Henini M. Handbook of Self Assembled Semiconductor Nanostructures for Novel devices in Photonics and Electronics

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Quantum Optics with Single CdSe/ZnS Colloidal Nanocrystals 719

greater than 5 nm, as with self-assembled quantum dots, cascades of emissions from bi- and

triexciton states have been evidenced by time-resolved spectroscopy [66] . For smaller nanocrys-

tals, the reduced distance between the charges enhances non-radiative multiexciton-to-exciton

Auger recombination, where the recombination energy is transferred to the remaining exciton

instead of photons ( Fig. 24.8b ). Multiexciton recombination occurs within tens of ps [67, 68] ,

faster than the radiative recombination.

(a)

(b)

Figure 24.8 Role of Auger energy transfer in the recombination of (a) an exciton and (b) a biexciton.

For small nanocrystals, although multiexcitons can be created, there is no cascade of photons,

and nanocrystal ﬂ uorescence shows complete antibunching. The possibility of observing multi-

excitonic emission by nanocrystals with large radii will be analysed in detail in section 24.7.

24.3.5 Fine structure of the emission level

In the conﬁ nement model described above, the emitting level 1 S

3/2

1 S

, having electron spin 1/2 and

hole momentum 3/2, is eight-fold degenerate. This degeneracy can be lifted by three pertur-bative

effects: the small gap between the light- and heavy-hole bands due to the hexagonal (not cubic)

crystalline structure, the slightly prolate shape of the nanocrystal [42] , and the exchange term in

the electron–hole Coulombian interaction, which is enhanced by conﬁ nement. The ﬁ rst two terms

just lift the degeneracy, both along the crystalline c -axis (labelled z ) as the nanocrystal elongation

follows the c -axis (for reasons related to the growth mechanism). The latter term couples the elec-

tron and the hole, so that the “ right ” quantum number is now the total momentum



A complete mathematical treatment can be found in [70] ; the result for a CdSe/ZnS nanoc-

rystal of diameter 3.5 nm is shown in Fig. 24.9a . The main result is that the lowest sub-level,

| F

 2 冭 (two-fold degenerate), cannot emit light through a linear process as it exhibits a 2 

momentum difference with the fundamental level. Emission arises mostly from two sublevels of

z -momentum   , which are around 10meV higher than the “ dark exciton ” state |  2 冭 . After

excitation, the system relaxes rapidly to the dark exciton state; it is later excited to the bright

state by the phonon bath, and emits a photon. This process is faster at higher temperature, which

explains the observed dependence of the excited-state lifetime T

as a function of temperature,

from 20 ns at 300 K to 300 ns at 10 K and 1 μ s below 2 K [72, 73] ( Fig. 24.9b ). At 2 K, the bright

level is not thermally populated at all, and there is only a weak phonon-assited ﬂ uorescence from

the dark state [74] .

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720 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

At room temperature, such an emission from two degenerate levels |  1 典 and |  1 典 shows two

original characteristics:

● It is an incoherent sum of two σ



and σ



emissions.

● It is oriented according to a peanut-shaped diagram directed along the c -axis.

Both aspects have been used to conﬁ rm that the emission of a nanocrystal does not correspond

to a linear dipole but to a two-dimensional degenerate dipole. *

On the one hand, the polarizations successively measured on 170 nanocrystals showed a

distribution which was not compatible with a randomly oriented one-dimensional dipole [76] ;

polarization measurements were used to determine the orientation of a nanocrystal relative to

the sample plane [77] .

Fine structure splitting (meV)

10

20

2

2

1

1

1

1

1 10 100

1000

100

R = 1.3 nm

R = 1.8 nm

R = 2.1 nm

Lifetime (ns)

Temperature (K)(a) (b)

Figure 24.9 (a) Fine structure of the emitting 1 S

3/2

1 S

level of a 1.8 nm radius CdSe nanocrystal, as cal-

culated from [70] and [71] , with the levels labelled by the z -component of the total electron–hole momentum



(b) Emission lifetime of CdSe/ZnS nanocrystals of various radii R , measured as a function of temperature (repro-

duced from [72] , authorization for reproduction asked).

