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Fundamentals of Optical Properties of
Nanostructured Materials and their
Growth.
L. M. Kukreja
Thin Film Laboratory
Raja Ramanna Centre for Advanced Technology
P.O. CAT, Indore 452 013,
Email: [email protected]
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Contents
1. Introduction
2. Fundamentals of Nanostructured Semiconductors
2.1 Energy states of carriers in free space of different dimensionality
2.2 Effect of confinement on carrier energy states
2.3 Density of states
2.3.1 Density of states for bulk materials
2.3.2 Density of states for two dimensional materials
2.3.3 Density of states for one dimensional materials
2.3.4 Density of states for zero dimensional materials
2.4 Applications of Nanostructured Semiconductors
2.5 Reference on growth methodologies
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1. Introduction
Later part of the twentieth century witnessed a spectacular turn in the way we understood
material science. Earlier notion that to change material characteristics we need to change its
chemical composition or we need to mix materials in different modes or conditions remained no
more isolated. A radically different approach to understand materials emerged which converged
to the finding that material characteristics could now also be changed by changing the size
keeping the chemical composition intact [1-10]. However the methodology of varying the
material characteristics with size was feasible only in a specific size regime, called the quantum
confinement regime and the variation in a specific material characteristic invariably affected the
other characteristics of the same material as well [11-23]. However despite these limitations this
possibility to tailor the material properties by varying the size alone provided the mankind a
revolutionary wisdom of materials science, which enabled the emergence of a new branch of
technology called Nanotechnology. The fundamental drive to scale the height of Nanotechnology
to the extent that we witness today was not only to make more compact, denser and more
integrated devices but also because newer characteristics of the materials were discovered. These
characteristics offered the possibilities to push the existing frontiers of the technology in leaps.
Today Nanoscience and Nanotechnology are not only highly developed arenas of human
endeavors but also some of the fastest growing branches of science and technology [24-36] with
ramifications into multitude of areas such as Nano-medicine [24-29,37-41], Nanostructured
Semiconductor lasers [30,42-62], Solar Energy [63-72], Nano-electronics and Devices [73-90]
and Quantum Computations [91-94] etc.
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Why do Nanostructured materials behave differently? As we know that the genesis of most
material properties such as optical, magnetic, mechanical and chemical lies in the states of
electrons in the constituting atoms, their configuration in the atomic structures and the way these
electrons form bonds with the neighboring atoms in a molecular system. These electrons
experience confinement effects when those are trapped in a nanometer sized structure of the
material. It is thanks to these confinement effects that the material characteristics are unusual
and generally size dependent. Our current understanding on the wave nature of the microphysical
particles such as electrons and the quantum mechanics provide a fundamental basis for
explaining the confinement effects. Since the typical de Broglie wavelength of electrons is in
the range on nanometers, the confinement effects are pronounced and observable when the
confining system is of the comparable size. Another equally important is the ratio of surface area
to the volume of a material structure, which tends to increase when the particle size is decreased.
As a result the surface area dependent material properties start dominating or anomalous
behaviors are observed in the nano-scale regime. For example, nano-particle catalysts are found
to be chemically more effective than the bulk catalysts. Similarly, electric field related effects are
also enhanced in the nano-particles due to their sheer small size. For example, metal nano-
particle X-ray targets have been found to produce harder X-rays with enhanced yields [95, 96].
One of the most significant aspects of any technology is to fabricate materials compatible to that
technology. This aspect is rather demanding so for as the Nanotechnology is concerned. This is
because fabricating material structures of nanometer size with comprehensive control is an
intricate problem. However a host of methodologies have indeed been evolved to grow
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nanometer size structures of different materials under variety of conditions with varying degrees
of controls on shapes, structural qualities and sizes. In fact Nanofabrication is a gargantuan and
contemporary field of research and technology development. But in this chapter, to provide
reasonably thorough and deep knowledge, we will confine ourselves to a narrow slice of this
field, which deals with the nanofabrication of semiconductors using a very specific methodology
of Pulsed Laser Deposition (PLD).
