Quantum confinement and ultrafast dephasing dynamics in InP nanocrystals

Article abstract of DOI:10.1103/PhysRevB.55.7059

The electronic level structure and dephasing dynamics of InP nanocrystals in the strong quantum-confinement regime are studied by two complementary techniques: nanosecond hole burning and the femto-second three-pulse photon echo. Hole burning yields the homogeneous electronic level structure while the photon echo allows the extraction of the linewidth of the band-gap transition. The congestion of electronic-levels observed close to the band-edge transition in the hole-burning experiments gives rise to a pulse-width-limited initial decay in the photon-echo signal. The level structure is calculated and assigned using a model which includes valence-band mixing. The homogeneous linewidth of the band-edge transition is approximately 5 meV at 20 K and is broadened considerably at higher temperatures. The temperature dependence of the linewidth is consistent with an intrinsic dephasing mechanism of coupling to low-frequency acoustic modes mediated by the deformation potential. Quantum-confinement effects in III-V semiconductor InP are compared to those of the prototypical CdSe II-VI semiconductor nanocrystal system.

Full text of DOI:10.1103/PhysRevB.55.7059

PHYSICAL REVIEW B
VOLUME 55, NUMBER 11
15 MARCH 1997-I
Quantum confinement and ultrafast dephasing dynamics in InP nanocrystals
U. Banin, G. Cerullo, A. A. Guzelian, C. J. Bardeen, and A. P. Alivisatos
Department of Chemistry, University of California, Berkeley, California 94720
and Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720
C. V. Shank
Department of Chemistry, University of California, Berkeley, California 94720;
Department of Physics, University of California, Berkeley, California 94720;
and Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720
͑
Received 26 June 1996; revised manuscript received 4 November 1996͒
The electronic level structure and dephasing dynamics of InP nanocrystals in the strong quantum-
confinement regime are studied by two complementary techniques: nanosecond hole burning and the femto-
second three-pulse photon echo. Hole burning yields the homogeneous electronic level structure while the
photon echo allows the extraction of the linewidth of the band-gap transition. The congestion of electronic
levels observed close to the band-edge transition in the hole-burning experiments gives rise to a pulse-width-
limited initial decay in the photon-echo signal. The level structure is calculated and assigned using a model
which includes valence-band mixing. The homogeneous linewidth of the band-edge transition is approximately
5
meV at 20 K and is broadened considerably at higher temperatures. The temperature dependence of the
linewidth is consistent with an intrinsic dephasing mechanism of coupling to low-frequency acoustic modes
mediated by the deformation potential. Quantum-confinement effects in III-V semiconductor InP are compared
to those of the prototypical CdSe II-VI semiconductor nanocrystal system. ͓S0163-1829͑97͒02111-5͔
the vibrations of the nanocrystals ͑optical, acoustic͒.9,10 The
electronic structure in such direct gap semiconductor nanoc-
rystals, which was investigated using size selective optical
techniques in the frequency domain, exhibits a complex set
I. INTRODUCTION
In a solid, the bands of electronic energy levels are con-
tinuous, and the electronic spectra are intrinsically diffuse. In
the atomic limit, the density of electronic states is discrete
and few broadening mechanisms exist, resulting in narrow
electronic transitions ͑1 kHz–100 MHz͒. In semiconductor
nanocrystals, comprised of tens to tens of thousands of at-
oms, the question of how discrete electronic transitions de-
velop and how the linewidths vary as a function of size is of
both theoretical and practical importance.1 In a simple
model of quantum confinement, the localization of the opti-
cally generated electron-hole pair in a semiconductor nano-
crystal arises from a superposition of bulk k states. The prin-
cipal effects are a shift of the band gap to higher energy in
smaller sizes and the development of a discrete electronic
structure. Localization is thus accompanied by the concen-
tration of oscillator strength into just a few transitions. Given
the large polarizability of nanocrystals, substantial enhance-
of electronic transitions which arises from the mixing of
multiple valence bands in finite size.1
1,12
The successful
spectroscopic identification of these transitions in CdSe is a
major accomplishment of quantum confinement theory.12
Recent advances in synthesis now make it possible to ob-
tain size selected, crystalline, monodisperse nanocrystal
,2
13–15
samples of the III-V semiconductor InP.
In this study we
present measurements of the homogeneous electronic level
structure and linewidths of these samples and compare them
to those of the prototypical II-VI nanocrystal system CdSe.
Compared to the II-VI semiconductor, the bonds in InP are
more covalent, so that the coupling of optical excitations to
the vibrations is expected to change in predictable ways. The
effective masses, the spin-orbit splitting, the deformation-
potential coupling, and other material parameters are signifi-
cantly different in InP. This leads to predictable changes in
linewidths and electronic structure, allowing us to separate
the influence of extrinsic, sample-dependent broadening
mechanisms and properties from intrinsic ones, and to test
models of confinement in quantum dots.
2
,3
ment of the nonlinear susceptibilities is predicted.
Semiconductor nanocrystals exhibit the expected shift of
the optical spectra to a higher energy, as well as the devel-
opment of discrete features in the spectra; however, in
nanocrystals of CdSe, which have been investigated exten-
sively due to the high sample quality and narrow size
Nanocrystal samples typically exhibit inherent inhomoge-
neous broadening primarily due to size variations in the en-
semble. In addition, a congestion of electronic transitions is
possible, as there may be several electronic transitions lying
close to the band edge. We use two complementary tech-
niques to study the homogeneous spectroscopy of InP nano-
crystals. In the frequency domain, transient ns hole burning
͑HB͒ is used to investigate the homogeneous electronic level
structure.6 In the time domain we use the femtosecond
4
,5
distributions, homogeneous absorption linewidths of the
band-gap transition of several meV have been observed in
6
,7
hole burning
measurements.
and in femtosecond photon-echo
In addition the linewidth considerably
broadens as the size is reduced. This broadening may be due
to extrinsic or intrinsic effects. Extrinsic effects would be
ultrafast scattering of charges at defects ͑surface or interior͒.