(a) (b)

Figure 24.10 Principle and experimental result of a defocused-microscopy orientation measurement for a CdSe/

ZnS nanocrystal with c -axis (a) perpendicular and (b) parallel to the sample plane [79] . The experiment consists in

measuring an image of the nanocrystal with a CCD camera at one micron from the focal plane.

* At cryogenic temperature (  4 K), a splitting of 1–2 meV has recently been reported, and attributed to a non-

perfect cylindrical symmetry along the c -axis [75] . We will discuss the inﬂ uence of this splitting in section 24.7.

On the other hand, emission diagrams and thus nanocrystal orientation were evaluated by

a defocused-microscopy technique, as explained in Fig. 24.10 : a nanocrystal was imaged on a

slightly (1 μ m) defocused CCD camera; for a nanocrystal with c -axis perpendicular to the sample

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Quantum Optics with Single CdSe/ZnS Colloidal Nanocrystals 721

plane, the defocused image was an isotropic ring; for a nanocrystal with c -axis parallel to the

sample plane, it was a ring with lobes oriented along the c -axis [78] . By varying the distance of

the nanocrystal to an interface (following the procedure detailed in Fig. 24.11 ), one generates a

change in the lifetime T

which depends on the nanocrystal orientation. Comparing this change

and the diagram observed by defocused microscopy, it could be proved, on a single nanocrystal,

that the dipole had to be in two dimensions [79] .

24.4 Nanocrystals as single-photon sources

Now that the fundamental optical properties of colloidal nanocrystals have been explained, we can

consider how well they qualify as single-photon sources, and how they compare to other sources.

24.4.1 Photon antibunching

A ﬁ rst requirement is of course to emit single photons. As described before, antibunched emis-

sion from colloidal nanocrystals was reported in 2000–2001 [24, 28, 69] , and after subtraction

of the background noise a g

(2)

(0) value smaller than 0.01 was found. Single-photon emission

was exhibited by other sources: atoms, ions, molecules, colour centres, self-assembled quantum

dots, etc.

It is worth mentioning that in fact self-assembled quantum dots do not emit single photons.

For these dots, which are larger than nanocrystals, the Auger effect is not fast enough to sup-

press radiative multiexciton recombination, so that single photons can only be obtained by spec-

trally selecting the one-exciton line. This is possible only at cryogenic temperature as the exciton

and biexciton lines are broadened and merge as the temperature increases. Thus, in order to

obtain single photons from a quantum dot, one needs to work at cryogenic temperature and to

add a spectrometer before detection. Colloidal nanocrystals, like terrylene molecules and colour

centres, directly emit single photons at room temperature, which makes them quite easy to use.

Moreover, working at room temperature allows the use of a high-NA immersion objective, lead-

ing to simple and efﬁ cient light collection.

Time (s)

(ns)

024681012

PDMS (1 mm)

PMMA (45 nm)

Nanocrystals

Glasscoverslip

(a) (b)

Figure 24.11 An application of the method demonstrated in [79] to measure the quantum yield of a single emitter.

One considers a sample with nanocrystals spin coated on a glass coverslip, and covered by a polymethylmetacrylate

(PMMA) 45 nm layer. PMMA has the same refractive index as glass, so that each nanocrystal is in a 1.5 index

medium, at 45nm from an interface with a medium of index 1 (air). At t  7 s, a droplet of polydimethylsyloxane

(PDMS) is deposited. PDMS is a liquid polymer of the same 1.5 index, so that after droplet deposition the nanoc-

rystal is in a homogeneous 1.5 index medium. (a) A measure of the lifetime change β  T

after

/ T

before

after deposi-

tion of the droplet (here: β  0.81). (b) A measure of the nanocrystal orientation (here: horizontal). Knowing the

orientation, the radiative decay rate Γ

can be calculated as a function of the interface distance, and the Γ

change

1/ α  Γ

after

/ Γ

before

is obtained. The quantum yield is then ( β  1)/( α  1) (here: 0.99).

