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 709

dots. Due to quantum conﬁ nement, the ﬂ uorescence of individual nanocrystals is similar to a

two-level system in terms of single-photon generation. Section 24.4 discusses the characteristics

of this single-photon source. Since colloidal quantum dots are solid states structures, they are

not isolated and interact strongly with their surrounding environment. The consequences on the

ﬂ uorescence of CdSe nanocrystals are reported in section 24.5. We detail the properties of ﬂ uo-

rescence intermittency and more generally show that emitter/environment interactions induce a

dynamics in many ﬂ uorescence characteristics. More speciﬁ cally, in section 24.6, we analyse the

diffusion of the wavelength emission in terms of time coherence and coalescence of single pho-

tons. The two ﬁ nal sections gather preliminary results concerning new quantum optics experi-

ments carried-out on CdSe colloidal quantum dots. In section 24.7, we discuss the possibility of

generating entangled photon pairs using biexcitonic emission. Section 24.8 presents the most

recent implementations of cavity quantum electrodynamics concepts applied to control the emis-

sion of CdSe nanocrystals. Finally, we conclude on the perspectives opened by colloidal quantum

dots in terms of realizing new quantum optics experiments.

24.2 Theoretical background on single-photon sources

24.2.1 Introduction: quantum key distribution and single photons

Among the ﬁ elds in which single-photon sources could be useful, quantum cryptography is the

closest to practical realization, and some commercial products are already available [1] .

The aim of quantum cryptography is to permit the secure distribution of a secret key between

a sender (conventionally named Alice) and a receiver (Bob). The idea of quantum key distribu-

tion (QKD) was ﬁ rst proposed by S. Wiesner in the 1970s [2] and by C.H. Bennett and G. Brassard

in 1984 [3] . It relies on the fundamental rule of quantum physics that any measurement per-

turbs a system. This allows Alice and Bob to detect the presence of an eavesdropper (Eve), over

whatever technical means Eve may have. For a more precise description of QKD protocols, one

can read the review by Gisin et al. [4] .

Photons, being quickly and easily transmitted over long distances, are the most efﬁ cient way to

encode the secret key. If each bit is coded on a single photon and the transmission and detection

are perfect, eavesdroppers are always detected. In a real system, however, Eve can perform a few

attacks which will be mistaken for transmission losses or detector dark counts. There is thus, for a

given detector efﬁ ciency and dark count rate, a maximum amount of losses above which QKD is

not secure (as discussed in [5] ). Losses being a function of distance, this corresponds to a maxi-

mum distance over which QKD can be securely performed.

The ﬁ rst realization of QKD based on single photons was demonstrated in 1992, over a 30 cm

distance in free space [6] . Since then, a great effort has been made on the transmission and detec-

tion devices, motivated by the perspective of creating a QKD network involving telecom-ﬁ bre net-

works and satellites [7–9] . Transmission distances up to 100 km of telecom ﬁ bre [10, 11] , 1 km

in free space at nighttime [8] and 75 m in free space under bright daylight conditions [7] can be

achieved.

In these experiments, single photons were approximately obtained by attenuating laser pulses.

These can be described by a coherent state with low average photon number: 〈 N〉  1. The prob-

ability of ﬁ nding N photons in such a coherent state follows the Poisson distribution:

pN e

()

.

〈〉

〈〉

(24.1)

Thus there is a non-zero probability (around 〈 N〉

/2) of having more than one photon cod-

ing a bit. Eve can then in principle select these bits, keep one photon in order to detect its state

and send the others to Bob without being noticed. To prevent such a “ photon number splitting

attack ” , the ratio of multiphoton- to one-photon-qubits 〈 N〉/ 2 can be made arbitrarily small by

choosing a large pulse attenuation ( 〈 N〉 → 0), but then most pulses are empty ( p (0) ⬃ 1) and

Bob measures mostly dark counts. Eventually, this leads to a reduced secure QKD distance [5] .