Intrinsic mechanisms of line broadening arise by coupling to
8
,9
,12
0
163-1829/97/55͑11͒/7059͑9͒/$10.00
55
7059
© 1997 The American Physical Society

7060
U. BANIN et al.
55
three pulse photon echo ͑3PPE͒, which provides a direct
probe of electronic and vibrational dephasing times.8 Both
techniques have been used previously in separate studies to
investigate the electronic level structure and linewidth issues
in CdSe nanocrystals, however this is the first time, to our
knowledge, that simultaneous measurements on the same
samples are conducted and used in conjunction to gain a
self-consistent picture of the dephasing dynamics in nano-
crystals from the fs to the ns timescale.
The paper is organized as follows: In Sec. II experimental
details of the sample preparation, the hole-burning setup, and
the three-pulse photon-echo experiment are given. In Sec.
III, results and discussion for the hole-burning experiment
are presented followed by results of the femtosecond three-
pulse photon-echo measurements. Both experiments are in-
terpreted using a three-level model for the near band-edge
absorption in the nanocrystals. A theoretical assignment of
the observed levels is presented and finally the dephasing
mechanisms are investigated by conducting a temperature
dependence of the echo decay rate. Conclusions are pre-
sented in Sec. IV.
tion ͑5-ns gating time͒ allowed suppression of the unwanted
contribution of pump scattering to the HB signal. The hole-
burning signal, corrected for residual pump scatter and
sample fluorescence, was obtained by using a sequence con-
trolled by shutters on both pump and probe pulses from the
following calculation:
,9
IwpϪIp
Ϫ⌬ODϭ
,
͑1͒
Inp
where Iwp is probe signal going through the sample with
pump, I is the contribution of the pump alone, and I is
p
np
probe signal going through the sample without pump.
C. Femtosecond three-pulse photon echo
The femtosecond laser system used for the three-pulse
photon-echo experiment has been described in detail
previously.16 Briefly, 60-fs pulses generated by a colliding
pulse mode-locked dye laser are amplified in a dye amplifier
pumped at 8-kHz by a copper vapor laser ͑CVL͒. A broad-
band continuum is generated by subsequently coupling the
pulses into an optical fiber. The spectrally broadened pulses
are then amplified by a broadband dye amplifier pumped by
a second CVL and compressed using a sequence of gratings
and prisms.
II. EXPERIMENT
A. Sample preparation
InP samples have been synthesized using the dehalosyla-
tion reaction of InCl and P„Si(CH ) … in the coordinating
3
3 3 3
solvent trioctylphospine oxide at temperatures between
For the photon-echo experiments on InP nanocrystals we
use a selected portion of the amplified pulse spectrum to
match the sample absorption onset ͓see inset of Fig. 3͑b͔͒.
The duration of the resulting pulses, which are close to being
transform limited, is 20 fs. The nanocrystal sample was
cooled by a closed cycle He cold finger cryostat and the
temperature was monitored by two diodes at the cold finger
and on the opposite side of the sample holder. The integrated
echo signal was detected in the phase matched direction
(Ϫk ϩk ϩk ) and was measured with a photomultiplier
1
4
2
40 °C and 270 °C as reported by Guzelian et al. The sol-
vent provides high-temperature medium for annealing the
nanocrystals and also acts to passivate the particle surface
limiting growth and imparting solubility to individual par-
ticles in a variety of organic solvents. The nanocrystals pro-
duced are discrete particles ranging in size between 20 and
60 Å in diameter. Following the initial reaction which pro-
duces a wide range of particle sizes, distinct distributions are
obtained by using size selective precipitation techniques.
Structural characterization shows that the particles are highly
crystalline with most consisting of a single, faultless crystal-
line domain and are spherical to slightly elliptical in shape.
Size distributions are approximately Ϯ20%. For the low-
temperature spectroscopic investigation, the particles were
embedded in a polymer film ͑poly-vinyl-butyral͒ with thick-
ness of less than 500 ␮m and typical optical density of 0.4–
1
2
3
tube using lock-in detection by chopping one of the beams.
All measurements were taken in the perturbative regime, as
checked by observing the expected cubic dependence of the
echo signal on excitation intensity.
0.7 at the inhomogeneous excitonic shoulder.
B. Transient nanosecond hole burning
A dye laser pumped by the second or third harmonic of a
Q-switched Nd-Yag laser served as the excitation source for
the ns HB experiments. A portion of the second or third
harmonic of the Nd-Yag was split off, optically delayed by
20 ns, and used to pump a jet containing a mixture of dyes to
provide a broadband probe source from the dye fluorescence.
Pump and probe were overlapped on the sample which was
placed in a continuous He flow cryostat. Typical pump ener-
gies of 50 ␮J with a spot size of 1 mm in diameter were
used, and the signal was checked to ensure linearity in the
pump intensity. The probe pulse was dispersed in a spec-
trograph, and detected by a gated multichannel detector ͑in-
tensified CCD or intensified diode array͒. The gated detec-
FIG. 1. ͑a͒ The absorption spectra (Tϭ6 K͒ for the series of InP
nanocrystal samples. Arrows designate the position of the nanosec-
ond excitation pulse for the hole-burning spectra presented in ͑b͒.

55
QUANTUM CONFINEMENT AND ULTRAFAST DEPHASING . . .
