Understanding Colloidal Quantum Dot Excitation with Solution Photon
Correlation Fourier Spectroscopy
MCHIVes
MASSACHUSETTS INSTITUTE
OF TECHNOLOLG Y
by
I
S. Leigh Heathcote
MAY 19 2015
B.A. Chemistry; Biochemistry & Molecular Biology
Cornell College, 2009
LIBRARIES
SUBMITTED TO THE DEPARTMENT OF CHEMISTRY IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN CHEMISTRY
ATTHE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
FEBRUARY 2015
© Massachusetts Institute of Technology. All rights reserved. 1.0f$
Certified by: _ __
Signature redacted
Signature redacted
Moungi G. Bawendi
Professor of Chemistry
Thesis Supervisor
Accepted by: _ __
Robert W. Field
Chairman, Department Committee on Graduate Students
Understanding Colloidal Quantum Dot Excitation with Solution Photon
Correlation Fourier Spectroscopy
by
S. Leigh Heathcote
Submitted to the Department of Chemistry
on January 30, 2014 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Chemistry
ABSTRACT
Colloidal quantum dots (CQDs) have useful absorption and emission properties but exist in
inhomogenous batches. Solution photon correlation fourier spectroscopy (S-PCFS) combines
interferometry with fluorescence correlation spectroscopy (FCS) to measure the spectral
correlation of the average single fluorophore in a solution. With S-PCFS, the intrinsic emission
linewidth of CQDs can be assessed. This linewidth information is related to phonon coupling and
gives insight into the separation of holes and electrons in excited CQDs. Experimental
considerations and recommendations for troubleshooting a S-PCFS apparatus are also presented
here.
Thesis Supervisor: Moungi G. Bawendi
Title: Professor of Chemistry
I.
Introduction
Colloidal quantum dots (CQDs) are roughly spherical semiconductor nanocrystals whose
excitons are confined in all three dimensions. The unique physics of these quantum confined
systems produces a distinctive set of promising characteristics. CQDs are solution processable,
possess high quantum yields, and have emission that is both narrow and tunable, making them
candidates for biological imaging, display technology and photovoltaics. In fact, CQDs have
very recently been incorporated into widely available consumer products which capitalize on
these properties to create vivid, energy efficient, economical color display screens (1).
As new structures and syntheses for CQDs develop, it will be important to monitor and
maintain key qualities such as emission width. Emission broadening of CQDs is caused by four
processes: lifetime broadening, phonon coupling, spectral diffusion, and sample inhomogeneity.
Lifetime broadening is an intrinsic property of any single emitter and comes from the energy-
time uncertainty relationship, where the lifetime broadening is given by h/(2T) for fluorescence
lifetime
T,
which is well-characterized. Phonon coupling broadens emission of a single emitter as
relaxation events couple to vibrational modes in the CQD and the local environment. Spectral
diffusion can be observed in single particles at low temperature, high flux and long timescales, a
result of the movement of trapped charges inducing Stark shifted peaks (2). As spectral
information is collected on timescales that incorporate shifted peak emissions, the total spectrum
is broadened to incorporate these features. On the ensemble level, CQD emission is broadened
by sample inhomogeneities such as size, shape and stoichiometric ratios which lead to a
distribution of peak emissions. Chen et al. (3) have recently developed a procedure that produces
nearly monodisperse CQDs, resulting in negligible inhomogenous line broadening.
4
To better understand the effects of different CQD structures on photoluminescence and
phonon coupling, a technique that reports on the emission linewidth of individual dots in solution
will be invaluable. With this information, it may be possible to improve our ability to fine-tune
absorption and emission properties as well as energy efficiency, and gain insight into the physics
of nanocrystal excitons.
II.
Background
II.a.
Background: CQD phonon coupling
An excited crystal structure experiences some deformation in the excited state compared
to the equilibrium atomic positions in the ground state. As the crystal relaxes, the deforming
forces are removed. Atoms are displaced from their ground state equilibrium positions, and
vibrate in response to this change. Phonon coupling is often characterized simply in terms of the
Huang-Rhys parameter, which is defined as Si = Ai 2/2, where Ai is the dimensionless difference
between ground and excited state potential energy minima in phonon coordinate space for the
ith
phonon mode (Fig.1). When the Huang-Rhys parameter for phonon mode i is larger, it is more
likely that emission will couple to this phonon.
