233
Journal of Luminescence 44 (1989) 233—246
North-Holland, Amsterdam
TFLE EXCITONIC OPTICAL STARK EFFECT IN SEMICONDUCTOR QUANTUM WELLS
PROBED WITH FEMTOSECOND OPTICAL PULSES
D.S. CHEMLA, W.H. KNOX and D.A.B. MILLER
AT&T Bell Laboratories, Holmdel, New Jersey 07733, USA
S. SCHMITL’-RINK
AT&T Bell Laboratories, Murray Hill, New Jersey 07974, USA
J.B. STARK
Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA
R. ZIMMERMANN
Zentralinstitut f Elektronenphysik d
Akademie d. Wissenschaften der DDR, Hausvogteiplatz 5— 7, 1086 Berlin, Germany
Received 8 August 1989
Accepted 21 August 1989
We review recent experimental and theoretical studies of the excitonic optical Stark effect in semiconductor quantum wells
probed with femtosecond optical pulses.
1. Introduction
Recent advances in laser spectroscopy have
permitted the observation of coherent light—matter
interactions in semiconductors, such as the optical
or AC Stark Effect, i.e. the renormalization of
optical transitions in an intense laser field. This
effect is readily observable in atomic vapours,
where it has been extensively studied in the past
[1,2]. In semiconductors, however, its observation
is usually obscured by the fast dephasing rates,
Two ways to circumvent this problem are: (1) the
use of ultrashort laser pulses, and (2) nonresonant
excitation of virtual excitons below the fundamental absorption edge, where the dephasing rates are
exponentially small.
Apart from early work on bi-excitons [3], the
first observation of the AC Stark effect in a semiconductor was reported by Froehlich et al., who
demonstrated a mixing of the is and 2p exciton
states in Cu20 by the electric field of an intense
CO2 laser pulse [4]. This work was later extended
to intersubband transitions in a semiconductor
quantum well (QW) [5].Although in these experiments one observes a shift or splitting of a level in
a semiconductor, this effect is directly related to
the AC Stark effect of two-level atoms and it
should be contrasted with the excitonic AC Stark
effect observed subsequently under intense, nonresonant excitation of virtual excitons in QWs,
bulk semiconductors and polymers, Following its
initial observation in 1986, by Mysyrowicz et al.
[6] and Von Lehmen et al. [7], the latter effect has
attracted much attention recently, both experimentally [8—17]and theoretically [18—42].
Because of the extended nature and strong interactions of the elementary excitations in semiconductors, the excitonic AC Stark effect is more
complex in character than that in atomic systems.
The theory reveals profound similarities between
coherent virtual excitons and Bose condensed gases
or superconductors [18—21].The excitonic optical
0022-23i3/89/$03.50 © Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
234
D.S. Chemla et a!.
/ Excitonic optical Stark effect in semiconductor Q Ws
nonlineanties originate from anharmonicities in
the exciton—photon, exciton—phonon and exciton—exciton interactions, which depend directly
on the polarization and density of virtual excitons.
In turn, the distribution of the latter over bound
and scattering (i.e. unbound) states is determined
by the pump intensity, I~,and pump detuning
from the fundamental absorption edge, E1~ hw~.
0 1
.
0.01
(b)
—
—
Thus, experimental studies of the excitonic optical
absorption under various conditions of nonresonant excitation provide information on a novel
coherent many-body state of matter.
In this article, we review our work on the
excitonic AC Stark effect in semiconductor QWs
probed with femtosecond optical pulses. In sect. 2
we discuss the experimental studies and in sect. 3
the theory.
2. Experimental studLes
—
—
I
752
I
775
799
822
WAVELENGTH (nm)
845
Fig. 1. (a) Spectrum of the ferntosecond continuum and (b)
absorption spectrum of the p-i-n quantum well sample at
T = 35 K (in arbitrary units).
The sample we investigated consists of a p-i-n
diode with 50 periods of 74 A QWs with 60 A
barriers in the intrinsic region. The GaAs substrate was removed by selective etching and the
sample was antireflection-coated. The absorption
spectrum, a, of the sample at T 35 K is shown
in fig. 1. The is heavy-hole exciton line is centered
at X1~ 780 nm. The femtosecond continuum
pro2) convides extremely
(up pulses
to 1 TW
tinuously
tunableintense
excitation
justcm
below this
resonance.
For pump and probe experiments, the intensity
ratio of pump and test beams, ‘p/It’ should be as
large as possible. The saturation density of QW
excitons, however, is so small [18,44] that it is also
important to make sure that the absolute intensity
of the test beam, i~,does not perturb the absorption of the sample by creating too many real
excitons or carriers. We have found that for 100
fs pulse duration I~ 100 kW cm2 produces
significant absorption saturation and that it is
necessary to keep I~below
10 kW cm2 to
avoid perturbation of the absorption by the test
beam itself. In all our experiments the test beam
intensity was kept below this value.
