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Relaxation and Dephasing of Multiexcitons in Semiconductor Quantum Dots
P. Borri, W. Langbein, S. Schneider, and U. Woggon
Experimentelle Physik IIb, Universita
¨
t Dortmund, Otto-Hahn Strasse 4, D-44221 Dortmund, Germany
R. L. Sellin, D. Ouyang, and D. Bimberg
Institut fu
¨
r Festko
¨
rperphysik TU, Hardenbergstrasse 36, D-10623 Berlin, Germany
(Received 15 February 2002; published 10 October 2002)
We measure the dephasing time of ground-state excitonic transitions in InGaAs quantum dots under
electrical injection in the temperature range from 10 to 70 K. Electrical injection into the barrier region
results in a pure dephasing of the excitonic transitions. Once the injected carriers fill the electronic
ground state, the biexciton to exciton transition is probed and a correlation of the exciton and biexciton
phonon scattering mechanisms is found. Additional filling of the excited states creates multiexcitons
that show a fast dephasing due to population relaxation.
DOI: 10.1103/PhysRevLett.89.187401 PACS numbers: 78.47.+p, 63.22.+m, 78.67.Hc
Semiconductor quantum dots (QDs) are atomlike ob-
jects that, with modern nanotechnology, can be fabri-
cated with high crystalline and optical quality. Their
application as an active part of novel optoelectronic nano-
devices [1] and in quantum information processing [2] is
becoming a reality. The knowledge of the homogeneous
broadening, inversely proportional to the dephasing time,
of an excitonic transition in a QD is of fundamental
importance for any of these applications.
Beyond the one exciton limit, multiexcitons in QDs
provide additional degrees of freedom for quantum com-
putation [3] and in tailoring quantum devices such as
lasers [4] and single photon emitters [5]. The optical
properties of multiexcitons in III-V QDs have been ad-
dressed experimentally mainly by isolating a single dot
from the inhomogeneously broadened ensemble. Pauli
blocking allows no more than two carriers in the lowest
lying twofold spin degenerate state of electrons (e
0
)and
holes (h
0
), which form the ground-state biexciton. Ad-
ditional carriers have to occupy higher states. At high
exciton occupancies, multiexciton spectra with a large
number of emission lines are observed [6] reflecting the
renormalization of the transition energies due to Coulomb
interaction between carriers [7]. However, the homoge-
neous broadening of multiexcitonic transitions in QDs
was hardly investigated.
We have recently reported on the homogeneous broad-
ening of the transition from the crystal ground state to the
e
0
h
0
exciton in InGaAs QDs from 7 to 300 K, using a
heterodyne four-wave mixing (FWM) technique that al-
lows direct measurement of the dephasing time in an
inhomogeneously broadened QD ensemble [8]. The inves-
tigated sample was a p-i-n ridge waveguide allowing
for electrical injection, containing three layers of self-
organized In
0:7
Ga
0:3
As QDs in the intrinsic region. The
measurements were performed without electrical injec-
tion in order to investigate the dephasing processes due to
exciton-phonon interactions.
In this Letter, we present measurements of the dephas-
ing time of multiexcitonic transitions under electrical
injection in the temperature range from 10 to 70 K. By
varying the electrical injection we progressively increase
the number of excitons in the QDs and investigate the
passage from a FWM response dominated by the one
exciton transition to a response dominated by multiexci-
tonic transitions. The measured dephasing is compared
with population relaxation dynamics obtained from dif-
ferential transmission spectroscopy (DTS).
In Fig. 1 a sketch of the investigated structure is shown.
In the presence of an injection current (I
C
), amplified
spontaneous emission spectra are measured (see also
Ref. [8]). By comparing with the emission spectra of a
sample with twice the length (1 mm), the spectral gain at
10 K was inferred using the stripe-length method [9].
FIG. 1. Spectral gain at 10 K for 2, 4, 6, 8, 10, 15, 20, 25 mA
injection current. The inhomogeneously broadened dot ground
state (GS) and first optically allowed excited state (ES) tran-
sitions are indicated. A sketch of the investigated structure is
also shown. In the inset the gain at 1.158 eV is plotted versus
injection current.
VOLUME 89, NUMBER 18 PHYSICAL REVIEW LETTERS 28OCTOBER 2002
187401-1 0031-9007=02=89(18)=187401(4)$20.00 2002 The American Physical Society 187401-1
With increasing I
C
the optical response goes from ab-
sorption to gain and shows two groups of inhomogene-
ously broadened transitions labeled GS and ES. The GS
group of transitions involves the recombination of e
0
h
0
excitons, with or without additional occupation of other
dot levels. The ES group involves the recombination of
electrons and holes in excited states. Grouped transitions
are observed in dots with good cylindrical symmetry [10]
where the ES group is dominated by the recombination of
electrons and holes in the first two excited levels e
1;2
h
1;2
.
