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1032 Part 5 Special Structures

They were named “zeolite” (“boiling stone”) in 1756 by

Cronstedt, a Swedish mineralogist, who observed their

emission of water vapor when heated. At the other size

limit, opals constitute another example of a naturally oc-

curring nanostructured material. These gems are made

up mainly of spheres of amorphous silica with sizes

ranging from 150 nm to 300 nm. In precious opals, these

spheres are of approximately equal size and can thus

be arranged in a three-dimensional periodic lattice. The

optical interferences produced by this periodic index

modulation are the origin of the characteristic iridescent

colors (opalescence).

Apart from these few examples, most nanostruc-

tured materials are synthetic. Empirical methods for the

manufacture of stained glasses have been known for

centuries. It is now well established that these methods

make use of the diffusion-controlled growth of metal

nanoparticles. The geometrical constraints on the elec-

tron motion and the electromagnetic ﬁeld distribution in

noble-metal nanoparticles lead to the existence of a par-

ticular collective oscillation mode, called the plasmon

oscillation, which is responsible for the coloration of the

material. It has been noticed recently that the beautiful

tone of Maya blue, a paint often used in Mesoamer-

ica, involves simultaneously metal nanoparticles and

a superlattice organization [3.1].

The chemistry and color changes of colloidal gold

solutions were observed by Faraday during the 19th

century [3.2]. These properties were due to highly size-

dispersed gold nanoparticles.

Improvements in the diffusion-controlled growth

technique opened up the possibility of growing nano-

crystallites with better-controlled sizes and densities

and permitted its extension to various semiconduc-

tors. The fabrication of colored long-wavelength-pass

glasses and of photochromic glasses provides well-

known examples of commercial technologies based on

such methods developed decades ago. Various tech-

niques for the production and assembly of cluster- and

nanoparticle-based materials are currently under intense

study.

More recently, important technological efforts have

been made, driven by the increasing needs of the elec-

tronics industry, in order to understand and control the

growth of semiconductors at the atomic level. The de-

velopment of molecular-beam epitaxy (MBE) permitted

the control of atomic-layer-by-atomic-layer growth of

semiconductors. It has become possible to create struc-

tures made up of an alternation of different layers,

each of which is only a few atomic layers thick. The

ﬁrst observation of the quantization of energy levels in

a quantum well in 1974 [3.3] opened the way to the tai-

loring of the electronic wave function in one dimension

on the nanometer scale, leading to the production of new

electronic and also magnetic materials. A new trend in

surface science is work aimed at the control of in-plane

nanostructuring, such as the formation of wire or dot

shapes, through self-organization.

In parallel to developments in the ﬁeld of electron-

ics, nanostructured materials have been developed by

materials scientists and chemists also. The concept of

nanocrystalline structures emerged in the ﬁeld of ma-

terials science, and polycrystals with ultraﬁne grain

sizes in the nanometer range have been produced. These

“nanophase materials” have been shown to have sig-

niﬁcant modiﬁcations of their mechanical properties

compared with the coarse-grain equivalent materials.

The huge surface area of nanoporous materials has at-

tracted much attention for applications in chemistry such

as molecular sieves, catalysis, and gas sensing. This

has motivated intense research aimed at the fabrica-

tion of materials with a well-controlled composition and

nanoscale structure, such as synthetic zeolites.

The scientiﬁc and technological domains of research

on nanostructured materials cover a range of disci-

plines, from biology to physics and chemistry. However,

their convergent aspects, as well as, to some extent,

a common type of approach, have been recognized re-

cently in the realm of nanoscience and nanotechnology,

under the term “nanostructured materials”, or simply

“nanomaterials”.

For an extended review on nanotechnology see the

recently published Handbook of Nanotechnology [3.4].

5.3.1.2 Deﬁnitions

In their broadest deﬁnition, nanostructured materials

show structural features with sizes in the range from

1 nm to a few hundred nanometers in at least one dimen-

sion. This very general criterion actually includes very

diverse physical situations.

First, as is apparent from the previous section, each

nanostructured material is associated with a speciﬁc

novel property or a signiﬁcant improvement in a spe-

ciﬁc property resulting from the nanoscale structuring.

As a consequence, the type of nanostructuring used

must be based on a spatial dependence of some param-

eter related to the property under consideration. This

parameter could be, for example, the material density,

transport parameters, or the dielectric constant. Another

consequence is that the upper size limit of the structural

features varies depending on the property considered,

Part 5 3.1

Mesoscopic and Nanostructured Materials 3.1 Introduction and Survey 1033

from a molecular size for molecular-sieve properties up

to the wavelength of light for the optical properties.