As discussed in sect. 3, numerical calculations
based on the theory of ref s. [19,20] have recently
predicted that in two dimensions and for pump
detunings of about 10 Rydbergs, low-intensity
nonresonant excitation should produce a pure AC
Stark shift of the is exciton resonance without any
loss of oscillator strength [23,24,30—33].We, therefore, first performed experiments under the condi=
The signature of the AC Stark effect is that it
lasts
only as longabsorption
as the sample
is Therefore
excited (below
the fundamental
edge).
it is
studied using time-resolved pump and probe techniques. For our experiments we used a newly
developed laser system which generates
100 fs
optical pulses of microjoule energies at a 8 kHz
repetition rate, at a photon energy in the vicinity
of the band gap of GaAs, with its center at
X 805 nm [43]. The pulses are focussed onto ajet
of ethylene glycol to generate an intense continuum, the spectrum of which extends from the
visible to the infrared (fig. 1). The continuum is
split into two parts. One is used as a broadband
test beam to probe the sample transmission, while
pump pulses of
100 fs duration are selected
from the other with interference filters. Differential transmission spectra are measured with an
optical multichannel analyzer and spectrometer
operating in differential detection mode while synchronously chopping the pump beam. The time
delay, L~t, between pump and probe beams is
varied continuously to obtain the full temporal
evolution. Kilohertz repetition rates allow us to
use differential detection techniques for small signal analysis.
=
0.001
=
—
D.S. Chemla eta!.
/ Excitonic optical Stark effect in semiconductor QWs
tions. When the sample is excited
50 meV below the is heavy-hole exciton with low intensity
2, the changes of
pump
pulses, I~
30 MW
the absorption
spectrum
are cm
very small, only a few
percent, i.e. about the width of the pen line of the
absorption spectrum shown in fig. 2a. They can be
seen only by measuring the change in sample
transmission induced by the pump beam, The
so-called differential transmission spectrum (DTS),
12
—
—
[T(I~ 0)
T(I~)]/T(I~ 0). In the
small-signal regime, The DTS is simply propor=
=
—
235
II
I
0
~
6
w
z
0
-6
zH
w
-12
=
UI
10
(a)
1.56
1.57
I
1.58
I
1.59
Fig. 3. Differential transmission spectrum at delay ~t
0 for
ENERGY (eV)
—50meV pump detuning and —100 MW cm2 pump intensity. The signal no longer matches the derivative (dashed line)
z
05
of the experimental absorption (fig. 2(a)) indicating a departure from pure AC Stark shift of the is exciton.
0
Cd)
sample, ~ T/ T
L~aX I. An example of DTS at
tional to the change in the absorption of the
delay z.~t 0 is shown in fig. 2(b) [14]. In the limit
of a pure AC Stark shift of the exciton theoretically predicted, the DTS around the exciton resonance should have exactly the same lineshape as
the
derivative
of theoflinear
absorption,
fig. 2a,
i.e.
aa/a~.
The profile
this derivative
is also
shown
—
0
—
=
J~
1.56
1.57
1.58
1.59
ENERGY (eV)
12
(b)
z
0
U)
z
~
IL
Ii.
—
in fig. 2(b). Around the is heavy-hole exciton
resonance the two lineshapes are almost identical,
in agreement with the theory. The origin of this
8
4
8T\1~
pure AC Stark shift of the is exciton without any
loss of oscillator strength will be discussed in
detail below.
_______________
1.56__________________
1.57 1.58
1.59
ENERGY (eV)
Fig. 2. (a) Detail of the absorption spectrum of the p-i-n
quantum well sample close to the exciton resonance at T= 35
K. (b) Differential transmission spectrum at delay ~.lt—0 for
— 50 meV pump detumng and
— 30 MW cm2 pump inten
sity. The derivative of the experimental absorption, fig. 2(a),
with respect to frequency, is also shown (smooth line), demonstrating that the is exciton exhibits a pure AC Stark shift at
low intensities,
and
broadens,
2,ofthe
solinear
that it
lobe
nooflonger
the DTS
matches
decreases
the
derivative
thenegative
absorption.
This
is shown
in As
fig.cm
4, where
the
at delay
is
the
pump again
intensity
isDTS
increased
to ~tI~, 0100
MW
compared to aa/a~ [14]. The DTS profile is
indicative of both a shift and a bleaching of the is
heavy-hole exciton absorption, as seen earlier [6,8].
At these intensities, the absorption still completely
recovers when the pump pulse is over, i.e. for
L~t> 150 fs, demonstrating that these effects are
=
—
still due to virtual excitons. We note that although
numerical calculations for monochromatic laser
236
D.S. Chemla et a!.