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

For this reason, as well as for further fundamental-optics and quantum-computing realiza-

tions, various “ true ” single-photon sources have been investigated: trapped atoms and ions

[13–15] , molecular impurities in a crystal [16–19] or in a polymer matrix [20] , some nitrogen-

vacancy defects in diamond called “ colour centres ” [21–23] , semiconductor self-assembled

quantum dots [24–26] , and the semiconductor colloidal nanocrystals [27, 28, 80] which are the

subject of this chapter.

In 2002, QKD with true single photons was realized, using either a colour centre [29] or a

quantum dot in a micropost cavity [30] . By simulating a variable transmission distance with an

attenuator, Waks et al . demonstrated experimentally that longer secure-transmission distances

could be achieved with a true single-photon source [30] .

24.2.2 Statistical properties of a light beam

Let us now set the basic elements of theory which are involved in single-photon characterization.

Demonstrations and further developments on this matter can be found in quantum optics text-

books such as [31] .

24.2.2.1 Quantized electromagnetic ﬁ eld

A mode of the electromagnetic ﬁ eld is deﬁ ned by its wave vector



, its angular frequency



and

its polarization



perpendicular to



. The operator which creates a photon in this mode is writ-

ten

†



(



being the operator which annihilates a photon in this mode).

In the Heisenberg picture, the electric ﬁ eld in a cavity of volume V writes, as a function of time

t and position



(,)

(,)Ert E rt E rt



(24.2)

where the positive-frequency component





decomposes into the different mode annihilations:



(,)

()

†

Ert i

ikr t







ω

∑

(24.3)

and the negative-frequency component





is its Hermitian conjugate – a sum of creation

operators.

For the following, we assume the ﬁ eld operator to be a scalar (meaning that the ﬁ eld is polarized).

24.2.2.2 Photodetection

Photodetectors in the optical range, such as photomultipliers or photodiodes, rely on a quantum

process in which a bound electron is excited by the incoming light to a continuum. The free elec-

tron gives rise to a photocurrent

which is measured by electronic means. The quantum analy-

sis of this problem, developped by Glauber in [32] , shows that the mean photocurrent is:

〈〉 〈 〉

()

()It E tE t



(24.4)

where the ﬁ eld is considered at the position of the detector * and α depends on the detector. If

one assumes a photodetector with 100% quantum efﬁ ciency, one can deﬁ ne the photocurrent

operator:

()

()It E tE t



(24.5)

which enables us to consider not only mean values but also photocurrent quantum ﬂ uctuations.

* One should actually sum over the whole photodetector area: let us neglect this spatial aspect.

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Quantum Optics with Single CdSe/ZnS Colloidal Nanocrystals 711

Let us for clarity discuss here the case of a single-mode ﬁ eld. The photodetection signal

is pro-

portional to the photon number

ˆˆ

†

aa N

. If the different photons are statistically independent,

the photodetections are distributed following Poisson statistics, so that the variance of the meas-

ured N is

ΔNN 〈〉

. Coherent states (deﬁ ned as eigenstates of

) exhibit such a Poissonian dis-

tribution. Usual laser beams, as previously mentionned, can be described by coherent states, so

that their photon number ﬂ uctuates as

〈〉N

. This noise, called “ shot noise ” , constitutes the low-

est limit for the light ﬂ uctuations of classical sources like thermal or discharge lamps.

Sub-Poissonian instensity ﬂ uctuations can be obtained for states called “ intensity squeezed

states ” at the cost of increased phase ﬂ uctuations. If one can produce a number state such as the

single-photon state |1〉 , N will be deﬁ ned exactly ( Δ N  0) but the phase will be random.

States showing sub-Poissonian intensity ﬂ uctuations can be used to enhance the sensitivity of

intensity measurements. For example, if one wants to measure a weak absorption signal, it is bet-

ter to send a single photon: the absence of photon detection at the output will correspond to one

absorption event [33] .