To make the chapter self contained we have discussed in brief the fundamentals of
Nanostructured semiconductors specifically those pertinent to the confinement effects,
dimensions of confinement, dependence of band-gap on the confinement size and the density of
states. Since PLD is the basic methodology of nanofabrication discussed in this chapter, it has
been dealt with in detail. In this chapter we have exhaustively presented all the fundamental
aspects, the available level of technology, its merits and demerits and a perspective of the
existing trends of this versatile and still esoteric fabrication methodology for semiconductors.
Finally we have presented a digest of the state of affairs of the nanostructured semiconductors
grown by PLD. Some of the novel optical, structural, electrical and magnetic properties of these
PLD grown nanostructured semiconductors are also presented and discussed in this chapter
upholding its brevity.
2. Fundamentals of Nanostructured Semiconductors
A large number of books [1-10], review articles [24-36], and research papers [11-23,37-150]
have been published on the subject of nanostructured semiconductors and related materials. Most
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of the reported papers are devoted to confinement effect [11-23], applications in the area of
biotechnology, lasers, electronic devices and solar energy [37-96]. Some of these papers are also
devoted to growth and characterization of nanostructured semiconductor [97-150]. A
semiconductor structure having at least one dimension in the range of 1 to few hundred
nanometers is defined as a nanostructured semiconductor. Semiconductor structure with reduced
dimensionality has, amongst others, optical properties that are quite different from the optical
properties of crystalline bulk matter. For example, silicon has a band gap of about 1.2 eV in the
bulk crystalline state whereas nanocrystalline dots of silicon have band gaps of 2ev or more.
Thus semiconductor nanostructure shows size dependent properties different from those of a
macroscopic semiconductor if one or more dimensions of the structure are comparable to
wavelength of light or wavelength of electrons and holes. The quantum effects arise in systems,
which confine electrons to regions comparable to their de Broglie wavelength. In other words if
the physical size of a material structure becomes smaller, the quantum mechanical effects
become observable. In the following sections we will discuss the confinement effect and its
implications on allowed energy bands and densities of sates.
2.1 Energy states of carriers in free space of different dimensionality
In a nanostructured semiconductor, the quantum confinement occurs when electrons or holes in
the material are trapped in one or more of the three dimensions within nanometer size regime. If
the confinement occurs in one dimension only (say in the z-direction), it is assumed that the
carriers like electrons and holes have free motion in other two x and y directions. This
assumption is valid because size of the material in x and y directions is very large compared to
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the size in z-direction. Such a structure would be two dimensional and the carriers confined in
one dimension will have two degrees of freedom. If the confinement is in two directions (say y
and z), the electrons have free motion in x-direction. Such a structure would be one dimensional
and the carriers confined in two dimensions will have only one degree of freedom. Confinement
in all the three directions (i.e. x, y and z) will result in no free motion of electrons or holes in any
direction. Such a structure would be zero dimensional and the carriers confined in all the three
dimensions will have no degree of freedom. The inherent size dependent properties of these
Nanostructured materials will be result of two basic mechanisms: free energy variation due to
free motion of carrier like electron and quantized energies due to quantum confinement. The
total energy of electrons (or holes) will be the sum of allowed energies associated with the
motion of these carriers along the confined direction and the kinetic energy due to free motion in
the remaining unconfined directions. Let us first consider the energy of electrons due to free
unconfined motion.