Exciton emission in CdSe CQDs predominately couples to longitudinal optical (LO) and
various acoustic (AC) phonons. In wurtzite CdSe, LO phonon frequency is near 210cm' (5) and
coupling occurs via the Frhlich mechanism while the low frequency AC phonons couple via
deformation and piezoelectric potential. The LO Huang-Rhys parameter (SLO) gives insight into
the extent of charge separation and Fr6hlich coupling in the nanostructure. However, reports of
SLO in CdSe CQDs vary by more than an order of magnitude. Comparing the different reports of
5
S~o highlights the broad range of values in the literature and demonstrates problems with current
techniques.
C
Phonon coordinate
Figure 1.(Adapted from (4).) The potential energy surface of a phonon vibration for the ground
and excited electronic states as a function of phonon coordinate. The ground and excited states
have different minima, separated by A. The Huang-Rhys parameter, S, is related to this
displacement: S = A 2/2. The reorganization energy is given by X.
Alivasatos et al. (6) used resonance Raman overtone ratios to assess this magnitude for
4.5 nm CdSe clusters and found Sw - 0.25. With resonance Raman, there is always a chance
that multiple transitions occur and interfere with each other. This possibility is strongly enhanced
by the fact that the samples used had significant size polydispersity (4), (7).
Later, Nirmal et al. (8) as well as Norris et al. (9) used fluorescence line narrowing
experiments to find SLO. Nirmal et al. (8) found both size and temperature dependence for 2.3
nm CdSe nanocrystals, but still found emission Sw - 1. Norris et al. (9) examined So for both
emission and absorption, which were -0.1 and -0.5, respectively. However, in these
6
experiments, constant exposure to continuous wave excitation may trap charges on the surfaces
of the quantum dots, which can result in significant increases to phonon coupling (2).
In contrast, Sagar et al. (7) found absorption SLO of similar particles to be -0.028 with Fs
pump-probe measurements. Not to be confused with emission SLO, this number still indicates
significantly less coupling than the experiments cited above. Although less prone to surface
trapping than the experiments by Nirmal et al. (8) and Norris et al. (9), this measurement could
probe interfering contributions from different transitions (4). High peak excitation intensities
could create multiple excitons that perturb phonon coupling, and perturbations from the probe
are also possible (4).
Generally, techniques for assessing the Huang-Rhys parameter do not clearly resolve
phonon features, expose particles to high excitation flux that may significantly alter local
environments and therefore exciton-phonon coupling, or are performed on samples that are less
than ideal. The discrepancies between the different reports of S illustrate that phonon coupling
and the resulting emission broadening in CQDs is not yet well understood. A better
understanding of phonon coupling and intrinsic linewidth in CQDs will allow design of particles
with more efficient downshifting.
II.b.
Background: Solution Photon Correlation Fourier Spectroscopy (S-PCFS)
We can circumvent many of the pitfalls listed above with our recently developed
technique, Solution Photon Correlation Fourier Spectroscopy (S-PCFS). This technique
combines fluorescence correlation spectroscopy (FCS) with interferometry to access the single
molecule lineshape under very low flux conditions and very short timescales.
7
In a simple FCS experiment, a fluorophore in dilute solution diffuses in and out of an
excitation focal volume, and emission intensity is measured as a function of time and correlated
with itself. The general normalized correlation of a function Ia(t) with another function Ib(t) is
defined as:
(2)(T)
-
(Ia(t)lb(t+T))
(Ia(t))(Ib(t+r))
where () indicate averaging over t and r is a displacement in t.
A correlation compares two functions and identifies their degree of similarity at different
displacements
T.
One can visualize the correlation as area of overlap of the two original functions
at a given displacement (Fig. 2). Imagining Ia(t) and Ib(t) as discretized functions, we can
understand the resulting correlation function as the probability of an event occurring in Ib given
an event in Ia after a t separation of T.