24.2.2.3 Second-order coherence

In order to characterize single photons and other sub-Poissonian behaviours, one usually meas-

ures the second-order ** correlation function g

(2)

( τ ) . When we restrict ourselves to a stationary

state of light, the g

(2)

function is symmetrical and deﬁ ned for τ  0 b y :

EtEt Et Et

EtEt

()

()()()()

() ()

ττ





  



〈〉

ˆˆ ˆ ˆ

ˆˆ

(24.6)

Equation 24.6 corresponds to the quantum expression of the classical normalized intensity

correlation 具 I ( t ) I ( t  τ ) 冭 / 具 I ( t ) 冭

Long-time correlations are simply measured by use of a photodiode, either by measuring I ( t )

and calculating g

(2)

( τ ), or, in the photon-counting regime, by plotting the histogram of the delays

between detected photons.

** The ﬁ rst-order correlation function is deﬁ ned by

gEtEtEtEt

()

( ) ( ) () () ()

ττ



〈〉〈〉

ˆˆˆˆ

and is the

Fourier transform of the emission spectrum (Wiener–Khintchine theorem). It is measured by interference

experiments, such as the Fourier spectroscopy protocol described in Fig. 24.19 .

Light source

Beam splitter

Photon

counters

Stop

Start

Figure 24.1 Hanbury-Brown and Twiss set-up (a 50/50 non-polarizing beam splitter and two detectors), with a

“ start–stop ” analyser and a delay D between the two photodetectors.

This method is not appropriate for short-time correlations in the photon-counting regime, since

the avalanche photodiodes used for single-photon detection exhibit a period of blindness ( “ dead

time ” ) of a few tens of nanoseconds after each detection. This difﬁ culty is overcome by use of

the set-up proposed by Hanbury-Brown and Twiss [34] , which measures the cross-correlations

between the intensities detected at the two outputs of a 50/50 beam splitter ( Fig. 24.1 ):

EtEt Et Et

EtEt

ab b a

()

()()()()

() ()

ττ





  



〈〉

〈〉〈

ˆˆ ˆ ˆ

ˆˆ

EEtEt

ˆˆ



() ()〉

(24.7)

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

In a quantum description, the ﬁ elds at the outputs are given by the same relations as in a clas-

sical one * **:

ˆˆ





(24.8)

so that the measured normalized cross-correlation g

is equal to the second-order correlation g

(2)

of the incident beam.

The electronic treatment of the data is then done by a “ start–stop ” time-interval analyser.

Since the analyser cannot work on strictly zero times, a delay D is introduced so that g

(2)

(0) 

( D ) can be measured.

Let us mention that a start-stop system plots the distribution of the delays between consecutive

photons. This is not strictly equal to the distribution of delays between all photon pairs which

would yield g

(2)

, but is approximately equal to it for delays shorter than the inverse of the photon

detection rate, which is typically 10 to 100 μ s (see [35] , p. 720).

Let us now consider, from a theoretical point of view, the second-order correlations of a state

of light. For a coherent state, one easily ﬁ nds:

()

() .

1τ 

(24.9)

Thus detection of one photon tells us nothing about the probability to detect the next one. This

enforces the Poissonian image of coherent states as a beam of independent photons.

Using a classical description of the electromagnetic ﬁ eld, Cauchy’s inequality 2 具 I ( t

) 〉

具 I ( t

) 〉  具 I ( t

) 〉

 具 I ( t

) 〉

leads to g

(2)

(0)  1. Since 具 ( I ( t

)  I ( t

))〉

 0, one can also show that

(2)

(0)  g

(2)

( τ ). These equations indicate that photons emitted by a classical source are more

likely to be detected together ( “ bunched ” ). These two inequalities become equalities in the case

of a coherent state: for intensity correlations, as for intensity ﬂ uctuations, the coherent state sets

the limit between super-Poissonian behaviour and non-classical sub-Poissonian behaviour.