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722 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

24.4.2 Single-photon emission on demand

Let us now examine the possibility of emitting single photons on demand. This requires absorp-

tion saturation, a quantum yield equal to 1, and 100% collection.

The broad absorption continuum renders nanocrystal excitation extremely easy, with any

source below 530 nm. An absorption cross-section of around 0.1 nm

was measured by ensemble

absorption measurement [54] or single nanocrystal saturation [80] , smaller than self-assembled

quantum dots (10 nm

derived from [25] ) but larger than coloured centres (0.002 nm

derived

from [81] ) and terrylene molecules (0.004 nm

[19] ). Ninety ﬁ ve per cent saturation of the

nanocrystal emission can be achieved with a standard pulsed laser diode with 5 pJ energy per

pulse [38] .

The ﬂ uorescence quantum yield Γ

/ ( Γ

 Γ

) is routinely measured by cuvette techniques,

which compare the light intensities that are absorbed and emitted by a solution of emitters. This

is not appropriate for nanocrystals as it gives a value (typically 40%) which is averaged on all

nanocrystals, while some of them may be “ off ” because of the blinking. An original method has

been developed for the measurement of single-emitter quantum efﬁ ciency [79] . It measures the

excited-state lifetime 1/( Γ

 Γ

) consecutively with a dielectric interface close to and far from

the nanocrystal, and uses theoretical calculations on the effect of a nearby interface on Γ

determine Γ

and Γ

( Fig. 24.11 ). An almost perfect quantum yield (95 to 100%) is derived.

A general consequence of refraction laws is that light remains mostly in the medium of higher

refractive index. Colour centres and quantum dots are embedded in a medium of index higher

than the objective components (⬃1.5), so that only a small fraction of their ﬂ uorescence is

extracted and collected ( Fig. 24.12a ).

For a nanocrystal or a ﬂ uorescent molecule at room temperature, the standard procedure is to

use an immersion objective. Nanocrystals are usually spin coated on a glass coverslip of index 1.5

and protected from oxidation by a polymer layer (usually polymethylmethacrylate – PMMA) of the

same index 1.5. An objective is approached from under the coverslip, and separated from it by a

droplet of immersion oil, of the same 1.5 refractive index ( Fig. 24.12b ). By use of a high numerical-

aperture objective (1.4), most of the ﬂ uorescence is then collected: a model including the two-

dimensional-degenerate emission dipole ﬁ nds that around 85% of the emission is directed into the

glass, and that approximately 72% of the emission is collected by the objective [38] .

The main obstacle to single-photon emission on demand lies in fact in the transmission and

detection step. Each optical component introduces 5 to 20% losses, and the photodiodes have

at best 60% detection efﬁ ciency. With the set-up of [38] , at 95% saturation and using an immer-

sion objective with 1.4 numerical aperture and 60% efﬁ ciency photodiodes, around 8% of the

pump pulses lead to photon detection, so that the transmission losses of this set-up are estimated

to be 81%.

In comparison, the rate of photon detection is typically 10

 4

per pump pulse for a quantum

dot [25] , 0.05 per pulse for a terrylene molecule [19] , 0.02 for a trapped atom [82] , and 10

 4

for

a coloured centre (derived from [21] ).

For self-assembled quantum dots and colour centres, methods using cavities have been explored

in order to improve the light extraction efﬁ ciency. Epitaxially grown dots have been inserted

in a micropillar cavity in order to redirect the emission, yielding a photon detection rate of

0.002/pulse [26] ( Fig. 24.12c ). Colour centres in diamond nanocrystals have been considered: as

these diamond structures are smaller than the wavelength, the low collection of light due to the

high refractive index of diamond is suppressed ( Fig. 24.12d ). A rate of 0.013 photon detection

per excitation pulse is obtained [23] .

24.4.3 Other characterisitics of a single-photon source

Some other properties can be taken into account to characterize a single-photon source. A key

parameter is the emitter stability. After a characteristic time of a few seconds, most ﬂ uorescent

molecules “ photobleach ” : they irreversibly become non-ﬂ uorescent. This problem is quite impor-

tant for terrylene molecules, although a small fraction of them ﬂ uoresce up to 30 mn [20] .