7061
FIG. 2. Nanosecond hole-burning data for the 34-Å InP sample
Tϭ6 K͒. The absorption spectrum is presented in the top frame
(
and the arrows designate the pump energies for the HB experiments
shown in the bottom frame. The extracted ‘‘single-particle’’ absorp-
tion spectrum ͑solid line͒ and inhomogeneous distribution ͑dashed
line͒ are shown in the top frame. The fits obtained using this model
are shown as dashed lines in the bottom frame.
FIG. 3. Three-pulse photon-echo experiments on the 29-Å InP
sample (Tϭ20 K͒. ͑a͒ Vibrational dynamics measured in the three-
pulse photon-echo configuration. The signal as a function of t13
with t ϭ30 fs is modulated by optical-phonon modes. The inset
III. RESULTS AND DISCUSSION
A. Transient nanosecond hole burning
1
2
The low-temperature absorption spectra for a series of InP
nanocrystals ranging in size from 22 to 41 Å are presented in
Fig. 1͑a͒. Shifts of the absorption edge due to confinement
are evident. The spectra are inhomogeneously broadened pri-
marily due to the size distribution and only shoulderlike ex-
citonic features are observed.
Transient ns HB spectra for the series of samples under
study are presented in Fig. 1͑b͒. Excitation with a narrow-
bandwidth ns laser pulse at the band edge optically selects
the largest nanocrystals in the ensemble. The resulting spec-
tral hole enables the extraction of the ‘‘single-particle’’ ab-
sorption spectra. In all cases, a narrow feature with a size-
dependent width ranging from 10 meV for the largest size to
shows the Fourier transform of the oscillatory component of the
signal. ͑b͒ Electronic dephasing measured in the three-pulse
photon-echo configuration. Echo decays as a function of t12 at three
t13 delays corresponding to 2␲/␻ ͑solid line͒, 3␲/␻ ͑solid line͒,
v
v
and 4␲/␻v ͑dotted line͒ conditions are presented. The effect of
mode suppression is evident. The inset shows the pulse spectrum
͑dashed line͒, along with the low-temperature absorption spectrum.
feature is gradually diminished, appearing as only a small
peak above a broad background for the bluest pump fre-
quency ͓in Fig. 2͑c͔͒. This background reflects the width of
the inhomogeneous absorption. The homogeneous absorp-
tion and position as well as the width of the inhomogeneous
distribution are extracted by fitting the wavelength-
dependent hole spectrum. In the limit of linear response to
pump energy, far from saturation, we use the following
formula:1
2
0 meV for the smallest size is accompanied by a broad
feature at higher energies.
The homogeneous spectra for the band-edge absorption of
these samples can be extracted by conducting an excitation
wavelength dependence of the hole-burning spectrum.6
This is presented in Fig. 2, which shows the evolution of the
spectral hole with excitation wavelength for the 34-Å InP
sample. Upon excitation at the absorption edge, particles are
selectively excited at the narrow-band-gap transition leading
to a spectral hole that closely resembles the ‘‘single-
particle’’ absorption spectrum. Aside from a narrow feature
at the excitation frequency, a weak phonon side band is ob-
served along with a broad feature spaced at 0.1 eV from the
band gap. As the excitation is tuned to the blue, a larger
fraction of the distribution is excited into the higher, broad-
ened excited electronic states. The contribution of the narrow
6,17
,17
ϱ
Ϫ⌬OD͑v͒ϭCЈ
͵
ϪϱH͑vϪvЈ͒H͑vpumpϪvЈ͒g͑vЈϪv ͒dvЈ
0
͑
2͒
where H(v) is the homogeneous ‘‘single nanocrystal’’ ab-
sorption line shape, g(vϪv0) is the inhomogeneous distri-
bution centered at v , and CЈ is proportional to the excitation
0
intensity.
The ‘‘single-particle’’ absorption extracted from the mod-
eling ͑see Fig. 2͒ consists of a Lorentzian band-gap transition
with a width of 10Ϯ2 meV, a phonon side band spaced at

7
062
U. BANIN et al.
55
3
30 cmϪ1 from the band gap, and a second excited transition
frequencies in bulk InP. Both modes are coupled to the op-
tical transition, while in II-VI nanocrystals only the LO
mode is observed.
The electronic dephasing in the 3PPE case is measured by
setting t13 at a fixed delay to suppress the contribution of LO
modes and detecting the scattered signal as a function of
t12 . Figure 3͑b͒ exhibits the echo signal for three different
t13 delays corresponding to t13ϭ2␲/␻v ͑95 fs͒,
spaced at 0.10 eV with a width of 0.10 eV. The extracted
oscillator strength of the second state is 1.5 times larger than
that of the band-gap transition. The inhomogeneous Gauss-
ian distribution has a width of 0.17 eV and is centered at
1.86 eV. We do not include in the model possible induced
absorption from the excited state͑s͒, and since we are in the
linear regime biexciton effects are neglected.
t ϭ3␲/␻ ͑135 fs͒, t ϭ4␲/␻v ͑190 fs͒. Data are pre-
13
v
13
sented only for delays t Ͻt . At longer delays, t Ͼt ,
the echo signal no longer represents a polarization decay but
measures the decay of the excited-state population and can-
12
13
12
13
B. Femtosecond three-pulse photon echo
The width of the band-edge transition extracted from the
HB experiments as discussed above is broadened consider-
ably compared with atomic linewidths. The HB measure-
ment takes place on a nanosecond time scale in which vari-
ous broadening mechanisms not strictly related to the
not be used to extract T . In the three cases, the echo signal
2
exhibits a fast pulse-width-limited initial response, followed
by a slower decay. As expected, the slow decay is observed
to be strongly dependent on the timing of the third-pulse
18
delay (t ) due to the mode suppression. For t ϭ2␲/␻ the
homogeneous linewidth may be active. We thus comple-
ment the frequency domain measurements with a direct time
domain probe of the electronic dephasing dynamics—the
13
13
v
modulation of the signal by the vibrational beat is sup-
pressed, for t ϭ3␲/␻v the modulation is maximal and a
13
faster decay is observed, finally for t ϭ4␲/␻ , the decay is
3
PPE which allows the investigation of linewidths and line
13
v
19–21
longer relative to the 3␲ case as suppression is effective
again. This is a manifestation of the effect of quantum beats
of the coupled phonon modes on the observed decay and
demonstrates the utility of 3PPE in extracting reliable T2
times when a vibrational mode is coupled to the electronic
transition. The t13 delays were chosen to suppress the more
broadening mechanisms on the fs timescale.