Second, in addition to the nanometer-scale struc-

turing, a larger-scale ordering of the unit patterns

may be necessary for the existence of the property

sought. For example, the particular optical prop-

erties of opals mentioned above require the silica

nanospheres to show a long-range order with a co-

herence length well beyond a micron. The same

considerations hold for quantum-well superlattices, for

example. In other cases, the nanosized building blocks

do not need long-range order to provide a speciﬁc

property, but still require some degree of short-range

organization. For example, an electrical conductiv-

ity appears only above a critical percolation density

of conducting particles. Finally, some properties of

nanostructured materials simply reﬂect a correspond-

ing intrinsic property of their individual building blocks.

This is the case, for instance, in nanoparticles embed-

ded in glass or polymer matrices for optical-ﬁltering

applications.

Two main technological approaches may be deﬁned:

•

The top-down manufacturing paradigm consists in

downscaling the patterning of materials to nano-

meter sizes. This allows the generation of materials

Table 5.3-1 Examples of reduced-dimensional material geometries, and deﬁnitions of their dimensionality and of the associated

type of conﬁnement

X,Y,Z

> L

No nanostructures No conﬁnement Bulk material

X,Y

> L

Two-dimensional (2-D) One-dimensional (1-D) Wells

nanostructures conﬁnement

> L

Y,Z

One-dimensional (1-D) Two-dimensional (2-D) Wires

nanostructures conﬁnement

> L

X,Y,Z

Zero-dimensional (0-D) Three-dimensional (3-D) Dots

nanostructures conﬁnement

which are coherently and continuously ordered from

macroscopic down to nanoscopic sizes.

•

The bottom-up paradigm is based on the atomically

precise fabrication of entities of increasing size. It

is the domain of macromolecular and supramolecu-

lar chemistry (dendrimers, engineered DNA, etc.)

and of cluster and surface physics (epitaxy, self-

assembly, etc.).

Mesoscopic materials form the subset of nanostruc-

tured materials for which the nanoscopic scale is large

compared with the elementary constituents of the mater-

ial, i. e. atoms, molecules, or the crystal lattice. For the

speciﬁc property under consideration, these materials

can be described in terms of continuous, homogeneous

media on scales less than that of the nanostructure. The

term “mesoscopic” is often reserved for electronic trans-

port phenomena in systems structured on scales below

the phase-coherence length Λ

of the carriers.

Most of the common nanomaterials can be classiﬁed

in terms of dimensionality, according to the number of

orthogonal directions X, Y, Z in which the structural pat-

terns referred to above have dimensions L

X,Y,Z

smaller

than the nanoscopic limit L

. This leads to the classical

deﬁnitions of dimensionality summarized in Table 5.3-1.

However, it should be noted that experimental situations

Part 5 3.1

1034 Part 5 Special Structures

can be encountered in which the dimensionality may

not be so obviously deﬁned. For instance, the structural

patterns may be formed by nonrectilinear wires which

occupy a surface or a volume or may be branched.

The structure of the density of states (DOS) in nano-

structures is strongly dependent on the dimensionality.

A free 3-D motion yields a band characterized by three

wave vectors k

, k

. The corresponding DOS de-

pends smoothly on the energy E,as(E − E

)

1/2

, where

is the energy of the bottom of the band. Conﬁne-

ment into a 2-D system splits the band into subbands,

and leaves only two continuously varying wave vectors

, k

in each subband. The DOS for each subband is

then constant above the energy E

of its bottom state.

The overall DOS is discontinuous, with a stepwise struc-

ture that is characteristic of quantum wells (Fig. 5.3-1).

Conﬁnement in one additional dimension splits each 2-D

subband further into a set of 1-D subbands. Each 1-D

subband is characterized by only one continuously vary-

ing wave vector k

, and two quantum numbers N, M.

The DOS corresponding to the subband N, M has a vari-

ation of the form (E −E

N,M

)

−1/2

, with a divergence at

the bottom of the subband E

N,M

. The DOS of a quan-

tum wire thus has a more pronounced structure than

does a 2-D well, with a larger number of subbands,

each one starting as a peak (Fig. 5.3-1). Finally, con-

ﬁnement in all three dimensions creates a completely

discrete, atom-like set of states. The DOS of a quan-

DOS

Fig. 5.3-1 Schematic illustration of the density of states for

3-D motion of a free electron (dashed line), a 2-D quantum

well (solid line), and a 1-D quantum wire (shaded area )as

a function of energy

tum dot thus consists of a series of δ-functions (not

represented in Fig. 5.3-1). The sharpening of the DOS

at speciﬁc energies induced by quantum conﬁnement is

the origin of many improvements in the properties of

nanostructured materials compared with bulk materials.