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Quantum Optics with Single CdSe/ZnS Colloidal Nanocrystals 723

Nanocrystals, like self-assembled quantum dots and colour centres, do not photobleach.

However, as previously mentioned, they blink from “ on ” to “ off ” states. This change is reversible,

but the duration of the “ off ” states exhibits very peculiar statistics and is on average almost inﬁ -

nite (see section 24.5). The explanation and eventual elimination of this “ blinking ” behaviour is

one of the main challenges in the development of nanocrystal-based single-photon sources.

The emission wavelength could also become a crucial parameter for practical implementa-

tion of quantum cryptography protocols. In contrast with molecules and colour centres, zero-

dimensional semiconductor structures offer the ability to tune the emission wavelength by

adjusting their size. In particular, the size of colloidal nanocrystals is determined, with less than

5% dispersion, by the synthesis parameters. Using CdSe or CdTe cores, their emission wave-

length can be tuned in the whole visible range. Commercial nanocrystals can be purchased from

companies like Invitrogen or Evident Technologies.

Research on nanocrystals has focused up to now on CdSe and other II–VI materials emitting

at visible wavelengths, because photodetectors and many other optics work better in this range.

Silicon avalanche photodiodes exhibit very good performance (such as an efﬁ ciency of up to

60%) only between 600 and 900 nm.For large-scale applications in quantum cryptography, how-

ever, it would be better to use the existing network of telecom optical ﬁ bres, functioning between

1350 and 1550nm. In this range, commercial avalanche photodiodes made of InGaAs layers on

an InP substrate are available [1] , but only have 10% efﬁ ciency [83] . Better performances have

been obtained by up-converting the 1.55 μ m photons to 700 nm photons and detecting them

with an Si photodiode; this system achieved a 45% efﬁ ciency [84] .

Synthesis of nanocrystals emitting in the near-infrared region is currently under investiga-

tion, IV–VI semiconductors like PbS or PbSe being the most promising materials. Such nanocrys-

tals would allow transmission through telecommunication ﬁ bres or coupling to state-of-the-art

photonic crystals (see section 24.8). In terms of applications of nanocrystals as labels in biology,

the near-infrared constitutes a “ window of biological transparency ” in which it would be inter-

esting to work. On a more theoretical level, PbS and PbSe have very large exciton Bohr radius

(20 and 46 nm, respectively) so that conﬁ nement effects should be important on both electron

and hole. The ﬂ uorescence properties of PbS nanocrystals are quite unsatisfying for the moment

(30% quantum yield, strong photobleaching), and should be improved by progresses in synthesis

reactions and addition of passivating shells [85] . Some groups are also starting experiments to

investigate the synthesis of colloidal III–V quantum dots such as InP .

Another parameter is the excited-state lifetime T

. Since the interval between two excitation

pulses must be larger than T

, the rate of single-photon emission is ultimately limited to approxi-

mately 1/ T

. Nanocrystals have a T

lifetime of around 20ns at room temperature. They can be

brighter than trapped atoms and ions ( T

⬃ 100–200 ns), but not as bright as terrylene molecules

( T

⬃ 4 ns), coloured centres ( T

⬃ 11 ns) or self-assembled dots ( T

⬃ 1 ns). A solution to decrease

could be to couple the nanocrystal with a cavity; this aspect will be developped in section 24.8.

Sample

Emitter

Objective

Collected light

(a)

(b) (c) (d)

Figure 24.12 (a) The extraction problem for an emitter in (or at the surface of) a high-index sample: most of the

emission remains in the sample. Three solutions: (b) the emitter is on a glass coverslip and an immersion objective is

used; (c) the emitter is in a micro-cavity; (d) the emitter is in a nanocrystal.

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724 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

In most quantum cryptography procedures, the information bit is encoded in the photon

polarization by means of an electro-optical modulator which shifts the photon from its ini-

tial polarization to the desired polarization. This requires that the initial photon polarization is

known: since nanocrystal emission is degenerate, one has to add a polarizer at the output, which

implies the loss of half of the signal. Improvements could also be obtained by coupling to a single

cavity mode of speciﬁ c polarization.

