Download Light Scattering of Semiconducting Nanoparticles

Light Scattering of
Semiconducting Nanoparticles
G. Irmer, J. Monecke
Institute of Theoretical Physics, Freiberg University of Mining and Technology,
D-09596 Freiberg, Germany
P. Verma
Department of Applied Physics, Osaka University, Osaka 565-0871, Japan
CONTENTS
1. Introduction
2. Quantum Confinement in Semiconductors
3. Electron–Phonon Coupling
4. Acoustic Phonons
5. Surface Phonons
6. Optical Phonons
7. New Instrumentation and Methods
Glossary
Acknowledgments
References
1. INTRODUCTION
The electronic and optical properties of semiconducting crystallites with a size of a few nanometers (often
called nanocrystals or quantum dots) differ considerably
from those of the corresponding bulk material. Systems
with nanoparticles represent a new class of materials
with promising properties due to their nonlinear optical,
photoconductivity, photoemission, and electroluminescence
behaviors. They have been thoroughly investigated during
the last two decades for their promising practical applications, for instance, in solar energy conversion, optical
processing devices, and photocatalytic processes.
The bulk crystalline structure is preserved in nanocrystals. However, due to quantum confinement nanocrystals
have molecule-like discrete electronic states which exhibit
strong size dependence. Semiconductor nanocrystal systems
are often synthesized in oxide glass hosts or prepared in the
form of dispersed colloids. Among numerous other methods
that have been developed we mention the preparation of
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nanocrystals in polymers, zeolites, and micelles. By electrochemical anodic oxidation of crystalline semiconductor substrates porous nanostructures or free standing columns can
be produced.
Most of the light scattered by a particle is elastically scattered without an energy shift. A small part, a few orders
weaker in intensity, is scattered inelastically. This inelastic
scattering carries information about energetic levels of the
particle. In a Raman scattering process with incident photon energy L the scattered photon S has lost (Stokes
scattering) or gained (anti-Stokes scattering) the energy of an elementary excitation (phonons, electronic transitions,
etc.) of the particle:
S = L ∓ (1)
The Raman scattering is sensitive for probing the local
atomic arrangement. Crystalline and amorphous phases or
the influence of strain can be detected in the Raman
vibrational spectra. Light couples to phonons intermediate through electronic exitations by either the deformation potential interaction or the Fröhlich interaction. The
deformation interaction is present in polar crystals such as
the compound semiconductors as well as in nonpolar crystals such as Si and Ge. In polar crystals the electric fields
induced by vibrations interact with electronic excitations via
the Fröhlich mechanism. When the Raman scattering process involves real intermediate electronic states by excitation
with energy near gaps of the electronic band structure, the
scattering cross section may be resonantly enhanced [1]. The
Raman scattering probes the electron–phonon interaction as
well as the confinement of the phonons and contains information about the confined vibrational and electronic states.
The light-scattering process in an infinite crystal obeys the
pseudo-momentum conservation relation
= k
L ∓ q
k
S
(2)
Encyclopedia of Nanoscience and Nanotechnology
Edited by H. S. Nalwa
Volume X: Pages (1–26)
2
S k
L denote the wavevectors of the plane waves
where k
of the scattered (incident) light, q is the wavevector transfered to the excitation. The energies of crystalline vibrations
probed by Raman scattering are about <01 eV and those of
the exciting visible laserlight are about 2.5 eV (corresponding to the wavelength L ≈ 500 nm). Therefore, from Eq. (1)
we get S ≈ L and the wavevector in a typical backscat ≈ 4n/L and
tering experiment will be on the order of q
much smaller than the wavevector q = 2/a at the boundary
of the Brillouin zone, a is the lattice constant, and n denotes
the refractive index of the sample. That means that only
excitations near the center of the Brillouin zone ( -point)
will contribute to the Raman scattering.
If the process of light scattering by excitations is limited
to the finite space occupied by the nanoparticle, an uncertainty of the transferred wavevector q occurs and excitations
with wavevectors from the whole Brillouin zone contribute
to the scattering. This effect will vary with size as well as
with the eigenfrequencies of acoustic and optical phonons
and the electron–phonon interaction in small particles. The
vibrational Raman spectra of nanoparticles can therefore be
used for size determinations. This is important because one
of the major goals in the field of preparation is the synthesis
of monodisperse nanocrystals with a narrow size distribution
and with well-defined surfaces. Size effects will be discussed
in Sections 3, 4, and 6.
In addition to acoustic and optical phonon modes,
surface-related modes appearing in polar crystals will be discussed in Section 5. They are observable for particle sizes
smaller than the wavelength of the exciting laser light inside
the particle.
The pioneering work was the first observation of confined acoustic modes with low-frequency Raman scattering
in nanocrystals embedded in glasses by Duval et al. [2] and
the unambiguous observation of surface-related modes in
semiconductor nanoparticles by Hayashi and Kanamori [3].
It is relatively easy to produce nanocrystals of II–VI compounds embedded in different media such as glasses, polymers, or solutions. Besides potential applications of such
systems the possibility of growing isolated nanoparticles in
glasses with desired size and narrow size distribution and
nearly spherically shaped particles makes such systems interesting for light-scattering investigations in comparison with
theoretical calculations.
The observation of strong visible photoluminescence at
room temperature from porous Si [4] with the possibility of
optoelectronic applications has initiated intensive research
on nanostructured silicon (e.g., reviewed in [5]).
Intensive light emission from Ge nanocrystals has also
been found.
Within the last few years there has also been progress
in the fabrication of nanostructured compound semiconductors with desired structures, especially porous III–V
semiconductors.
Nanostructuring provides important degrees of freedom
for phonon engineering. It may change considerably the density of phonon states, induce surface-related vibrations, and
spatially confine the bulk phonon modes.
Although not expected by use of the term “quantum dots”
for nanoparticles, many properties of confined vibrations
Light Scattering of Semiconducting Nanoparticles
and surface modes can be understood by applying classic
physics developed about 100 years ago [6, 7].
Inelastic light scattering also has been very intensively
investigated in two-dimensionally layered semiconductor
structures exhibiting one-dimensional confinement in growth
direction. Readers may refer to reviews in Refs. [8, 9].
2. QUANTUM CONFINEMENT
IN SEMICONDUCTORS
Semiconductor crystals with sizes of a few nanometers typically contain several hundreds to several thousands of atoms;
they are too small to have bulk electronic properties. Quantum confinement in such crystals arises due to the finite
size of the crystal, which limits the motion of electrons,
holes, and excitons. In a small nanocrystal with size comparable to the Bohr radius of the corresponding bulk material, spatial confinement effects on the electron–hole system
become significant. At very small crystal sizes, the character of the exciton starts disappearing and the electrons
and holes become individually confined, because the kinetic
energies of the electrons and holes become dominant and
the Coulomb interaction between electrons and holes starts
losing its significance. Theoretical models predict [10] that
the energy spectrum of such a material consists of a series
of discrete lines, which change their positions with particle
size, and a blueshift of the band gap with respect to the
corresponding bulk material is observed. This blueshift gives
a measure of the confinement, which can be observed, for
example, in absorption [11] or photoluminescence (PL) [12]
experiments.
For a bulk crystal, the electron wavelength can extent
to take all values including zero
to infinity, facilitating k
and, hence, the band structures are parabolic. Electrons
(holes) can occupy any position on the parabolic conduction (valence) band. The band gap is defined as the energy
difference between the conduction and the valence band at
= 0. On the other hand, in the case of a nanocrystal, the
k
electron wavelength is confined to the size of the crystal and,
takes only nonzero discrete values, depending on
hence, k
the particle size. The electrons and the holes with minimum
= 0, and the band
energy can only take a position where k
gap for a nanocrystal has to be defined as the energy difference between the conduction and the valence band at this
which is higher than the band gap of
nonzero value of k,
the corresponding bulk. This can be understood from the
illustration presented in Figure 7 of Ref. [13]. The band gap
for nanocrystals can be expressed as
E = Eg + E
(3)
where Eg is the bulk band gap. The energy difference E
is referred as the confinement energy. The value of E
depends on the crystal size and becomes zero for bulk. The
confinement can be classified into two categories, depending
upon the size of the crystal. If the Bohr radius of the exciton
is aB and R is the particle radius, then the system is said to
have strong confinement if R aB and weak confinement if
3
Light Scattering of Semiconducting Nanoparticles
R aB . In the simplest approach, the confinement energy
in the case of strong confinement is given by [10, 14–17]
E =
2
2
2R2
with
1
1
1
+
=
me
mh
(4)
In the case of weak confinement the confinement energy is
E =
2
2
2MR2
with
M = me + m h (5)
Here, me and mh are the electron and the hole mass,
respectively, and are roots of the spherical Bessel functions; the first root for the lowest energy transition is = .
The simple effective mass model was improved by empirical
pseudopotential [18] and tight-binding [19, 20] methods with
application to II–VI semiconductors. The first-principles
pseudopotential method was also applied to silicon [21]. For
more details see, for example, the review in [22].
The II–VI nanoparticles are the most investigated
materials, and PL spectroscopy is widely used to measure
the confined energy [12, 23]. A typical PL spectrum from
nanoparticles contains a sharp structure corresponding to
the band edge, and some broad structures related to shallow
and deep traps. The structure corresponding to the band
edge is usually asymmetrically broadened, partially due to
the temperature and the shallow traps very close to the band
edge. However, under careful experimental conditions, the
luminescence at the band edge can still be used with sufficient accuracy to estimate the band gap of the material.
As discussed above, the effective band gap of a nanoparticle system is increased due to confinement. This can be
easily observed in PL spectra, in the form of a blueshift
of the band-edge luminescence. Because this shift depends
on particle size, and the samples usually have an asymmetric particle size distribution, the band-edge luminescence
is also partially asymmetrically broadened due to the particle size distribution. It has been observed [13] that in
some cases the peak position of the band-edge luminescence
strongly depends on the probing laser power and, hence, it
is important to measure the luminescence at very low probing power-density to avoid other causes, which may shift
the luminescence peak. By measuring the peak position of
this band-edge luminescence, one can estimate the effective band gap E = Eg + E of the material. By substituting
the bulk band gap Eg and by using the above-mentioned
equation for the confinement energies, one can estimate the
particle size. Although the absolute particle size estimation
using PL experiments alone may not give a very accurate
value of the particle size, it gives a good estimation of the
confinement energy, especially when one wants to compare
the confinements due to changing particle sizes. A comparison with Raman scattering also shows satisfactory agreement
for the particle size estimation.
3. ELECTRON–PHONON COUPLING
Electron–longitudinal optical (LO) phonon coupling, governed by the Fröhlich interaction, is very important in the
electronic and optical properties of a nanoparticle system,
and its investigation has attracted a lot of attention. The
coupling between free charge carriers and vibrational excitations plays a central role in determining the transport
properties and the energy relaxation rates of the excited
carriers. One of the interesting points is that the electron–
phonon interaction in small nanoparticles can be investigated experimentally beyond the bulk approximations. In
this system, the confinement effects influence not only
the electronic and vibrational states, but also the coupling between them. Although the effect of size dependence on the electron–phonon coupling has inspired a
lot of work, the initial studies have produced contradicting results. In the early work, Schmitt-Rink et al. [24]
theoretically suggested that, if electron and hole charge
distributions are identical, then exciton–LO phonon coupling should vanish in small nanoparticles. Klein et al. [25]
predicted size-independent coupling, if the dimensions of
the charge distribution scales as the particle size. However, other authors [26–28] suggested that electron–phonon
coupling increases with decreasing particle size for small
particles. Here, the electron–hole correlation, the valence
band degeneracy, the conduction band nonparabolicity, and
proper confined phonon wavefunctions were considered.
Experimentally, absorption spectroscopy [29] and photoluminescence [30] measurements indicated that the electron–
phonon coupling increases with decreasing particle size. On
the other hand, in some resonant Raman scattering experiments, in which in principle the electron–phonon interaction can be directly probed, either a size-independent
coupling [25] or a decrease of electron–phonon coupling
with decreasing particle size was detected [31, 32]. Later,
Scamarcio et al. [33] unambiguously demonstrated that the
electron–LO phonon coupling increases with increasing confinement in the strong confinement region. This was experimentally demonstrated through the size dependence of
the ratio between the two-phonon and the one-phonon
Raman cross sections in the resonant Raman scattering of
a prototype system. These experiments were done under
improved conditions, with constant resonance conditioning
being maintained for all the particle sizes investigated. It was
experimentally shown that, as the nanoparticle size changes
from about 3 to 2 nm, the electron–phonon interaction
increases more than two times. They also suggested that the
electron–phonon coupling governed by the Fröhlich interaction increases stronger than that governed by the deformation potential with decreasing particle size.
Resonant Raman scattering up to the third order was
studied by Rodríguez-Suárez et al. [34]. The observed relative intensities of the overtones were found to be very sensitive to the particle size, in accordance with calculations
considering the Fröhlich interaction between excitons and
phonons.