This spectral concentration enhances all resonant effects

and increases the energy selectivity. The preservation of

these effects when one is dealing with an ensemble of

quantum-conﬁned systems requires a high homogeneity.

5.3.1.3 Speciﬁc Properties

The speciﬁc properties of nanostructured materials can

have two different possible origins:

•

Size effects, which result from the spatial conﬁne-

ment of a physical entity inside an element of the

nanoscale structural pattern. Such an element is

called a low-dimensional system. An example is the

conﬁnement of electron wave functions inside a re-

gion whose size is smaller than the electron mean

free path. This class of effects may give birth to

completely new properties.

•

Boundary effects, which are a consequence of the

signiﬁcant volume fraction of matter located near

surfaces, interfaces, or domain walls. Processes that

take place only at such locations may be highly

favored, and properties speciﬁc to structural bound-

aries may also be greatly enhanced.

For a recent comprehensive review of the basic prin-

ciples of the origin of the properties of nanostructured

materials, see [3.5].

5.3.1.4 Organization of this Chapter

Many classiﬁcation schemes for nanostructured ma-

terials exist. These may be based on their chemical

composition, on the technique for their manufacture,

or on their dimensionality. These schemes, however, are

often suitable only for a subset of materials. Moreover,

they generally address only one particular scientiﬁc or

technological approach and its associated community

of specialists. Since, as noted above, most nanostruc-

tured materials are associated with a speciﬁc property,

we have chosen a presentation based on properties. This

scheme allows us to include all nanostructured materials

and is accessible to the largest possible readership.

Consistently with the materials approach of this

chapter and with its limited size, only the proper-

ties of statistical ensembles of nanostructures will be

considered. The speciﬁc behaviors of individual nano-

Part 5 3.1

Mesoscopic and Nanostructured Materials 3.2 Electronic Structure and Spectroscopy 1035

sized objects or devices are not included. Since surface

and interface properties are treated in another section,

emphasis will be put here on size effects. Properties

related to electronic conﬁnement and its consequences

for spectral properties are addressed in Sect. 5.3.2. Ef-

fects of the conﬁnement of electromagnetic ﬁelds are

treated in Sect. 5.3.3. Magnetic size effects are addressed

in Sect. 5.3.4. Finally, we list and brieﬂy describe in

Sect. 5.3.5 some generic methods for preparation of

nanostructured materials.

5.3.2 Electronic Structure and Spectroscopy

5.3.2.1 Electronic Quantum Size Effects

Conﬁned electronic systems are quantum systems in

which carriers, either electron or holes, are free to move

only in a restricted number of dimensions. In the con-

ﬁned dimension, the sizes of the structural elements

are of the order of a few de Broglie wavelengths of

the carriers or less. Depending on their dimensionality,

these structures can be quantum dots (0-D), quantum

wires (1-D), or quantum wells (2-D). Quantum wells

are typically produced by the alternate epitaxial growth

of two or moredifferent semiconductors. Quantum wires

are less commonly encountered, since their fabrication

procedures are much more complicated (Sect. 5.3.4).

One of the most dramatic effects, called the quan-

tum size effect, consists in a redistribution of the energy

spectrum of the system, the density of states becom-

ing discrete along the conﬁnement direction. In the

most simple “particle in a box” model of a quan-

tum well, the energies of the corresponding eigenstates

are

N,k

= N

2 2

2md





(3.1)

where m is the effective mass of the carrier, d is the

conﬁnement dimension, and N is a quantum number.

The ﬁrst term appears because of the quantized motion

in the z direction, whereas the second term represents

the energy of the free x, y motion, characterized by

wave vectors k

, k

. Each value of N deﬁnes a semi-

inﬁnite subband of energy levels. Although the “particle

in a box” model is simplistic compared with real sys-

tems, this limiting case is often successful in describing

the essential features of quantum size effects [3.6].

The importance of the quantum size effect is mainly

determined by the energy differences E

N+1

− E

Quantum size effects become observable when this sep-

aration exceeds the thermal energy of the carriers, so

that adjacent subbands are differently populated. Since

the energy difference E

N+1

− E

increases with N,

it could be anticipated that quantization effects would

be more important for processes involving higher sub-

bands. However, the practical observation of such effects

in systems with a large Fermi energy (E

 E

), i. e.

in metals, is often made difﬁcult because of con-

tributions from many levels and the appearance of

collective phenomena such as plasmon oscillations,

which wash out quantization effects. Most of the spec-

tral data revealing quantum well states in metal ﬁlms

a few monolayers thick have been obtained by angle-

resolved photoemission for Cs on Cu(111) [3.7] and for

Ag on Au(111) [3.8]. The Fabry–Perot-like regularly-

spaced spectrum of energy levels appears particularly

clearly in a study of Ag ﬁlms on Fe(100) for thick-

nesses up to more than 100 monolayers (Fig. 5.3-2).