The GS transitions reach transparency (i.e., one e
0
h
0
exciton on average) at lower injection current than the
ES transitions, and the GS gain saturates for I
C
> 15 mA
(see inset). The FWM experiment is performed using
Fourier-limited 150 fs pulses at 76 MHz repetition
rate, with a wavelength of 1070 nm at the center of the
GS group, and a heterodyne detection [8]. The pulse
intensities excited 0.1 electron-hole pair per dot, and
the FWM signal was in the third-order regime.
The time-integrated FWM field amplitude is shown in
Fig. 2 versus the delay time between the exciting pulses at
10 K for different injection currents, as indicated. Without
electrical injection (I
C
0), the excitonic transition from
the crystal ground state to e
0
h
0
(labeled 0-X) is probed in
the experiment and shows an initial fast decay followed
by a long exponential decay over several hundreds of
picoseconds. As discussed in Ref. [8], these dynamics
correspond to a non-Lorentzian line shape with a sharp
zero-phonon line (ZPL) and a broad band from elastic
exciton-acoustic phonon interaction. At 10 K the ZPL
dominates the line shape. With increasing I
C
afaster
exponential decay of the FWM is observed, i.e., the
ZPL broadens due to Coulomb interaction with the in-
jected carriers. For I
C
> 14 mA, the majority of dots is
occupied by two e
0
h
0
excitons and the biexciton to ex-
citon transition (labeled XX-X) is probed [11]. The ho-
mogeneous broadening of the ZPL shows a linear increase
versus I
C
(see inset). At low temperature a carrier which
is captured into a dot level is not reemitted and changes
the oscillation frequency of the GS polarization. As this
capture and the subsequent intradot carrier relaxation are
statistical processes not linked to the phase of the polar-
ization driven by the laser pulses, they lead to a polar-
ization decay in the measured ensemble average. Thus the
capture rate per dot
c
, which can be estimated using the
equilibrium between the total capture rate and the total
recombination rate, also leads to a polarization decay rate
c
. At the transparency current I
tr
5:5mAthere is on
average one exciton per dot, i.e., the total number of
excitons is equal to the number of dots, so that
c
r
I
C
=I
tr
using the exciton recombination rate
r
0:67 eV obtained from the exciton lifetime of 1 ns
measured by DTS [12]. The corresponding homogeneous
broadening 2
c
is much less than the measured homoge-
neous broadening due to electrical injection. Electrically
injected carriers in the barrier and wetting layer regions
thus interact efficiently with the excitonic states in the QD
without being captured, and give rise to a pure dephasing
of the 0-X and XX-X transitions [14].
By fine variation of the electrical injection near trans-
parency, the passage from a FWM dominated by dots in
the 0 state to the one dominated by dots in the XX state
occurs. For a dot in the 0 state, the 0-X has a population
inversion of 1, while for a dot in the XX state the XX-X
transition has a population inversion of 1. Therefore,
the optically driven polarizations of these two transitions
have a relative phase shift of and thus show a destruc-
tive interference. This interference is visible in Fig. 3 as a
dip in the FWM signal versus delay. If the amplitude of
the 0-X and XX-X contributions are similar, which is
expected close to transparency, the dip occurs near zero
delay. With increasing amplitude of the XX-X contribu-
tion the dip shifts to positive delays, as shown in Fig. 3.
The rise and decay of the FWM signal after the dip can
be fitted by the difference of two exponential decays
(dotted line in Fig. 3) and shows that the 0-X transition
has a longer dephasing time than the XX-X transition.
Since the dephasing of the 0-X and XX-X transitions due
to I
C
are equal within errors (see slope in the inset of
Fig. 2), the difference in the dephasing rates is a sensitive
measure of the XX-X dephasing by radiative pro-
cesses and phonon interaction. In the inset of Fig. 3 the
temperature dependence of the 0-X and XX-X ZPL line-
width extrapolated to I
C
0 is shown. The XX-X broad-
ening is larger than the 0-X broadening by a factor
that varies with temperature, and tends to one at high
0 50 100 200 300 400
0 5 10 15 20 25 30
0.00
0.05
0.10
XX-X
0-X
14mA
20mA
25mA
5mA
3mA
2mA
1mA
0mA
30mA
TI FWM field amplitude (a.u.)
delay (ps)
0-X
XX-X
20 x
γ
c
I
C
(mA)
γ (meV)
FIG. 2 (color online). Time-integrated four-wave mixing at
10 K and different injection currents, as indicated. The depen-
dence of the homogeneous broadening on injection current
deduced from the signal decay at long delays is shown in the
inset (squares), together with 20 times the estimated carrier
capture rate per dot (line). Sketches of the 0-X and XX-X
transitions probed in the experiment are shown (curly arrows
are the interacting photons).