We now turn to the number states | n 冭 , corresponding to exactly n photons in a certain mode.

We have already shown how these states exhibit sub-Poissonian behaviour in terms of ﬂ uctua-

tions. Here we ﬁ nd:

()

() .

1





(24.10)

This effect can be understood just by considering photons as particles. Since the total number

of photons is known, detection of one of them makes it less likely to detect some others: the emis-

sion is called “ antibunched ” . In particular, for the single-photon case |1 冭 , there is no photon left

to detect after the ﬁ rst one and g

(2)

(0)  0. Measuring this complete antibunching is the stand-

ard way to evidence single-photon sources. Partially antibunched emission was ﬁ rst observed in

1977 from a beam of sodium atoms [12] . Complete antibunching was later demonstrated from

a single trapped ion [13] and then from other sources such as molecules and semiconductor

nanostructures.

The variation of g

(2)

corresponding to different kinds of light emitters is plotted on Fig. 24.2 .

24.2.2.4 Two-level system

In a ﬁ rst approximation, single-photon sources are usually modelled by a two-level system: a fun-

damental state | f 〉 and an excited state | e〉 . One can easily understand that, because of the ﬁ nite

duration of its excitation–emission cycle, this system cannot emit two photons at the same time.

* ** The vacuum ﬁ eld impinging on the other side of the photodiode is omitted here as it has no effect on the

calculated correlations.

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Quantum Optics with Single CdSe/ZnS Colloidal Nanocrystals 713

This is conﬁ rmed by basic quantum optics considerations. It can be shown that the emitted

ﬁ eld, far from the source, is proportional, in the source domain, to the dipole operator. More pre-

cisely, detection (annihilation) of a photon (term



) corresponds to a relaxation of the system

(term



||fe〉〈

) – and in the same way



is associated to



|

ef〉〈

. The effect on expec-

tation values is, for instance:

〈〉〈 〉

()

()Et E t w t t

  

 SS

(24.11)

where the coefﬁ cient w depends on the source and the collection efﬁ ciency, and similar relations

hold for higher-order correlations. Given that

ˆˆ



 0

, it is easy to show that the emission

from a two-level system is completely antibunched ( g

(2)

(0)  0 ) .

If one considers the emission from

independent dipoles, it can be shown that g

(2)

(0) 

(

 1 ) /

, so that the emitted light is bunched for any

 2 number of dipoles. With a

large number of dipoles we ﬁ nd the classical case: g

(2)

(0)  2. In order to obtain a single-photon

source, it is thus required to collect the ﬂ uorescence of a single emitter.

In order to calculate the theoretical g

(2)

( τ ) function and compare it with experimental results,

one must introduce a more speciﬁ c model. Spontaneous emission has to be described in a den-

sity-matrix formalism. * The elements of the density matrix

are deﬁ ned as:

ρρ

ee ff



 





〈〉

(24.12)

where

ee f f || | |〉〈 〉〈

is the population inversion operator, the diagonal components ρ

and

are the populations of levels | f 〉 and | e〉 , and the off-diagonal components ρ

and ρ

are the

coherences of the dipole.

We ﬁ rst consider the case of an off-resonance excitation, which is the most common in prac-

tice (as photoluminescence cannot be spectrally separated from scattered excitation if excitation

is resonant). By deﬁ ning  ω

, Γ

 1 / T

, Γ

 1 / T

and r being, respectively, the energy between

| e 〉 and | f 〉 , the excited-state decay rate, the emitting-dipole decoherence rate and the pumping

rate ( Fig. 24.3 ), one writes:



ρρρ ρρ

ρω ρρρ

ff ee ff ef fe

fe fe ff ee

  

 

;

(); .

(24.13)

0 246

(a)

(b)

(c)

(d)

Second-order correlation

function g

(2)

(τ)

τ (arb. units)

Figure 24.2 Second-order correlation g

(2)

function in the cases of (a) a classical lamp with a collision broadening

of 1, (b) a coherent laser beam, or a single-photon source of lifetime T

 1, (c) pumped off-resonance at a rate 0.1,

or (d) excited at resonance with a Rabi frequency 3 (arbitrary units).