/ Excitonic optical Stark effect in semiconductor Q Ws
beams have predicted pure AC Stark shifts of the
is exciton up to very large values, the finite bandwidth of ultrashort pulses modifies these results
especially at higher intensities [23—25,31—36,39,
41,42]. For example, it leads to an “inhomogeneous broadening” of the absorption spectrum,
—
proportionality to the pump intensity, I~IX E1, 12,
and to the inverse of the pump detuning, E1~
have been verified experimentally [7]. The
dependence on the dipole matrix element, j.L, can
be studied by employing the quantum-confined
stark effect (QCSE) [45,46]. It is known that, when
an electrostatic field is applied perpendicular to
the plane of the QW, the is exciton resonance
experiences a red shift and loses oscillator strength,
because the electron—hole overlap is reduced. This
is easily achieved by applying a reverse bias to the
p-i-n diode sample. The AC Stark effect in biased
QWs have
has been
predicted
to have specific
features
that
potential
applications
in high-speed
—
(b)
‘
0
6
0)
z
___________________
I—
F-
~
—6
UJ
IL
-12
w
IL
0
1.56
1 57
1.58
1 .59
ENERGY (eV)
Fig. 4. Differential transmission spectrum at delay ~‘.5
t = 0, (a)
for — 50 meV pump detuning at 0 V applied DC bias, and (b)
for the same pump wavelength and intensity, but with —10 V
DC bias applied to the sample. The AC Stark effect is reduced
in the presence of a —iO~V cm~ DC bias field perpendicular to the quantum wells.
our experiment the decrease of the electron—hole
overlap, and consequently of the excitonic transition matrix element, with F dominates.
We have also explored another regime not yet
investigated experimentally nor theoretically.
When the pump intensity and detuning are significantly increased, a completely new regime is found
for both the DTS profile and its dynamics. For
example, fig. 5 shows a typical DTS at delay2
zXt
0, for detuning
a high pump
and large
E intensity I~ 3 GW cm~”
1~
80 meV. The is
=
—
—
optoelectronics [47,48]. Investigations of some of
these effects have been initiated recently in heterostructures biased parallel to the plane of the QWs
[49], but they will not be discussed here. An
applied electrostatic field, F iO~ V cm~, redshifts the heavy-hole exciton resonance by
6
(a)
U)
ent detuning
which
andwith
amplitude
intensityofas
[32—36].This
each
effect
corncan
be increases
understood
intuitively
due Founer
to the differponent of the pump pulse. We have not tried to
make a detailed comparison with the published
numerical calculations, which were performed for
simplified exciton and pump pulse lineshapes. Our
results are, however, in qualitative agreement with
theory.
As discussed in sect. 3, theory predicts that in
leading order the is exciton AC Stark shift varies
2/(E
roughly like ~E1~~ 2 I
I
1~ hw~). The
12
.~.
—
08
z
0
—
—
meV in our sample. In fig. 4 we show the DTS
obtained at moderate pump intensity and for
50
z
<
—
meV
with
—10
biasfind
voltage
at
zero
applied
bias,
without
the p-i-n
and
sample
[14].
We
that
the
applied
DC
field
reducespump
the Vdetuning
AC
Stark
effect,
even to
though
the
detuning has significantly decreased due to the
QCSE. This is in contrast with the theoretical
prediction of Hiroshima et al. [50], who predicted
an increase of
I with DC field. Obviously, in
z
IL
w
IL
0
J~L~
_____________________________
020
1.57
1.59
1.61
1.63
1.65
ENERGY (eV)
Fig. 5. Differential transmission spectrum at
= 0 for
2 delay
pumpz.~t
intensity.
— 80 meV pump detuning and — 3 GW cm
D.S. Chem!a eta!.
/ Excitonic optical Stark effect in semiconductor QWs
HH
Z
(a)
(b) ~t+264
~
LH
0
237
DTS around the is heavy-hole exciton resonance
at delay L~t 264 fs. The AC Stark effect does not
=
contribute to the signal at this delay, since the
pump pulse is only 100 fs long. The DTS profile is
remarkably symmetric, in contrast to the DTS
profile at delay t 0, shown in fig. 5. Curve (b)
shows the integrated DTS around the is heavy-hole
exciton resonance at delay i.~t 264 fs. The DTS
integrated over the exciton line is nearly zero,
which corresponds to a pure broadening without
any loss of oscillator strength. This lineshape is
reminiscent
of that
produced by
screening/collisional
broadening
in dynamical
an electron—hole
rilasma [18 44
~
=
=
ENERGY (eV)
1.58
1.60
1.62
~
—
Fig. 6. Time course of differential transmission spectrum for
2 pump intensity.
— 80 meV pump detumng and — 3 GW cm
Signals at the heavy-hole (HH) and light-hole (LH) exciton are
simultaneously observed. The non-zero signal at late times
(i.St> 264 fs) is due to two-photon generation of an
electron—hole plasma. Inset: curve (a) shows the differential
transmission spectrum around the HH exciton and curve (b) its
integral, demonstrating the area conserving broadening of the
2D exciton by the 3D electron—hole plasma.
heavy-hole exciton resonance mainly displays
bleaching, while the is light-hole exciton resonance displays both bleaching and shift. This
high-intensity regime is clearly more complicated,
This is further demonstrated by the observation
that the DTS does not recover at times long cornpared to the duration of the pump pulse. This is
shown in fig. 6, where the full time course of the
1~’
L
It has been shown that, due to a large contnbution from nonresonant processes, the two-photon
absorption (TPA) cross section is nearly independent of photon energy in the energy range ~ E~<
h~<Eg [53,54]. Since we are exciting just below
the gap, it is thus possible to generate an
electron—hole plasma by TPA at energies close to
2Eg. Initially, such a plasma will not occupy states
at the band edge, because the relaxation of carriers by emission of phonons takes many ps [55,56].