VOLUME 89, NUMBER 18 PHYSICAL REVIEW LETTERS 28OCTOBER 2002
187401-2 187401-2
temperature. Both dependencies are fitted with the fol-
lowing expression:
0
aT b=expE
A
=kT1.
We used
0
equal to 0:67 and 2 eV for the 0-X and the
XX-X transition, respectively. For the linear temperature
dependence we obtain a
0X
0:22 0:02 eV=K and
a
XXX
0:37 0:04 eV=K. The thermally activated
behavior is identical within error for both transitions,
with an activation energy E
A
16 1 meV and b
1:1 0:2 meV.
0
0:67 eV is taken from the mea-
sured exciton lifetime of 1 ns and 2 eV, i.e., 3 times this
value, for the XX-X transition corresponds to a fully
uncorrelated radiative decay of the three involved exci-
tons. If a
0X
is due to transitions between the different
spin states of the e
0
h
0
exciton as we suggested in Ref. [8],
a
XXX
and a
0X
should be equal, since XX is a singlet
spin state. Alternative explanations could be the broad-
ening of the ZPL due to the elastic exciton-acoustic
phonon interaction expected if virtual excitations of the
excited states or a finite phonon lifetime are considered.
Finally, the activated part of the 0-X broadening was
attributed to a phonon-assisted transition of the e
0
h
0
into the e
0
h
1
exciton state [8]. The identical activated
part of the XX-X transition indicates fully correlated
exciton and biexciton scattering via phonon absorption,
similar to observations in InGaAs/GaAs and GaAs/
AlGaAs quantum wells [15] and in CdSe/ZnSe quantum
dots [16].
With increasing electrical injection above 15 mA, satu-
ration of the e
0
h
0
excitons and filling of the e
1;2
h
1;2
excitons occurs (see Fig. 1). In the presence of e
1;2
h
1;2
excitons the probed GS transition in the FWM experiment
goes from a multiexciton ground state to a multiexciton
excited state with only one e
0
h
0
exciton. This transition
has a strong final state damping due to the relaxation of
the excited multiexciton into its ground state with two
e
0
h
0
excitons, resulting in a fast FWM decay (see fast
initial dynamics for I
C
> 15 mA in Fig. 2). We have
probed this relaxation dynamics in a degenerate DTS
experiment on the GS transition utilizing the same het-
erodyne technique as the FWM experiment. A strong
pump pulse leads a weak probe at positive delay time,
and the heterodyne detection selects the transmitted
probe instead of the FWM signal [17]. DTS data were
taken for varying I
C
from 0 to 30 mA.
The pump-induced change of the gain, deduced from
the probe differential transmission, is shown in Fig. 4 at
10 K for transparency of the GS transition (I
C
5:5mA)
and strong filling of the e
1;2
h
1;2
excitons (I
C
30 mA). In
a QD ensemble, a macroscopic configuration for a given
electrical injection is a superposition of microstates [13].
A microstate of a QD is defined by the occupation of the
dot levels with integer numbers of carriers. The probabil-
ity of finding a specific microstate in the macroscopic
configuration depends on I
C
. Each microstate has a
given internal dynamics while only the capture rate is
proportional to I
C
. Good fits to the DTS data required at
least four time constants. In Fig. 4 the fourfold exponen-
tial fit function has been decomposed into the contribu-
tions for the different time constants, to explain the role
of different microstates in the DTS signal. Moreover, an
instantaneous contribution from nonresonant two-photon
absorption and coherent artifacts was included [17].
The amplitudes A
1
...A
4
of the time constants versus
-0.5
0.0
0.5
1.0
-1 0 1 20 40
-3
-2
-1
0
0 102030
-0.5
0.0
0.5
τ
4
τ
3
τ
2
τ
1
TPA
+
+
delay(ps)
5.5mA
G(dB)
G(dB)
30mA
A
1
A
2
A
3
A
4
amplitude (dB)
I
C
(mA)
FIG. 4 (color online). Differential transmission spectroscopy
at 10 K and different injection currents, as indicated. The
measured change of the gain in decibels is shown (open circles)
together with its best fit (solid line). A decomposition of the fit
into four exponential contributions, and an instantaneous re-
sponse from two-photon absorption (TPA) is also shown. In the
inset the dependence of the amplitudes of the four exponential
contributions versus injection current is shown.