The superposition at half-integer coverages (27.5and

42.5 monolayers) of two sets of peaks corresponding

to the two closest integer thicknesses emphasizes the

extreme sensitivity to surface homogeneity. Quanti-

tatively, the characteristic roughness of well barriers

must be much below the de Broglie wavelength of

the quantum states observed. This is another reason

why quantum size effects have been observed mainly

in semiconductor nanostructures, which have large

Fermi wavelengths, rather than in metals, which have

Fermi wavelengths comparable to a few crystal lattice

periods.

The existence of discrete electronic states of elec-

trons conﬁned in a small metal cluster has been observed

to inﬂuence the thermodynamic stability of the system,

in particular during the production of sodium clus-

ters in supersonic beams composed of the metal vapor

and an inert gas. The statistics of the relative abun-

dances of different particle sizes reveal the existence

of “magic numbers” for the number of atoms in the

cluster, N = 8, 20, 40, 58, 92,... [3.9]. This has been

interpreted in terms of the existence of degenerate en-

ergy levels in a spherical well with inﬁnite-potential

walls. Particularly stable structures are obtained when

the number of valence electrons is such that it leads

to a closed-shell electronic structure, i. e. a structure

with a completely ﬁlled energy level and an empty up-

Part 5 3.2

1036 Part 5 Special Structures

119

42.5

27.5

Photoemission intensity (arb. units)

Binding energy (eV)

= 16 eV

Ag / Fe (100)

Thickness

(ML)

Fig. 5.3-2 Dots: experimental normal photoemission spec-

tra for Ag quantum wells deposited on Fe(100), with various

thicknesses. Solid curves: ﬁts and background functions.

(After [3.12])

per level. The electronic shell structure induces similar

size dependences in the work function and the electron

density, through Friedel-like oscillations [3.10]. The se-

ries of “magic numbers” is discernible up to N = 1430.

For larger particles, the relative abundance of different

sizes is controlled instead by the stability and relative

sizes of the various facets of the surface of the par-

ticle. It is noteworthy that 2-D quantum well states are

also predicted to produce oscillatory variations of the

energetics of a ﬁlm as a function of thickness, with

similar “magic numbers” for the number of deposited

monolayers [3.11].

5.3.2.2 Breakdown of the Momentum

Conservation Rule

As a result of the Heisenberg uncertainty principle, lo-

calization of carriers in a conﬁnement volume spreads

their wave functions in reciprocal space. The possible

overlap in k space of electron and hole wave functions

breaks the rule of conservation of crystal momentum.

In other words, the momentum needed for a transi-

tion at a different k may be provided by scattering

from the boundaries. This consequence of quantum

conﬁnement is particularly meaningful for nanostruc-

tures based on indirect-band-gap semiconductors, for

which luminescence from recombination of the lowest-

energy electrons and holes is forbidden in the bulk

material, but becomes increasingly possible for sys-

tems of decreasing size, with a scaling law of d

−6

for dots [3.13]. The observation of bright luminescence

from nanoporous silicon samples [3.14] was, from the

start, interpreted in terms of quantum conﬁnement into

silicon nanowires [3.14,15]. This observation triggered

an important researchactivity motivated by theindustrial

prospect of being able to incorporate silicon photonic

elements into the current silicon technology. Although

it has been shown that other phenomena such as sur-

face effects and phonon modes (discussed later) come

into the play, and that more complicated geometries

including both quantum dots and quantum wires have

to be considered, the role of quantum conﬁnement in

the luminescence of porous silicon is generally con-

sidered to be central, at least for the “red” part of the

luminescence. The studies of nanostructure-induced lu-

minescence have been extended to porous GaP [3.16,17]

and SiC [3.18], two other semiconductors with an indi-

rect band gap, and the results parallel those for porous

silicon. Quantum-conﬁned luminescence has been ob-

served in indirect-gap semiconductor nanostructures

with well-controlled geometries such as silicon and

germanium 2-D wells [3.19] and quantum dots (see

e.g. [3.20]), which may possibly be integrated into light-

emitting devices [3.21]. The observation of optical gain

in silicon nanocrystals [3.22] opens up new prospects

for the creation of silicon-based lasers [3.23].