(a)
sctral oneblaon
Is(w)s(w+(dw
p( =
(0
(b)
(b
)
sW+
(a)()
0
Figure 2. (Adapted from (10).) The correlation of two functions (here, s(w) with itself) can be
visualized as the area of overlap between the two functions at different displacements ;.
8
For FCS, the fluorescence intensity is correlated with itself, so in equation (1), a(t) =
Ib(t). For clarity we will call the correlation of the FCS intensity signal with itself gFCS(T). A
typical gFCS(T) curve is shown in figure 3. Essentially, for each photon detected at time t,
gFCS(T)
is the conditional probability of detecting a second photon at time t + r. There are two
types of photon pairs: those coming from the same emitter and those coming from different
emitters.
To understand the contributions of the different types of photon pairs to gFCS(T),
consider a solution containing just one diffusing emitter. Assume the lone particle diffuses
through the focal volume just once, so the conditional probability of detecting a second emitted
photon at long times is zero. When the particle is in the focal volume, it emits photons in rapid
succession, causing an increase in this conditional probability at time separations below the
dwell time. At extremely short times, this probability decreases as emission events are spaced by,
on average, the fluorescence lifetime - that is, the average time for a fluorescence cycle to occur.
Now consider a sample with not one but many diffusing fluorophores. The probability of
detecting photons from different emitters is now present at all times, but there is still a chance of
detecting photon pairs from the same emitter. The resulting increase in gFCS (r) does not cut off
at a distinct time but rather reflects the distribution of dwell times of diffusing particles, with a
characteristic diffusion time equal to the average dwell time (Fig. 3). The more concentrated the
solution, the less likely photons are to come from the same emitter. The amplitude of this
increase in autocorrelation, therefore, is inversely proportional to the concentration of
fluorophores (Fig. 3).
9
Antibunching
Characteristic
diffusion timne
0.
0
I4
101
102
1o
102
r (Ms)
Figure 3. (Adapted from (11).) In FCS, photon pairs from different emitters contribute to a
constant background. Photon pairs from the same emitter contribute to a portion of the signal that
rises as antibunching diminishes, then decays with the characteristic timescale for the diffusion
of emitters through the focal volume.
S-PCFS differs from FCS in that emission light is sent through a Michaelson
interferometer (Fig. 4). In an interferometer, the emission beam is split and then directed along
two paths of variable distance. Beams are retroreflected, recombined, and directed in varying
ratios to two detectors depending on the interference of the two beams.
laser
correlator
Figure 4. (Adapted from (11)). The S-PCFS setup combines the confocal microscopy setup used
in FCS with an interferometer and digital correlator.
10
Consider emission of a single emitter sent through the interferometer. When the arms of
the interferometer are the same length, interference effects direct light preferentially to one
detector. The intensity at one detector as a function of optical path-length difference (6), called
the intensity interferogram, and can be expressed as follows:
Ia orb (8) =
where
'a
or b(
6)
(1 + Re{FT[s(w)],s })
(2)
is the intensity at detector a or b, Itotal is the total intensity detected, and s(w)
is the emission spectrum. If we increase the optical path length of one arm by a half wavelength,
the recombined beams will be directed evenly to the two detectors. Increasing the optical path
length of that arm by another half wavelength, light will be preferentially directed to the other
detector. As we continue to increase optical path length difference, light oscillates from one
detector to the other, but the effect diminishes as the breadth of the emission spectrum causes
dephasing over many wavelengths (Fig. 5). While these intensity oscillations, or fringes, occur at
intervals determined by the emission wavelength, all emission colors will be directed to the same
detector at exactly 0 optical path length difference. This interferometer position is therefore
referred to as the "white fringe."
Because intensity differences depend on the energy differences between pairs of photons,
we can cross-correlate the signals from the two detectors to gain insight into these relative
energy differences:
9'(6,T) = 1 - 1Re{FT[p(, T)]4s }
(3)
where gX (6, T) is the cross correlation of the two intensity signals and p(, T) is the correlation of
s(w) with itself as a function of displacements in time (T) and displacements in frequency().