Nevertheless, it would be able to produce immediately some form of nonequilibrium collisional
broadening before relaxing to the band edges. To
demonstrate the presence and origin of the plasma,
we studied the photocurrent in the p-i-n diode
with a reverse bias of 10 V at E
15 ~
80 meV.
As shown in fig. 7, we find indeed a photocurrent
with a quadratic dependence on the pump intensity, I~,which unambiguously proves the presence
of real carriers in the sample [14]. We estimate,
using the known TPA coefficient
of bulk
GaAs
2 the
number
of
[53,54], that
I~, would
3 GWproduce
cm a photocurrent
generated
e—hatpairs
of
2 ptA, which is about iO times higher than
we measure. There are many explanations for this
discrepancy. The uncertainty in the TPA cross
section is rather large, and the collection of carriers may be inefficient in our unoptimized sample
and difficult to evaluate, since the photoconduction quantum efficiency in perpendicular low-ternperature QW transport is not yet well understood.
Our estimates of the expected TPA photocurrent
are, however, in reasonable agreement with our
measured values.
—
DTS is presented [i4]. The plot is obtained by
overlaying a grid on top of ten measured DTS. At
negative delays ~ t < 0, i.e. when the probe precedes the pump, an oscillating DTS is seen, in
agreement with theory and other measurements on
molecules
[51] and~ semiconductors
[il—13,52]
(see
sect.
3). Around
t = 0 a rapid transient
with
a
duration close that of the pump-probe cross correlation is observed, both at the heavy-hole and
light-hole excitons. The profile of this initial transient corresponds to a combination of both shift
and bleaching, as discussed earlier. The DTS,
however, does not recover after the pump pulse is
over, but stabilizes to a very long-lived component
(many ps), the magnitude of which is only about
two times smaller than that of the maximum at
z~
t
0. The lineshape of the DTS evolves as well,
as shown in the inset of fig. 6. Curve (a) shows the
=
—
—
238
/ Excitonic optical Stark effect in semiconductor Q Ws
D.S. Chemla et a!.
1000
700
500
—
—
—
one-photon processes, and (2) the real 3D plasma
—
—
generated by two-photon processes. This explains
______
•
/
the
lineshape
and dynamics
of the
DTSvery
for complex
which no
quantitative
description
is
available at the present time.
300
~200
———-—-—l
—
/
3. Theoretical studies
(
100
0
—
70
““
—
50
—
—
~
30
—
SLOPE
—
The excitonic AC Stark effect is a pure nonequilibrium effect and, consequently, its correct
theoretical requires nonequilibrium Green’s function techniques [20,29,32,33], as introduced by
...
2
-.—.-
2C
1
235710
2)
INTENSITY (GW/cm
Fig. 7. Photocurrent measured with 10 V reverse bias applied
on the p-i-n quantum well sample, showing a quadratic dependence on the pump intensity.
The effects of the nonequilibrium electron—hole
plasma on the excitomc absorption change qualitatively with time [i8,57]. Initially, as mentioned
above, the carriers excited by TPA have excess
energy 2hca~ Eg i.5 eV, far larger than the
confinement potential in our QW
200 meV).
Therefore, right after their generation, they are
free to move in 3D and able to screen the 2D
excitons effectively. This prevents the full recovery
of the excitonic absorption, shown in fig. 6. At
early times, they do not occupy states in the QW,
and it will take a few ps for them to relax down to
the band edge [55,56]. They will then become
confined and their screening of the 2D excitons
will qualitatively change. Eventually they will occupy the states out of which the 2D excitons are
constructed and bleach the exciton lie in such a
way that the integrated DTS will no longer be zero
[18,57,58]. Finally, they will recombine in a few
ns, and the excitonic absorption will fully recover.
The spectra shown in fig. 6 contain the combined
effects of (1) the virtual 2D excitons generated by
—
—
(—
Keldysh [59] and Kadanoff and Baym [60]. However, for nonresonant excitation of semiconductors, a considerable simplification arises, if the
detuning and thus the Rabi frequency are larger
than any other energy scale in the problem (other
than the exciton binding energy), in particular de
dephasing/collision rate. It is then possible to
neglect collisions altogether and derive a closed,
collisionless kinetic equation for the equal-time
density matrix. This equation, based on the
Hartree—Fock (HF) approximation, was first discussed by Schmitt-Rink et al. [18—20] and has
been rederived many times [26,29,32,33,37,40].
Later work considered in addition phenomenological [23,24,36,42] or Golden Rule [26,40] collision terms. In this context, it should be noted,
however, that the concept of a local (in time)
collision term becomes meaningless on short time
scales comparable to the collision duration time,
where the Markov assumption breaks down and
retardation effects become important. This is a
well-known problem in “ballistic” transient transport in small semiconductor structures [61]. If
collisions are to be included, there is only one
correct way of doing it, the nonequilibrium Green’s
function formalism.
Below, we will discuss the HF theory of the
excitonic AC Stark effect in pure two (2D) and
three (3D) dimensions. For reasons that will become clear, there is as yet no quantitative theory
for real QWs. Such a theory would involve infinite
sums over the complete set of QW states. Whether
QWs behave like 2D or 3D systems depends then
on the ratio of the detuning or Rabi frequency to
the electron—hole (e—h) zero-point energy due to
.