In the 3PPE the electronic dephasing can be measured
while circumventing the complications arising from coupling
to vibrational modes, which leads to modulation of the echo
decay by vibronic quantum beats.8,9,16 A sequence of three
pulses is used in this experiment. The first pulse generates a
coherent polarization in the material, the second pulse inter-
acts with the polarization forming a population grating, and
the third pulse is scattered off the grating in a phase-matched
direction. Proper tuning of the third-pulse delay to 2␲/␻v ,
Ϫ1
strongly coupled LO-phonon mode at 350 cm . The TO
mode is close in frequency and thus is also effectively sup-
pressed.
The magnitude of the LO coupling, quantified by the
Huang-Rhys parameter S, can be extracted from a fit to the
where ␻ is the frequency of the coupled vibrational mode,
v
9
effectively suppresses the contribution of the vibronic quan-
tum beats to the echo decay. The mode suppression is par-
ticularly important for systems in which the electronic
dephasing is on the order of a vibrational period. In these
cases, the two-pulse photon-echo response without mode
suppression is strongly modulated by the vibrational beat
resulting in an extremely rapid decay which does not accu-
rately reflect the true dephasing of the system.8
2␲, 3␲, and 4␲ data. Such a fit yields a value of Sϭ0.08
for this sample. This is lower than the value reported for
CdSe nanocrystals of similar size (Sϭ0.18͒.9 This can be
explained by considering that coupling to LO modes occurs
,22
via the Fro
¨
hlich mechanism—the electronic excitation alters
the charge distribution and distorts the lattice. As expected,
the coupling is weaker in the more covalent III-V material,
,9,16
since the Fro
¨
hlich mechanism is polar in nature.
The 3PPE experiment was conducted on the 29-Å InP
nanocrystal sample, using 20-fs pulses centered at 1.95 eV so
as to overlap with the absorption onset ͓see inset of Fig.
An intriguing aspect of the echo response in the electronic
dephasing measurement ͓Fig. 3͑b͔͒ is the presence of a
pulse-width-limited initial decay. For an inhomogeneously
broadened two-level system, a calculation of the echo re-
sponse for the 3PPE case using real pulses within a third-
order perturbation expansion with all terms predicts a single
exponential decay with a time constant of T2/4 where T2 is
the electronic dephasing time.20
3͑b͔͒. All pulses are linearly polarized with parallel polariza-
tion. Two different experiments have been performed to ex-
amine in the first case the vibrational dynamics, and in the
second case the electronic dephasing.
The vibrational dynamics are studied by setting the first
and second pulse at a fixed delay Ϫt ϭ30 fs and detecting
12
the scattered signal as a function of the third-pulse delay
C. Interpretation of the echo response
t13 ͓Fig. 3͑a͔͒. In this case (t Ͼ0, t Ͼt ) a population
12
13
12
grating is formed and the signal is modulated by coherent
optical-phonon modes. By increasing the fixed delay t12 the
modulation depth is increased but the overall signal de-
creases as the polarization dephases in the t12 interval. The
delay t ϭ30 fs is chosen to yield the best signal to noise
To understand the observed echo response we need to
consider the HB spectra for the same sample which show
that a two-level system is inadequate to describe the band-
edge absorption of the nanocrystals. The HB data for the
29-Å InP sample is presented in Fig. 4͑a͒. As for the 34-Å
sample shown in Fig. 2, the HB data shows that above the
lowest band-gap transition lies an additional electronic tran-
sition.
1
2
ratio for the oscillatory component. The Fourier spectrum of
the oscillatory component shows two modes at 350 and 310
Ϫ1
cm ͓inset of Fig. 3͑a͔͒. Similar modes are observed in the
14
resonance Raman spectrum of InP nanocrystals. The fre-
We model both HB and 3PPE results within this three-
Ϫ1
quencies of the two modes, assigned to the LO ͑350 cm
͒
level picture ͑ground state
͘ ͘
͉a and two excited states ͉b and
Ϫ1
and TO ͑310 cm ͒ phonons, correlate well with the phonon
͉c separated by ⌬E ). For HB we repeat the procedure
͘
bc

55
QUANTUM CONFINEMENT AND ULTRAFAST DEPHASING . . .
7063
and electric dipole approximations. Coupling of the transi-
tion to optical phonons, which is effectively suppressed in
the experiment by the timing of the third-pulse delay, was
excluded. We assume different homogeneous linewidths
⌫baϭ1/(␲T2ba) where T2ba represents a polarization de-
phasing time for the
⌫caϭ1/(␲T2ca) were T2ca represents a polarization dephas-
ing time for the transition. Negligible population
͉
a
͘
→
͉
b
͘
transition, and
͉
a
͘
→
͉c
͘
decay is assumed in the time window of the experiment. The
equations were solved within the perturbative approach,8
using real pulse widths, to calculate the third-order matrix
,9,19
(
3)
(3)
ca
elements ␳ (t,t12 ,t ) and ␳ (t,t12 ,t ). In the phase-
ba
13
13
matched direction Ϫk ϩk ϩk , the radiated polarization is
1
2
3
a superposition of the contribution of the two transitions
and , and the echo response is calculated
as the square of this polarization. The fit is presented in Fig.