* Or in the more recent “ quantum jumps ” formalism [36] .

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

Quantum correlations for an operator

()O t

are calculated by use of the quantum regression

theorem, which states that [37] :

if, for

then

τττ





0, ( ) ( ) ( ) ,

() ( )

〈〉 〈〉

〈

∑

OO O

ˆˆ

ˆˆ ˆ

tft

(() ( )( )().tf t

kikj

〉〈〉

∑

 τ OO O

ˆˆ ˆ

Solving Eqs 24.13 leads to:

〈〉 〈〉

⎛

⎝

⎜

⎞

⎠

⎟

〈〉

()











te t

(() ()

() ()



 



ee t

ττ

〈〉

⎛

⎝

⎜

⎞

⎠

⎟

(24.14)

and, by direct application of the quantum regression theorem, one obtains:

()



 兩

兩

(24.15)

which corresponds to a Lorentzian emission spectrum of width 2 Γ

, and:

() ( )

()

ττ



 兩兩

(24.16)

which indicates a complete antibunching dip of width 1/( Γ

 r ).

As an experimental example, Fig. 24.4 shows the measured intensity correlations for a

colloidal nanocrystal. They clearly agree with the present theoretical description. The excita-

tion being performed well below saturation ( r  Γ

), a value T

 20 ns can be measured, in

accordance with direct lifetime measurements. A 95% antibunching is obtained. The non-zero

value of g

(2)

(0) is due to the background noise, which was not included in our model. When the



Figure 24.3 Model for a system with two levels |f 〉 and |e〉 excited off-resonance: h

–

, Γ

and r are, respectively,

the energy between |e〉 and |f 〉 , the excited-state decay rate, the emitting-dipole coherence and the pumping rate.

Coincidence counts

800

600

400

200

Delay (ns)

100 100 200 3000

Figure 24.4 Second-order correlation function of the emission of a CdSe/ZnS colloidal nanocrystal of diameter

3.5 nm, excited well below saturation, and measured by use of a Hanbury-Brown and Twiss set-up with start–stop

detection and a delay D  150 ns, from [28] (authorization for reproduction asked). The solid curve is a ﬁ t by an

exponential function a [1  b exp(  |τ| / τ

)] , with a  577.5, b  0.95 and τ

 20.1 ns.

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Quantum Optics with Single CdSe/ZnS Colloidal Nanocrystals 715

background is correctly substracted, one ﬁ nds an antibunching greater than 99%. Other single-

photon sources were evidenced in the same way.

24.2.2.5 Single photons on demand

Single-photon emission has been evidenced for a few sources by antibunching experiments.

However, when the source is under continuous excitation (as on Fig. 24.4 ), single photons are sep-

arated by an interval which cannot be controlled. A better way to proceed is to excite the source

by a pulsed laser, with a pulse duration t

pulse

much shorter than T

(so that only one emission

occurs during one pulse), and intervals between pulses T

pulse

much longer than T

(so that two

photons corresponding to consecutive excitation pulses are well separated in time) (see Fig. 24.5 ).

The delay between two photons is then a non-zero multiple of T

pulse

: nT

pulse

 2 T

, as evidenced by

intensity-correlation measurements [38] .

pulse

>> T

pulse

<< T

Figure 24.5 Realization of an on-demand single-photon source by pulsed excitation of a nanocrystal, with a pulse

duration t

pulse

shorter than the excited-state lifetime T

, and an interval between two pulses T

pulse

longer than T

Although photon antibunching is sufﬁ cient for quantum key distribution, it would be preferable

to generate on demand single photons: one photon detected for each excitation pulse ( Fig. 24.5 ).