II
0.8-
4.6-
.-
-30
-20
-10
0
10
20
30
Optical Path-length Difference
Figure 5. (Adapted from (11).) Modeled intensity of light incident on one detector in an
interferometer as a function of path-length difference. Rapid oscillations occur with periodicity
determined by the average wavelength of the light source. The envelope decays as the breadth of
the spectrum causes dephasing over many wavelengths. The coherence length characterizes the
decay of this envelope.
When we combine this analysis with FCS, we gain insight into the average relative
energy differences of photons in many single dot spectra. If emission comes from many diffusing
particles as in FCS, the signal is modulated by diffusion of particles through the focal volume:
gX(6, T) = gFCS(T)
(1.
-
Re{FT~p(4 T)]s6
})
(4)
Recalling from FCS that single particle contributions can be distinguished from ensemble
contributions, we can see that p(4 T) is composed of ensemble and single particle contributions
12
(Fig. 6). At long T, we know that single particles do not contribute photon pairs. We also know
that the ensemble contribution is constant for all T. Therefore, we can express p((, T) as a sum of
single and ensemble contributions, with the single contribution weighted by gFCS:
(T) = pensemble (T)+psingle(4,)(g
(5)
FCS.)
=p(C, FCS() )
and when we insert this expression for p( , u) into the governing equation for single emitter
PCFS (eq. 3), we find
(6)
9'(6,T)=
(1
FCS (T)
-Re{FT
[pensembe g
+ (gFCS(Q)
-
1)Re {FT [psingle( , T)]_}]
Here, we disentangle the single emitter component from the ensemble component for diffusing
fluorophores in solution. Now if we recall equation (4), we can relate measured quantities to the
emission spectrum:
2 (1
-
= 1FT[s(o)]2
FC
(7)
and name a new function:
9S-PCFS(8,T)
=
IFT[s(w)]1
2
(8)
Here we relate our measured quantities and the emission spectrum. However, there is no
direct conversion from p(,T) to s(w). Any imaginary components of FT[s(o)] are lost. If we
simply work from the assumption that FT[s(o)] is real, we can find the spectral lineshape.
FT[s(o)] is almost certainly not real and this lineshape is not entirely accurate, so we refer to
this lineshape as the Effective Spectral Lineshape (ESL). While the ESL does not capture
asymmetries in the spectral shape, its Full Width at Half Maximum (FWHM) has proven to be a
reliable gauge of the spectral FWHM (12).
13
Walstycerrelation at dffret4"tefeoonwftr PostOdm
1.8
1.6
1.4
1.2
1
0
1s100s
.1ns
us
00
I
1
OOms
1S
t=700Ps
T=10PS
S=600nsi
l10ns
s
T=1Oms
single
ensemble
40 -. -200
0
mV
200
400 -M0
200
0
200
mneV
400 -W0
-200
0
meV
200
400 -40W
-
0
200
meV
400
Figure 6. (Adapted from (11).) S-PCFS cross correlations are shown in (a). Each blue line is a
cross correlation measurement near a given interferometer path length difference 6, while
gFCS (r) is shown in black. For a given T (red), gX(8, r) is corrected for gFCS(r) and the Fourier
transform is taken to produce the spectral correlation function. This result shows that the
ensemble spectral correlation can be measured at long t while the single-nanocrystal spectral
correlation can be measured at short T.
Ultimately, S-PCFS measures the average spectral correlation of individual fluorophores
in solution. Note that this is not the same as the ensemble spectral correlation, which will
incorporate any inhomogeneous broadening. From the spectral correlation, we can draw the ESL
and obtain a reliable spectral FWHM for the average single dot at fast timescales (Fig. 6) (12).
Il.
Phonon coupling in CdSe/CdS
CdSe/CdS CQDs allow the study of enhanced Frohlich coupling. The relative band gaps
of CdSe and CdS are such that an excited electron in a CdSe/CdS nanocrystal delocalizes
throughout the particle while the excited hole remains largely confined to the core (Fig. 7).