.
D.S. Chemla eta!.
/ Excitonic optical Stark effect
in semiconductor QWs
239
confinement [48]. Also, part from a second order
perturbation calculation [27], no theory has yet
been developed which, for small detuning or Rabi
frequency, comparable to the biexciton binding
energy or optical phonon energy, links the HF
theory to the respective theories of the biexcitonic
AC Stark effect [3] or “phonoritons” [16,62,63].
The HF theory treats on equal footing the
effects of the laser field and the Coulomb interaction. For a simple two-band model of a semiconductor and within the rotating wave approximation, the equal-time density matrix satisfies the
Liouville equation (h = i) [18—20]
renormalizes the individual e and h energies (diagonal elements in eq. (3)),
~ Vkk’n,k’(t),
i c, v.
(i)
but also a significant internal one, the “molecular”
field associated with e—h pairs created at k’. At
(2)
each k, external and Coulomb fields combine to
give an effective self-consistent “local” field to
which the system responds. The problem is thus
similar to that of a paramagnet in an external
magnetic field.
In order to describe pump and probe experi-
iaflk(t)
=
[~~(t),
hk(t)],
where
=
mnCk(t)
[~(t)
and
(k(t)
=
t)j
sPk(t)
n vk (
1
[
_~E(t)]
~°vk
~ck
_~*E*(t)
—
v~k~n,1~(~).
~
=
(3)
Eg/
and
~vk
—Eg/2—k2/(2mh)+
—
=
and (2) the e—h Coulomb attraction renormalizes
the Rabi frequency at resonance (off-diagonal elements in eq. (3))
~E(t)
~E(l) + ~
—+
Both changes express the fact that in the presence
of Coulomb interaction an e—h pair with a given k
does not only experience the external field, E(l),
ments, pump
we split
the gives
laser rise
field
twotest
parts,
strong
field,
E(t),
and to
ainto
weak
field,a
6E(t).
The
former
a renormalized
k’ k 2/(2 me)
2+
ck
~
semiconductor “ground state”, described by eqs.
(i)—(3),thewhile
the linearrenormalized
response to“excitation
the latter
yields
corresponding
spectrum”, which exhibits the AC Stark effect.
Linearizing eq. (1) with respect to ÔE(t), we find
[18—20]
~.Vk,k’
i~6nk(t)
are the conduction and valence electron energies,
n ck and n vk are the corresponding nonequilibrium
distributions. ~.t is the interband dipole matrix
element; Vkk’ is the Coulomb interaction,
(2i~e2)/(oIk—k’I) in 2D and (4ire2)/(~oIk—
k’ 12) in 3D. Finally,
is the e—h pair amplitude induced by the laser field, E(t), related to
the total induced polarization through
=
[~~(t),
flk(t)]
+
kk(t),
Mk(t)1,
(5)
where
0
_~t*~E*(l)
&k(t)j
s/Ik(t)
—
~
~“k,k’~k’@)’
~t~E(t)1
0
j
(6)
k’
P ( t)
=
2if1’ ~
4~k(
)‘
(4)
The total polarization induced by the test field is
k
Were it not for the second term in eq. (3), eqs.
(i)—(4) would be identical to the Bloch equations
for an inhomogeneously broadened two-level atom.
The Coulomb interaction modifies this picture in
two ways: (i) the e—e and h—h Coulomb repulsion
i~P(t)
=
2!s*~6%pk(t).
(7)
k
In typical experiments, after passing the sample,
the test beam is sent through a monochromator
and registered by a time-integrating detector. In
240
D.S. Chem!a et al.
/ Excitonic optical Stark effect in semiconductor QWs
this situation, we may define an effective optical
I
(8)
)
where only that part of ~P(W) which propagates
in the test beam direction is to be considered.
At low pump intensities, eqs. (i) and (5) reduce
to the Ginzburg—Landau-like equations [i8—20]
/ /
~
I
~Yk.k’~k’(t)
\
..
‘
I
~1t~
~Eg~
\
__.
~
I.
I
I
~
~~
2D
_[i_2I~k(t)I2j~E(t)
=
+2~Vkk’[
-
IlPk(t) 125]Ik’(t)
•
I~k’(t)I 2~(~)]
jj[
(9)
4i3
-2
-1
0
and
0
—
Eg
k2
—
~
~s~k(t)
m
I
=
+ ~
Fig. 8. Absorption spectra for a pump detuning of Eg —
= 10
E
0 and increasing pump intensities, IE~I2 The full lines show
the unperturbed absorption (after refs. [30,31]).
J’~.~’(t)
~iPk’(t)
k’
_[i_2I~k(t)I2}~E(t)
+2s(4(t),.tE(t)~Pk(t)
2&iIk’(t)
+2~ Vkk’[ I ‘Pk(t) 1
—
For a monochromatic pump field, E(t)=
E~exp(
w,~,t),the
fullnumerically
kinetic equations
(5)
have ibeen
solved
by Ell(1)et and
al.