24.5 Beyond the isolated emitter – QD/environment interactions

24.5.1 Introduction

Besides atomic-like properties such as photon antibunching and discrete energy levels, QDs also dis-

play a range of dynamical effects the “ isolated artiﬁ cial atom ” picture cannot account for. Indeed,

QDs signiﬁ cantly interact with their local environment due to their high surface-to-volume ratio, * and

dynamical effects in their surroundings directly translate into random ﬂ uctuations of QD properties.

Environmentally induced ﬂ uctuations ﬁ rst appeared on the scene of single QD photolumines-

cence spectroscopy in the form of ﬂ uorescence intermittency and spectral diffusion [86, 87] . As

will be shown, these effects never escaped researchers curiosity because of the surprising prop-

erties they exhibit (non-ergodicity, aging) and the window they opened on condensed matter

dynamics at the nanoscale.

First considered as speciﬁ c to a small number of pathological emitters embedded in glassy

media or relaxing crystals, environmentally induced ﬂ uctuations have since been reported in a

variety of systems (silicon nanocrystals [88] , single molecules [89] , ﬂ uorescing polymers [90] ,

light harvesting complexes [91] , green ﬂ uorescent proteins [92] , carbon nanotubes [93] ) and,

incidentally now appear as the norm rather than an exception.

Compared to other blinking emitters, QDs have the distinct property of combining size tunabil-

ity and intense brightness with a high immunity against photobleaching. These characteristics

naturally make them ideal for measurements over long durations in order to accumulate statis-

tics on their ﬂ uctuations, and better understand their interactions with the outside world in a

variety of optical, chemical and temperature environments.

In the following, we introduce environmentally induced effects on single QDs through the phe-

nomenon of ﬂ uorescence intermittency, i.e. any isolated QD randomly blinks between a bright

“ on ” state and a dark “ off ” state, where the QD is virtually non-radiative ( Fig. 24.13 ). Existing

data are presented and their signiﬁ cance discussed. We explain how single QD blinking handles

the trapping/detrapping dynamics of charges around a given QD, and highlight some counter-

intuitive effects blinking has on ensemble quantities. Further examples are provided to illus-

trate how, quite generally, charge motion and changes in the environment of a single QD can be

tracked in the ﬂ uctuations of its optical properties (excited-state lifetime, phonon coupling and

emission wavelength). Finally, current challenges and perspectives in the elucidation of environ-

mentally induced ﬂ uctuations are presented.

24.5.2 Quantum dot ﬂ uorescence intermittency

Among the properties of single-QD ﬂ uorescence, ﬂ uorescence intermittency is probably the most

striking one, seen simply by looking through a high numerical aperture microscope at the ﬂ uo-

rescence of a dilute sample of QDs dispersed on a glass substrate. Blinking is a surprisingly ubiq-

uitous phenomenon, routinely observed on any QD sample, and so far largely indifferent to any

form of control [94] .

24.5.2.1 On states, off states

Recorded over long durations with an avalanche photodiode or a CCD camera, the intensity time

trace of each individual nanocrystal forms a binary “ random telegraph ” signal ( Fig. 24.13 ).

* Twenty ﬁ ve per cent of the core atoms lie on the core/shell surface for a spherical 2 nm CdSe QD.

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Quantum Optics with Single CdSe/ZnS Colloidal Nanocrystals 725

Figure 24.13 Fluorescence intermittency of single CdSe nanocrystals under continuous laser excitation (Ar



–

488 nm) at room temperature [95] . (a) Fluorescence intensity trajectories as recorded on a CCD camera (10 ms

integration time) at various laser excitation intensities k

. The higher the excitation power, the shorter the on events.

(b) Log–log plot of the on and off state statistics of an ensemble of 106 QDs at low (red, k

 5 0 W cm

 2

) and high

excitation powers (blue, k

 150 Wcm

 2

). The short timescale behaviour follows a power-law decay with μ  1 / 2

(dotted line). The truncation effects seen on the off states at longer timescales are mostly due to the ﬁ nite observation

window Θ  600 s of the experiment (solid line). The ﬁ nite value of the acquisition time Θ , however, cannot account

for the truncation effects seen with the on states – these being truncated at earlier timescales as their statistics are

intrinsically truncated, truncation occuring at shorter timescales at higher excitation powers.