14
Varying the core size and shell thickness alters the degree of charge separation and the resulting
Fr6hlich coupling in these particles.
(a)
(b)
-----
CdSe
CdS
CdSe
CdS
---- bulk
bl
--- confined core
confined shell, thin
---
confined shell, thick
radius
Figure 7. Band edge states for a CdSe/CdS CQD. The bulk band edges are shown in solid black.
The confined states in the core are shown for small and large cores in (a) and (b), respectively.
The confined states for thin and thick shells are shown in blue and red, respectively.
CdSe cores were synthesized in three batches with peak emission at 542 nm; 563 nm; and
605 nm, corresponding to diameters of roughly 2.9 nm; 3.3 nm; and 4.8 nm, respectively. Each
batch of cores was then overcoated with a CdS shell. Shells were synthesized with slow,
continuous injection of reactive precursors and aliquots were taken at regular time intervals, and
shell size is accordingly classified by relative shell volume. Shell volumes - both within and
between core size series - are relative to each other, not the core.
For each core size, as the shell gets thicker, the fluorescence lifetime gets longer, though
this increase is faster for smaller cores (Fig. 8). The fluorescence lifetime relates to the
separation between the excited hole and electron: the closer the hole and electron, the stronger
the attractive Coulomb force; the stronger the attractive Coulomb force, the faster the hole and
electron recombine. Consider the band structure of CdSe and CdS (Fig. 7) in a quantum dot. An
excited electron is free to delocalize throughout the particle while the excited hole remains in the
15
CdSe core. As the shell grows thicker, the electron spreads farther from the hole, increasing
charge separation and fluorescence lifetime.
200
605 lifetime
--- 543 lifetime
200 E
-a-. 605 spectral correlation FWHM
*--+.543spectral correlation FWHM
150
E 150.
UL
..
--.
0'
o
0
~
100
S100
T-
-50
50
0L
0
(b)
(a)
2
4
6
relative shell volume
8
1
3
5
0
CL
a
shell thickness (nm)
Figure 8. Fluorescence lifetimes in ns and average single dot spectral correlations for CdSe/CdS
CQD series with core emissions at 605 and 542 nm. Each is plotted as a function of relative shell
volume (a) and shell thickness (b).
Why should this lifetime increase be faster for smaller cores? Consider the confined
energy levels of the core and shell (Fig. 7). For both core sizes, the hole remains well-confined
as the shell grows. And for both core sizes, the shell becomes more favorable for the electron as
the shell grows. However, the favorability of the shell is more dramatic for the smaller core.
The exciton charge separation will also result in relatively high nanocrystal polarizability
which can increase Frohlich coupling to the LO phonon and manifest as broader fluorescence of
single particles. S-PCFS measurements of these particles indicate that there is indeed an increase
in the spectral correlation FWHM as the shell thickness increases (Fig. 8). Note, too, that the
increase in FWHM levels off as the shell volume increases. The excited electron is loosely bound
by Coulombic forces to the confined hole. Though there is not a clear cutoff at which increasing
16
shell thickness ceases to affect the single dot spectral correlation FWHM, this increase does taper
off around 5 nm, a little more than the 3 nm CdS exciton Bohr radius.
Additionally, this FWHM is larger for smaller cores (Fig. 8). For this difference, consider
a small core and a large core, each with the same shell volume. In each case, the hole is localized
in the core and the electron is delocalized throughout the nanocrystal. However, the hole in the
smaller core is confined to a smaller percent of the total nanocrystal volume than the hole in the
larger core. Thus there is more overlap between the hole and electron in the larger particle, and
there is less Fr6hlich coupling and a narrower emission and spectral correlation in the larger
particle.
IV.
Experimental Considerations
In the course of gathering the data presented in the above section, several concerns arose
whose solutions had not yet been documented. Here I present those concerns for future reference.
IV.a. The Interferometer: When is the Alignment Good Enough?
When checking the alignment of the interferometer, it is obviously important to have high
contrast; that is, a large (>0.3) value for gX (6 = 0). But, if the translation stage does not travel
parallel to the beam path (Fig. 4), it is possible to have high contrast at zero path length
difference, but then see rapid contrast decay as the stage translates. In this case, the light
returning from the translation stage would not recombine with light returning from the dithering
stage, and contrast would plummet.