For exciton–acoustic phonon interaction the deformation
potential coupling mechanism was shown to predominate in
semiconductor nanocrystals [35, 36].
4. ACOUSTIC PHONONS
4.1. Confinement
The finite size of nanoparticles restricts the motion
of phonons inside a nanocrystal; hence the confinement effects on the phonons can be observed. For a bulk
4
2 s/t 2 = vt2 s + vl2 − vt2 grad div s
(6)
under the corresponding boundary conditions. Here, s is
the lattice displacement vector, and vt and vl are the transverse and the longitudinal sound velocities, respectively. By
classifying the eigenfrequencies according to the symmetry
group of the sphere, two types of vibrational modes can be
obtained, spheroidal modes S and torsional modes T . The
torsional modes are dilatation-free vibrations without a volume change of the sphere and the spheroidal modes are
curl-free.
The eigenvalues are characterized by the angular quantum number and by another quantum number p, which
gives the order of the zeroth of the radial part of the
wavefunction. The torsional mode is only defined for ≥ 1
because the mode with = 0 has no displacement. The
eigenvalues determine the quantized vibrational frequencies
Sp and Tp for the particle with the radius R from the
relations
Sp =
S
p
R
vl
Tp =
T
p
R
vt (7)
where and are the eigenvalues of Eq. (6).
The “wavevectors” of acoustic phonons in nanoparticles
S
are also quantized and are described by the two sets p
/R
T
/R. Figure 1a illustrates the deformations of the
and p
vibrating sphere for the eigenfrequencies with lowest indices.
The total symmetric spheroidal mode or the “breathing”
mode is denoted by S01 , S21 indicates the quadrupolar symmetric spheroidal mode, and T21 is a torsional mode. Only
the lowest modes Sp with p = 1 have large amplitudes
s 2x2
S
ω01
S
ω02
S
ω21
0
T
a
S
ω03
Proofs Only
s2x2
crystal, the phonons can extend up to infinity, which allows
q to take any value in the reciprocal lattice, including zero.
Owing to the scattering selection rules, phonons at the
-point (q = 0) are the only observable phonons in the firstorder Raman scattering. Therefore, the acoustic phonons,
with vanishing energy at the -point, are not observed in the
first-order Raman scattering of a bulk crystal. On the other
hand, because of the uncertainty of momentum transfer for
small particles, acoustic phonons in nanocrystals become
observable in first-order Raman spectra. They are called
“confined acoustic phonons.” The vibrational frequencies of
these phonons are very close to zero. They appear in the
low-frequency range of the Raman spectra, with frequencies
inversely proportional to the diameter of the nanoparticles.
One way to understand these acoustic phonons is to consider the possible vibrations of the whole particle. To calculate these vibrations, the particle can be considered as
an elastic sphere and elastic continuum theory can be used,
along with the proper boundary conditions at the surface
of the nanoparticle. More than 100 years ago, Lamb [6]
discussed the vibrations of a homogeneous elastic body of
spherical shape under stress-free boundary conditions. This
model, which was extended later (see, e.g., Tamura et al.
[37]), is valid if the wavelengths of the acoustic phonons sufficiently exceed those of the lattice constants. It has been
proved to be useful as an approximation for the calculations
of vibrational confinement in nanoparticles. The eigenfrequencies of the elastic body can be obtained by solving the
differential equation
s 2x2
Light Scattering of Semiconducting Nanoparticles
ω2 2
b
S
ω01
0 .2 0 .4 0 .6 0 .8
x /R
1
Figure 1. Some vibrational modes of an elastic isotropic sphere.
(a) Spheroidal “breathing” mode S01 with = 0 and p = 1, spheroidal
quadrupolar mode S21 , and torsional mode T21 . (b) Radial distribution
of the energy density s 2 x2 of the spheroidal breathing mode S01 (surface mode) and the first two overtones S02 and S03 (inner modes) for
spherical particles with free surface.
near the surface of the nanocrystals; the higher modes with
p > 1 correspond to inner modes. Figure 1b shows the radial
distribution of the energy density s 2 R2 of the spheroidal
breathing mode S01 (also called surface mode) and the first
two overtones S02 and S03 (inner modes).
As shown by Duval [38], only the spheroidal modes with
angular quantum numbers = 0 and = 2 are Raman
active. The total symmetric mode = 0 is polarized, and
the quadrupolar symmetric mode = 2 is depolarized. The
simplest case = 0 with the eigenvalue equation
sin = 42 j1 (8)
can be analyzed easily. Here, j1 x = sinx − x cosx/x2
is the spherical Bessel function of first order. Eigenvalue
equations for higher quantum numbers are given, for example, in Refs. [37, 39, 40].
For a free particle, the parameter in Eq. (8) is given
by the ratio of the transverse and longitudinal sound velocities vt /vl . In the case of an embedded particle, the relations
for the continuity of the displacement and of the stress vectors at the spherical surface result in a generalized eigenvalue Eq. (41). The parameter then additionally depends
on the sound velocities in the matrix and on the ratio of
the mass densities of the particle and the matrix. Further,
is obtained as a complex number, sound emission into
the matrix occurs, and therefore the particle vibrations are
damped.
4.2. Raman Scattering
4.2.1. Scattering Efficiency
The Raman scattering is caused by fluctuations in the dielectric susceptibility due to the elastic waves. The Raman
scattering efficiency can be expressed as a function of the
5
Light Scattering of Semiconducting Nanoparticles
elasto-optic coefficients [1, 42]. The wavevector dependence
of an acoustic mode is given by
V
L −k
S − q·
r dV
exp −ik
2
(9)
where n = exp/kT − 1−1 is the Bose–Einstein
population factor. For bulk materials, corresponding to large
dimensions of the scattering volume V in comparison with
1/q, the integral in Eq. (9) can be replaced by the function
− q.
L − k
This expresses the momentum conservation:
k
S
S and of the excitThe wavevectors of the scattered light k
ing light kL are very small. Therefore, only phonons near
the center of the first Brillouin zone with wavevectors q =
∼ 0 contribute to the light scattering which hinders
L − k
k
S
the observation of Raman scattering by acoustic phonons in
bulk material. By assuming nanoparticles to be spherically
shaped with radius R and by integrating over the sphere
volume, the Raman scattering efficiency results in
d/d ∼ n + 1qSq R
(10)
Intensity (arb. units)
d/d ∼ n+1q
2
1
-1 00
3
0
1 00
4
Proofs Only
c
b
a
0
40
80
120
Raman shift (cm -1)
with
Sq R =
3j1 qR
qR
2
(11)
The scattering function Sq R replaces the function. In this case, the wavevector selection rules are
relaxed. Phonons with larger wavevectors contribute to the
Raman scattering, and the observation of the confined
acoustic phonons is allowed. For particles embedded in a
matrix, the elastic excitations are exponentially damped due
to complex eigenvalues. Therefore, we assume for the
phonon modes Sp Lorentzian bands L p bp ∼
1 + 2 − p /bp 2 −1 with the maximum at p and
half-width blp describing the damping. Further, we take into
account the density of states ∼q 2 and a size distribution
cR. Then the Raman scattering intensity is obtained as
d/d ∼ n+13 SqRcRV RL Sp bi dR
(12)
with the wavevector q = Sp /vl and the particle volume
V R.
The homogeneous line broadening due to the matrix
effect is considerable and must be taken into account if
one wants to deduce the particle size distribution from the
Raman spectra [41].
4.2.2. II–VI Semiconductors
Figure 2 represents as an example low-frequency Raman
spectra of CdSx Se1−x nanoparticles with a mean radius of
R = 31 nm embedded in a silicate glass matrix. The spectra were excited with a Ti:sapphire laser operating at a
wavelength far from resonance. The curves (a) and (b)
correspond to polarized and depolarized configurations,
respectively. For comparison, curve (c) shows an unpolarized
spectrum of the base material, which contains no nanoparticles. The broad band with maximum at about 50 cm−1 , which
Figure 2. Low-frequency Raman scattering spectra of CdS067 Se033
nanocrystals embedded in borosilicate glass excited at the wavelength
= 857 nm below the absorption band. The mean particle radius is
3.1 nm. The quadrupolar mode is indicated by 1; the phonon modes 3
and 4 are overtones of the breathing mode 2. (a) Polarized spectrum;
(b) depolarized spectrum; (c) base material of the sample. In the inset
the polarized spectra in the anti-Stokes and in the Stokes regions are
shown.
is typical for glass vibrations and is known as the Boson
peak [43, 44], is also present in the spectra (a) and (b).
Band 1 in curve (a) at 8.5 cm−1 corresponds to the depolarized spheroidal vibration with = 2 and p = 1 and band 2 at
17.5 cm−1 corresponds to the polarized spheroidal vibration
with = 0 and p = 1. Weaker bands 3 and 4 at about 40 and
55 cm−1 can be assigned to the inner modes with = 0 and
p = 2 and p = 3, respectively. The frequencies of the bands
scale with 1/R, as can be seen in Figure 3 where results for
CdSx Se1−x nanoparticles with different sizes are shown. Such
doped semiconductor glasses are used as sharp cutoff filters,
the absorption edge being adjusted by changing the particle
size. The value obtained experimentally agree well with calculations based on the continuum theory. The influence of
the composition x is small as can be seen by the three curves
for each phonon mode calculated for x = 08 05, and 0.2.
Nanocrystals of II–VI semiconductors are conventionally
synthesized either in oxide glass hosts or by colloidal precipitation. The synthesis of quantum dots in polymer films has
also been reported. For more information see, for example,
Woggon [22], Herron [45], Kamat et al. [46], and references
therein.
In the case of semiconductor-doped oxide glasses,
precipitation of the semiconducting dots from the solid
solution is obtained by annealing the glass after quenching. Silicate glasses containing CdSx Se1−x mixed crystals
have been extensively investigated. Particle sizes have
been obtained by Raman scattering of acoustic modes in
6
Light Scattering of Semiconducting Nanoparticles
R (n m )
10
5
4
3
ω03
2
ω02
R a m a n s h ift (c m - 1 )
50
Proofs Only
ω01
30
ω22
ω21
0 .0
0 .2
0 .4
1 /R (n m - 1 )
O G 5 30
O G 5 15
O G 55 0
R G 63 0
R G 66 5
R G 645
R G 695
O G 590
10
0 .6
Figure 3. Measured low-frequency peaks in dependence on 1/R for various filter glass samples. p indicates various polarized modes, depolarized modes, and their overtones. The experimental values for the
corresponding phonons are shown by full circles. The dashed, solid, and
dotted lines correspond to the theoretical values calculated with x =
08 05, and 0.2, respectively. It can be seen that the influence of the
composition x on the phonon frequencies is small. Reprinted with permission from [41], P. Verma et al. Phys. Rev. B 60, 5778 (1999). © 1999,
American Physical Society.
CdSx Se1−x [41, 47, 49–52], CdS [49, 53, 54], CdSe [39, 55],
Znx Cd1−x S [49] crystals in silicate glass matrix, CdS [56], and
CdSe [39] crystals in GeO2 glass matrix.
Other methods used to determine the particle sizes were
transmission electron microscopy (TEM) [23, 47, 56–61],
optical absorption [47, 49, 51, 59, 62], small angle scattering
of neutrons (SANS) [63], small angle scattering of X-rays
(SAXS) [13, 50, 54, 64, 65], and photoluminescence [58, 66].
In addition to size-dependent shifts of the photoluminescence peaks, the discrete acoustic phonon modes could also
be observed in photoluminescence spectra [67].
Saviot et al. [53, 54] observed size and excitation dependence of resonant low-frequency Raman scattering in CdS
nanocrystals embedded in glass. They reported size-selective
excitation, when the exciting laser wavelength was changed
within the absorption band. With variation of the excitation
within the absorption band, they observed dependence of
the low-frequency band position and changes of its depolarization. The low-frequency band was 100% polarized for
excitation below the absorption edge while it was depolarized when excited above the absorption.
etching in hydrofluoric acid solution of heavily boron-doped
silicon wavers. With the assumption of spherical particles, on
the basis of Eq. (7), 2R = 65 nm is obtained. However, due
to the columnar morphology of the porous structure a model
with particles of elongated nonspherical shapes is more realistic. For long cylinders this model yields 2R ≈ 41 nm, 2R
corresponds to the diameter of the columns. A similar value
was deduced from the observed downshift and broadening
of the 520 cm−1 one-phonon peak (see Section 6).
Liu et al. [65] succeeded in preparing Si clusters embedded in a porous Si skeleton. The Si clusters were produced by evaporation and inert gas condensation [69, 70],
and the cluster size was controlled by adjusting the evaporation temperature, the Ar gas pressure, and the distance
between the substrate and the W boat containing the Si powder. The cluster size was determined with TEM measurements. The strongest peaks in the low-frequency region of
the Raman spectra were observed at 29 cm−1 (particle size
3.8 nm) and at 28 cm−1 (particle size 3.9 nm), respectively.