[30,3i] and Schaefer et al. [32,33], and their results
—
I ~k ~‘t~’ I 2~pk\(t\1~
are shown
in fig.
8. The full
lines
give
unperturbed
exciton
absorption
in 2D
and
3D,the
while
the
I
+2~ Vk
dashed and dotted lines correspond to increasing
pump intensities, I E
1, I 2 for a pump detuning of
k’[~Pk(t)tPk’(t)~Pk(t)
/
Eg
iOE0,
E0 is the
3D isexciton
berg. w1,
Strong
ACwhere
Stark shifts
of the
excitonRydare
where m is the reduced e—h mass, m~ m~ +
m ~ Without the nonlinear corrections, eqs. (9)
and (10) correspond to inhomogeneous Wannier
equations, driven by the laser fields, E(t) and
6E(t), respectively. In this limit, they describe the
linear response of an unperturbed exciton. The
nonlinear corrections in eq. (9) describe the effects
on the exciton of the Pauli exclusion principle,
through phase space filling/exciton—photon interaction (first term on the rhs of eq. (9)) and exciton—exciton exchange interaction (second term on
the rhs of eq. (9)). The additional nonlinear corrections in eq. (10) are due to exciton exchange
between pump and probe beam.
found, like those of a two-level atom. However, in
striking contrast to two-level atoms and in agreement with our low-intensity data, the is exciton
oscillator strength is nearly constant in 2D, while
in 3D it increases slightly with pump intensity.
(The absorption spectrum integrated over all Irequencies decreases, of course.)
In order to understand these theoretical results,
it is useful to consider the limit of low pump
intensities, eqs. (9) and (iO), which contains already much of the important physics. This limit
has been studied in detail by Zimmermann [22,23],
Zimmermann and Hartmann [24], and Ell et al.
[30,311, who evaluated the corresponding renor-
—
i * (t~
It\ô
~
‘i’k ~ / Vk\ / ‘t’k ‘~ ~
‘
=
~
—
=
/ Excitonic optical Stark effect in semiconductor QWs
D.S. Chemla et aL
derived formally by Schmitt-Rink et al. [i8—20]
and rederived by Combescot and Combescot [27],
using the Coulomb Green’s function. Parametrizing the results in terms of those of a two-level
malized exciton energies and oscillator strengths,
atom with level spacing E15, one finds for the
12 energy
change 2j~E
in exciton
p
~ E1~—w~
(ii)
~,
and for the relative change in exciton oscillator
strength
2
2~.tE~~
(i2)
fn
(Ei~—cap)
—
241
.
I
68 _
I
I
3D—’
1
/I
~nd9I
b
2D~~
2
=
—
—
—
I
I
=
—
—
—
—
—
—
~15
a,~and p, are functions of the pump detuning,
E
1~ W~, and are displayed in figs. 9—u [22—24].
The dashed lines show the contribution of phase
space filling/exciton—photon interactions, while
the full lines include the contributions of
exciton—exciton interactions as well. For large detunings, ; and p~ are dominated by the
exciton—photon interaction and approach unity,
while for small detunings the exciton—exciton in—
8
Is exciton shift
I
I
I
I
_I~
I
I
I
I
—~
I
I
I
I
0
Fig. 10. AC Stark shift of2/(E
the band gap, normalized to that of a
two-level atom, 2 jsE~I
1~— ~
versus pump detuning,
E1~— w.~,.Full lines: exciton—photon and exciton—exciton interaction; dashed lines: exciton—photon interaction only.
teraction dominates a~and the resonant HF values of o,~and p~are recovered [64]. Since for small
detunings there exist other exciton—exciton contributions, of the same order in the pump intensity
should be viewed with caution. (For systems, in
which no is exciton shift is observed under resonant
it is thus interaction
not unreasonable
to
neglectexcitation
the exciton—exciton
altogether.
-
This point of view was first adopted by Schmitt-~IIii
/
—
—
-15
—
—
I
—
I
I
-10
I
I
I
I
I
I
I
—
2D
—
—
I
I
.,~
(w~-Ei~)fEoFig. 9. AC Stark shift of 2/(E
the is exciton, normalized to that of a
two-level atom, 21 ,.iE~I
5, — ~,
versus pump detumng,
E1~— o.~,. Full lines: exciton—photon and exciton—exciton interaction; dashed lines: exciton—photon interaction only.
but neglected
Rink
et
al. [i8—20]
here,comparison
and
the later
corresponding
by
Lindberg
results
and
Koch
[40,42]).
The
of figs.
9 and
iO
shows that the AC Stark shift of the band gap is
always larger than that of the is exciton. The
origin of this behavior is the larger spatial extent
of scattering states, which makes exclusion principle (“overlap”) effects more effective. This increased AC Stark shift of the band gap corresponds to an effective increase in is exciton binding energy and thus in oscillator strength, which
can overcome the decrease due to phase space
filling and which is responsible for the behavior
shown in figs. 8 and ii. In the case of 2D excitons
the two effects cancel almost exactly over a large
range of detunings. This explains why only a shift
242
/
D.5. Chemla et a!.