A properly aligned interferometer will record g'(6) that does not decay artificially
rapidly. The 514 line of an Argon ion laser was measured to assess interferogram decay (Fig. 9).
17
While not all measurements give the same g X(6) decay, the width appears to be a function of
current supplied to the laser. Larger currents should activate more lasing modes, producing
broader emission and narrower interferogram envelopes. The emission FWHM modelled from
interferometer data falls in the range 4.8 - 11.4 peV, consistent with up to 21 peV expected for
this laser line. The smallest FWHM measured represents an upper limit on the instrument
function. Since the CQD samples measured typically have emission FWHM around 60 meV, the
interferometer is aligned well enough for these samples.
0.35
O
0.3-
0.3 -
-
.
0
,A
W.%
-.
.
0
.
*
0.2*
*B
A
-'A
9 *
0.1
Amps
Amps
Amps
Amps
Amps
'.*
*. *.
**.*,*.
yg
0.1'.
-,.
*
0.05-
-,
*..*
*
0.25 -
17
20
-23
26
32
0f
%OA 0
-50
0
50
100
150
interferometer path length difference (mm)
Figure 9. g' (6) of an Argon ion laser, measured with different amounts of current supplied to
the laser.
IV.b. Interferometer Instability: Diagnosing and Addressing the Problem
IV.b.i.
Identifying instability
After determining the interferometer was well-aligned, another problem arose wherein
spectral correlations measured with S-PCFS were narrower than the autocorrelation of the
18
measured fluorescence spectrum by up to 8 meV. In contrast, S-PCFS measurements of the same
sample taken months earlier differed by only 1 meV. As expected, g'(5) for the problematic
measurement was broader than gX (6) for the older, more accurate measurement (Fig. 10a).
However, the values for gX (6) here are actually an average of three measurements as part of a
protocol to improve the accuracy of gX (6).
0.35
0.3
(a)
_
-+-May 2014
- December 2014
(b)
0.25[
0.2
)0.15
0.1
0.05
07
-0.005 0 0.005
-0.005 0 0.005
interferometer path-length difference (mm)
Figure 10. Averaged ensemble gX(6) for the same CQD sample measured in May 2014 and
December 2014 (a). Note that the newer g(
6
) is broader than the older g'(6) in (a). The raw
ensemble g X(6) measurements are shown in (b). The 11 center positions were measured three
times. Note that triplicate measurements from May are more consistent than those from
December.
Positions near the white fringe were measured sequentially, beginning farthest from the
beamsplitter. After these points near the white fringe were measured once, the translation stage
moved back and repeated the sequence twice, for a total of three times (Fig. 10b). Positions far
19
from the white fringe were measured only once. The points measured in triplicate were then
averaged before further analysis (Fig. 10a). Typically, this arrangement efficiently measures an
extensive baseline and an accurate peak for the interferogram. That is, this protocol will ensure
an accurate g X (6) peak if the position of the white fringe is constant and variations in g X(6) are
the result of random noise.
However, closely examining problematic S-PCFS data - in which the ensemble spectral
correlation is artificially narrow - reveals significant variations in gX (6) for each of the three
measurements and a shift in the position of the white fringe (Fig. 10b). Averaging peaks with
different centers will result in a broader peak (Fig. 10a). An artificially broad interferogram
explains the artificially narrow spectral correlation.
IV.b.ii.
Diagnosing the cause of instability
What caused this drift in the position of the white fringe? Consider the S-PCFS setup
(Fig. 11). The magnitude of gX (6) is given by the relative distance between the beamsplitter and
translation retroreflector; and the beamsplitter and dithering retroreflector (Fig. 11). The most
reasonable explanation for changes in this relative distance is position instability in either the
dithering retroreflector or translation retroreflector.
To determine whether the dithering or translating retroreflector is unstable, both
retroreflectors were moved as close as possible to the white fringe and held stationary (Fig. 12a).