Raman scattering from acoustic phonons confined in Si
nanocrystals which were dispersed in SiO2 thin films have
been observed by Fujii et al. [71]. The samples were prepared by a radiofrequency cosputtering method [72, 73]. The
sizes of the particles were determined from high-resolution
electron microscopic observations. It was found that the
Si nanocrystals are single crystals with good crystallinity.
Raman spectra are shown in Figure 4. As the size decreases,
the peaks shift to higher frequencies. The intensity ratio of
the peaks in the polarized and depolarized spectra is about
0.25 for samples with d = 31 and 3.5 nm; for d = 55 nm a
distinct peak was not observed in the depolarized spectrum.
However, the quantitative comparison between the experiment and the theory based on the continuum model is not
successful. This will be discussed in Section 4.3.
Low-frequency Raman measurements on Ge microcrystals
with sizes of 5 nm and greater embedded in a GeO2 glass
matrix have been performed by Ovsyuk and Novikov [40, 74].
The crystals were grown using the reaction 2GeO(gas) =
Ge + GeO2 [74]. The low-frequency Raman scattering data
4.2.3. Si and Ge
Low-frequency Raman modes of nanometric Si crystallites
were first observed in porous Si [65, 68]. Gregora et al. [68]
found a well-defined peak at 37 cm−1 in the low-frequency
spectra of porous silicon layers produced by electrochemical
Figure 4. Low-frequency Raman scattering of Si nanocrystals embedded in SiO2 matrices. (a) Polarized spectra; (b) depolarized spectra.
Reprinted with permission from [71], M. Fujii et al. Phys. Rev. B 54,
R8373 (1996). © 1996, American Physical Society.
7
Light Scattering of Semiconducting Nanoparticles
were used to show that the vibrations of the Ge nanocrystals differ considerable from those of free particles and that
the influence of the matrix is considerable. Using the theory of Tamura et al. [37] the influence of the matrix on
the spheroidal and torsional modes of nanocrystals was analyzed. It was found that surface vibrations were caused by
the restoring force which limits the free rotation of microcrystals.
4.2.4. Other Compounds
While in semiconducting nanoparticles the spheroidal
breathing mode with = 0 and p = 1 is the strongest in
the Raman spectra, the scattering intensity of this mode is
very low for metallic nanocrystals, for example, consisting
of Ag or Au, and the light scattering of the quadrupolar
mode dominates the Raman spectra. The reason is that in
the latter case the inelastic light scattering is induced by
the phonon–plasmon coupling. The electric dipole moment
of the surface plasmon is modulated by the vibrational
modes. As noted by Gersten et al. [75], the breathing mode
( = 0) will produce no significant modulation; however,
the quadrupolar vibrations ( = 2) will be strongly coupled
to the dipolar plasmon. A depolarization ratio of 0.75 was
determined for the = 2 modes [76].
Light scattering is sensitive to the size and the shape
of the nanoparticles. For ellipsoidally distorted shapes,
the threefold degenerate dipolar plasmon splits into a
lower-energy plasmon oscillating along the long axis of the
ellipsoid and a twofold degenerate higher-energy plasmon
oscillating perpendicular to this axis. Further, the fivefold
degeneracy of the = 2 mode (m = 0, ±1, ±2) is lifted.
The observed large inhomogeneous width of the plasmon
absorption of Ag clusters may be explained by ellipsoidal
distortions because the plasmon energy very weakly depends
on the size [77]. The shape of the clusters can even be
selected by Raman scattering: Palpant et al. [78] investigated
thin films consisting of small Ag clusters embedded in a
porous alumina matrix. Most clusters were roughly spherical, but a noticeable amount of clusters, generally the larger
ones, were found with a prolate ellipsoidal shape, as shown
by TEM investigations. By excitation with the 457.9 nm Ar
laser line close to the absorption maximum at 420 nm, the
low-frequency peaks of spherical nanoclusters were observed
with a depolarization ratio close to 34 . By shifting the excitation to the red region, the low-frequency peak was shifted
to lower frequency with a depolarization ratio close to 13 .
This can be explained by selective enhanced excitation of
the nondegenerate plasmon along the long axis of the ellipsoidally distorted particles by excitation in the low-energy
wing of the absorption band.
Silver nanocrystals embedded in SiO2 [79], alkali halide
[76], and porous alumina matrix [78] were investigated too.
It was shown that the observed low-frequency peaks agree
fairly well with calculations based on Lamb’s theory if the
particle size is less than about 4 nm [79]. Courty et al.
[80] performed low-frequency Raman measurements on silver nanocrystals self-organized on a hexagonal compact
array on highly oriented pyrolytic graphite substrate. Besides
the quadrupolar vibration mode, two weaker bands were
observed at larger frequency shifts. They were attributed to
the spheroidal breathing modes = 0, and p = 1 2, and
their observation was made possible by the high crystallinity
and the narrow size distribution of the particles. Recently,
low-frequency Raman measurements of Au particles supported on CeO2 and Fe2 O3 catalytic matrixes were performed [81, 82] with the aim of measuring their grain size.
Supported by measurements of the depolarization ratio, it
was concluded that the observed bands are due to the = 0
and = 2 spheroidal modes [82].
Confined acoustic phonons were investigated in Sn
nanoparticles embedded in a SiOx thin film, the inelastic
scattered light was measured by a Sandercock interferometer [83]. The observed low-frequency modes scale with the
inverse of the particle size in accordance with Lamb’s theory.
The vibrational modes of the smallest particles were overdamped. A similar result was obtained with femtosecond
pump-and-probe spectroscopy measurements, where oscillations with comparable frequencies were detected for the
big particles but not for the smaller ones. By increasing the
temperature, clear effects of particle melting on the acoustic
and the optical modes could be observed [84].
For SnO2 nanoparticles grain size and distribution
deduced from Raman scattering were found in agreement
with TEM measurements, especially for the smaller particles [85].
4.3. Size Determination
In this section we will discuss other methods that are used
for particle size determination and often correlated with
Raman scattering measurements.
The average sizes calculated from the frequencies at the
low-frequency peak maxima in the Raman spectra are overestimated if a size distribution is present because the scattering intensity is proportional to the particle volume. This
is true for perfect lattice structure; however, lattice defects
may reduce the scattering intensity due to spatial coherence
effects as discussed by Duval et al. [86].
(a) Transmission Electron Microscopy. TEM is often used
to deduce a mean particle size and size distributions
from a limited number of particle images. Preparation
techniques are often pretentious and may affect the
observation; the particle sizes are underestimated.
The lattice images of nanocrystals observed with
high-resolution TEM are produced by interference of
diffracted beams; different types of lattice images are
found, depending on the orientation of the particle
relative to the electron beam. They are visible when
the crystal thickness exceeds a threshold value t0 . For
the case of CdSe in glass, Champagnon et al. [47] estimated a threshold thickness of t0 ≈ 3 nm. Therefore,
the lattice is visible only in particles larger than the
threshold thickness. Further, for larger particles the
lattice image is produced from the central parts but
may be invisible (below a threshold intensity) near
the surface over a width w ≈ t02 /4R, depending on the
particle radius R. The deviation of the apparent size
from the real size, therefore, is much more important
at small sizes. This effect could be one possible reason
for discrepancies in the observed sizes of very small
8
Light Scattering of Semiconducting Nanoparticles
waves. The scattered waves are all in phase when the
scattering direction is the same as that of the incident
beam. As the scattering angle increases, the difference in phase between the various scattered waves
also increases and the intensity of the resultant scattered wave decreases because of destructive interference. The intensity becomes zero when there are as
many waves between 0 and as between and 2;
this will occur for a scattering angle of about 2 ≈
/L, where L is an average size of the particle. The
scattering cross section for spherical particles with
radius R and size distribution cR embedded in a
matrix can be expressed as [94, 95]
d
q E ∼ AE Sq RcRV 2 R dR
d
(13)
where q = 4/ sin is the magnitude of the scattering vector, E is the X-ray energy, and the scattering
function Sq R was defined in Eq. (11). The expression A depends on the electronic contrast between
the particle and the matrix. Figure 5 shows the scattering curve of a sample with CdS067 Se033 nanocrystals. The corresponding Raman spectrum is given in
Figure 2. The solid line has been fitted by assuming
spherically shaped particles with asymmetric size distribution as shown in the inset. The dashed line was
calculated for a size distribution with a -function-like
profile.
From Eq. (13) it can be seen that particles with
larger radius contribute much more to the scattering
cross section than smaller ones. Therefore, average
size values could be overestimated if Guinier plots
and not a complete fit procedure are used.
In Figure 6 results of Raman measurements of nanocrystals in glass matrices are plotted against the inverse radius
A S A X S intensity (e.u./S iO 2 m ol.)
particles determined with different methods [47]; see
Figures 6 and 7.
(b) Absorption, Luminescence. The energetically lowest
optical transition can be used to determine the particle size. The corresponding absorption is often not
a well-defined narrow peak. Some authors used the
second derivative of the absorption spectrum [61] to
locate the band edge; other authors took the energy
at which the absorption appears to extrapolate to
zero [87]. A clear inverse square size dependence in
both the absorption edge and in the photoluminescence was observed in most cases, at least for II–VI
compounds. Potter and Simmons [58] studied CdS
crystals at sizes between 4 and 40 nm and obtained
good agreement with the case of strong confinement
[Eq. (4)]. The bulk Bohr radius in CdS is 2.8 nm. In
general, the determination of size values from absorption and luminescence measurements depends on the
model used and requires knowledge of parameters
such as exciton masses.
(c) X-Ray Diffraction (XRD). The finite crystallite size
leads to a broadening of the peaks in the XRD
spectra. This can be understood if one considers
that in a finite crystal the number of rays reflected
from successive lattice planes that add up in constructive or destructive interference is finite. Therefore, they do not reinforce or cancel out completely.
The determination of the crystalline size from the
XRD peaks is discussed in several books [88–90],
mostly the Scherrer formula is used. The peak broadening gives a weighted average of the mean grain
size, but the grain size may be considerably less than
the particle size [91]. Variations in the lattice constant, for instance, from one crystallite to another
and structural defects will also broaden the peaks.
Several methods have been described to separate
crystalline size- and distortion-induced broadenings.
Further the XRD spectra can be influenced by particle shape effects. For prolate nanoparticles with
more planes contributing to the diffraction in the
direction of the prolate axis the corresponding XRD
peaks can be increased in intensity and reduced in
width as shown for CdSe nanocrystals [92] with sizes
of about 8 nm and for Si nanocrystals [93] with
sizes of about 50 nm. In the last case the peakdependent linewidth broadening was used as evidence
for pressure-induced diamond-to-hexagonal structure
phase transition accompanied by shape changes from
spherical to prolate.
(d) Small Angle Scattering (SAXS and SANS). The size
as well as the size distribution can be obtained with
SAXS or SANS. We will discuss here as an example SAXS. The central scattering of X-rays due to
the presence of nanoparticles having dimensions from
several tens to several hundred times the X-ray wavelength is analogous to the well-known phenomenon
of optical diffraction, where a halo is produced by
the passage of a light ray in a powder with grain
dimensions on the order of a hundred times the light
wavelength. Let us consider a particle in a beam of
X-rays; then all the electrons are sources of scattered
10 3
1
0 .8
10 2
Proofs Only
0 .6
0.8 nm
0 .4
10
1
0 .2
0
10 0
0
2
4
6
R (n m )
0.1
1
q (1/nm )
10
Figure 5. Small angle X-ray scattering curve of the sample with
CdS067 Se033 nanocrystals described in Figure 2. The solid line has been
fitted by assuming spherically shaped particles with asymmetric size distribution as shown in the inset. The dashed line was calculated for a
size distribution with a -function-like profile. Reprinted with permission from [95], G. Irmer et al. J. Appl. Phys. 88, 1873 (2000). © 2000,
American Institute of Physics.
9
Light Scattering of Semiconducting Nanoparticles
R (nm )
10
5
4
3
R (n m )
2
10 5 4 3
30
a
2
1
80
R a m a n s h if t (c m -1 )
20
R a m a n s h ift (c m - 1 )
10
0
b
30
20
60
Proofs Only
40
20
10
0
0
c
0
0 .4
0 .8
1 .2
1/R (n m -1 )
30
20
10
0
0
0 .2
0 .4
1 /R (nm - 1 )
0 .6
Figure 6. Position of the spheroidal mode S01 as a function of the
inverse of the particle radius as observed by other scattering methods or TEM, respectively. (a) Nucleated cordierite glass: •, SANS [2].