5.3.2.3 Excitons

in Quantum-Conﬁned Systems

A very general effect of quantum conﬁnement in semi-

conductors is a widening of their optical band gap. For

the model system of an inﬁnite-barrier 2-D quantum

well, (3.1) shows that the lowest energy(N =1) of a con-

Part 5 3.2

Mesoscopic and Nanostructured Materials 3.2 Electronic Structure and Spectroscopy 1037

duction electron is increased compared with the bottom

of the conduction band. A similar consideration applies

to the lowest energy of a hole in the valence band. Both

effects contribute to an increase in the minimum ex-

citation energy of the system compared with the bulk

band gap. This blueshift has a comparable magnitude in

2-D and 0-D structures [3.24]. This constitutes the lead-

ing contribution to the characteristic blueshifts in the

optical spectra of strongly quantum-conﬁned semicon-

ductor systems. However, the conﬁnement of oppositely

charged carriers at reduced separations also has dramatic

effects on the electron–hole Coulomb energy and thus

on exciton formation.

Two regimes of exciton conﬁnement must be distin-

guished, depending on the conﬁnement size d compared

with the Bohr radius a

∗

of the Mott–Wannier exciton in

the bulk semiconductor [3.25]. The weak-conﬁnement

regime corresponds to sizes such that d ≥ 4a

∗

.Inthis

regime, therelative electron–holemotion, and in particu-

lar its binding energy, is essentially left unchanged. The

exciton can still be considered as a quasiparticle, but its

center-of-mass translational motion becomes quantized.

A simple model [3.26] of a single-point quasiparticle

with mass M

equal to the total mass of the exciton,

= m

∗

, explains the observed frequency shifts.

The translational state with the lowest energy, N = 1,

which is the only optically allowed state, has a nonzero

kinetic energy. Compared with an inﬁnite-sized system,

the lowest excitation energy in the weak-conﬁnement

regime is thus increased by

∆E

Weak

2 2

. (3.2)

Like that for electron states, this shift has comparable

magnitudes for various dimentionalities. In the weak-

conﬁnement regime, the exciton wave function still

involves combinations of several conduction-band elec-

tron states and valence-band hole states, and not only the

lowest-energy states. The quantization of carrier states

does not change signiﬁcantly the average energy of all

of the states involved in exciton formation. As a con-

Table 5.3-2 1s exciton Bohr radius for various semiconductors

Direct-band-gap semiconductors

Semiconductor CuCl Diamond CdTe CdS GaN CdSe GaAs

∗

(nm) 0.7 0.85 2.8 2.9 3.6 5.6 12

Indirect-band-gap semiconductors

Semiconductor GaP SiC Si Ge

∗

(nm) 1.17 2.7 4.9 17.7

sequence, in the weak-conﬁnement regime, the exciton

energy is not affected by the band gap increase which

results from the increase in the energy levels of the

lowest electron and hole states discussed above. The

weak-conﬁnement model is very suitable for describing

spectral variations in nanoparticles of semiconductors

with low exciton Bohr radii, such as CuCl (Table 5.3-2),

as shown in Fig.5.3-3.

The strong-conﬁnement regime corresponds to the

opposite limit, d ≤2a

∗

. In that case the relativeelectron–

hole motion is strongly affected by the barriers. In the

direction of restricted motion, the kinetic energies of the

lowest-energy electron and hole states determined by

the quantum conﬁnement are larger than the Coulomb

0.005

0.01

0.05

0.1

0.5

1.0

R/a*

02468101214

Energy shift (E*

)

= ∞

= 1

= 5

= 10

Fig. 5.3-3 Observed values of energy shifts in the lu-

minescence peak of CuCl nanocrystals in NaCl (circles)

and in the absorption peak of CdS nanocrystals in sili-

cate glass (triangles), compared with theoretical models:

weak-conﬁnement model (dashed-dotted lines), strong

conﬁnement model (short-dashed line); σ =m

∗

is con-

sidered as a ﬁxed parameter. (After [3.25])

Part 5 3.2

1038 Part 5 Special Structures

attraction. The exciton states are thus formed from un-

correlated electron and hole states of the “particle in

a box” model in the conﬁnement direction, whereas spa-

tially correlated electron–hole bound states can still be

formed in the unrestricted directions. This yields 2-D,

1-D, and 0-D excitons in quantum wells, quantum wires,

and quantum dots, respectively. The binding energy of

a 2-D exciton can be deduced from the two-dimensional

hydrogen atom problem [3.27]. The exciton Rydberg

series changes from E

∗

3-D

=−R

∗

in the 3-D case

to E

∗

3-D

=−R

∗

/(n −1/2)