As explained in the beginning of this section, only a certain amount of losses are acceptable to per-

form secure QKD, which results in a limited distribution distance. If only some of the excitations

lead to single photon detection, this is equivalent to losses and reduces the secure-QKD distance.

Realization of on-demand single-photon sources is one of the challenges in single-photon stud-

ies. It requires:

● A pulse intensity sufﬁ cient to saturate the transition.

● A near-unity quantum yield (deﬁ ned as Γ

/( Γ

 Γ

), Γ

and Γ

being, respectively,

the radiative and non-radiative components of the decay rate Γ

● 100% collection and detection efﬁ ciency.

We now outline the optical properties of colloidal nanocrystals (typically, 3.5 nm diameter

CdSe/ZnS nanocrystals). Then, we will compare them to other single-photon sources, in terms of

on-demand emission and other considerations.

24.3 Optical properties of a nanocrystal

24.3.1 Colloidal semiconductor nanocrystals

The chemical synthesis of quasi-spherical nanocrystals with monodisperse nanometre size is

delicate as it requires the control of the kinetics of the nanocrystal growth in order to avoid fast

agglomeration. One possibility is to perform the reaction in a vitrous medium at high tempera-

ture. This method was used in the Middle Ages to fabricate coloured glasses, and more recently by

companies like Corning to make optical ﬁ lters. The ﬁ rst studies on quantum conﬁ nement by A.I.

Ekimov in Leningrad in the early 1980s considered such samples [39, 40] . At the same time, the

group of L.E. Brus in Bell Labs synthesized semiconductor nanocrystals in solution, by a careful

control of the reagents and stabilizing agents, of the temperature and duration of the reaction,

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

and the same conﬁ nement effect was evidenced [41] . This last method appeared to be the most

promising as nanocrystals in solution could be more easily handled (see [51] for more informa-

tion on nanocrystal history and synthesis).

Synthesis techniques are ideal for CdSe nanocrystals, using typically spherical 3.5 nm nanocrys-

tals of hexagonal (wurtzite) crystalline structure, emitting at 585 nm at room temperature. A size

dispersion as low as 5% is obtained. In order to passivate the unbound electrons on the surface and

improve the quantum yield, a shell of one to two monolayers of another semiconductor is added.

This other semiconductor must have a larger band gap, so that the charges remain conﬁ ned in the

core. Usually, one uses ZnS; such nanocrystals are labelled “ core-shell CdSe/ZnS ” [42, 43] .

For the moment, we will only describe the optical properties that are related to the conﬁ ne-

ment of the charge carriers in a small volume – that is, the nanocrystal can be considered as a

sphere. In section 24.5, we will describe phenomena that are caused by the passivating shell and

the local environment of the nanocrystal. Two of these phenomena are:

● “ Blinking ” : the nanocrystal switches randomly between an “ on ” ﬂ uorescent and an “ off ”

non-ﬂ uorescent state.

● “ Spectral diffusion ” : the emission wavelength changes randomly in time.

24.3.2 Elements of theoretical description

Electrons and holes in a nanometre-sized semiconductor are described either (i) by means of

quantum chemical calculations (pseudo-potential method, tight binding method, etc.) – it is then

easier to include a complex shape of the nanocrystal [44] , surface reconstruction [45] , addition

of shells or ligands [46] or surface defects [47] – or (ii) in the framework of the envelope function

[48] , which is used here as it gives a more physical picture.

The formalism of the envelope function relies on the assumption that the nanocrystal diameter

2 R , although small, is still much larger than the crystalline lattice parameter (0.43 nm for CdSe),

so that the notion of valence and conduction bands, respectively containing holes and electrons,

is still relevant. The wavefunction Ψ of a charge is of the form:

Ψ() () ()



rrur

(24.17)

The function u

has the same period as the crystalline lattice and reﬂ ects the local behaviour of

the charge. It is the same function as in the bulk material, with μ indicating to which (conduction

or valence) band it corresponds. Φ is an envelope function reﬂ ecting the charge distribution over the

nanocrystal. If one assumes that the energy bands are parabolic, Φ is the wavefunction of a free par-

ticle (electron or hole) with effective mass

eh/

. The further assumption that this charge is conﬁ ned

in the nanocrystal by an inﬁ nite potential leads to the easy boundary condition: Φ ( r  R )  0 (more

elaborate models have been developed in [49] and [50] ). Thus the result of “ quantum conﬁ nement ”

is that only a few discrete energy levels compatible with Φ ( R  0) are accessible ( Fig. 24.6 ).