I
I
3-
I
I
I
I
I
I
I
I
I
Excitonic optical Stark effect in semiconductor Q Ws
I
I
-
Is exciton bleaching
2-
1
—
-
—
—
—
—
I
-
I
0
/
much larger than the inverse time, E0, it takes to
form an exciton, so that Coulomb effects become
unimportant [22_24,27,30,3i]. By the same token,
at very large detunings, every solid behaves like
isolated atoms; and coherence, band or quantum
confinement effects play no role in the nonresonant nonlinear optical response [65]. In the case of
/
-I
-
...—
—
-
—
—
-2
These results show that, contrary to claims in
the literature, the excitonic AC Stark effect can in
general not be understood in terms of a two-level
or free e—h pair model, except for huge detunings,
exceeding the exciton binding energy by more
than
an order
of magnitude.
the latter limit,
oscillate
between
valence andInconduction
band,the
is
Rabi frequency, E5_w~, with which electrons
-
I
—
-15
I
I
—
I
—
—
I
I
-10
I
-
I
I
-5
(Wp
I
I
I
0
E15 ) /E0—~
Fig. 11. Relative change in oscillator strength of the is exciton,
normalized to that of a two-level atom, —2 I ~ I 2/( E1~—
versus pump detuning, E1~— w1,. Full lines: exciton—
photon and exciton—exciton interaction; dashed lines: exciton—photon interaction only.
QWs, we therefore expect qualitative changes when
the detuning exceeds the e—h zero-point energy
due to confinement by some unknown numerical
factor [48].
6C
of the is exciton is seen experimentally. It should
be noted, however, that a larger shift of the gap
~
uJ
JU,
-2
cannot
There
be
(2s,
masked.
3s,
be
many
identified
First,
form
reasons
the
a quasi-continuum
inexcited
the
why
DTS
suchspectra
states
anof
effect
that
theofmaking
exciton
merges
fig.
could
2.
into
theare
real
continuum
of
unbound
states
20-4-
...)
the exact position of the gap difficult to assign.
Each of these excited bound states as well as the
gap experience shift and eventually some loss of
oscillator strength occurs. The excitation with ulbutions by introducing the broadening discussed
earlier. Altogether
thismixes
produces
a verycontricomtrashort
pulses further
these various
distinct features, as shown numerically by Schaefer
plicated change in absorption close to E5, without
et al. [32,33]. If in addition one considers that in
real QWs all the lines are inhomogeneously broadened (in fact, in real QWs the 2s and higher bound
states are almost never resolved), it is not surprising that the gap shift is not directly seen. The
important point is that experimentally the is exciton shifts but does not lose oscillator strength, in
agreement with theory.
-
0
2
4
30C
t
200
0
I
.100
c
I
-
1
0
1
(weFig. 12. AC Stark shift of the is exciton, normalized to twice
the Rabi frequency at resonance, 21 ~sE~ 2, versus pump detuning, Ei~— i~. Even for negative detumngs the shift is
positive due to the dominance of exciton—exciton interactions.
Arrows indicate the three lowest exciton states.
D.S. Chemla et a!.
/ Excitonic optical Stark effect in semiconductor QWs
Finally, it is amusing to note that, if one takes
the HF results literally, a blue shift of the is
exciton occurs even for negative pump detunings,
as shown in fig. i2, where the absolute is exciton
AC Stark shift, 6is a1~/(E15 wy), is displayed.
=
—
For a two-level atom, one would expect a red
shift. The origin of this behavior is simply the fact
the close to resonance exciton—exciton interactions dominate. Within HF, these are always repulsive.
The polarization dependence of the excitonic
AC Stark effect (for light propagating in the QW
growth direction) has been discussed by Joffre et
al. [iO] and Combescot [28]. In agreement with
previous measurements on QWs, they do not find
any pronounced dependence of the is exciton
shifts on the linear polarizations of pump and
probe beams, but do see one for circularly
polarized light. In bulk GaAs, unlike in QWs, the
is exciton shift is found to depend on the linear
polarizations [10].They analyze their data starting
from a (2 + 6)-level model, describing the conduclion and valence bands of GaAs at the F point,
and employed earlier in studies of optical orientation [66].The dependence of the problem on the
ratio of QW zero-point energy to pump detuning
is neglected, and the reader may decide for
him/herself whether such a description is appropriate.
The dynamics of the AC Stark effect has been
studied for two-level atoms by Brito Cruz et al.
[25], Lindberg and Koch [41], and Balslev and
Stahl [34,35]; and for excitons by Schaefer et al.
[32,33], Balslev et al. [36], Zimmermann [23], Zimmermann and Hartmann [24], and Lindberg and
Koch [42] (the latter authors neglected exciton—exciton interactions), who integrated the kinetic
equations for ultrashort pump and probe pulses.
As pointed out first by Schaefer et al. [32,33] and
Balslev and Stahl [34,35], the large spectral width
of an ultrashort pump pulse leads for example to
an “inhomogeneous broadening” of the absorption spectrum, which increases with pump intensity and which is the origin of part of the is
exciton bleaching seen at higher intensities,
Equally important for the analysis of experiments
is the actual temporal evolution of the AC Stark
effect. As mentioned above and shown, e.g., in fig.