in the 2-D case, where R

∗

is the Rydberg constant for the exciton. This effect is op-

posed by the increased energy required for the creation

of an electron–hole pair owing to the conﬁnement of the

carrier motion. The variation of the excitation energy of

the ﬁrst 1s-like exciton state in the regime of strong 1-D

conﬁnement is then

∆E

2-D

Strong

2 2

2µd

−3R

∗

, (3.3)

where µ is the electron–hole reduced mass, such that

−1

= m

∗

−1

∗

−1

. The ﬁrst term is larger than the

term due to the weak conﬁnement of exciton motion

(3.2) because of the difference between µ and M

.The

0-D case is treated in detail in [3.25]. The variation of

the minimum excitation energy in the regime of strong

3-D conﬁnement is

∆E

0-D

Strong

2 2

2µd

−1.786

4πεd

−0.248R

∗

. (3.4)

The divergence as d

−1

of the second term for small sizes

results from the increased Coulomb interaction in this re-

stricted geometry compared with the 2-D case. As could

be expected, the case of a 1-D exciton in a quantum wire

is intermediate between 0-D and 2-D, with a divergent

Coulomb interaction which is sublinear in d

−1

[3.28].

For all dimensionalities, in the strong-conﬁnement

regime, the increased kinetic energy of quantum-

conﬁned carriers is dominant, and a blueshift of the

optical gap is observed that varies continuously between

the weak- and the strong-conﬁnement regimes. How-

ever, the considerably increased exciton binding energy

in strongly conﬁnedsystems yields more pronounced ex-

citon effects. For the most widely used semiconductors,

excitonic effects are not discernible in room tempera-

ture spectra without quantum-conﬁnement. However,

in strongly conﬁned geometries of any dimensionality,

many semiconductors exhibit well-resolved excitonic

peaks since the exciton binding energy can then ex-

ceed the thermal excitation energy k

T . Both the

blueshift of the spectra and the increasingly pronounced

excitonic absorption are very evident in II–VI semi-

conductor nanoparticles, as exempliﬁed in Fig. 5.3-4

for CdSe [3.29]. Semiconductor-doped glasses, formed

from a dispersion of II–VI semiconductor crystallites

in a silicate glass matrix, are the basis for some com-

mercially available yellow-to-red long-wavelength-pass

optical ﬁlters [3.30]. The glasses with a cutoff wave-

length in thevisible contain CdS

1−x

nanocrystallites,

and the glasses used in the infrared ﬁlters contain es-

sentially CdTe nanocrystallites. Their abrupt absorption

band edge can be tuned by adjusting the composition,

the size [3.31], and, as will be discussed later, the

concentration of the particles. In other widely used semi-

conductors, such as Si, Ge, and III–V compounds, the

valence bands may have complex features and contain

light- and heavy-hole branches. The two corresponding

types of excitons will have different characteristics, such

as different binding energies.

The second important consequence of the squeez-

ing of excitons in the strong-conﬁnement regime is

an increase in the optical matrix element associated

with their excitation. A detailed discussion of this ef-

fect can be found in [3.32]. Its origin is the stronger

overlap of electron and hole wave functions in reduced-

dimensional excitons. This effect, associated with the

concentration of oscillator strength into a few discrete

transitions, has found many applications in optoelec-

tronic devices. Certainly the most important is the

fabrication of electrically powered multiple-quantum-

well (MQW) lasers in In

1−x

As and Al

1−x

(infrared), and Al

1−x−y

P (red). These systems

are widely used today in data storage systems, bar code

scanners, laser pointers, and printers. The physics and

technology of MQW lasers are far beyond the scope

of this section. Details may be found in [3.33], for ex-

ample. 1-D and 0-D systems are also of interests for laser

systems [3.34]. Laser applications of quantum-conﬁned

systems exploit the enhanced stimulated-emission coef-

ﬁcients that result from the increased transition matrix

elements and spectral sharpening in quantum-conﬁned

systems. Besides MBE-grown devices, optical gain in

chemically synthesized CdSe nanocrystal quantum dots

has been achieved [3.35]. The operation of an op-

tically pumped solid-state distributed-feedback (DFB)

laser based on a CdSe-doped titania matrix gain medium

has been demonstrated [3.36]. The output color of this

laser can be selected by choosing appropriately sized

nanocrystals and tuning the DFBperiod accordingly (see

Fig. 5.3-5).