100

200 300

Time (s)(a)

400 500 600

 2 W

2

 8 W

2

 32 W

2

 128 W

2

Intensity (a.u.)

1

(b)

5

4

3

2

1

τ (s)

Off

on,off

(τ

on,off

> τ)

Hidden in previous ensemble-averaged measurements, such intermittent ﬂ uorescence traces

come as an unforeseen effect raising questions on the microscopic nature of their on and off

states, and the underlying mechanism at the origin of their switching dynamics.

Nirmal et al . ﬁ rst suggested that the QDs ’ intrinsic radiative recombination observed in the on

state would switch off, due to efﬁ cient Auger recombinations, as soon as a charge carrier escapes

from the core, leaving the QD ionized – emission ﬁ nally resuming when an incoming charge

restores the QD neutrality [86] .

This turned into a reality with three sets of experiments: (i) Klimov ﬁ rst demonstrated an

experiment showing that Auger recombinations in a charged QD are much faster than radiative

recombination [67] , indicating that charging a QD with a single electron is enough to turn off its

ﬂ uorescence, i.e. to put it in a off state; (ii) data in electrostatic force microscopy from the group

of Louis Brus later conﬁ rmed this scenario, indicating that the charge of isolated QDs under con-

tinuous excitation ﬂ uctuates with time [96] ; and (iii) plasmon enhancement of the dark-state

ﬂ uorescence revealed that dark states are red shifted by ⬃25 meV compared to the bright states,

in good agreement with the fact that a dark QD possess an excess charge  e [97] .

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726 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

24.5.2.2 Environmentally induced ﬂ uctuations

Studied as an isolated phenomenon, blinking simply appears as a binary random telegraph proc-

ess. Experiments correlating blinking events with the measurement of other QD properties (e.g.

its excited-state lifetime, emission wavelength or lineshape), however, revealed a more complex

pattern, as the ionization/neutralization dynamics underlying QD ﬂ uorescence intermittency is

accompanied by charge reorganization in or around the QD. The QD therefore ﬁ nds itself sur-

rounded by a time-varying electric ﬁ eld, abruptly changing each time a blinking event happens.

As a result, QD on/off blinking dynamics fully correlates with sudden temporal ﬂ uctuations

of many of its ﬂ uorescence properties such as spontaneous Stark shifts in its ﬂ uorescence spec-

tra, as expected for QDs experiencing strong, time-varying internal ﬁ elds under the inﬂ uence of

electrons relaxing in their vicinity [98, 99] ( Fig. 24.14a ). Similarly, time-resolved spectroscopy

and photoluminescence decay measurements on single QDs reveal that blinking also correlates

with spontaneous changes in the excited-state lifetime ( Fig. 24.14b ) or in the exciton–phonon

coupling strength [100] .

Time (s)(b)

(a) Time span  9 s

Blinks and shifts

20 40 60 80 100 120

Fluorescence

intensity (kHz)

Excited state

lifetime (ns)

Emission

wavelength

Figure 24.14 Examples of blinking-induced ﬂ uctuations in the properties of single QDs. (a) Blinking-induced

spontaneous spectral jumps, reproduced from [99] . (b) Blinking-induced ﬂ uctuations of a QD excited-state ﬂ uores-

cence lifetime – the excited-state lifetime appears to have changed during the off state shown by the arrow [95] .

QD ﬂ uorescence intermittency and its related spontaneous sudden emission properties ﬂ uctua-

tions therefore appear as a pace clock of the QD dynamics. The remainder of this section focuses

on this QD clock, its tempo, and the mechanisms accounting for its peculiar motion.

24.5.2.3 Blinking kinetics: measurement and basic properties

Numerous strategies have been developed to study the temporal dynamics of ﬂ uorescence inter-

mittency each representing a different compromise between high temporal resolution – to resolve

short on and off events and high sensivity – meaning long integration times in order to detect QD

ﬂ uorescence under low excitation.