(b) CdS in glass matrix: , SAXS [53]; , [55]; , [241]. (c) CdSx Se1−x
in glass matrix: , anomalous SAXS [13]; , SANS [62]; ×, SAXS [47];
+, TEM [47].
as determined by other scattering methods (SAXS, anomalous SAXS, or SANS) or TEM, respectively. The straight full
lines are calculated with Lamb’s theory for spheres with free
surfaces, and the dotted line in Figure 6c takes into account
the matrix effect. The agreement between various scattering
methods is good; however, for smaller particles deviations
from results of TEM measurements can be observed [47].
Large discrepancies between experimental results and the
theory based on the continuum model were obtained for
Si nanocrystals, as shown in Figure 7. Because the formulae in Section 4.1 are based on the isotropic elastic medium
approximation, mean values for the sound velocities vl and
vt were used for the calculation of the size dependence
of the polarized spheroidal mode 01 and the depolarized
spheroidal mode 21 of a free particle. The calculated values are much larger than those of the measured peaks, even
if the large anisotropy of the elastic constants in Si is taken
into account. The influence of the SiO2 matrix [72] is discussed as one possible reason for the deviation between
experiment and theory. However, quantitative estimations
are still lacking and further theoretical studies are necessary.
For instance, opposite predictions were made for the influence of SiO2 matrices on Ag particles. Montagna and Dusi
[96] predicted very small and negligible influences, while
Ovsyuk and Novikov [40] reported that the phonon modes
Figure 7. Position of the spheroidal mode S01 as a function of the
inverse of the particle radius of Si nanocrystals. The radii were obtained
by •, TEM measurements [71]; , TEM measurements [65]; , estimation of the size analyzing the shift and shape of the optical phonon band
assuming a columnar structure (column radius R) [64]. The straight
lines were calculated based on Lamb’s theory for particles with free surfaces: solid line, spheres; dotted line, columns, assuming isotropic elastic
medium. The hatched region indicates the range of theoretical values
when the elastic anisotropy of Si is taken into account. For comparison
theoretical results (+) obtained with lattice dynamical calculations on
Si clusters are included [166].
due to matrix effects shift to lower frequencies by considerable amounts.
5. SURFACE PHONONS
5.1. Surface Modes of Small Particles
The classical electromagnetic theory for the absorption and
scattering of a sphere was given a century ago by Mie [7].
About 60 years later Mie’s theory was applied to spherical
ionic crystallites [99–102] characterized by the frequencydependent dielectric function which also describes the
optical behavior of a polar semi-insulating cubic semiconductors in the infrared (IR) region:
2
− 2 − i
= 2LO
(14)
TO − 2 − i
Here, TO and LO are the frequencies of the transverse
and longitudinal optical bulk phonons, respectively, is
the dielectric constant at high frequencies, and the constant
takes damping into account. The bulk phonons in small
particles have properties similar to those of the corresponding phonons in an infinite crystal; however their wavefunctions are adapted to the geometry of the small particle.
For spherical geometry, for example, the phonon wavefunctions can be described by orthogonal eigenfunctions consisting of products of spherical Bessel functions j qn r with
spherical harmonics Ym , classified by the quantum numbers
10
Light Scattering of Semiconducting Nanoparticles
m, and n. The radial wavevectors qn are size-quantized
due to the boundary conditions with qn = n /R, where n
are the th order spherical Bessel function roots [103].
In addition to the bulk phonons in small particles, there
exist surface phonon modes, which have no counterpart
in infinite crystals. Their frequencies are intermediate
between TO and LO . They can be obtained by solving the
equation
+ 1 · + 1/ = 0
= 1 2 3 (15)
where 1 is the dielectric constant of the embedding medium
and is an angular quantum number. Unlike bulk phonons,
the surface phonons have nonvanishing electrical fields outside the particle; inside the electrical field changes with r −1
radial dependence. The surface mode with the lowest index
= 1 has a constant amplitude inside the sphere (therefore,
the restriction of the notation “surface mode” to modes with
> 1 is also in use). This mode of homogeneous polarization
has a frequency F given by the solution of + 21 = 0
and is often called the Fröhlich mode [104]. With each surface mode a surface charge distribution is associated, which
generates the polarization field inside the sphere. The Fröhlich mode with = 1 corresponds to a dipolar charge distribution, the mode with = 2 to a quadrupolar, and the inner
modes with higher to higher multipolar surface charge distributions. For particles that are small compared with the
wavelength, the electrostatic approximation can be used and
the mode with = 1 dominates [105].
Up to now we have considered a spherical particle. However, the shape has an important influence on the surface
mode frequencies. As an example, we consider an ellipsoid
of homogeneous polarization (Fröhlich mode). Three frequencies are obtained instead of one, given by the solutions
of the equations
+ 1 · 1/ni − 1 = 0
i = 1 2 3 of longitudinal type are purely radial with the mechanical
displacement vector
s=0 n = Aj1 n r/Rer (17)
where n is the nth zero of the spherical Bessel function j1 .
Assuming a quadratic negative bulk LO phonon dispersion,
the eigenfrequencies of the modes with = 0 are given by
2n = 2LO − 2L n /R2 (18)
Here, 2L is a parameter describing the dispersion of the
LO phonon in the bulk [109, 111]. For large R, n → LO is
obtained. For example, for GaAs quantum dots embedded
in AlAs, the n = 1 mode was calculated to occur basically at
the LO frequency of bulk GaAs for radii >25 nm.
≥ 1: The frequencies of these surface modes are more
difficult to obtain [109]. However, for large R the frequencies are again solutions of Eq. (15), meaning that the effect
of mechanical boundary conditions becomes important only
for small R.
The theories mentioned above concern the optical
response of an isolated particle embedded in a homogeneous medium characterized by a dielectric constant 1 ; particle interactions are not included. However, in practice the
particles are often not separated and light-scattering experiments have to be performed on collections of particles like
powders or nanosized porous media. As mentioned above,
the surface modes in polar semiconductors are accompanied by electrical fields. For instance, the Fröhlich mode of
an particle with its dipole field will interact with those of
other particles via dipole–dipole interaction. A theory that
accounts approximately for the interactions is the effective
medium theory.
5.2. Effective Medium Theory
(16)
where ni are the so-called depolarization factors corresponding to the ith axis of the ellipsoid. The depolarization factors
obey the relations 0 ≤ ni ≤ 1 and n1 + n2 + n3 = 1. The
inverse of ni is approximately proportional to the ith axis.
Note, that for n1 = n2 = n3 = 13 the case + 21 = 0
of the sphere is obtained; for an infinite long cylinder with
n1 = n2 = 21 n3 = 0 the surface frequency is determined by
+ 1 = 0. The electromagnetic theory was extended to
cylindrical [106], cubic [107], and other nonspherical particles; reviews are given in Refs. [102, 105, 108].
Roca et al. [109] and Chamberlain et al. [110] extended
the treatment of polar optical phonons in quantum dots
by including mechanical boundary conditions. Phonon dispersion up to quadratic terms in the wavevector and the
coupling between the mechanical displacement and the electrostatic potential were taken into account. Their model
gives (1) uncoupled transverse optical (TO) modes with
purely transverse character and (2) coupled modes with in
general mixed LO–TO character.
= 0: The most important contribution to Raman scattering corresponds to the modes with = 0. These LO modes
For visible light interacting with semiconducting nanoparticles (characteristic size L and dielectric function 2 ) distributed in a medium with the dielectric constant 1 in the
limit L the heterogeneous composite can be treated as
a homogeneous medium, and the so-called effective medium
theory applies.
The effective dielectric function eff is defined by
= eff E
D
(19)
and E
denote spatial averages of the dielectric
where D
displacement and the electric field, respectively. For a composite of two materials with the dielectric functions 1 and
2 we get
1 − f 1 E 1 + f2 E 2 = eff 1 − f E 1 + f E 2 (20)
where the averages are taken over the volumes occupied by
the material with the dielectric function 1 or 2 , respectively. The parameter f is the relative volume V2 /V occupied by the material with 2 . If the fields have the same
directions in the two media (which has to be proved), the
11
Light Scattering of Semiconducting Nanoparticles
dielectric function can be expressed as a function of the field
ratio = E2 /E1 :
eff =
1 − f 1 + f2 1 − f + f
(21)
The field ratio can be obtained by solving the electro = 0 under the approstatic equations rot E = 0 and div D
priate boundary conditions. Exact solutions exist only in a
few cases:
(a) The medium consists of parallel slabs of both materials; the applied electrical field is parallel to the slabs
(perpendicular to their common normal). Continuity
of the tangential electric field components requires
= 1 and we get from Eq. (21)
eff = upp = 1 − f 1 + f2 (22)
the so-called upper Wiener limit [112], because it can
be proven to be the absolute upper limit for eff for a
two-material composite.
(b) The medium consists of parallel slabs of both materials; the applied electrical field is perpendicular to
the slabs (parallel to their common normal). Continuity of the normal displacement field components
requires = 1 /2 and we get from Eq. (21) the socalled lower Wiener limit [112] as the absolute lowest
limit for eff for a two-material composite:
1
1
1 − f f
=
=
+ eff
low
1
2
(23)
(c) An exact solution is also possible for the case of
an external electrical field applied along an axis of
an ellipsoid (e.g., dielectric function 2 ) embedded
into an infinite matrix (of dielectric function 1 ); see,
for example, Ref. [105]. Here, we restrict ourselves
to the special case of a sphere. The field ratio is
then = 31 /2 + 21 , often derived in textbooks
of electrodynamics. From Eq. (21) we obtain the
Maxwell–Garnett (MG) I result [113]; see Table 1,
three-dimensional case. The result is exact for the
f → 0 limit (and for the trivial case f = 1). Therefore,
the application of the formula is justified if the matrix
contains only a few well-separated spheres (f → 0),
but often it is used as interpolation formula for all f .
The formula, for example, describes the case of semiconducting spheres with concentration f and dielectric function 2 = embedded in air (1 ). The
replacements 1 ↔ 2 and f ↔ 1 − f result in the
Maxwell–Garnett II formula eff = MGII (see Table 1)
corresponding to the “swiss cheese” case of spherical air inclusions (1 ) in the semiconducting material
with 2 = . It can be shown that (if 1 < 2 MGI
and MGII are the absolute lowest and the absolute
upper limit, respectively, for the effective dielectric
function eff for isotropic or cubic two-material composites [114]:
low < MGI ≤ eff ≤ MGII < upp The replacements 1 ↔ 2 and f ↔ 1 − f do
not result in MGI = MGII ; such a topology is called
a matrix topology. If one of the phases is strongly
diluted, the Maxwell–Garnett formulae are good
approximations. Examples are semiconductor doped
glasses with filling factors f of about 0.01 discussed in
the next sections.
If phases 1 and 2 are equivalent, a so-called aggregate topology has to be described. An example is the
Bruggeman expression [115] (see Table 1, first row).
The solution eff of the Bruggeman formula is invariant under the replacements 1 ↔ 2 and f ↔ 1 − f ,
characteristic of an aggregate topology. The Bruggeman formula can be derived by generalizing Eq. (21)
for three materials, spherical inclusions 1 and 2 with
volume fractions f1 = V1 /V and f2 = V2 /V embedded
in a matrix m :
eff − m
− m
− m
= f1 1
+ f2 2
2m + eff
2m + 1
2m + 2
Table 1. Effective dielectric functions.
Topology
Matrix
Aggregate
3D
2D
Maxwell–Garnett I
21 − f 1 + 1 + 2f 2
eff = MGI = 1
2 + f 1 + 1 − f 2
Maxwell–Garnett I
1 − f 1 + 1 + f 2
⊥eff = 1
1 + f 1 + 1 − f 2
Maxwell–Garnett II
3 − 2f 1 + 2f2
eff = MGII = 2
f1 + 3 − f 2
Maxwell–Garnett II
2 − f 1 + f2
⊥eff = 2
f1 + 2 − f 2
Bruggeman
Bruggeman
− eff
− eff
0 = 1 − f 1
+f 2
2eff + 1
2eff + 2
Monecke
+ 4f1 2 − 1 /21 + 2 eff = 1
1 + f 2 − 1 /21 + 2 2f 2 2 − 1 2 /21 + 2 +
1 + f 2 − 1 /21 + 2 (24)
0 = 1 − f 1 − eff
− eff
+f 2
eff + 1
eff + 2
(25)
12
Light Scattering of Semiconducting Nanoparticles
If, in a self-consistent way as the matrix material m the
effective medium eff itself is choosen, with f1 = 1 − f and
f2 = f , the Bruggeman expression is obtained. It is often
considered to be the best possible interpolation formula
for an aggregate topology. It has, however, the following
essential shortcoming. Let us consider a few semiconducting
inclusions with the dielectric function 2 in a matrix of
constant 1 . In the IR region between the transverse and the
longitudinal optical phonon, 2 is negative and 21 + 2
may become zero at the Fröhlich frequency F . For f →
0 the Maxwell–Garnett I formula is correct and results in
a pole at 21 + 2 = 0, giving rise to the Fröhlich mode
[104]. This exact result is not reproduced for f → 0 by the
Bruggeman formula. To overcome this deficiency, another
interpolation formula for an aggregate topology has to be
found. An isotropic aggregate topology is given, for example, by spherical 2 inclusions in a matrix 1 for f → 0
interpolated to spherical 1 inclusions in a matrix 2 for
f → 1. In both limits the field ratios are exactly known as
f →0 = 31 /21 + 2 or f →1 = 32 /22 + 1 −1 , respectively. The simplest interpolation for is given by
=
31 − f 1 + f 22 + 1 3f2 + 1 − f 21 + 2 (26)
By inserting into Eq. (16), a new expression for eff is
obtained [116, 117]; see Table 1.