243
6, both in molecules [Si] and semiconductors
[11—i4,52], an oscillating DTS is seen when the
probe precedes the pump. In order to understand
this result, it is again sufficient to consider the
limit of low pump intensities, eqs. (9) and (iO),
and the origin of the various nonlinear terms in
these equations. The exciton Hartree terms (nonlinear corrections in eq. (9)) are due to the population induced by the pump, while the exciton Fock
terms (additional nonlinear corrections in eq. (10))
are due to the population modulation induced by
both the pump and the probe [23—25,41].The
former dominate at positive time delays, and the
latter at negative time delays. This is illustrated in
figs. 13(a, b), which show time-resolved DTS of a
two-level atom excited (a) 8/T2 below resonance
_—__.
~
120 fs
~—‘----—--
80 ts
40 fs
—~
~
~
-
~ofs
so ts
-
120 is
-
i~o
fs
~
—‘\~..._--—-—
—‘N----—.~~-~----—
I
I
I
-15
-10
-5
I
I
0
1w
-
w~)
5
-200fs
.~~_~-___
10
15
T2
•
~
iso ts
120 fs
__________
___________
_________
~
40fs
_________
____________
-80 Is
___________
—j\~..----—~A~-—-I
-is
I
-10
~
-5
0
-120 fs
-
I
I
5
10
160 fs
-200 Is
15
1w - Sip) ‘
Fig. 13. Time-resolved differential transmission spectra for a
two-level atom excited (a) 8/T2 below resonance, and (b) on
resonance. The longitudinal and transverse relaxation times
used in the calculation were T = i40 fs and T = 70 fs. Pump
1
2
and probe pulses were assumed to have hyperbolic secant
squared intensity profiles with durations of 50 and 6 fs, respectively (after ref. [25]).
244
D. S. Chem!a et a!.
/ Excitonic optica! Stark effect in semiconductor Q Ws
and (b) on resonance [25]. The longitudinal and
transverse relaxation times used in the calculation
were T1 140 fs and T2 70 fs, and pump and
=
=
probe pulses were assumed to have hyperbolic
secant squared intensity profiles with durations of
50 and 6 fs, respectively. For positive time delays,
longer than about half the pump pulse duration,
both figs. i3(a) and (b) show symmetric DTS,
decaying with the longitudinal relaxation time T5
and characteristic of absorption saturation due to
the pump-induced population. For negative time
delays, i.e. when the probe precedes the pump, fig.
13(a) shows an oscillatory behavior of the DTS.
With decreasing time delay, this oscillatory DTS
evolves continuously into an asymmetric feature,
characteristic of the AC Stark effect. Oscillations
in the DTS are also seen under resonant excitation, fig. 13b, and in both cases the rise time of the
DTS is basically given by the transverse relaxation
time, T2, while the period of the oscillations is
related to the inverse time delay between pump
and probe. This identifies these precursor effects
as being due to the perturbed free induction decay
of the probe-induced polarization [25,41].As noted
by several authors, precursor effects in the DTS
are expected whenever a dipole is perturbed on a
time scale faster than T2, in dependent of the
nature of the perturbation [11—13,23—25,36,39,4i,
42,5i,52]. For example, we have shown recently
that similar effects occur in excitonic electroabsorption sampling [67].
4. Discussion
In this article we have reviewed the present
status of our expenmental and theoretical investigations of the AC Stark effect in semiconductor
QWs. There are still a number of issues which
have not been solved to date. Experiments are
performed on real QW samples, with all the imperfections that this implies, using time-varying
pump and probe pulses from real fs light sources,
whereas theory has been developed for pure 2D
and 3D systems excited by monochromatic or
perfect transform limited pulses. Even the linear
absorption spectrum of QWs is not well described
by the present theories. For example, the height of
the exciton peaks relative to that of the continuum
is greatly theoretically overestimated, and furthermore for real QWs a good description of excitonic
lineshapes including inhomogeneous broadening
has not yet been developed. Experimentally, no
shift of the continuum edge is observed and a pure
shift of the exciton peak is only seen at very low
intensities. At high intensities we have found that
TPA becomes important and changes the exciton
lineshape. Theoretically, correct inclusion of dephasing processes is yet to be made. For example,
we know from two-level model calculations that
dephasing can have a very strong effect on the
differential absorption lineshape. Under these circumstances a direct quantitative comparison of
experimental data and theoretical results is very
difficult at present.
What have we learned during this short period
of time following the initial observation of the
excitonic AC Stark effect? A lot, in the sense that
we known what it is about, and what makes it
different from the AC Stark effect in atomic
vapors. However, the outstanding question is exactly the same as in “ballistic” transport in small
semiconductor structures, namely the transition
from the coherent to the incoherent regime [6i].
The limiting cases are established, and we have to
join them now.
Note added: After completion of this manuscnpt we learned that Balslev and Hanamura have
recently attempted to extend the Hartree—Fock
theory of the excitonic AC Stark effect to account
in addition for bound bi-exciton formation close
to resonance [68]. As for the latter, their treatment
is similar to earlier work by Ivanov and Panahchenko [69].
.
.
.
.
.
.
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