24.5.2.3.1 Fluorescence intensity time traces

In the vast majority of experiments, the determination of single QD blinking statistics consists

in recording the ﬂ uorescence intensity of many individual QDs. Due to the binary nature of the

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Quantum Optics with Single CdSe/ZnS Colloidal Nanocrystals 727

switching process, the time trace of each QD is considered as a sequence of on and off times

{,,,,,,}

()

ττττ ττ

off

…

from which their corresponding statistical (cumulative * ) dis-

tributions P

( τ ) and P

off

( τ ) can be determined ( Fig. 24.13b ). First experimental observation by

Nirmal et al . in 1996 revealed that unlike most atomic-like single emitters, CdSe QDs have on and

off state durations following non-exponential statistics [86] . Further studies by Kuno et al. con-

ﬁ rmed this observation, and showed that off states are actually power-law distributed, meaning

that their distribution can be written as:

off

() .ττ

∝ 1

The exponent μ

off

usually peaks around a value of 1/2 for any QD sample. The correspond-

ing power-law behaviour is observed over a broad range of timescales extending from 10

 4

s to

tens of minutes in both cryogenic (10 K) and room temperature conditions, independently of the

power of the continuous optical excitation and of the QD ’ capping layer [101] .

The statistics of on states, on the contrary, strongly depend on the experimental conditions

(capping layer thickness, temperature, excitation power, cf. Fig. 24.13a ,b). For capped QDs under

vanishing excitation, on states also have a statistical cumulative distribution P

( τ ) following a

power-law decay:

()ττ

∝ 1

with μ

⯝

off

 1/2. Shimizu et al. later showed that raising either the excitation intensity or the

QDs ’ temperature, or thinning their cap, results in the appearance of a truncation of the power-

law statistics at shorter and shorter timescales [102] . As will appear below, these seemingly

uncomplicated experimental facts are actually enough to make QD blinking a subtle and fasci-

nating puzzle, as much for its exotic statistical properties as for anyone attempting to determine

its microscopic origin.

24.5.2.3.2 Intensity autocorrelation measurements

While ﬂ uorescent time traces suffer from low time resolution, ** several strategies have been

developed to investigate the blinking kinetics at a faster timescale. Intensity correlation experi-

ments, for example, consist in recording correlations in the detection times of photons and offer

the same time resolution as photon antibunching experiments, i.e. down to the excited state life-

time of the emitter. Although the approach itself does not allow the determination of the on and

off event statistics, it provides some insight on the short timescale intensity ﬂ uctuations [103] ,

and shows, indeed, that blinking occurs down to timescales as short as microseconds [28, 104] .

Variations of this approach have been reported, which consist either in a direct measurement

of the Mandel parameter Q on single QDs [105] , or in experiments performed in the frequency

domain, measuring the Fourier transform of the intensity correlation function (i.e. the intensity

noise power spectra) of ensembles of QDs [106] .

24.5.2.3.3 Intensity decay measurements

As mentionned above, the on and off waiting time distributions P

( τ ) and P

off

( τ ) coincide in

the limit of low excitation intensities and low temperatures, but differ otherwise. Assuming the

sample excitation starts at time t  0, the fraction of QDs in the on state at later times directly

depends on the detailed shape of the distribution P

( τ ) and P

off

( τ ). As a result, the overall time

* The cumulative distribution P ( x ) of a random variable X is the probability of observing the variable X with a

value X  x. P ( x ) is directly related to its histogram h ( x ) of the random variable X by

Px hxdx

() ()

∞

∫

. The

cumulative distribution P ( x ) must be preferred to the histogram when studying experimental data. Indeed, the

histogram of a random variable is always more noisy than its corresponding cumulative distribution, as (i) the

binning process suppresses all information on the exact ordering of the random variables belonging to a partic-

ular bin and (ii) the histogram h ( x ) is always dependent of the set of bins chosen for its calculation. Cumulative

distributions, in comparison, do not suffer these limitations, as their calculation does not involve any bins.