1.7 eV3.6 eV

6.6 eV

ZnS

CdSe

Figure 24.6 Schematic of a CdSe/ZnS nanocrystal, and the corresponding three-dimensional potential well. The

dotted lines symbolize the discrete set of accessible energy levels.

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Quantum Optics with Single CdSe/ZnS Colloidal Nanocrystals 717

As conﬁ nement and Coulombian effects scale, respectively, as 1/ R

and 1/ R

R , R being the

nanocrystal radius and R

the exciton Bohr radius [51] , the former effect is more important than

the latter for nanocrystals smaller than R

(5.6 nm for CdSe). In describing this “ strong conﬁ ne-

ment ” regime, the electron and hole are ﬁ rst considered separately, and Coulomb interaction is

added later as a perturbation. *

For the electron, the problem is then reduced to the well-known free particle in an inﬁ nite

well, which is treated in [52] . A solution in spherical coordinates is of the type

lnl ll

() (,)θφ

where the radial component

is expressed in terms of Bessel functions,

is a harmonic func-

tion, and n , l and l

are the quantum numbers as deﬁ ned for an atom. By analogy with hydrogen

states, these states are labelled nL

, where L is S , P , D … when l  0, 1, 2 … . The discrete acces-

sible values of k

are given by

( k

R )  0, and are thus proportional to 1/ R . The conﬁ nement

energy



2km

nl e

then varies as 1/ R

For the hole, the problem is more complex as it involves the degenerate structure of the valence

band [48] . A complete treatment can be found in [49] . Let us just mention that the correct quan-

tum number is then the sum of the hole spin and the envelope momentum. It is 3/2 for the low-

est hole state (labelled 1 S

3/2

One can then consider the electron–hole wavefunction as the product of an electron and a

hole wavefunction. The effect of the direct component of the Coulomb interaction (the smaller

exchange component will be discussed later) is simply to shift the energy levels. For instance, the

ﬁ rst 1 S

3/2

1 S

level is stabilized at 1.8 e

/4 π ε

R (around 150 meV in CdSe for R  1 . 7 n m ) .

24.3.3 Absorption spectrum

The ﬁ rst decade of experiments on nanocrystals was mostly dedicated to evidencing these dis-

crete features in the ensemble absorption spectra or photoluminescence-excitation (PLE) spectra

[40, 41] ( Fig. 24.7 ), and comparing the measured spectra with theoretical predictions. Reduction

of the inhomogeneous broadening by synthesis of better samples and use of size-selective pump–

probe techniques has revealed up to ten absorption lines, motivating the development of precise

theoretical models including the complex valence band structure [49, 50, 53] .

300 400 500 600

(b)

(a)

Figure 24.7 (a) Photoluminescence spectrum (with excitation at 300 nm) and (b) PLE spectrum (with photolu-

minescence detected at 585 nm) of a solution of 3.5 nm diameter CdSe/ZnS nanocrystals in toluene.

* This does not mean that it vanishes for R → 0; on the contrary, Coulomb interactions are enhanced by the

conﬁ nement, as it brings the charges closer to one another.

More than 1.2 eV above the lowest 1 S

3/2

1 S

level, spectrum discretization cannot be observed.

The levels tend to merge into a continuum because (i) at higher energies, the dispersion rela-

tion is no longer parabolic and becomes concave, and (ii) higher levels are more degenerate, and

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

their degeneracy is lifted by perturbative effects. At these excitation energies, the absorption

cross-section is proportional to the nanocrystal volume [54] , and can be described in the

framework of the Mie theory [55] : the nanocrystal behaves like a macroscopic sphere.