Fluorescence from Rhodamine 610 was directed through the interferometer and each detector
recorded count rates. Figure 12b shows the percent of the emission directed to one of the
detectors.
20
laser
instability?
(
I
N
CIu U
corr
instability?
lator
Figure 11. Irreproducibility of gX (6) is likely due to position instability of one of the
retroreflector stages in the interferometer.
T-\
(a)
laser
'I
"a
0
I
O.Sr
still
correlator
0.31
-
10Cr
WC
dithering
U
s
correlator
10
laser
still
(b)
80
1
(d)
0.290.28
0.27-
60:
10.26
.o
t0
40
0.250.24
20
0.23
0
0
20
40
60
time (min)
80
0.22'-
20
40
time (min)
60
Figure 12. Both retroreflector stages were positioned at the white fringe and held stationary (a)
while each detector measured count rates. The percent of the total signal directed to detector 0 is
shown in (b). In another experiment, the translating retroreflector was positioned near the white
fringe and held stationary while the other retroreflector dithered (c). Ensemble g ()
measurements were made every 3.5 minutes (d).
21
Consider that the interferogram, near the white fringe, approximates a sine wave as a
function of path-length difference with wavelength given by the color of the light (Fig. 5) - in
this case, Rhodamine 610 emits at 564 nm. There is nearly equal signal at each detector, so the
phase on that sine wave is approximated by a, where So is the normalized intensity at detector 0:
sin(a) = So
(9)
a= 0
Signal changes by no more than 20% during this 90 minute measurement, and the phase
change given by change of that magnitude is approximated by b:
sin(b) = 0.2
(10)
b = 0.06Tu radians
and because the wavelength is 564 nm, this oscillation accounts for changes in the path-length
difference around 1.9 nm
Now consider the difference between points in g X(6) near the white fringe for a CQD
(Fig 10). This difference is as high as -0.08, and the change in path-length difference here is
1pm. With a linear interpolation, the interferogram envelope difference given by a 1.9 nm
oscillation in path-length difference is given by c:
0.08
1 pm
1.9 nm
(11)
c = 1.5(10-4)
The problematic variations in the interferogram envelope are much larger: -0.02 (Fig.
10b). The drift observed when there is neither dither nor translation does not account for the
variations observed in gX(s).
Another diagnostic experiment was performed in which the translation stage remained
stationary at the white fringe but the second stage was dithering (Fig. 12c), again using
22
Rhodamine 610 as a fluorophore. During a typical S-PCFS measurement, the time between
triplicate measures at a given position is 7 minutes, and correlation measurements were taken
every 3.5 minutes.
The resulting g X(6) measurements show significant variations in g X(6) up to
approximately 0.02 between measurements made 7 minutes apart. Using a linear extrapolation
similar to equation (11), with gX(6) for Rhodamine 610 (not pictured), the variations pictured in
Figure 12d correspond to variations in path-length difference of about 200 nm. For the CQDs
examined here, this corresponds to variations in gX(6) of about 0.016. This uncertainty is on the
order of the problematic variations observed, -0.04 (Fig. 10b). Generally, the degree of variation
was not constant, so the fact that the variations observed when the translation stage was
stationary are smaller than those present in Figure 10b is not problematic.
Dithering produces inconsistent results. The dither is controlled by piezoelectric
components which can be unreliable. Slight variations in the distance travelled during the dither
can affect gX (6).
IV.b.iii.
Addressing the instability
To circumvent this problem, S-PCFS can simply be performed without a piezoelectric
dither; rather, the translation stage can be programmed to both translate and dither. Avoiding the
problem by avoiding piezoelectronics is a most reasonable solution. But, is there a way to get
accurate S-PCFS data simply by performing a slightly different analysis?
Averaging three g X (6) measurements with different centers will create a broader
interferogram (Fig. 10). However, each gX (6) can be assessed separately: each g'(6) can be fit,
then Fourier transformed into p, and these spectral correlations can be averaged. Analyzing each
23
g9 (6) separately avoids the problem of artificially broadening gX(6). An example of many
spectral correlations measured this way is shown in figure 13. There is variation in the spectral
correlations, but they do stay within 20 meV of each other. Averaging several spectral
correlations corrects random noise and changes in the shape of g'(6) due to white fringe drift.