Often two-dimensional (2D) effective functions more
adequately describe materials with nanosized structures.
Examples are porous semiconductor structures produced by
electrochemical etching with cylindrical pores, porous membranes with columnar semiconductor structures, and materials with quantum wires.
A typical porous polar semiconductor can be described
as a heterogeneous material consisting of, for example, airfilled (1 ≈ 1) cylindrical pores in the z direction, which
are randomly distributed in the xy plane and embedded in
the semiconducting matrix with 2 = . The symmetry of
the material is reduced. For a diatomic semiconductor with
zincblende structure, the cubic point group Td is changed
to an uniaxial one [D2d for a (100) surface and C3v for a
(111) surface]. In both cases the effective dielectric function
becomes a tensor
 ⊥

0
eff 0


0 ⊥
0 
˜ eff = 
(27)
eff


0
0 eff
II formula. Single columns (f → 0) would then result in a
Fröhlich mode given by F + 1 = 0.
Any dielectric function has to obey the Bergman spectral
representation [118]. It can be shown that the effective
dielectric functions given above fulfill the corresponding
conditions. Additionally, in the two-dimensional case the
Keller theorem [119] has to be fulfilled. This is the case
for both Maxwell–Garnett formulae and for the Bruggeman
expression.
The behavior of the effective dielectric function in the
region of the optical phonons is illustrated in Figure 8.
Parameters for GaP (TO = 366 cm−1 , LO = 402 cm−1 , =
1 cm−1 , = 85, and f = 07) and the model interpolating
between I and II were used. Phonons of transverse (longitudinal) character appear at the poles (zeros) of the real
part Re() of the effective dielectric function; see Figure 8c
(the poles are smoothed due to the finite damping constant used). Between the pole corresponding to the TO
phonon and the zero corresponding to the LO phonon an
additional pole and an additional zero appear. They can be
assigned to the Fröhlich mode. The unusual sequence, the
longitudinal Fröhlich mode being below the transverse one,
results from the fact that in any dielectric function poles
should alternate with zeros in the sequence T → L → T →
L, etc., with increasing frequency. The Raman scattering
efficiency is ∝ Im() for TO phonons and ∝ Im(−1/) for
LO phonons; the corresponding imaginary parts are plotted
in Figure 8a and Figure 8b in arbitrary units. Although a
strong Fröhlich mode phonon of longitudinal character can
be seen, the transverse Fröhlich mode at F−T is very weak
(not shown in Fig. 8b; its intensity is <1% of that of the TO
c
b
Im (-1 / ε )
R e ( ε)
Proofs Only
x100
LO
F -T
F -L
d
R (ω )
For the electric field component parallel to the pores, eff
is given exactly by the upper Wiener limit [Eq. (22)]. For ⊥
eff
a two-dimensional variant of the Maxwell–Garnet model I
can be used (see Table 1). The formula describes the morphology of the porous structure as an interpolation between
two boundary cases: cylindrical voids in the semiconducting matrix (large material concentration) and a skeleton
of intercrossing semiconductor plates (large void concentration). Being correct in both limits f → 1 and f → 0, it can
be used in the entire range f ∈ 0 1.
For columnar nanostructures it is more appropriate to
use the two-dimensional equivalent of the Maxwell–Garnett
Im ( ε)
a
TO
340
360
380
400
420
ω (c m -1 )
Figure 8. Effective dielectric function model: interpolation between
three-dimensional (3D) MGI and 3D MGII [117]. Parameters: TO =
367 cm−1 , LO = 402 cm−1 , = 1 cm−1 , and f = 074 (closely packed
spheres). (a) Imaginary part Im(; (b) Im(−1/; (c) real part Re(;
(d) reflectivity R(.
Light Scattering of Semiconducting Nanoparticles
phonon) and should be not observable in the Raman spectra. However, the L–T splitting of the Fröhlich mode gives
rise to a reflectivity minimum; see Figure 8d. The reflectivity
is easy to calculate according to the standard formulae
n − 12 + 2
(28)
n + 12 + 2
√
√
with n =
a2 + b 2 + a, =
a2 + b 2 − a, a =
Re⊥ , and b = Im⊥ .
R =
5.3. Raman Scattering
5.3.1. III–V Semiconductors
Hayashi and Kanamori [3] were the first to observe Raman
spectra by surface modes in polar semiconductors unambiguously. The GaP microcrystals they investigated were
prepared by a gas evaporation technique. Figure 9 shows
Raman spectra of samples with different particle sizes in
the range between 51 and 430 nm. The surface peak can
be clearly seen between the TO and the LO phonon peaks,
when the microcrystals are about 1 order of magnitude
smaller than the wavelength of the incident laser. For the
samples used the surrounding medium was air. A frequency
shift of the surface mode for different sizes was not reported.
The frequency shifted to lower frequencies when the dielectric constant of the surrounding medium was increased.
Proofs Only
Figure 9. Light-scattering spectra of the surface mode of GaP nanoparticles. d¯ is the average particle size. Reprinted with permission from
[3], S. Hayashi and H. Kanamori, Phys. Rev. B 26, 7079 (1982). © 1982,
American Physical Society.
13
This was realized by embedding the microcrystals in the liquids nujol (1 = 200), aniline (1 = 256), and methylene
iodide (1 = 310). The lineshape of the peaks attributed to
an ellipsoidal particle shape was calculated using effectivemedium theory [120]. It was observed that the intensity of
the surface phonon peak increases as the average particle
size decreases. An explanation was given based on calculations of Ruppin [103] for Raman scattering efficiencies of
bulk and surface phonons in very small spherical crystals.
Watt et al. [121] investigated Raman spectra from arrays
of GaAs cylinders fabricated by lithographic techniques with
radii in the range of 30–100 nm and heights of 140–570 nm.
Those with radius ≤ 40 nm showed a well-defined feature
between the TO and LO phonons. The position of the
additional peak was found to be in agreement with calculations based on the model developed by Ruppin and
Englman [102] for the case of an infinite cylinder embedded
in a dielectric medium. Wang et al. [122] reported Raman
measurements of GaAs dots, fabricated by electron beam
lithography and dry etching. It was found that the surface
phonon intensity increased dramatically with decreasing dot
sizes. da Silva et al. [123] observed the surface phonon at
287.6 cm−1 in 6–10 m long wire-like crystals of GaAs with
radii of 30 nm. The GaAs crystals were epitaxially grown on
porous Si. They did not observe a dependence of the Raman
intensity on the angle of the incident light, in contradiction
to the observations of Watt et al. [121].
In recent years, in addition to porous silicon, increasing attention has been paid to porous III–V semiconductor materials [124–130] due to their potential applications
in the field of electronics and photonics [131]. The nanosized structures of such polar porous media (not in the
case of the nonpolar Si) allow investigations of the surface modes. Their properties strongly depend on the morphology (i.e., symmetry, structural geometry, effective size,
etc.) and on the constituents. The analysis of these modes
helps to characterize the porous structure by the comparison
of experimental results with theoretical calculations based
on the effective-medium theory. Although for powders or
nanocrystals embedded in homogeneous media 3D models
of the effective-medium theory are successfully applied, 2D
models are more appropriate to describe the morphology
of porous semiconductors. The morphology of the porous
structures fabricated by electrochemical etching of III–V
materials depends mainly on the doping level of the bulk
material, on the defect density on the surface and on the
crystal orientation for given etching conditions [126, 127,
132–134]. A surface-related phonon mode in porous GaP
at 397 cm−1 observed by Tiginyanu et al. [135] was interpreted in terms of a Fröhlich mode on the basis of an
effective medium approach [128]. It was shown that its frequency decreases with increasing anodization current. The
fabrication of free-standing porous membranes, detached
from the substrate, described in [136], enabled the investigation of the surface modes in porous membranes filled with
liquids [137] and infrared reflectance measurements [134].
Figure 10 shows infrared reflectance spectra (a) and Raman
scattering spectra (b) of a porous (111) GaP membrane.
The pores of about 30 nm diameter exhibit a column-like
shape, stretching into the sample perpendicularly to the initial crystal surface. The experimental spectra (upper row)
14
a
1
por-G aP
G aP
bulk
0 .8
0 .6
R e fle c tiv ity
0 .4
0 .2
0
0 .8
b
phonon and a surface mode approximately 10 times smaller
in intensity were observed, in agreement with calculations
based on the Fröhlich interaction and taking into account
a degenerate valence band. Varying the excitation energy
below the direct gap the LO phonon band decreases dramatically, whereas the intensity of the surface mode band is
not much changed. This could be evidence that besides the
Fröhlich interaction other scattering mechanisms like deformation potential scattering contribute to the surface mode
scattering. However, up to now calculations on the electron–
phonon interaction concentrated only upon the Fröhlich
interaction.
por-GaP
G aP
bulk
Proofs Only
surface
m ode
0 .6
0 .4
0 .2
0
R a m a n s c a tte r in g in t e n s it y ( a r b . u n its )
Light Scattering of Semiconducting Nanoparticles
ωΤ O ω
LO
300 400
300
ω( c m )
-1
400
surface
m ode
ωL O
380
400
5.3.2. II–VI Semiconductors
380 400
420
ω( c m )
-1
Figure 10. Surface modes in GaP with columnar nanosized pores. (a)
Fourier transform infrared reflectivity spectra and (b) Raman spectra of
porous and of bulk material. The lower row shows results of calculations
based on the effective dielectric function corresponding to the 2D MGII
model.
can be interpreted theoretically (lower row) using a twodimensional effective dielectric function 2DII; see Table 1,
which describes the morphology of the porous structures as
an interpolation between two cases: cylindrical voids in the
semiconducting matrix (f → 1) and a skeleton of intercrossing semiconductor plates (f → 0). The wave-like structures
in the reflectance spectra of the porous membrane are interferences of light reflected at the membrane boundaries. The
effect of hydrostatic pressure on the surface modes in porous
GaP was investigated in [138] and the temperature influence
in [139].
Sarua et al. [133, 134] studied the influence of free charge
carrier in porous GaAs and porous InP on the surface
phonons. It could be shown that the Fröhlich mode, like the
LO phonon mode, couples to the plasmon modes induced
by vibrations of the free carrier plasma of electrons. As the
LO phonons, the Fröhlich modes were shifted to higher
frequencies with increasing carrier density. In porous InP
this effect could be observed both by light-scattering and
by infrared reflectance measurements; in porous GaAs the
F-T–F-L splitting is too small to be observed in the Fourier
transform infrared experiment.
In GaN columnar nanostructures a new mode near
716 cm−1 was found by Raman scattering measurements
[140]. The columnar structures with the thickness of about
1.5 m were fabricated by electrochemical dissolution of
bulk material. Scanning electron microscope images proved
that the columns with transverse dimensions of about 50 nm
or less were oriented perpendicular to the initial surface. By
Raman lineshape analysis based on the effective dielectric
function for a composite of GaN columns in an air matrix
(2D MGI; see Table 1) the new mode could be attributed to
Fröhlich vibrational modes.
Electron–phonon coupling due to Fröhlich interaction has
been considered by Efros et al. [26]. They investigated resonance Raman scattering of GaP nanocrystals embedded in
glass. With laser excitation near the direct gap a strong LO
Surface modes in II–VI compounds were first observed
by Scott and Damen [141] in cylindrical microcrystallites
( 1 m) of CdS in a polycrystalline film using Raman scattering. Pan et al. [142] studied Raman scattering of CdS
nanocrystals (size <30 nm) by surface modes in different
organic media. Most Raman scattering experiments were
performed with nanocrystals embedded in a glassy matrix:
CdS [143], CdSe [25], CdSx Se1−x [34, 144–146], and ZnS
[147]. As an example, we discuss here the case of CdSx Se1−x
nanoparticles.
It is known that in the mixed crystal CdSx Se1−x the optical phonons show the so-called two-mode behavior, meaning
that a CdS-like mode (TO1 and LO1 ) and a CdSe-like mode
(TO2 and LO2 ) can be attributed to two oscillators which
contribute with different oscillator strengths depending on
the composition x.