͑b͒ as the dashed line. Aside from the linewidth of the
͉
a
͘
→
͉
b
͘
͉
a
͘
→
͉c
͘
4
band-gap transition (⌫ ), all the parameters used to fit the
ba
echo response are identical to those extracted from the fit to
the HB data. This analysis provides the following interpreta-
tion: the fast pulse-width-limited initial decay in the echo
response is associated with the contribution of the excited
transition. The slow component of the decay which follows
͑decay constantϭ70Ϯ10 fs͒ yields a T2ba time of 280Ϯ40 fs
for the band-gap transition. The transition line-
͉
a
͘
→
͉b
͘
width extracted from this dephasing time is 5Ϯ1 meV com-
pared to 11Ϯ2 meV extracted from HB. This disparity in
linewidth can be explained taking into account the six orders
of magnitude difference in the time scales of the two mea-
surements. The HB experimental linewidth is additionally
broadened by processes occurring on the ns time scale such
as spectral diffusion,18 and possibly by the presence of an
electron and hole pair trapped on the excited nanocrystals
which exert an electric field acting to Stark shift and broaden
the transition.24 It should be noted that both the HB and the
3
PPE results are also consistent with the broad high-energy
transition being composed of a manifold of overlapping
states. In this case, the broad excited transition in the model
represents the manifold of states.
FIG. 4. Simultaneous modeling of HB and 3PPE data for the
9-Å InP sample. ͑a͒ HB data ͑solid lines͒ and fits ͑dashed lines͒.
Arrows designate the position of the pump pulse. The top frame
shows the extracted ‘‘single-particle’’ absorption spectrum ͑solid
line͒ and inhomogeneous distribution ͑dashed line͒ with the follow-
2
The appearance of an instantaneous fast decay component
in 2PPE experiments has been reported previously in quan-
tum well samples.2
5,26
Two primary mechanisms have been
ing parameters; inhomogeneous distribution: ⌫ ϭ0.17 eV,
in
E ϭ2.0 eV; homogeneous absorption: ⌫abϭ11 meV, ⌬Evibϭ330
cm , ⌬Ebcϭ0.09 eV, ⌫acϭ0.11 eV, ␮ca /␮baϭ1.9. ͑b͒ 3PPE
electronic dephasing measurement for the 2␲ delay ͑solid line͒ and
model ͑dashed line͒. See text for details.
shown to contribute to the appearance of the instantaneous
response—the first is the interaction induced mechanism
which is a many-body effect resulting from coherent exciton-
exciton interaction.25 This mechanism can be ruled out in the
present case as coherent exciton-exciton interactions cannot
be present since the average distance between the nanocrys-
tals is estimated to be hundreds of angstroms, and in the
intensity regime of this study the excitation density is much
less than one exciton per nanocrystal. In this respect, the
nanocrystal response is analogous to that of a molecule.
The second mechanism, reported by Kim et al. who con-
ducted time and spectrally resolved four-wave-mixing ex-
periments on quantum well samples26 is assigned to the free-
induction decay of off-resonant laser fields, which is
enhanced by the strong pulse overlap and the resulting popu-
lation grating near zero delay in the 2PPE case. This mecha-
nism is ruled out in the present study as our experiments are
conducted on resonance with no detuning ͓laser bandwidth
smaller than the inhomogeneous absorption width, refer to
in
Ϫ1
followed for the 34-Å sample and extract a ‘‘single-particle’’
absorption spectrum which consists of a band-gap Lorentzian
transition (
phonon sideband ͑spacing ⌬Evib), and a second broad ex-
cited transition ( , width ⌫ca , dipole moment ␮ ),
see Fig. 4͑a͔͒. Additional parameters that are included in the
model are the linewidth (⌫ ) and center (E ) of the inho-
͉
a
͘
→
͉b , width ⌫ba , dipole moment ␮ ), a
͘
ba
͉
a
͘
→
͉c
͘
ca
͓
in
in
mogeneous distribution g(␻) ͑see caption of Fig. 4 for the
exact parameters͒. The presence of the higher-lying transi-
tion in InP nanocrystals is confirmed by theoretical level
structure calculation ͑Sec. III D͒.
The 3PPE data was modeled using a Bloch equation for-
malism. The optical Bloch equations23 were written for the
three-level system described above within the rotating wave

7064
U. BANIN et al.
55
inset of Fig. 3͑b͔͒ and the off resonance contribution is ex-
pected to be minimized. Furthermore, in other systems such
as CdSe nanocrystals a similar fast decay was observed to
depend systematically on size with diminishing contribution
in the smaller nanocrystals.8 Finally, the consistency of the
,9
3PPE and the HB experiments and the ability to model both
of them simultaneously within the three-level picture de-
scribed above provides conclusive evidence for our assign-
ment of the echo response.
This case presents an interesting limit for the four-wave-
mixing response in a correlated inhomogeneously broadened
three-level system.27 In the limit of two narrow transitions
with similar values of T , the short pulse will create an elec-
2
tronic quantum beat. However, for InP nanocrystals the sec-
ond state homogeneous width is comparable to the energy
splitting between the two states and the electronic quantum
beat is overdamped. Thus in our measurements we observe a
pulse-width-limited initial response arising from the contri-
bution of the second broad transition, followed by a slow
decay which arises from the band-gap state. In the following
section we will present results of a calculation of the size-
dependent level structure in InP nanocrystals consistent with
the observed transitions.