For a single nanocrystal, direct acquisition of the absorption spectrum has never been reported.

For a diffraction-limited spot, the excitation area is around 1 μ m

, while the absorption cross-sec-

tion is 0.1 nm

[54] . The necessary 10

precision could be achieved by acquiring long enough,

but the acquisition time is limited as the nanocrystal blinks from “ on ” to statistically longer “ off ”

states (see section 24.5).

The absorption spectrum has been measured indirectly by photothermal absorption spectros-

copy, a technique that was ﬁ rst developed to detect non-ﬂ uorescing particles such as metallic

nanocrystals. The principle is to measure, by use of a probe beam, the changes in the refractive

index of the surrounding matrix caused by pump absorption-induced heating. The results were

similar to those of ensemble measurements [56] .

The PLE of a single nanocrystal is very delicate due to the presence of spectral diffusion. The

only PLE data available to date used a constant switching between the scanning laser and a refer-

ence laser. They show the lowest 1 S

3/2

1 S

level separated by a 50 meV “ minigap ” from the other

levels, which form a dense quasicontinuum [57] .

To sum up, because of quantum conﬁ nement, nanocrystals exhibit a discrete energy spec-

trum. Nanocrystals are often referred to as “ artiﬁ cial atoms ” . At this point, one might infer that

this quantum-conﬁ nement discretization causes antibunched emission. In fact, other mecha-

nisms, also related to the small size of the particles, play a major role in nanocrystal single-

photon emission.

24.3.4 Exciton and multiexciton relaxation mechanisms

In a nanometre-sized structure where both charges and phonons have discrete energy spectra, a

charge should a priori not interact with a phonon, as there is a priori no ﬁ nal state with the right

energy. This “ phonon bottleneck ” effect [58] was expected to drastically slow down both intra-

band relaxation and decoherence mechanisms. It should be even more important for structures

of a few nanometres like colloidal nanocrystals, as the separation between levels 1 P

and 1 S

is of

a few hundred meV, while phonon energy is at most 25 meV.

Femtosecond transient absorption experiments showed the opposite: intraband phonon-

assisted relaxation (thermalization) occurs within 500 fs [59] and is faster in smaller nanocrys-

tals [60] . The explanation is that:

● Holes relax as quickly as they do in the bulk [61] since they have a large density of states,

because (i) the valence band is degenerate and (ii) the separation between states varies as

the inverse of the effective mass, and holes are heavier than electrons.

● The electrons transfer their energy to the holes by a Coulombian mechanism called the

Auger effect [62] . The Auger effect is enhanced by charge conﬁ nement, since it depends

on the distance between the charges. Fig. 24.8a illustrates how an excited electron–hole

pair decays to its lower energy state before radiative recombination.

Intraband relaxation to the lowest excited state 1 S

3/2

1 S

is thus much faster than radiative

in-terband relaxation ( T

⬃ 20–2000 ns), so that ﬂ uorescence emission occurs only by recombi-

nation of the exciton from its 1 S

3/2

1 S

state, and photoluminescence (PL) spectra show a single

peak ( Fig. 24.7 ).

Thus, in terms of emission, a nanocrystal behaves the same as a two-level system: a funda-

mental (no exciton) and an excited state (the 1 S

3/2

1 S

exciton). Excitation to a higher level fol-

lowed by intraband relaxation is equivalent to the incoherent pumping which was included in

the model of section 24.2. According to this model, such a system should emit single photons.

However, there is an important point which was not included in that model: it is possible, by

intense excitation, to create more than one electron–hole pair in a single nanocrystal. A biexci-

ton peak appears in absorption spectra of nanocrystal ensembles, shifted from the exciton peak

by the biexciton binding energy – around 30meV [63–65] . With single nanocrystals of radius

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