The dielectric function of the mixed crystal can be
described by
2L1 x − 2 − i1 2T1 x − 2 − i1 2
L2 x − 2 − i2 ×
2T2 x − 2 − i2 x = x
(29)
with L1 x = 266 + 55x − 19x2 , T1 x = 266 − 28x,
L2 x = 211 − 25x, T2 x = 168 + 17x, and x = 53x +
631 − x. The parameters were taken from bulk measurements [148]. The phonon frequencies that depend on the
composition x are plotted in Figure 11. The symbols correspond to measurements on CdSx Se1−x nanoparticles embedded in glass. Below both LO phonons additional modes can
be observed (Fig. 12). In comparison with calculations on
the basis of the effective dielectric function according to
the 3D MGI model (dotted curve in Fig. 11), they can be
attributed to Fröhlich modes.
To study the influence of the particle size on the optical phonons and the surface modes, a semiconductor doped
glass slab was annealed in a furnace with an applied temperature gradient with temperatures between 850 and 1100 K.
With increasing temperature the CdS06 Se04 particles grow
to larger sizes. The composition x = 06 was estimated from
the position of the LO phonons [148]. The positions of the
LO phonons and the surface mode were obtained by fitting
with Lorentzian bands. The dotted lines in Figure 13a correspond to the calculated frequency of the Fröhlich modes
15
Light Scattering of Semiconducting Nanoparticles
C d S 0.6 Se 0.4
5 4
320
a
TO1
24 0
L O2
240
LO 2
200
F-L 2
20 0
Proofs Only
F -L2
T O2
160
40
b
0
0 .2
0 .4
0 .6
0 .8
1
x
Figure 11. Dependence of the optical phonons and surface modes on
the composition x of CdSx Se1−x nanoparticles embedded in borosilicate
glass. The lines were calculated with the effective dielectric function of
the 3D MGI model.
30
ω (c m - 1 )
16 0
LO1
F -L 1
T (K )
ω (c m - 1 )
ω (cm -1 )
280
2
875
F -L 1
28 0
R (nm )
3
1000
975
LO1
C d SxS e1-x
92 5
900
850
32 0
20
10
0
0
assuming spherical particles. Contrary to the acoustic vibrations (Fig. 13b), the optical phonons and the Fröhlich modes
are nearly not size dependent. The effect of small shifts due
to confinement effects for small particle sizes is discussed in
Section 6.2.2.
The electron–phonon coupling in the case of the Fröhlich
interaction between the field induced by the vibrational
motion and the charge distribution was studied by several authors theoretically with different approximations and
partly contradicting results (see Section 3). Klein et al. [25]
observed surface modes in CdSe. Their calculations were
based on a classical dielectric model. They concentrated
their investigations on the 1s–1s electronic transition and
assumed that only the electron wavefunction is confined
Cd S 0.6 Se 0.4
R = 3.7 nm
In tensity ( a rb. units)
LO1
F-L 1
LO 2
F-L 2
1 60
2 00
2 40
2 80
3 20
3 60
-1
R ama n sh ift (cm )
Figure 12. Surface modes in CdS06 Se04 nanoparticles embedded in
borosilicate glass.
0 .2
0 .4
0 .6
1 /R (n m -1 )
Figure 13. Size dependence of the (a) optical phonons and surface
modes and (b) acoustic modes of CdS06 Se04 nanoparticles embedded
in borosilicate glass.
whereas the hole resides near the center of the sphere. Their
result is that the radial charge distribution in the spherical box does not couple to surface modes. As a possible
reason for the observation of surface modes in the Raman
spectra they assume deviations from the spherical particle
shape. Efros et al. [26] included holes in a four-fold degenerate valence band in their calculation and showed that
surface modes with = 2 are allowed due to a mixing of
the valence bands even for perfectly spherical nanocrystals.
Chamberlain et al. [110] considered the mechanical as well
as the electrostatic boundary conditions in their model. In
addition to the dipole approximation they considered the
quadrupole contribution of the Fröhlich interaction, proportional to the wavevector. They showed that this term
dominates for intersubband scattering and occurs only via
= 1 phonons. Because they used a one-band effective mass
model, the phonon modes are restricted to = 0 when the
dipole approximation is used. The published calculations are
based on the Fröhlich interaction, but other scattering mechanisms could contribute to the light scattering by surface
modes, too. More detailed theoretical investigations are necessary and more experiments should be performed, analyzing the dependence on the excitation wavelength (near and
far of resonance), on the polarization of the scattered light,
and on the particle size to bring to light the properties of
the surface modes.
The selection rules for Raman scattering of optical
phonons in a homogeneous sphere are similar to those
obtained for the acoustic vibrations discussed by Duval [38].
The vibrational modes must belong to the irreducible reprel
l
sentations Dg and Du of the three-dimensional rotation–
inversion group O(3) (g and u mean even or odd upon
inversion). The coupled optical (and the spheroidal acoustic)
16
Light Scattering of Semiconducting Nanoparticles
0
2
modes transform according to Dg , Du , and the uncou1
pled optical (torsional) modes transform according to Dg ,
2
3
Du , Du , . The Raman transition operator for dipoleallowed scattering is the polarizability tensor. Its compo0
2
nents transform like Dg and Dg . The IR absorption is
electric dipolar; the dipole operator transforms according to
1
Du . Therefore, the uncoupled transverse optical (torsional
acoustic) modes are not expected to be optically active.
It should be noted that the considerations above do
not reflect the crystal symmetry inside the sphere. For
zincblende-type materials belonging to the point group Td
or wurtzite-type materials with point group C6v the inversion
symmetry is absent. For binary compounds such as GaAs,
InP, and GaP, upon lowering of the symmetry from O3
0
to Td the representation Dg becomes 1 [109]. Scattering
by optical phonons due to the so-called dipole forbidden
Fröhlich interaction with diagonal Raman tensor belongs to
this representation and will be visible. However, both repre2
1
sentations Dg and Du convert into 15 . The scattering by
optical phonons due to deformation potential and electro1
optic mechanisms belongs to this Raman activity and Du
surface modes are thus Raman active. The question of when,
upon increasing R, the sphere will behave as a piece of bulk
material, is discussed in more detail in [109], considering the
coherence length of phonons.
particles nearly spherically shaped with an average diameter of about 30 nm as proved by XRD measurements. In
the Raman spectra from these particles in air below the LO
phonon a band at 937 cm−1 was observed (Fig. 14c). Its position was found to be in agreement with calculations based on
the effective medium theory. The observed half-width of the
band is much larger than the half-width of the LO phonon.
One reason could be local changes in the particle packing
density. The theoretical curves (dotted lines in Fig. 14) were
obtained with the 3D Mo model (Table 1), assuming a distribution of f values around f = 07 with standard deviation
= 03. The sensitivity of the surface mode to the environment of the particles was proved by embedding them into
amorphous silica (1 ≈ 2). This was achieved by heating the
powder in a laser beam. The temperature was controlled by
measurement of the intensity ratio of the Stokes and antiStokes TO and LO phonons and by the frequency shift of
the phonons. At about 1200 C the oxidation of the SiC
crystals starts, resulting in a SiO2 oxide layer around the
particles [154, 155]. The SiO2 observed in the Raman spectra could be assigned to amorphous silica [156]. The corresponding peaks at about 500 and 600 cm−1 are shown in
Figure 14a. The small peak below the strong TO phonon is
a weak phonon of TO character, indicating the 6H polytype
of SiC.
5.3.3. Other Compounds
LO
TO
S iC
93 7
LO
In te n s ity (a r b . u n its )
A reflectivity gap corresponding to surface modes in a polar
porous semiconductor was first observed by Danishevskii
et al. [149] in porous SiC and interpreted by McMillan et al.
[150] on the basis of the effective medium theory. Sasaki
et al. [50] measured Raman spectra from SiC fibers consisting of microcrystalline SiC about 50 nm in size embedded
in amorphous SiC and carbon. They attributed features in
the spectra between the TO and LO phonon frequencies to
surface modes. The features observed were very weak and
very broad, perhaps due to a high free carrier concentration and high plasmon damping of the SiC microcrystals and
therefore difficult to interprete.
DiGregorio and Furtak [151] investigated commercially
available 6H-SiC particles with an average size of about
3 m. The particles were irregular in shape with an average aspect ratio of about 2. They obtained a weak band in
the Raman spectra below the LO phonon. However, scattering by surface modes is expected to be negligible in crystals that are larger than the wavelength of the exciting light.
The appearance of weak surface modes was qualitatively
explained as being caused by a decrease of the effective particle radius in irregularly shaped particles, enhancement of
the intrinsic electric fields, and localization of surface modes
near sharp corners and facets on the particles [152]. The
observed mode frequencies 947 cm−1 for the particles in air
and 928 cm−1 for particles embedded in KBr (1 = 233)
are near the solutions 932.3 and 948.8 cm−1 of the equation
+ 1 = 0 for cylinder-like particles.
Raman scattering spectra in [50, 151] were obtained from
SiC microcrystals of irregular shape and/or large size and,
hence, weak surface phonon structures. This suggests the
presentation here of Raman measurements of nanosized
c
F -L
Proofs Only
9 09
b
F -L
840
900
960
S iO 2
a
0
400
800
ω (c m -1 )
Figure 14. Influence of the embedding medium on the surface modes
of SiC nanoparticles. (a) Raman spectrum of SiC nanoparticles embedded in an amorphous silica matrix. (b) Detail of spectrum (a) with surface mode F-L and phonon. (c) Raman spectrum of the SiC nanoparticles embedded in air. Calculations (dotted lines) were based on the
effective dielectric function interpolated between MGI 3D and MGII
3D [153].
17
Light Scattering of Semiconducting Nanoparticles
The surface mode is located at about 909 cm−1 between
the TO and LO phonons of SiC, shown in Figure 14a and
in the inset (Fig. 14b). The theoretical curve in Figure 14b
was calculated with f = 065 and = 015.
In the optical phonon Raman spectra of SiC nanorods
(diameters range from 3 to 30 nm; lengths are >1 m) with
a high density of defects, bands at 791, 864, and 924 cm−1
were observed [157]. By comparison with calculations it was
found that the Fröhlich enhanced effective density of states
of the optical phonons, which becomes important because
of the broken translation symmetry induced by defects, is
responsible for the features in the Raman spectra.
Observations of surface phonon modes have also been
reported for ZnO [158], SiO2 [159], and Si [65].
(30)
has the periodicity of the lattice. The vibrawhere uq0 r
tional wavefunction of the nanocrystal is approximated by
6.1. Confinement Models
In the Raman spectra of nanocrystalline materials with
particle sizes of a few nanometers, peak-position shifts,
broadenings, and asymmetries of optical phonon bands
appear. In most cases the Raman peak shifts to progressively
lower energies and the lineshape gets progressively broader
and asymmetric (on the low-energy side) as the particle size
gets smaller. This effect can be used to determine the particle size. A typical Raman spectrum of the optical phonons of
nanocrystals with about 3 nm diameter is shown in Figure 15,
Si
nano cluster
509
R a m a n in te n s ity (a r b . u n it s )
5 09
Proofs Only
520
c-S i
0
Confinement Model CM-I, Phenomenological Phonon
Confinement Model To characterize and interpret the
observed bands, a phenomenological phonon-confinement
model is very widely used; therefore, it is briefly outlined
below. This model, originally proposed by Richter et al.
[160], was later extended by Campbell et al. [161]. In a
Stokes scattering process in an infinite crystal the wavevec S is transferred to a phonon with
=k
L − k
tor difference k
wavevector q0 . The wavefunction of a phonon with wavevector q0 in an infinite crystal is
= uq0 re
−iq0 r
q0 r
6. OPTICAL PHONONS
440
together with the Raman spectrum of a single crystal. The
upper dashed curve in the inset shows a calculated profile.
Different confinement models are in use to determine particle sizes and size distributions from the measured spectra.
480
520
400
800
12 0 0
R a m an s hift (c m - 1 )
16 0 0
Figure 15. Raman scattering of confined optical phonons in nanocluster
Si with crystals of about 3 nm size. The inset shows for comparison the
Raman spectrum of an optical phonon in bulk Si. The dashed curve
was calculated with the confinement model CM-II for spheres.
= W r
Lq0 r
= q0 ru
q0 r
q0 r
(31)
where the localization of the phonon in a microcrystal
is taken into account by the weighting function W r L.
A Gaussian weighting function W r R ∼ exp− r 2 /R2 ,
introduced in [160] is mostly used in the traditional
approach to bandshape analysis. The parameter is fixed
arbitrarily to get a small value at the sphere boundary R.