Another consequence ofthe large oscillatorstrengths

associated with conﬁned-exciton transitions is the

Part 5 3.2

Mesoscopic and Nanostructured Materials 3.2 Electronic Structure and Spectroscopy 1039

700 600 500 400

115

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

Wavelength (nm)

D (Å)

–∆

67Å

Intensity

67Å

PLE

Intensity

10 K

62Å

PLE

Energy (eV)

Absorbance (arbitrary units)

enhancement of nonlinear optical and electrooptical co-

efﬁcients. Nonlinear absorption and electroabsorption

correspond to changes in absorption coefﬁcients in-

duced by high-intensity optical ﬁelds and large DC

Fig. 5.3-4 (a) Low-temperature (10 K) absorption spectra

of CdSe dots dispersed in a polymer matrix, for vari-

ous mean diameters.

(b) Nanosecond saturated-absorption

spectra for 67 Å dots. The pump beam had a 7 ns FWHM.

The excitation photon energy, 1.984 eV, is prolonged as

a dashed vertical line from

(a) for easier comparison.

The horizontal dashed line corresponds to ∆OD = 0.

(c),(d) Photoluminescence(PL) and photoluminescence-

excitation(PLE, observed at 2.022 eV) spectra for 67 Å

dots

1.8 1.9 2.0 2.1 2.2 2.3 2.4

700 650 600 550 500

Energy (eV)

Wavelength (nm)

Fig. 5.3-5 The photoluminescence emission spectra of

CdSe nanocrystals with different radii (dashed and dotted

lines, for radii of 2.7nm,2.4nm,and2.1nm)andof1.7nm

CdSe/ZnS core–shell nanocrystals (the radii are given in

order of increasing photon energy) at low pump power, and

at a pump power above the laser threshold (solid lines).

The nanocrystals were dispersed in a titania ﬁlm used in

a distributed-feedback laser conﬁguration at 80 K. (After

[3.36])

electric ﬁelds, respectively. Exciton–exciton interactions

increase the energy of doubly excited states in quantum

dots, which results in large saturable absorption effects

(see Fig. 5.3-4b). The absorption saturation intensities

may be as low as those of saturable dye solutions,

but these systems have an increased robustness and

the possibility of including them in solid-state media.

Part 5 3.2

1040 Part 5 Special Structures

PbS-nanocrystal-doped glasses with an absorption satu-

ration intensity of 0.2MW/cm have been realized [3.37]

and used as intracavity passive Q-switches in picosec-

ond [3.37] and femtosecond [3.38] pulsed near-infrared

lasers. The Frantz–Keldysh effect is also strongly en-

hanced in quantum-conﬁned systems [3.39]. Moreover,

the well-resolved excitonic transition shows quantum-

conﬁned Stark effects (QCSEs) with particularly large

magnitudes. In quantum wells, this effect usually con-

sists of a redshift of the absorption and a strong decrease

of its oscillator strength, and has been exploited for the

realization of fast, low-drive-voltage electrooptic mod-

ulators [3.40], which could possibly be integrated with

semiconductor lasers [3.41].

5.3.2.4 Vibrational Modes

and Electron–Phonon Coupling

The description in terms of phonons of the vibrational

properties of solids assumes a translational invariance.

It can be used only inside domains of homogeneous

mechanical properties with sizes larger than the spa-

tial extension of the phonon considered, typically given

by its wavelength λ

. Similarly to the case of vi-

brational modes of impurities in crystals [3.43], the

conﬁnement of phonons in structures smaller than λ

leads to the localization and the quantization of the

vibrational modes. For typical solids, the phonon fre-

quencies affected by conﬁnement sizes on the 1–100 nm

scale have frequencies on the order of 0.1–10 THz, i. e.

thermal phonon energies and higher, and both acous-

tic and optical phonons are affected. The frequencies of

the quantized modes, as well as their damping rates,

depend on the size, shape, and nature of the envi-

ronment of the nanostructure and its interfaces. The

electron–phonon coupling may be strongly modiﬁed

by phonon conﬁnement, with manifestations in the

transport properties and in the optical absorption and

luminescence.

In 2-D systems, two-dimensional phonons appear,

with properties and consequences similar to those of

surface phonons. The role of in-plane propagating

phonons in electron–phonon scattering rates has been

demonstrated in metal ﬁlms [3.44] and in Si/Ge quan-

tum wells [3.45]. Phonon propagation perpendicular to

a quantum-well superlattice produces Bragg-like selec-

tive transmission [3.46]. The formation of phonon band

gaps results from the zone folding of the LA and TA

phonons propagating perpendicular to the layers, and

is well described by a mesoscopic model of a periodic

series of homogeneous layers [3.47].

When we move to 0-D systems, the phonon spec-

trum changes to a set of size-dependent discrete lines.