Our findings point out the complementary nature of the
time domain 3PPE approach and the frequency domain hole-
burning measurement in examining the homogeneous optical
properties of nanocrystal samples. Hole burning yields the
homogeneous electronic level structure. However, the homo-
geneous linewidth can be extracted only from the more direct
fs 3PPE experiment, since on the ns time scale additional
broadening mechanisms are present.
D. Assignment of the observed transitions
The zero-order effective-mass approach for the calcula-
tion of energy levels in quantum dots takes into account only
single valence and conduction bands confined within infinite
FIG. 5. Calculated excitonic transitions for InP nanocrystals ver-
sus size ͑lower frame͒. ⌬E between the higher-energy transitions
2
8,29
spherical wells.
In this case, both hole and electron wave
and the lowest 1S3/21S transition is shown. Crosses designate the
e
functions are a product of a spherical harmonic and a Bessel
function. Simple selection rules of ⌬nϭ0, ⌬lϭ0 ͓n is the
electron ͑hole͒ level number, 1 is the electron ͑hole͒ orbital
angular momentum͔ are obtained. This zero-order treatment
predicts the confinement related blueshift of the gap as well
as the appearance of discrete levels. However, while the
single-band description is sufficient for the conduction-band
states, Efros and co-workers have demonstrated that in the
measured position of the excited transition as extracted from the HB
data in Fig. 1. A similar calculation for CdSe is presented in the top
frame for comparison. The 1P3/2 hole level is shown as a dashed
line for both materials. We use bulk literature parameters for both
materials ͑Ref. 34͒.
designated by nL where n is the level number, L represents
F
the orbital angular momentum of the lowest contributing
quantum dot regime substantial mixing takes place among
hole sublevel, and F is the total hole angular momentum;
the valence-band levels.1
1,30
We find that this is also the ex-
electron levels are designated by nl where n is the level
e
planation for the multiple transitions observed at the band
edge of the InP nanocrystals. The proper description of the
hole levels is obtained by applying spherical boundary con-
ditions onto the Luttinger Hamiltonian. The starting point of
the treatment is the usual three-band Luttinger description
for the valence bands. After applying the spherical boundary
conditions, the only good quantum numbers are the total hole
angular momentum FϭL ϩJ (L is the orbital angular mo-
number and l is the orbital angular momentum. The selec-
e
tion rules are then (L Ϫl ϭ0).
h
e
The results of calculation of the size-dependent energies
of the first few transitions relative to the band-edge transition
between the 1S3/2 hole and the 1S electron levels are pre-
e
sented in Fig. 5 for InP and CdSe for comparison. The rela-
tive positions of the second excited state extracted from the
HB spectra for the InP samples shown in Fig. 1͑b͒ is marked
by crosses on Fig. 5 ͑bottom frame͒. This allows us to assign
the observed transitions; the lowest excitonic transition is
h
h
mentum of the lowest contributing hole level, J is the total
angular momentum before mixing͒, and the parity. This
treatment was previously applied to the II-VI systems, as
between the 1S3/2 hole level and the 1S electron level. Two
e
1
1,12,31
well as to other quantum dots.
We adopt the notation
transitions are energetically consistent with the second ex-
1
1
of Ekimov et al. for quantum dot states; hole levels are
cited state; 2S3/21S and 1S1/21Se .
e

55
QUANTUM CONFINEMENT AND ULTRAFAST DEPHASING . . .
7065
FIG. 6. Electronic dephasing measurements in the 3PPE con-
figuration (t ϭ95 fs, 2␲/␻ conditions͒ for the 29-Å InP sample
at three temperatures.
FIG. 7. Temperature dependence of the echo decay rate for
29-Å InP ͑circles͒ and CdSe nanocrystals ͑crosses͒. The solid lines
are the simulated decay rates for the model described in the text.
13
v
A comparison of the level structure of InP with CdSe
reveals that the small value of the spin-obit splitting ⌬ in InP
leads to a congestion of electronic transitions close to the
band-gap. The transitions do not fan out with size as reported
for CdSe; in fact, in InP nanocrystals the energy of the
nanocrystals, the low-temperature dephasing rate is slower in
InP, but the slope of the temperature dependence is consid-
erably steeper.
An intrinsic dephasing mechanism which is consistent
with the temperature dependence of the dephasing rate for
1
S1/21S transition gradually gets closer to the band-gap as
e
both materials is the coupling to low-frequency acoustic
modes which serve as a heat bath.1
0,32
Such coupling is me-
the size is reduced. An additional difference in the predicted
level structure of InP and CdSe is in the ordering of the
lowest hole levels. In CdSe the lowest hole level is 1S3/2 and
the second hole level 1P3/2 always lies energetically higher
diated by the deformation potential. The coupling strength,
g , to a mode of frequency ␻ can be extracted from the
a
a
6
following formula:
͓
see Fig. 5͑b͒, dashed line͔. In InP, however, the 1P3/2 hole
͑DeϪDh͒2
level is energetically lower than the 1S3/2 level. Although
this is not expected to affect the absorption of the nanocrys-
tals ͑the 1P3/21S level is optically dark͒, it may have an
gaϷ
,
͑3͒
3
␲
r C ប␻
1
1
a
e
effect on the luminescence properties of InP, as the hole can
float up to the 1P3/2 level affecting the radiative recombina-
tion of electron and hole.