Some other weighting functions were also applied [161–163].
can be expanded in a Fourier series, the Fourier
q0 r
coefficients being
1 3
=
−iq0 r
(32)
d r q0 re
Cq0 q
2 3
The Raman scattering intensity is then described by a continuous superposition of Lorentzian curves with bandwidth
centered at the wavenumbers q of the phonon disper 2 factor. By assumsion curve and weighted by the Cq0 q
ing a spherical Brillouin zone, isotropic dispersion curve,
L − k
S ≈ 0 for one-phonon scattering, the Raman
and q0 = k
intensity can be written as
2
Cq
(33)
I ∝ d 3 q
− q2 − /22
The Fourier coefficients for the Gaussian weighting function are Cq ∼ exp−q 2 L2 /4. This formula has been
often used to fit Raman spectra to obtain mean particle
sizes. However, there is no physical reason to use a Gaussian
weighting function or in the assumption that the phonon
wavefunction amplitude differs from zero at the crystalline
boundary. For the fit parameter the value 21 was originally used by Richter et al. [160]. Campbell et al. [160]
found the best agreement with experiments setting = 2 2 ;
this value has been used by the most authors afterwards.
With W 2 r R ∼ exp−4 2 r 2 /R2 strong confinement of
the phonons to the core is obtained: W 2 r R decreases
from 100% at r = 0 to 5% at r = 028R which means that
the phonon energy is concentrated in only 2% of the sphere
volume. Although the phonon confinement model is practicable to estimate particle sizes, it is theoretically not well
founded.
18
Light Scattering of Semiconducting Nanoparticles
Confinement Model CM-II This model uses the scattering function, which is well established in the analysis of
SAXS experiments on small particles [164] (see Section 4.3).
The application of similar expressions to light-scattering processes on small particles was also proposed in [48, 49, 96]. To
derive the structure factor we follow Nemanich et al. [165],
who presented an evaluation of the susceptibility function
over a limited spatial extent. The results indicate that spectral changes are related to the phonon dispersion, and the
photon wavevector uncertainty is accounted for.
The light-scattering cross section is proportional to the
space-time Fourier transform of the correlation function of
the fluctuations in the polarizability r t [42]:
∝ d 3 r d 3 r dt
∗ r
t r 0
I k
· r − r − it
× exp−ik
(34)
The fluctuations r t can be expressed by the atomic
displacements r t = ur t.
u
In the absence of interaction between different phonon
eigenmodes the displacements of modes with different q are
uncorrelated, and we obtain
uq
u∗ r t ur 0 = 1/V u∗ q
q
× expiq · r − r + i0 t (35)
describes the dispersion of the phonon mode.
where 0 q
The correlation function in Eq. (35) depends only on the
difference r − r . Therefore, the integrals can be factorized.
Further, the displacement correlation function is expressed
in terms of the Bose–Einstein factor n, and Eq. (34) can
be written
∝
I k
n + 1 − 0 q
Sq
k
Aq
q
(36)
where the scattering function S is obtained by
2
= 1/V d 3 r expiq − k
· r k
Sq
scattering [10] in opaque semiconductors. However, because
we have to deal with very small scattering volumes of particles with characteristic lengths L 1/, the absorption can
be neglected in general.
For finite scattering volumes the coupling of light to exci is
tation modes with wavevectors q in a range about k
k describes the range of
allowed and the function Sq
wavevectors that take part in the light-scattering process.
For optical phonons we assume small frequency changes
along the dispersion curves and we will use the approxima . The finite lifetime of the
tion of a constant term Aq
phonons is taken into account by replacing the -function
by a Lorentzian function and the sum is approximated by
=k
an integral over a spherical Brillouin zone. With k
L
S ≈ 0 we obtain
−k
(37)
V
is a measure of the uncertainty of
k
The function Sq
For an infinite transparent crystal with
the wavevector q.
∼ q − k
wavevector conservation is obtained.
k
Sq
Before we discuss the scattering function for small particles, some remarks concerning absorption are needed. For
a crystal that is opaque to light, the effective scattering volume is limited by the penetration depth, which is on the
order of 1/, where is the absorption coefficient. The
effect of absorption can be described by assuming complex
= k
+ ik
and k
S = k
+ ik
. By intewavevectors k
L
L
L
S
S
gration of Eq. (37) over the half-space z ≤ 0 we obtain
∝ qz − kL − kS 2 + 2 −1 ,
a Lorentzian function Sq
where we have used ≈ kL + kS . This means that the
wavevector component qz has a Lorentzian distribution centered at k = kL − kS with a half-width of 2. Lorentzian
distributions were successfully used to interpret Raman scattering of LO phonon–plasmon excitations [9] and Brillouin
∝
I k
d3 q
Sq
− q2 − /22
(38)
instead of
an expression similar to Eq. (33) but with Sq
2 . Table 2 presents the function Sq for some differCq
ently shaped particles and for a layer. In Figure 16 these
functions are shown in dependence on the dimensionless
parameter : = q · 2R for spheres with radius R, = q · 2R
for long cylinders (length L radius R), and = qx · L for a
cube (edge length L). The function S describes the range
of wavevectors that take part in the light-scattering process.
For example, for spheres the curve Sq diminishes from the
value 1 at = 0 (q = 0) to a weight of 21 at about = 4,
corresponding to a wavevector of q = 2/R; wavevectors with
less weight contribute up to about = 8 corresponding
to q = 4/R. The wavevector corresponds to a momentum
p = q; therefore, we can discuss these equations in terms
of Heisenberg’s uncertainty relation x · p ≥ /2 correlating the uncertainties of the coordinate x and the momentum p. A Gaussian distribution of x values corresponds to
a Gaussian distribution of p values. For Gaussian distributions only the lower limit p = /2x of Heisenberg’s
uncertainty relation is obtained [168]. In our case of x values strongly restricted to the particle volume the uncertainty
is larger. Using the example of the sphere we estimate that
p ≈ 4/x, setting x = 2R and q ≈ 2/R.
For comparison two Gaussian weighting functions used in
the phonon confinement models CM-I are also shown ( = 1
Table 2. Scattering functions Sq.
Shape
Sphere
Sq
9
j12 qR
qR2
R: radius
j1 : spherical Bessel function
Cylinder
Rhomboeder
4J q R sin qz L/2
·
R: radius
q R2
qz L/22
L: length in z direction
J1 : Bessel function
i=x y z
Layer
2
2
1
sin2 Li qi /2
Li qi /22
sin2 Lqz /2
Lqz /22
Li : lengths of the edges
L: layer thickness in z direction
19
Light Scattering of Semiconducting Nanoparticles
Raman intensities was applied to different porous Si structures with connected or isolated Si structures [174]. Comparing the experimental spectra with those simulated by models
the results were consistent with cylindrical nanocrystals of
3:1 length-to-diameter ratio and diameters of ∼ 5 nm, measured by TEM.
1 .0
s phe re
0 .6
c ylind er
0 .4
e xp (-κ2 )
0 .2
6.2. Size Determination
by Raman Scattering
cu b e
e xp(-κ /2 )
2
0 .0
6.2.1. Si and Ge
2
4
6
8
10
κ
Figure 16. Scattering functions S for light scattering of differently
shaped particles. The parameter is the dimensionless product of a
characteristic length L (2R for spheres and cylinders; edge length for
the cube) times the corresponding wavevector component.
and = 1/4. The two formulae [Eq. (33) and Eq. (38)] look
similar, but their interpretations are different. It should be
noted that the model CM-II does not involve arbitrary fitting
parameters.
To take the phonon confinement into account, we should
consider that the largest possible wavelength of the phonons
in the particle is restricted by their size. For example, in a
sphere the corresponding smallest wavevector is 4.49/L, the
wavevectors allowed are approximately 4.49/L, 7.72/L, and
10.90/L · · · 2/a, where a is the lattice constant. For L a
the approximation of the sum over q values by an integral
in Eq. (33) or (38) is justified.
The phonon dispersion curves often are approximated
on
the basis of a linear chain model by q = −A +
A2 − B1 − cosaq. The parameters A and B are chosen to fit neutron scattering data or theoretical calculations,
and a is the lattice constant. Other approximation formulae
are given in Ref. [109, 111, 145, 169].
Other Confinement Models Raman shifts of nanocrystals versus size were studied theoretically by a bond polarizability model [170]. In the first step the vibrational
properties of clusters of some hundred atoms were calculated. The force constants and the atom positions were
taken to be the same as those in the bulk. The force constants were considered up to the fifth order; a partial density
approach [171] was adopted to calculate them. The vibrational eigenfrequencies and eigenfunctions were calculated
for Si spheres and Si columns consisting of up to 657 atoms.
Then the Raman spectra were calculated by a bond polarizability model [172, 173] as a sum of independent contributions from each bond. It was found that the Raman
shifts due to the confinement effect could be described by
= 0 − L = Aa/L , where L is the phonon frequency of the nanocrystal with size L, 0 is the frequency
of the optical phonon in bulk material at the Brillouin zone
center, and a is the lattice constant. The fitted parameters
were A = 47412092 cm−1 and = 144108 for spheres
(cylinders), respectively.
A simple microscopic calculation using a one-dimensional
linear chain model with bond polarizabilities to calculate the
The confinement of optical phonons has been investigated
by Raman scattering in Si nanocrystals prepared by the
plasma transport method [159, 175], the gas-evaporation
technique [176], the reactive sputtering technique [177],
molecular beam deposition [178], laser annealing of a-Si:H
samples [179], plasma-enhanced chemical vapor deposition
[180], and surface modification by implantation [181] and
in porous Si layers [64, 162, 163, 169, 174, 181–187]. From
the observed phonon shifts the particle sizes have been estimated using the confinement model CM-I [Eq. (33)] with
Gaussian weighting functions [169, 179, 180, 181, 184, 187]
or with the weighting function W r L = sinr/r, by
analogy with the ground state of an electron in a hard sphere
[162, 163]. In some cases the particle sizes were determined
independently by other methods. In Figure 17 observed frequency shifts and half-width broadenings are plotted versus the particle size determined by TEM [169] or XRD
[177, 180]. For comparison calculated results were included,
obtained with the phonon confinement model CM-I using
Si
100
a
1
2 ,s
∆Γ (c m - 1 )
0
2,c
10
Proofs Only
1
b
∆Γ (c m - 1 )
S ( κ)
0 .8
3 ,s
10
1
3 ,c
1
2 ,c
1
2 ,s
10
L (n m )
Figure 17. Observed frequency shifts and half-widths of the Raman
bands of the optical phonon in nanocluster Si ([188], [177], [180])
and porous Si (• [169]) depending on the particle size L. The curves are
results of calculations: (1) confinement model CM-I, parameters taken
from [169]; (2) confinement model CM-II; s, spheres; c, long cylinders;
(3) microscopic calculations [170]; s: spheres; c: cylinders.
20
a Gaussian weighting function (parameters taken from
Ref. [169]), with the model CM-II (spheres and cylinders),
and results calculated with the bond polarizability model
[170] (spheres and cylinders). For the following reasons the
confinement models cannot be used without taking some
care.
(i) The particle shapes are often not well defined and
the particles have a size distribution.
A lineshape analysis of Raman spectra of porous
Si by assuming that the crystallite sizes obey a lognormal distribution has been performed in [163],
assuming two nanocrystalline components with Gaussian distributions in [187].
(ii) Strain-induced changes in the phonon frequencies
have to be taken into account [161, 180, 184]. It is
known that in nanoporous Si the Raman peak can be
shifted by strain to lower energies (≤2 cm−1 [189,
190]. Stress inhomogeneities on a nanometer scale
can be expected to broaden the Raman signal.
(iii) The structure of the particles is not perfect
crystalline.
In some cases spectral features were observed which consist of two or more bands [169, 176, 180, 187, 191–193].
Although a broad band at about 475 cm−1 can be assigned
to amorphous Si, we supposed that the origin of two bands
between this band and 520 cm−1 (bulk phonon frequency)
is due to a splitting of the optical phonon (degenerate for
q = 0 in the Brillouin zone center) in TO and LO modes
for larger q values [191] or to a combined contribution of
two types of nanocrystallites [184, 187, 193, 194] or to a shell
structure of the particles with modified vibrational contributions of the near-surface atoms. Xia et al. [180] observed
one band for crystalline sizes L ≤ 2.2 nm and L ≥ 5.3
nm in hydrogenated nanocrystalline Si films. For samples
with 2.2 nm ≤ L ≤ 5.3 nm the Raman spectra were composed of two bands: one peaked at 505–509 cm−1 and the
other at 512–517 cm−1 . A spherically shaped cluster with two
shells was used to model the nanocrystals, and the observed
Raman spectra were analyzed by combining the confinement
model CM-I and strain effects. The size dependence of the
relative intensity of the two bands corresponds to the ratio
of the near-surface region and the crystalline core. It should
be noted that a hexagonal phase of Si with Raman peak at
about 500 cm−1 has also been found in pressure-treated Si
[196] and in wear debris from dicing of silicon wafers [197].
For L ≤ 1.5 nm (corresponding to about 200 atoms)
reported Raman spectra of Si clusters are to some extent
similar to those of amorphous Si films. A new aspect is the
observation of Raman bands around 550 cm−1 extending
up to 600 cm−1 , well above the highest peaks of bulk Si in
Si-rich SiO2 films with clusters [198]. The observed spectral shapes are in qualitative agreement with the density of
states spectra calculated by Feldman et al. [199] for Si33 and
Si45 clusters. Raman spectra of molecular clusters Si4 , Si6 ,
and Si7 have been successfully measured by Honea et al.