0 8 0 160
0 204060
10
-3
10
-2
10
-1
7mA
10mA
TI FWM field amplitude (a.u.)
delay(ps)
XX-X
0-X
γ
(meV)
T(K)
FIG. 3. Time-integrated four-wave mixing at 10 K and 7, 8, 9,
10 mA injection current. The dotted line is a fit to the data at
9 mA. The inset shows the temperature dependence of the 0-X
(closed square) and XX-X (open square) homogeneous broad-
enings for I
C
0, together with fits (solid lines).
VOLUME 89, NUMBER 18 PHYSICAL REVIEW LETTERS 28OCTOBER 2002
187401-3 187401-3
injection current are shown in the inset of Fig. 4. All the
microstates recover, after the interaction with the pump
photons and the internal relaxation dynamics, on a long
time scale, determined by the interplay between radiative
recombination and capture [13], represented by
4
which
varies from 1 ns to 200 ps going from I
C
0 to 30 mA.
At low temperature before the optical excitation, the
carrier configuration of each microstate is in its ground
state (see sketches in Fig. 4). Microstates with no carriers
in the excited states do not have internal relaxation dy-
namics and thus evolve only with
4
. After the removal of
a e
0
h
0
by the pump photons, microstates with a high
number of carriers in the excited states undergo a fast
relaxation dynamics, modeled by
1
0:33 0:05 ps,
due to the high number of final states available for the
phonon-assisted relaxation. However, microstates with
only one carrier in the first excited state have only one
relaxation channel. We suggest that the slow time con-
stants
2
4 0:3ps and
3
35 4ps refer to this
relaxation for a hole or an electron. This is supported by
the fact that A
2
and A
3
have similar values versus I
C
,as
expected from the overall charge neutrality of the dot
ensemble.
We can finally compare the measured relaxation dy-
namics of the multiexcitons with the FWM decay. From
the inset of Fig. 4 we observe that at 30 mA the DTS
response is dominated by the microstates with more than
one carrier in the excited states, thus allowing for a
simple comparison between the measured time constant
1
and the FWM decay. Apart from the initial rise, the
time-integrated photon echo versus delay is a probe of the
time evolution of the first-order polarization and its
Fourier transform is similar to the homogeneous line
shape [8]. In Fig. 5 the calculated power spectrum of
the TI FWM field amplitude at 30 mA is shown with a
zoom of the FWM versus delay time in the inset. The line
shape of the power spectrum is well represented by a sum
of two Lorentzians, with the widths indicated in the
figure. The sharp Lorentzian is the Fourier transform of
the long exponential decay, corresponding to the broad-
ening of the XX-X transition due to pure dephasing from
the injected carriers in the barriers. Conversely, the width
of the broad Lorentzian corresponds to the broadening
given by
1
, showing that the dephasing of the probed
multiexcitonic transition is dominated by population
relaxation.
In conclusion, we found that electrically injecting car-
riers decreases the dephasing time of excitonic transitions
in InGaAs quantum dots at low temperature. In particu-
lar, a pure dephasing due to carriers injected in the
barrier material and a dephasing of multiexcitonic tran-
sitions by population relaxation are distinguished. These
results might be of considerable importance for future
applications based on a coherent optical control of the
excitonic transitions in quantum dots and suggest that
barrier engineering could reduce the dephasing time by
one order of magnitude to the limit given by the capture
rate. Additionally, the temperature dependent dephasing
of the biexciton to exciton transition revealed an exciton
and biexciton correlated phonon scattering, which should
stimulate further theoretical work.
This work was supported by DFG (Wo477/17-1 and
SFB296). P. B. is supported by the European Union
(MCFI-2000-01365).
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-4-20246
30mA
1.4meV
≈
h
/
τ
1
0.1meV >>
γ
c
|E(
)|
2
(a.u.)
Energy(meV)
010
delay(ps)
TI FWM (a.u.)
FIG. 5. Power spectrum of the four-wave mixing field am-
plitude at 30 mA and 10 K. The full widths at half maximum of
the two Lorentzian contributions to the line shape are indi-
cated. In the inset the initial fast dynamics of the four-wave
mixing versus delay time is given.
VOLUME 89, NUMBER 18 PHYSICAL REVIEW LETTERS 28OCTOBER 2002
187401-4 187401-4
