wave packet resulting from the biexciton cascade emis-
FIG. 1 (color online). (a) Level schemes showing the XX-X
cascade. The solid (dashed) line represents the decay channel
that yields H (V) polarized photons. Across generation color
coincidence of X and XX (E
X
¼ E
XX
) can be achieved by
applying tensile (compressive) stress to a QD with positive
(negative) E
B
ðXXÞ. (b) Schematic drawing of the experiment
and optical microscopy image of a 200 nm-thick GaAs mem-
brane (inset). (c) Low temperature PL spectra of QDs with
negative (QD1) and positive (QD2) XX binding energy.
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0031-9007=10=104(6)=067405(4) 067405-1 Ó 2010 The American Physical Society
sion. This novel concept is currently under vivid discussion
[15–18]: On one hand the two-photon wave packet may
suffer significant dephasing after the time reordering, on
the other hand the emission energies E
X
and E
XX
in the as-
grown QDs are usually different because of pronounced
interactions between charge and spin carriers in a QD.
In this Letter we show that an external biaxial ten-
sile (T) or compressive (C) stress can be used to achieve
E
X
E
XX
. Stress is provided in situ by placing a thin
GaAs membrane containing self-assembled InGaAs
QDs on top of a piezoelectric actuator, made of
½PbðMg
1=3
Nb
2=3
ÞO
3
0:72
-½PbTiO
3
0:28
(PMN-PT) [see
Fig. 1(b)]. With increasing compression, the relative spec-
tral positions of different excitonic species (X, XX, and
X
þ
) show subtle, but systematic changes: The binding
energies E
B
of both X
þ
and XX, defined as E
B
ðXX=X
þ
Þ¼
E
X
-E
XX=X
þ
increase in all studied QDs. Based upon
million-atom empirical pseudopotential many-body calcu-
lations of realistic InGaAs=GaAs QDs, we ascribe this
phenomenon to the increase in electron-hole Coulomb
interactions due to the increase in confinement of electrons
and slight decrease in confinement of holes upon compres-
sive biaxial stress. Finally, different from the behavior
observed under in-plane uniaxial stress [12], biaxial strain
does not appreciably affect the FSS, a behavior which is
also expected from our calculations.
We fabricated 200 nm-thick GaAs membranes with
embedded self-assembled InGaAs QDs, and then trans-
ferred them onto a 300 m-thick PMN-PT actuator via
PMMA resist [20]. A bias voltage V applied to the PMN-
PT results in an out-of-plane electric field F which leads to
an in-plane strain "
k
in the GaAs membrane and the QD
structure [see Fig. 1(b)]. The PMN-PT was poled so that
V>0 (< 0) corresponds to in-plane compressive (tensile)
strain "
k
< 0 (> 0). Figure 1(c) shows low-excitation
power photoluminescence (PL) spectra of two QDs with
negative E
B
ðXXÞ (QD1) and positive E
B
ðXXÞ (QD2). The
neutral exciton X and the biexciton XX are identified by
power- and polarization-dependent PL. (The latter allows
us also to determine the FSS of X and XX). An unpolarized
line lying at the higher energy side of X and XX is attrib-
uted to positive trion X
þ
emission. The assignment is
supported by the background p-type doping of our struc-
tures and by the correlation between E
B
ðX
þ
Þ and E
B
ðXXÞ
observed in all the studied dots [see Fig. 2(e)].
Figure 2(a) shows the color-coded PL intensity of QD1
as a function of emission energy and voltage V applied to
the PMN-PT actuator. V is swept several times between
0 and 1100 V with steps of 20 V, to demonstrate the
reversibility of the tuning. For V>0 the QD experiences
an in-plane compression (C). The emission energies of
different lines show roughly linear blueshifts with V.At
the maximum reached bias, E
X
shifts by 1:8 meV, with-
out appreciable deterioration of the emission linewidth and
intensity [20]. Similar energy shifts for a given V are
observed for different dots in the same device. However,
different devices show different maximum shifts, which
may be due to slight variations in the composition or the
mechanical contact of the PMN-PT to the coldfinger. More
interestingly, the E
B
ðXXÞ and E
B
ðX
þ
Þ show a linear in-
crease with E
X
, see Fig. 2(b), with slopes
XX
¼ 51:6 and
X
þ
¼ 62:7 eV=meV for QD1. The results of a similar
experiment performed on QD2 are shown in Figs. 2(c) and
2(d). In this case V is swept to positive and negative values.
Under tension (V<0) the QD emission lines redshift and
the linear dependence between binding energies and E
X
extends also to the tensile (T) strain region.