As in the case of 2-D systems, most data on the var-

ious vibrational modes of metal nanoparticles [3.48]

and semiconductor nanocrystals (see e.g. [3.49] and

references therein) embedded in solid matrices have

been obtained by spontaneous-Raman-scattering spec-

troscopy. The lowest frequencies are well described

by the simple Lamb theory [3.50] of the vibration of

a homogeneous, elastic sphere. The breathing mode,

the mode of lowest energy, is usually the best resolved

one. A detailed calculation of the mesoscopic acoustic

modes in quantum dots of various compositions em-

bedded in various matrices can be found in [3.51]. The

availability of ultrafast pulsed lasers, with a pulse du-

ration shorter than the period of an acoustic vibration

mode, offers the possibility to monitor the vibrations

in the time domain rather than the frequency domain, as

shown in Fig. 5.3-6 for 3 nm PbS quantum dots in a poly-

mer matrix. Similar experiments have been performed

on Ag particles of various sizes in a glass matrix [3.52].

These experiments offer a unique opportunity to ana-

lyze the damping of these vibrations, which is generally

attributed to radiation of acoustic waves into the matrix.

The modes excited by ultrafast laser pulses are

usually not the same as those detected by spontaneous-

Raman spectroscopy, mainly for symmetry reasons

[3.53] (see Fig. 5.3-7). Moreover, the symmetry se-

lection rules depend on the mechanism of excitation

by the laser pulse. A thermal mechanism provokes

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

012345

∆T/T

Delay (ps)

Fig. 5.3-6 Time-domain breathing-mode acoustic oscilla-

tions in 3 nm PbS quantum dots embedded in a polymer

matrix, observed by transient saturated absorption. (After

[3.42])

Part 5 3.2

Mesoscopic and Nanostructured Materials 3.2 Electronic Structure and Spectroscopy 1041

54 3 2 1

0 0.5 1

Frequency (THz)

1/radius (nm

–1

)

R (nm)

Fig. 5.3-7 Frequency of the coherent-phonon and Raman

peaks as function of PbSe dot size. The solid lines repre-

sent the calculation on the assumption that the stress at the

dot boundary is zero, and the dashed line represents the cal-

culation with a rigid boundary condition. The matrix was

a phosphate glass. The inset represents the displacement

in the S

and S

spheroidal modes. The coherent-phonon

and Raman frequencies are assigned to the S

and S

modes, respectively. (After [3.53])

breathing-mode (S

) oscillations, whereas electrostric-

tive excitation excites the elongation mode (S

Figure 5.3-7 also illustrates the parallel dependences

of the modes excited by Raman scattering and by ultra-

fast laser-pulse on particle size. These dependences are

consistent with a Lamb model with boundary conditions

corresponding to the absence of stress.

The conﬁnement also inﬂuences the electron–

phonon coupling. For example, in GaAs, each quantized

optical-phonon mode produces a Huang–Rhys ladder in

the phonon sidebands in the spectrum of the exciton

transition [3.55]. A scaling law in d

−3

is predicted for

the effective coupling strength [3.55]. It has been shown

that the luminescence spectra of silicon nanocrystals

exhibit not only a no-phonon peak but also TO- and

TA-phonon-assisted peaks. Hence, besides the break-

down of the momentum conservation rule discussed

above, an increased electron–phonon interaction also

favors luminescence in this indirect-band-gap semicon-

ductor. This coupling prevails for moderate conﬁnement

energies, below 0.7 eV [3.56]. The enhanced electron–

phonon coupling results not from phonon quantization

but rather from an increased overlap factor [3.57].

In metal particles, the main consequence of changes

in the electron–phonon couplings is a variation of the

electron relaxation rates. This effect depends strongly

on the intensity regime of the excitation, but exper-

iments carried out at weak excitation on Ag and Au

nanoparticles in various matrices show a strong increase

of the relaxation rate, i. e. a decrease of the electron–

lattice energy exchange time, with quantum conﬁnement

(see Fig. 5.3-8) [3.54]. In this case also, the increased

0.9

0.8

0.7

0.6

0.5

1.2

1.0

0.8

0.6

0.4

0 5 10 15 20 25 30

τ (ps)

Nanoparticle diameter (nm)

Fig. 5.3-8 (a) Size dependence of the measured electron-

lattice energy exchange time τ

e–ph

for silver clusters in

BaO

−

(circles), in Al

(open squares), in MgF

(diamond), in a polymer (up triangles), and deposited on

a glass substrate (down triangle). The solid line is the result

of the theoretical model proposed by the authors of [3.54].

The dashed line is a guide to the eye.

(b) As (a), for gold

clusters in Al

(squares), in colloidal solutions prepared

using a radiolysis technique (triangles), and in commer-

cially available colloidal solutions (circles). (After [3.54])

Part 5 3.2