where (D ϪD ) is the deformation potential for the bulk
e
h
33
34
material, C11 is the bulk elastic modulus, and r is the
nanocrystal radius. Using this formula and the material pa-
rameters for InP and CdSe, the temperature dependence of
the echo decay rate in both cases is reproduced ͑Fig. 7, solid
lines͒. The only free parameter is an offset which represents
the temperature-independent contribution to the echo decay
rate resulting from elastic scattering at surface or interior
E. Line-broadening mechanisms
An issue of considerable importance is to examine the
various contributions to the homogeneous linewidth of the
lowest transition in the nanocrystals. One wishes to assess
whether the observed width is an intrinsic size effect or
whether it is an extrinsic effect of scattering from surface or
interior defects which may be defeated by improved sample
preparation. The nature of the dephasing mechanisms can be
investigated by conducting a temperature dependence of the
linewidth. The 3PPE data for three temperatures for the 29-Å
InP nanocrystal sample are presented in Fig. 6. Shown are
electronic dephasing measurements of the echo response as a
Ϫ1
Ϫ1
defects ͑0.0044 fs in InP, 0.018 fs in CdSe͒. The larger
deformation-potential coupling ͑1.0 versus 0.4 for InP and
CdSe, respectively͒ and higher average acoustic frequencies
calculated for modes of an elastic 29-Å sphere35 in InP with
respect to CdSe lead to a considerably steeper temperature
dependence for InP nanocrystals. In our measurements we do
not access the temperature regime in which broadening due
to LO phonons is expected to contribute due to pulse width
8
and S/N limitations ͑the LO-phonon frequency in InP is 350
Ϫ1
function of t12 with t13 set at the fixed delay of 2␲/␻ . The
cm and the mode will not be significantly thermally popu-
v
slow decay component assigned to the band-gap transition is
clearly becoming faster as temperature is increased. Linear
fits to the slow component in the 3PPE case allows us to
extract the temperature dependence of the linewidth of the
band-gap transition. The extracted decay rates for the slow
component are shown in Fig. 7. Multiple points from several
separate sets of measurements are presented, along with the
echo decay rate for a CdSe sample of the same size reported
by Mittleman et al. in an earlier work ͑see Ref. 9 and refer-
ences therein for experimental details͒. Compared with CdSe
lated at Tϭ150 K͒.
The intrinsic acoustic-phonon-coupling mechanism ac-
counts for most of the observed zero-temperature linewidth
of the InP sample. Negligible in the bulk, the deformation-
potential coupling strength is enhanced with decreasing size.
This arises from the enhanced deformation, averaged over
fewer bonds, induced by the transition of the electron from a
bonding to an antibonding orbital. The average acoustic bath
frequencies are also larger for smaller sizes. The overall ef-
fect is considerable broadening at small diameters, which is

7066
U. BANIN et al.
55
qualitatively observed in the hole-burning spectra for the se-
ries of samples shown in Fig. 1͑b͒.
280Ϯ40 fs. The temperature dependence is consistent with a
line-broadening mechanism which results from coupling to
low-frequency acoustic modes. This intrinsic mechanism
limits the linewidth in highly confined quantum dots.
Compared to the CdSe nanocrystal system, we find that
the Fr o¨ hlich coupling to LO phonons is smaller in InP. The
TO mode is also coupled to the electronic transition unlike
the more polar CdSe case. The observed and calculated level
structure is found to be more congested in InP primarily
because of the smaller spin-orbit splitting in the bulk. The
low-temperature linewidth for 29-Å InP is smaller than that
observed for similar size CdSe, but the temperature depen-
dence of the broadening is considerably steeper indicating
stronger deformation-potential mediated coupling to acoustic
phonons in InP nanocrystals.
The deformation-potential coupling mechanism predicts a
5
1
/r dependence of the linewidth on size. This suggests that
while in the very small sizes considerable intrinsic broaden-
ing is present, there is a size regime in which the simple
confinement pictures suffice and the spacing of the electronic
transitions greatly exceeds the linewidth. This size regime is
material dependent. In InP nanocrystals with a radius of 50
Å, the predicted intrinsic low-temperature linewidth is 0.002
meV, and even at 300 K the linewidth is predicted to
broaden only to 0.06 meV by this particular mechanism.
IV. CONCLUSIONS
InP nanocrystals in the size range below 50-Å in diameter
exhibit an electronic structure of a band-edge transition
along with close-lying excited states, assigned within a
model of a particle in a spherical box including the effects of
valence-band mixing. The excited transitions, as observed by
ns hole burning, are broadened and congested leading to a
pulse-limited fast decay component in the fs 3PPE measure-
ments. The band-gap transitions have linewidths on the order
of a few meV. Direct 3PPE measurements for 29-Å samples
show that the low-temperature electronic dephasing time is
ACKNOWLEDGMENTS
We thank Dr. Bob Schoenlein for helpful discussions. We
are grateful to Dr. Al. L. Efros for assistance in the energy-
level calculations. U.B. thanks the Rothschild and Fulbright
foundations for support. G.C. acknowledges support of
NATO. This work was supported by the U.S. Department of
Energy under Contract No. DE-AC0376SF00098.
*
Permanent address: Dipartimento di Fisica del Politecnico, Piazza
L. Da Vinci 32, 20133 Milano, Italy.
A. P. Alivisatos, Science 271, 933 ͑1996͒.
L. E. Brus, Appl. Phys. A 53, 465 ͑1991͒.
and J. R. Heath, J. Phys. Chem. 100, 7212 ͑1996͒.
1
1
5
6
H. Geissen, B. Fluegel, G. Mohs, N. Peyghambarian, J. R. Spra-
gue, O. I. Macic, and A. J. Nozik, Appl. Phys. Lett. 68, 304
͑1996͒.
1
2
3
S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, Phys. Rev. B
C. J. Bardeen and C. V. Shank, Chem. Phys. Lett. 203, 535
3
5, 8813 ͑1987͒.
͑
1993͒.
D. J. Norris and M. G. Bawendi, J. Chem. Phys. 103, 5260
1995͒.