** This time resolution rarely exceeds milliseconds due to the integration time needed to collect enough photons

and determine the state (on/off) of the QD at any given time.

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728 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

evolution of the intensity I ( t ) of an ensemble of QDs reﬂ ects the competition occurring between

the on and off state under the inﬂ uence of their different waiting time distributions. This

approach, ﬁ rst used in [107] ( cf . Fig. 24.15c ) and later championed by Chung and Bawendi

[108–110] , appears as an efﬁ cient strategy for studying the long time (  2 hours) behaviour in

the on and off states ’ statistical distributions and determine the dependence of their respective

cut-offs against temperature and excitation intensity on large QD samples.

24.5.3 Blinking kinetics

Exploration into QD blinking kinetics has a two-fold purpose; it aims both at a broader understand-

ing of the statistical properties of samples formed from large number of emitters undergoing power-

law blinking statistics, and at the physical elucidation of the microscopic mechanisms producing

power-law blinking statistics. In the following, we ﬁ rst review some basic properties of power-law

statistics, and later investigate their inﬂ uence on the temporal dynamics of collections of QDs.

24.5.3.1 Power-law blinking statistics – broad distributions

The power-law distributed waiting times are at the core of peculiar emission properties of

these nanoscale emitters. To see this, let us ﬁ rst consider the simple situation corresponding to

a QD observed at low temperature (4 K) under very low excitation power. As explained above,

in this case, both its on-and off-state blinking statistics have a pure power-law behaviour with

 μ

off

 μ  1/2 over all accessible timescales, i.e. from the integration time τ

of the photo-

detector ( τ

⬃ 1 ms) to total duration Θ over which blinking is recorded (up to a few hours).

The important point is that for distributions P

on,off

( τ ) with a power exponent μ

on,off

 μ ,

all moments of order higher than μ diverge [111] . For QDs, we have μ  1. Hence, in the low-

temperature/low-excitation regime, the average duration of the on and off states becomes formally

inﬁ nite, e.g. implying that even if a number of commutations of the order of N  ( T / τ

)

1/ μ

 1 0

occurred during the course of the QD ﬂ uorescence trajectory, time averages computed by inte-

grating a single QD trajectory over the whole acquisition time Θ will not converge to any deﬁ -

nite value. This situation completely contrasts with exponential blinking kinetics, for which time

averages are deﬁ ned within a relative accuracy of the order of

110

/ N ⬃



after N  1 0

com-

mutations! We see that compared to standard emitters with exponential blinking statistics, QDs

undergoing ﬂ uorescence intermittency converge extraordinarily slowly (if ever * ) towards their

statistical equilibrium.

24.5.3.2 Power-law statistics in single molecule spectroscopy

Appearing in ﬁ elds ranging from complex system physics (chaos [112] , earthquakes [113] and

sandpiles [114] ) to random walks [115] , anomalous diffusion and transport problems [116] ,

broad power-law distributions – also called Lévy statistics – appear to be generally associated to

unusual statistical properties and dynamics, and must be handled with special care as common

intuition mostly derives from experiences involving regular distributions, i.e. with a variance or

a mean value [111] . Statistical physics, for example, is largely based on the ergodic principle, i.e.

on the hypothesis that ensemble and time averages of an observable coincide [117] . This hypoth-

esis therefore implicitly assumes that systems possess a ﬁ nite correlation time. How does it apply

in systems driven by broad distributions of times, and therefore lacking any ﬁ nite typical times-

cale over which ﬂ uctuations could be integrated to yield a time average? What are the dynamical

properties of systems with arbitrary long correlation times?

Single-QD spectroscopy indeed provides unprecedented opportunities to investigate such ques-

tions. In condensed matter systems, answers are signiﬁ cantly complicated as observations proceed

from macroscopic measurements leaving the detailed elementary processes unresolved due to the

* In practice, the long-time cut-off observed in the on-and off-state waiting time distributions however, ensures

convergence of time averages to a well-deﬁ ned value [110] . This asymptotic “ standard ” behaviour however,

does not change our conclusion on the anomalously long durations required for convergence of the blinking

process to reach its statistical steady state.

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