In order to verify whether these findings are affected by
QD structural fluctuations, we have determined the values of
for 14 different dots. The results are shown in Fig. 2(f).
In all the studied dots we observe that both
X
þ
and
XX
are positive and that for a given dot
X
þ
’
XX
. The former
observation is interesting as it opens up the possibility to
tune E
B
ðXXÞ to zero in a controllable way. The latter
suggests a common underlying physical mechanism re-
sponsible for the changes in binding energy of XX and X
þ
.
In order to understand the results and estimate the mag-
nitude of the in-plane strain achieved in the experiment, we
performed calculations on realistic InGaAs=GaAs QDs
containing 3 10
6
atoms using the empirical pseudo-
potential and the configuration interaction (CI) approaches
[21]. The excitonic states are calculated at the correlated CI
level, including all configurations generated from 12 elec-
tron and 12 hole states (spin included). We model the QD
as a lens shaped In
0:8
Ga
0:2
As structure with a height of
FIG. 2 (color online). Color-coded PL intensity of QD1 (a) and
QD2 (c) as a function of emission energy and voltage applied to
the actuator. The binding energies of XX and X
þ
show a linear
increase with E
X
for both QDs, as shown in (b) and (d).
(e) Binding energies at V ¼ 0 and (f) slopes
XX
and
X
þ
for
14 different QDs.
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2.5 nm and elliptical base of major (minor) axis of 10
(7.5) nm along the ½1
10 ([110]) crystal direction.
Figure 3(a) shows the calculated emission energies of X,
XX, and X
þ
as a function of biaxial strain "
k
¼
½ða a
0
Þ=a
0
, where a (a
0
) is the lattice constant of
strained (unstrained) GaAs. As in the experiment, all emis-
sion lines blueshift for "
k
< 0 and redshift for "
k
> 0.In
analogy to the results shown in Figs. 2(b) and 2(d), we plot
in Fig. 3(b) the relative binding energies of XX and X
þ
as a
function of E
X
. Although the initial values of E
X
and
binding energies are somewhat different from those ob-
served in the experiment [Figs. 1(c) and 2(e)], the calcu-
lation is able to reproduce the linear increase of E
B
ðX
þ
Þ
and E
B
ðXXÞ with E
X
. For the modeled QD structure the
slope of E
B
ðXXÞ is
XX
¼ 114 eV=meV, comparable
with the experimental values. By repeating similar calcu-
lations on a larger QD with a circular base and a height of
3.5 nm with a lower In concentration (60%), we find
XX
¼
28:3 eV=meV. Although the values of ’s depend on the
actual QD structure and size the linear increase of binding
energies of XX and X
þ
relative to X upon biaxial com-
pression is a rather general phenomenon as we observed in
the experiments.
From our empirical pseudopotential calculations, we
find that the effect of biaxial strain on the correlation
energy is very small. The main changes in the binding
energies of XX and X
þ
are due to changes in the direct
Coulomb interactions between electron and holes and can
be approximated by
E
B
ðX
þ
ÞJ
eh
J
hh
;
E
B
ðXXÞE
B
ðX
þ
Þ½J
ee
J
eh
;
(1)
where J
ee
, J
hh
, and J
eh
are the Coulomb integrals between
lowest electron (e) and hole (h) states. Figure 3(c) shows
that J
eh
and J
ee
increase with compressive strain, with
only small deviations from each other. Interestingly, J
hh
shows the opposite behavior, but its magnitude is substan-
tially smaller than those of J
eh
and J
ee
. We thus con-
clude that the increase in binding energies of XX and X
þ
upon compression is mainly a consequence of the increase
in the electron-hole attraction term. The fact that (J
ee
J
eh
) is small, qualitatively explains the similar values of
E
B
ðXXÞ and E
B
ðX
þ
Þ, as shown in Fig. 2(f).
To understand the increase in J
ee
, J
eh
, and decrease in
J
hh
upon compression, we plot in Fig. 3(d) the strain
modified conduction band minimum and the upper two
valence bands. For the latter bands we used circles propor-
tional in size to the fraction of heavy-hole character. In the
unstrained region, far from the dot, heavy- and light-hole
bands are degenerate; close to the dot (inside the dot), the
light (heavy)-hole band forms the valence band maximum.
Since the QD hole states have up to 92% heavy-hole
character, we define the valence band offset (VBO) as the
offset between the heavy-hole bands [Fig. 3(d)]. No such
complication arises for the conduction band offset (CBO).
In Fig. 3(e) we show a linear increase by 35 meV for the
CBO upon change in biaxial strain from 0.1% to 1%.