D. W. Peck, L. R. Narasimhan, and M. D. Fayer, J. Chem. Phys.
2, 4125 ͑1990͒.
S. Mukamel, in Principles of Nonlinear Optical Spectroscopy
Oxford, New York, 1995͒.
4
5
6
7
C. B. Murray, D. J. Norris, and M. G. Bawendi, J. Am. Chem.
Soc. 115, 8706 ͑1993͒.
J. E. Bowen Katari, V. L. Colvin, and A. P. Alivisatos, J. Phys.
Chem. 98, 4109 ͑1994͒.
A. P. Alivisatos, A. L. Harris, N. J. Levinos, M. L. Steigerwald,
and L. E. Brus, J. Chem. Phys. 89, 4001 ͑1988͒.
M. G. Bawendi, W. L. Wilson, L. Rothberg, P. J. Carrol, T. M.
Jedju, M. L. Steigerwald, and L. E. Brus, Phys. Rev. Lett. 65,
17
18
19
͑
9
͑
2
2
0
1
A. M. Weiner, S. De Silvestri, and E. P. Ippen, J. Opt. Soc. Am.
B 2, 645 ͑1985͒.
C. Bardeen, G. Cerullo, and C. V. Shank, in Ultrafast Phenomena
X, edited by P. F. Barbara, W. H. Knox, J. Fujimoto, and W.
Zinth ͑Springer-Verlag, Berlin, 1996͒, p. 197.
J. J. Shiang, Ph.D. thesis, University of California, Berkeley,
California, 1995.
N. Bloembergen, in Nonlinear Optics ͑Benjamin, New York,
1965͒.
Y. Wang, A. Suna, J. McHugh, E. F. Hilinski, P. A. Lucas, and R.
D. Johnson, J. Chem. Phys. 93, 6927 ͑1990͒.
D. S. Kim, J. Shah, T. C. Damen, W. Schafer, F. Jahnke, S.
Schmitt-Rink, and K. Kohler, Phys. Rev. Lett. 69, 2725 ͑1992͒.
D. S. Kim, J. Shah, T. C. Damen, W. Schafer, L. N. Pfeiffer, and
K. Kohler, Phys. Rev. B 50, 15 086 ͑1994͒.
M. Kock, J. Feldmann, E. O. Goebel, P. Thomas, J. Shah, and K.
Koehler, Phys. Rev. B 48, 11 480 ͑1993͒.
Al. L. Efros and A. L. Efros, Fiz. Tekh. Poluyprovodn. 16, 1209
͑1982͒ ͓Sov. Phys. Semicond. 16, 772 ͑1982͔͒.
L. E. Brus, J. Chem. Phys. 80, 4403 ͑1984͒.
1
623 ͑1990͒.
8
9
R. W. Schoenlein, D. M. Mittleman, J. J. Shiang, A. P. Alivisatos,
and C. V. Shank, Phys. Rev. Lett. 70, 1014 ͑1993͒.
D. M. Mittleman, R. W. Scshoenlein, J. J. Shiang, V. L. Colvin,
A. P. Alivisatos, and C. V. Shank, Phys. Rev. B 49, 14 435
2
2
2
2
3
4
͑
1994͒.
1
0
1
T. Takagahara, Phys. Rev. Lett. 71, 3577 ͑1993͒.
1
1
1
A. I. Ekimov, F. Hache, M. C. Schanne-Klein, D. Ricard, C.
Flytzanis, I. A. Kudryavtsev, T. V. Yazeva, A. V. Rodina, and
A. L. Efros, J. Opt. Soc. Am. B 10, 100 ͑1993͒.
D. J. Norris, A. Sacra, C. B. Murray, and M. G. Bawendi, Phys.
Rev. Lett. 72, 2612 ͑1994͒; D. J. Norris and M. G. Bawendi,
Phys. Rev. B 53, 16 338 ͑1996͒.
O. I. Micic, C. J. Curtis, K. M. Jones, J. R. Sprague, and A. J.
Nozik, J. Phys. Chem. 98, 4966 ͑1994͒; O. I. Micic, J. R. Spra-
gue, C. J. Curtis, K. M. Jones, J. L. Machol, A. J. Nozik, H.
Geissen, B. Fluegel, G. Mobs, and N. Peyghambarian, ibid. 99,
2
3
25
2
2
2
6
7
8
7754 ͑1995͒.
1
4
29
A. A. Guzelian, J. E. B. Katari, A. V. Kadavanich, U. Banin, K.
Hamad, E. Juban, A. P. Alivisatos, R. H. Wolters, C. C. Arnold,
3
0
G. B. Grigoryan, E. M. Kazaryan, Al. L. Efros, and T. V. Yazeva,

55
QUANTUM CONFINEMENT AND ULTRAFAST DEPHASING . . .
7067
3
3
Fz. Tverd. Tela ͑Leningrad͒ 32, 1772 ͑1990͒ ͓Sov. Phys. Solid
State 32, 1031 ͑1990͔͒.
A. Blacha, H. Presting, and M. Cardona, Phys. Status Solidi B
126, 11 ͑1984͒.
Physics of Group IV Elements and III-V Compounds, edited by K.
H. Hellwege, Landolt-B o¨ rnstein, New Series, Group III, Vol.
17, Pt. a ͑Springer-Verlag, Berlin, 1982͒, pp. 281–297.
H. Lamb, Proc. London Math. Soc. 13, 189 ͑1982͒.
3
3
1
34
T. Richard, P. Lefebvre, H. Mathieu, and J. Allegre, Phys. Rev. B
3, 7287 ͑1996͒.
S. Nomura and T. Kobayashi, Solid State Commun. 82, 335
1992͒.
5
2
3
5
͑