1-
1
-
(a)
o 0. 8
(b)
from emission ensemble
from S-PCFS
0.6
0
0.4-
M.
0.2
CD
0
T
-200 -100
0
/
L@
100 200
-200 -100
energy (meV)
0
100 200
Figure 13. Ensemble spectral correlations from S-PCFS for the same CQD sample without
gx(6) averaging, compared to the ensemble spectral correlation measured with a fluorimeter (a).
The corresponding single dot spectral correlations are shown in (b).
IV.b.iv.
Checking ensemble spectral correlations
While trying the new analytical protocol described above, ensemble emission spectra
were collected from samples diluted to S-PCFS level concentrations. The ensemble spectral
correlation collected with S-PCFS reflects the emission of that very dilute sample. Emission
from the S-PCFS setup was directed to a spectrometer with a charge-coupled device instead of
the interferometer and APDs.
24
1
-0
II
- from emissic n uncorrected
****'from emissic n corrected
(a)
0.8
(b)
1
0.8
-.- from S-PCF
0.6
0.6
0.4
0.4
0 0.2
0.2
C
E
C
m----
1.6
'0
--
'
0
1.8
2.0
2.2
2.4
-0.4 -0.2
energy (eV)
0
0.20 0.4
Figure 14. Emission spectrum for a CdSe/CdS CQD sample, and the same spectrum with a
curved baseline subtracted (a). The sample dilution, excitation and optical filters are the same as
those used in S-PCFS. The ensemble spectral correlation from the corrected and uncorrected
spectra are shown in (b), along with the ensemble spectral correlation from S-PCFS.
In these spectra, a gently curved baseline is always observed - for an example, see
Figure 14a. Sometimes other broad features and sharp features from excitation light or Raman
scattering are also observed. The purpose of this measurement, though, is to collect the spectrum
of the light detected in the S-PCFS experiment, so it should not be necessary to subtract these
-
features. However, the correlations of these spectra do not match those measured with S-PCFS
again, see Figure 14b for an example. Subtracting the baseline from the emission spectrum
produces a spectral correlation in very good agreement with the S-PCFS measurement (Fig.
14b).
But, why should it be appropriate to subtract this baseline from the emission spectrum
when no such subtraction occurs in S-PCFS? An extremely broad feature, such as this baseline,
will have an extremely narrow g'(6). It is possible that gX(8) may simply be too narrow to
register on the detectors.
25
Regardless of the interferometer's ability to detect these broad features, the single particle
spectral correlation may remain unaffected. A likely cause of these broad features is solvent
impurities, which are likely to be organic compounds with low quantum yields. Thus the fact that
emission is visible suggests the concentration of these emitters is quite high. With a very high
concentration, there will be a negligible FCS amplitude for these particles. That is, the broad
feature contribution to g X (8) is identical for all relevant timescales. These broad features may
affect S-PCFS ensemble calculations, but single particle calculations should not be affected.
The low importance of broad emission features should be considered when collecting
spectra to compare to S-PCFS measurements in the future.
IV.c.
Recommendations
Recommendations for future work with S-PCFS can be summarized as follows.
Interferometer alignment can be quickly assessed by measuring 514 nm emission from an Argon
ion laser, or another light source whose emission is much narrower than the sample. To improve
interferometer stability, do not control retroreflector movements with piezoelectrics; instead, use
a more consistently controllable device. In the event that S-PCFS ensemble measurements
produce narrower ensemble spectral correlations than expected when g X(8) is being measured in
triplicate and averaged, consider the possibility of interferometer instability. Refer to above
section V.B. InterferometerInstability: Diagnosingand Addressing the Problem for suggestions
on troubleshooting and analysis.
26
V.
Acknowledgements
Igor Coropceanu synthesized the CQDs described here and conducted fluorescence
lifetime measurements. Andrew Beyler provided frequent and thorough support and suggestions
for this work. Moungi Bawendi has been a supportive and understanding advisor.
VI.
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