This represents an increased confinement and localization
of wave function: the intuitive picture of a compressed
wave function obtained by a compression of the sample,
is valid. For the VBO, however, we find a decrease by
3 meV for the same range of strains. Upon compression,
the wave functions tend to become more delocalized. This,
rather counterintuitive behavior can be observed directly
on the wave functions in Fig. 3(f), where we display the
lowest electron state (LUMO) and highest hole state
(HOMO) at two different strains. This localization or
delocalization gives rise to the increase in J
ee
, J
eh
, and
decrease in J
hh
shown in Fig. 3(c). Similar results are
obtained also by eight-band kp calculations combined
with the CI model [20].
The fact that the biexciton binding energy monotoni-
cally increases upon compression provides a controllable
strategy to tune E
B
ðXXÞ to zero. If XX is initially located at
the high (low) energy side of X, a biaxial compression
(tension) allows us to tune E
B
ðXXÞ so that the color coin-
cidence between XX and X is achievable, depending on the
available tuning range, the value of
XX
and the initial
value of E
B
ðXXÞ. Figure 4(a) demonstrates this capability
for a QD where E
B
ðXXÞ is initially 270 eV. The total
shift of E
X
in this device is 11 meV, which according to
the calculations correspond to strain values exceeding
0:3%. Figure 4(b) shows gray scale coded PL intensity
plots of X and XX as a function of linear polarization angle
and energy for various values of V. As expected, X and XX
FIG. 3 (color online). (a) Calculated emission energies of X,
XX, and X
þ
as a function of in-plane biaxial strain "
k
. (b) Rela-
tive binding energies of X
þ
and XX vs E
X
. (c) Changes in Cou-
lomb integrals with biaxial strain. (d) Calculated band offset dia-
gram at "
k
¼ 0; for the valence band, the size of the circles is
proportional to the heavy-hole character of the bands. (e) Changes
in CBO (VBO) with strain "
k
. (f) HOMO and LUMO wave
functions at "
k
¼ 0:1% and 1%. The red color encloses 75% of
the charge density, while light gray color represents the outline
of the QD.
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show anticorrelated shifts as we rotate the analyzer. The
FSS for this QD, i.e., the energy separation between the
H and V components of X or XX is 48 5 eV. As the
voltage increases the magnitude of E
B
ðXXÞ decreases,
while stays constant within the measurement uncertain-
ties. This finding, which is reproduced by our calculations,
is not surprising, because the symmetry of the structure is
not appreciably changed by biaxial strain. At 1100 V
E
B
ðXXÞ!0 and dominates the energy scale in the
problem: only two peaks with the splitting of can be
observed. We have now practically reached the color co-
incidence of (V2, H1) and (H2, V1). For future entangle-
ment measurements, we envision the use of a Michelson
interferometer. That will allow the ‘‘red’’ (V2, H1) and the
‘‘blue’’ (H2, V1) photons to be directed to the two outputs
of the interferometer, and hence spatially separated [22].
In conclusion, we have studied in detail experimentally
and theoretically the effect of biaxial strain on the binding
energies of different excitonic species confined in single
InGaAs=GaAsð001Þ quantum dots. The most intriguing
finding is that biaxial strain is a reliable tool to engineer
the QD electronic structure and reach color coincidence
between exciton and biexciton emission. While other tech-
niques may be used to reach this goal, such as rapid
thermal annealing [13,23] or lateral electric fields [24],
strain engineering is advantageous as it can be performed
in situ and does not produce any appreciable degradation of
the emission, which usually occurs at large electric fields
[25]. Furthermore, biaxial strain does not alter the exci-
tonic FSS, which should remain sufficiently large for ex-
perimental tests on the viability of a newly proposed
concept for the generation of entangled photon pairs
[14]. Finally, the employed method can be used to study
the effect of stress and tune the properties of a broad range
of nano- and microstructures, such as optical microcavi-
ties [26].
We acknowledge N. Akopian, U. Perinetti, P. Klenovsky
´
,
C. C. Bof Bufon, R. Hafenbrak, S. Mendach, and
P. Michler for fruitful discussions and the financial support
of the DFG (FOR730), BMBF (No. 01BM459), NWO
(VIDI), CAS-MPG and NSFC China (60625402).
*f.ding@ifw-dresden.de
†
g.bester@fkf.mpg.de
‡
a.rastelli@ifw-dresden.de
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FIG. 4 (color online). (a) Biexciton binding energy of QD3 as
a function of E
X
. Several spectra are shown, demonstrating the
decreasing distance between X and XX with increasing E
X
.
(b) Polarization-resolved PL map for the X and XX lines at
several voltages. At 0 V E
B
ðXXÞ is much larger than , while
E
B
ðXXÞ vanishes (limited by the system resolution) and
dominates at 1100 V.
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