[200] by applying a technique of surface-enhanced Raman
scattering. Si cluster ions were produced by laser vaporization, size-selected in a quadrupole mass spectrometer and
co-deposited with an N2 matrix onto a liquid helium-cooled
Light Scattering of Semiconducting Nanoparticles
substrate. By comparison of the observed very sharp vibrational lines with ab initio calculations they could determine
the cluster structures.
In comparison with Si nanoparticles of the same size the
observed frequency shifts and half-width broadenings in Ge
nanoparticles are much smaller. A smaller shift is expected
due to the position of the optical phonon in c-Ge at the
lower frequency of about 300 cm−1 , the main reason could
be the smaller dispersion of the optical phonons, which was
observed by neutron scattering for Ge [201] compared with
measurements for Si [202–204]. Kanata et al. [205] investigated gas-evaporated Ge particles with sizes of 10–60 nm,
as determined by Raman measurements and by XRD measurements. Good agreement was obtained by applying the
CM-I model ( = 25/8) and assuming spheres with a singlecrystalline core and a coating amorphous shell. The confinement effect in Ge dots in the size range L = 2.6–13 nm
has been studied systematically by Bottani et al. [206] with
Raman scattering, TEM, and absorption measurements. The
samples were grown by an evaporation-condensation selforganization technique that allows obtaining nearly spherically shaped nanoparticles with a narrow size dispersion
[207]. As the size decreases an inhomogeneous broadening, a redshift and a softening of the optical phonon peak
were observed. The measurements were found to be in
agreement with confinement model calculations (CM-I),
taking into account the size distribution as measured by
TEM. Sasaki and Horie [208] investigated gas–evaporated
Ge nanocrystals in the size range 2.6–13 nm with resonant Raman scattering. They found that the size as deduced
from the frequency shift and half-width broadening is about
half of that obtained from the lowest electronic excitation
energy measured in the resonant Raman scattering. Additional anharmonicity is assumed to be associated with a
decay process including surface vibrational states. They used
a confinement model with phonon waves confined to a cube
and vanishing amplitudes at the surface.
Intensive photoluminescence around 2 eV has been
observed by formation of Ge nanocrystals in SiO2 [209–211].
The phonon confinement measured by Raman scattering
was used to show that the crystalline phase formed consists
of Ge nanocrystals. If Si substrates are used, the analysis of
the experimental results has to be done with extreme care,
because an observed peak at about 300 cm−1 could originate from the two-phonon peak of the Si substrate and not
from Ge ([211–213]; see also Fig. 15). To tackle this problem resonant Raman scattering with excitation laser energy
at 2.4 eV and polarized Raman scattering can be used [214].
6.2.2. II–VI Semiconductors
In polar semiconductors the confinement effect of the optical phonons in particles of a few nanometers in size can be
more difficult to observe due to the simultaneous appearance of surface modes between the LO and the TO phonons
[145].
Small frequency shifts (up to about 3 cm−1 of the LO
phonon of small quantum dots of II–VI compounds in comparison with the bulk value were observed for decreasing radii (down to ≈2 nm) in glass matrices. Redshifts for
CdSe dots [39, 54, 110, 146] blueshifts [60] as well as redshifts [52, 145] for CdSx Se1−x dots, and size-independent LO
Light Scattering of Semiconducting Nanoparticles
phonon frequency for CdS dots [32] are reported. Obviously
the shifts depend also on the glass matrix used [215]. To
explain the observed phonon shifts and broadenings some
authors, in addition to the confinement models, introduced
lattice contraction caused by the mismatch of the thermal expansion between the quantum dots and their host
material and surface tension increasing with decreasing dot
size [60, 215]. Shiang et al. [32] estimated with a confinement model the observed asymmetric band shape, the
increase in bandwidth, and the frequency downshift with
decreasing size. They assumed that surface pressure effects
nearly compensate for the confinement effect in such a way
that no size dependence of the LO frequency results. For
small CdSe nanoparticles embedded in glass, an asymmetrical broadening of the LO phonon band toward the lowfrequency side for nanocrystals with R = 21 and 1.8 nm and
a decrease to lower frequencies was observed [111]. This was
attributed to confinement of the LO phonons. The dispersion of the observed LO phonon frequencies was compared
with ab initio calculations [111].
6.2.3. Other Compounds
Phonon confinement effects were also studied on selfassembled GaN quantum dots of 2–3 nm height [216],
crystalline As precipitates in GaAs [217], in CeO2−y
nanoparticles [218], diamond nanoparticles [219, 220] and in
BN micro- and nanoparticles [165]. The latter case is interesting because an upward shift of the high frequency mode
E2g was observed with decreasing particle size similar to that
for small particles of graphite. Calculations were performed
with the confinement model CM-II using phonon dispersion
curves reported in [221].
Redshift or blueshift in dependence on the coating was
observed in coated TiO2 nanoparticles [222].
7. NEW INSTRUMENTATION
AND METHODS
High-Throughput Devices Many applications of Raman
spectroscopy benefit from developments to measure the
usually weak signals more efficiently. For recording scattered light, charge-coupled devices (CCD) have become very
widespread during the last 10 years and are now used in
the most multichannel Raman spectrometers. They have a
very high quantum efficiency over a broad spectral range
and a very low dark count level when cooled. The separation of the Raman signals from the much stronger elastically
scattered light has been achieved in the past by use of double or triple monochromators. Another technical solution,
now used more and more, are Raman spectrometers consisting of a holographic notch filter for stray light suppression
and a single grating spectrograph. Such spectrometers have
up to 10 times larger optical throughput, are more compact with fewer optical components, and are less expensive
than triple monochromators. The notch filters suppress the
incident laser light by a factor of more than 106 while the
rest of the spectrum is transmitted with a high efficiency of
about 80%. The small edgewidth of less than 4 nm allows
measurements of frequencies as low as 50 cm−1 . The application of this new generation of Raman spectrometers is
21
attractive for measurements on nanoparticles with low signals, too. However, for low-frequency Raman measurements
on acoustic modes closer to the laser line, better stray light
rejection is required. This can be realized by application of
a triple monochromator and a photomultiplier as detector
or by using a double monochromator in the configuration as
a cutoff filter followed by a single spectrograph and CCD as
detector.
Surface-Enhanced Raman Scattering An enhancement
of Raman scattering intensity was observed for molecules
deposited on silver surfaces. A large increase in the local
electric field intensity occurs when the wavevector of the
surface plasmon mode of the silver film is matched to that
of the incident laser [223–226]. Vibrational Raman spectra
of small Si clusters, size-selected in a quadrupole mass spectrometer and co-deposited with an N2 matrix onto a liquid helium-cooled silver film were measured with an signal
enhancement of about 50 at the surface plasmon–polariton
resonance [200].
The Raman scattering intensity of molecules can be
enhanced in SERS processes by 5 to 6 orders of magnitude
when they are adsorbed on a roughened silver surface.
Although all details of the origin of the signal enhancement are not yet understood, two main mechanisms are
usually assumed to be responsible. The so-called electromagnetic effect is considered by most authors to be the
major contributor to SERS. The electromagnetic effect was
explained in terms of the excitation of localized surface plasmons in silver nanostructures (and nanostructures of a few
other metals), which enhances the electromagnetic fields
near the nanoparticle’s surface. Resonance of the exciting/scattered fields with the surface plasmons is assumed.
The second mechanism discussed, the so-called chemical enhancement effect, is attributed to resonant Raman
processes involving charge transfer between the adsorbed
molecule and the metal surface. Although the average
enhancement factor has been normally measured to lie in
the 105 –106 range, it has been shown recently that the
enhancement is laterally highly inhomogeneous and concentrated at hot spots at the surface of the metallic nanostructures where the SERS enhancement can be as large as 1010
[227, 228]. The lateral resolution is thus not determined by
the diffraction limit, but by the spatial confinement of the
local fields. Over the last 5 years it was shown that SERS can
be observed even from single molecules [229–233]. The fact
that enhancements large enough to allow single-molecule
spectra to be detected have only been observed for some
adsorbates (generally dye molecules) suggests that resonance processes are involved.
The target species are in local electromagnetic fields of
high intensity and, in particular, also very strong field gradients. These effects could explain some observed results of
SERS measurements on single carbon nanotubes, not seen
with conventional Raman scattering, for example, exchange
in scattering intensity between phonon modes and changes
of the depolarization ratios [234, 235].
Near-Field Optical Microscopy In conventional Raman
microscopy a pixel size no smaller than 1 m is common.
Resolutions of 100 nm or even substantially smaller can be
realized by combining Raman scattering with the near-field
22
optical microscopy [236]. A laser beam is coupled into an
optical fiber that is sharpened to a very small tip with conical shape at the end. The dimensions of the tip are much
smaller than the wavelength of the exciting laser light. Initially, the major obstacles for obtaining Raman spectra were
arising from the weak light intensity given by the excitation
and the detection through small apertures and by the Raman
scattering process. The principle of the method was demonstrated, for example, by the measurement of the phonon
spectra of diamond with a fiber probe of 100 nm diameter
[237]. In measurements of Raman spectra from liquid CCl4
the sampling volume of 70 nm diameter and 10 nm depth
contained only about 250,000 molecules [238]. For applications with weaker scatterers a considerable improvement
of the signals obtained is necessary. One way to enhance
the Raman cross section is resonant excitation, which was
already used in some cases. Another technique is the combination of near-field optical microscopy with SERS, with
which strong enhancement of the Raman signals by many
orders of magnitude was obtained [239, 240].
Tip-Enhanced Near-Field Raman Spectroscopy Tipenhanced near-field Raman scattering is one of the forthcoming characterization techniques, which can be effectively
used to understand the vibronic and electronic properties of a material at nanoscale. This technique can also
be used even for a single molecule detection, because it
allows for the detection of molecular vibration spectra with
a resolution smaller than the diffraction limit of the light.
It has recently been demonstrated [241] that the nearfield detection of Raman scattering using a metallized tip
achieves high spatial resolution beyond the known limits of
surface-enhanced Raman scattering and reveals new weak
vibrational modes, which cannot be seen in usual far-field
measurements. It has been demonstrated that because of the
local field enhancement, the Raman scattering is enhanced
by a factor of 40 and the spectra could be measured at a
spatial resolution of several tens of nanometers. By using
this technique, single nanoparticle detection could be possible. Also, single molecule detection could be much more
precise and accurate compared to SERS, where the spatial
resolution is diffraction limited. For near-field detection, it
is possible to observe the dynamics of a molecule with spatial resolution on a molecular scale. In these experiments,
a metallic tip with a size of a few tens of nanometers is
approached to the sample where the source light is tightly
focused. The localized surface plasmon–polariton is excited
at the tip apex. The area of the field localized at the tip corresponds to the diameter of the tip, which works as the nearfield light source and attends the super-resolving capability.
Molecules soaked in this localized field around the tip scatter inelastically through Raman scattering. The Raman spectra recorded contain useful information about molecular
vibrations at the nanoscale level, which cannot be detected
through other techniques. In this way, it is possible to
understand the special behavior of molecules, excitons, and
electron–phonon interaction inside a nanoparticle, which
can reveal new understanding about the confinement and
the effect of confinement on the electron–phonon system.
Light Scattering of Semiconducting Nanoparticles
GLOSSARY
Effective medium theory The properties (e.g., the dielectric function) of a composite are described by averaging over
local inhomogeneities introducing suitable effective media
models.
Quantum confinement In nanometer-sized structures
(quantum wells, wires, and dots) the allowed energy states
and the density of states are modified in comparison with
the corresponding bulk material. Onset occurs when one or
more dimensions of a structure become comparable to the
characteristic length scale of an elementary excitation.
Raman scattering process Inelastic light scattering. The
scattered photo has lost (Stokes process) or gained (antiStokes process) the energy of an elementary excitation (e.g.,
a phonon) compared with the energy of the incident photon.
Rayleigh scattering process Elastic light scattering. The
scattered and the incident photon have the same energy.
Small angle X-ray scattering (SAXS), small angle neutron
scattering (SANS) Scattering at small angles in the vicinity of the primary beam. The scattering features at these
angles correspond to structures ranging from nanometers to
submicrometers.
X-ray diffraction (XRD) Technique for characterizing
crystalline materials. It provides informations on structures,
phases, average grain size, crystallinity, strain, crystal
defects.
ACKNOWLEDGMENTS
The authors acknowledge fruitful cooperation with Professor I. M. Tiginyanu (Technical University of Moldova),
Dr. A. Sarua (University of Bristol), and Dr. W. Cordts
(Technical University Bergakademie Freiberg) who contributed to many illustrative results presented in this chapter
and to express their gratitude to them for stimulating discussions.
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