Acoustic vibrations of semiconductor nanocrystals in doped glasses
Prabhat Verma, W. Cordts, G. Irmer, and J. Monecke
Institute of Theoretical Physics, Freiberg University of Mining and Technology, Bernhard-von-Cotta-Strasse 4, 09596 Freiberg, Germany
Received 23 February 1999
Polarization-dependent low-frequency off-resonant Raman scattering has been studied from various com-
mercially available filter glass samples, which contain CdS
x
Se
1-x
nanoparticles embedded in a glass matrix. In
order to distinguish the confined acoustic phonons from the glass background, the spectra have been compared
with those obtained from the base material, which does not contain nanoparticles. Polarized and depolarized
scattering from confined acoustic phonons was distinctly resolved near the laser line and overtones of the
polarized modes were observed. A theoretical treatment, which establishes a relation between the particle size,
the frequencies, and the widths of various phonons, taking into account the matrix influence on the vibrational
spectrum and on its damping, is presented. The material-dependent generalized form of this model enables one
to use it for any given combination of particle and matrix materials. A good agreement between the experi-
mental and the theoretical results is found. The nanoparticle sizes obtained from Raman scattering agree well
with those obtained from transmission electron microscope and anomalous small angle x-ray scattering experi-
ments. S0163-18299907231-8
I. INTRODUCTION
In recent times, the optical properties of quantum con-
fined electronic systems such as quantum wells, quantum
wires, and quantum dots have attracted considerable atten-
tion, because they differ strongly from those of the corre-
sponding bulk crystal. Especially, the electronic and optical
properties of semiconductor nanocrystals embedded in solid
matrices such as CuCl in NaCl Refs. 1 and 2 and CuCl,
CdS, CdSe, CdS
x
Se
1 -x
, and Ge in glass
3–13
have been stud-
ied in the past decade. Specific attention has been paid
8–13
to
silicate glasses in which CdS or CdS
x
Se
1 -x
nanoparticles are
grown by special thermal treatment. These glasses are the
basis for commercially available sets of yellow to red sharp
cut filters, which are very attractive from a fundamental
point of view as well as for their potential use in the field of
the nonlinear optics, where it is critical to have an accurate
knowledge of the size distribution of the particles. In a small
nanocrystal with size in the range of a few nanometers to a
few tens of nanometers, spatial confinement effects on the
electron-hole system and on the propagation of phonon be-
come significant. Due to this confinement, nanoparticles ex-
hibit distinct physical properties which have been studied in
the recent past
8–19
both experimentally and theoretically.
However, at present the information on the fundamental
physical properties of such a system is insufficient, and much
more information and knowledge is required in order to use
them as optical processing devices. A deeper knowledge of
the vibrational properties is necessary and it is indispensable
to consider the confinement effects on the vibrational prop-
erties of such a system.
Raman scattering, which is sufficiently influenced by sur-
face conditions, particle sizes, and size distribution of a
nanoparticle system, is one of the most important and non-
destructive techniques to obtain information about the vibra-
tional and electronic states in a confined system. In previous
studies,
9–19
special attention has been paid to the low-
frequency Raman scattering from elastic spherical nanopar-
ticles, which vibrate with frequencies inversely proportional
to their diameter. A peak in the low-frequency range was
observed in Raman scattering from symmetric and quadru-
polar acoustic vibrations of these spherical particles, and the
particle size was deduced from the energy of this peak. Fol-
lowing the treatment of Lamb
20
from the end of the 1800s,
these vibrations are usually described as the eigenfrequencies
of a homogeneous elastic sphere under stress-free boundary
conditions, and are classified into two categories, torsional
and spheroidal, the torsional modes being Raman inactive.
These modes can be classified according to the symmetry
group of the sphere by the angular quantum number l
( =0,1,2, . . . ), which measures the number of wavelengths
along a circle on the surface of the particle. The l =0 sphe-
roidal modes are purely radial with spherical symmetry, and
at higher values of l an angular corrugation appears. Another
index p ( =1,2,3, . . . ) distinguishes the lowest-order mode
( p =1) from its overtones ( p 2) in the Raman spectra.
Duval
21
has shown that the spheroidal modes with l =0 and
l =2 are the only Raman active modes. However, the l =1
mode also becomes Raman active under resonant
conditions.
22
The l =0 mode is completely polarized and the
l =2 mode is depolarized for a perfect sphere. After Duval,
this problem has been studied by some other authors
8–13
for
semiconducting nanoparticles embedded in glass. In most
cases,
8,11–13
only one structure in the low-frequency range
was observed and no specific polarization properties were
reported, however. Tanaka, Onari, and Arai
10
have discussed
polarization properties, but for most of their samples they
could observe only one structure, which they identified as a
combination of polarized ( l =0,p =1) and depolarized ( l
=2,p =1) modes. They could observe a trace of the depo-
larized ( l =2,p =1) mode in the cross-geometry configura-
tion only for the smallest particle in the form of a shoulder
on the Rayleigh background. Overtones of the confined
phonons ( p 1) were not reported in the past.
The frequency positions of elastic vibrations are inversely
proportional to the particle size
9
and, therefore, they are very
close to the laser line for samples with larger particles. A set
of very careful experiments is needed to observe and to re-
PHYSICAL REVIEW B 15 AUGUST 1999-II VOLUME 60, NUMBER 8
PRB 60 0163-1829/99/608/57788/$15.00 5778 ©1999 The American Physical Society