In the case of bulk material an electron can be treated as unconfined. Therefore the following
Schrdinger equation for unconfined electron in free space can be used [2, 3]
=
+
+
E
zyxm 2
2
2
2
2
22
2
h (1)
Where m is the mass of the electron and E represents the allowed energy of such an
electron. The solution of this equation gives:
( )( )
( rikrk
.exp2
13
= ) (2)
Which are plane waves having wave vector kwith its components kx, kyand kzalong the
x, y and z directions respectively and can thus be represented as:
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( zyx kkkk ,,= ) (3)
Solution of equation (1) gives the kinetic energy E, which can be written as:
(222
222
22zyx kkkmm
kE ++==
hh
) (4)
For confinement in one dimension (say z), free motion of an electron is allowed only in two
dimension (x and y directions). The kinetic energy of an electron in two directions (x and y) can
be written as:
( 22222
22yx kk
mm
kE +==
hh ) (5)
For confinement in two dimensions (say z and y directions), the free motion of an electron is
allowed only in one direction (x direction). Therefore the kinetic energy of an electron due to
free motion in x-direction can be written as
m
k
m
kE x
22
2222hh
== (6)
So far we considered the kinetic energy of the electrons with different degrees of freedom. But
what happens to the energy states due to confinement? To understand this we need to solve the
Schrdinger equation in a potential well created by the confinement instead of the free space
case dealt with here. For the sake of comprehending the fundamentals concisely we will consider
an ideal square potential well, in different dimensions. For the confinement along one dimension,
it is the case of a quantum well, for confinement in two dimensions, it is quantum wire and
finally for the confinement in all the three dimensions it is the case of a quantum dot.
2.2 Effects of confinement on energy states
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A confined material structure is the one, which has one or more of its dimensions reduced to the
size of confinement regime i.e. in the range of nanometers. For example a semiconductor
quantum well is a two dimensional structure, which in practice consists of a low band gap
semiconductor confinement layer flanked by two layers of a higher band gap semiconductor on
either side. These layers are called the barrier layers and are invariably thicker than the
confinement layer, which is of thickness in the nanometer regime. Although such a structure has
finite depth of the potential well equal to the difference of the band gaps of the confinement and
the barrier layers, but for a theoretical understanding we can take the square potential well of
infinite depth. A schematic of such an infinite square well is illustrated in figure 1, where a is
the width of the well, which is equal to the thickness of the confinement layer. Now we will see
what happens to the electrons, which are confined in one direction in such a well.
V =
0 a
VZ a
V
Z
V= V=
V V = VZ a
0
0
0
V
Figure 1. One-dimensional infinite square potential well.
The electrons will have zero potential in the region < z
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An electron within this quantum well behaves as a free particle and its time independent
Schrdinger equation can be written as:
( )( )zE
z
z
m=
2
22
2
hfor az
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0 a
Energ
y(E)
Distance (Z)
n z = 1
nz = 2
Figure 2. Quantized energy levels for n = 1 and n = 2 and the corresponding wave functions for
infinite square potential well.
From equations (5) and (12) we can write total energy of the electron which is the sum of
quantized energy (in confined direction z) and the kinetic energy due to its (x, y) motion (in
unconfined directions). Thus the total energy of electronETotalis
( )222
2yxzTotal kk
mEnE ++=
h
or, by substituting for Enzfrom equation (12)
( 22222
22yx
zTotal kk
ma
n
mE ++
=
hh ) (13)
From this equation one can draw two inferences. One, the energy states are now quantized due to
the presence of the integers nz and second, the energy states are now size i.e. a dependent.
Thus quantum confinement results in the discreteness of the energy levels, which are now
dependent on the size of the confinement dimension.
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In similar way we can calculate the quantized energies and total energy for a quantum wire and a
quantum dot. In case of quantum wire two dimensions (say z and y) are confined, therefore total
energy for quantum wire can be written as
( )22
2xyzTotal k
mEnEnE
h++=
Or, by substituting for Enz,
( 22
2222
222x
yzTotal k
mb
n
ma
n
mE
hhh+
+
=
) (14)
The total energy for a quantum dot having all three dimensions (z, y and x) confined can be
written as
222
222
222
+
+
=
c
n
mb
n
ma
n
mE x
yzTotal
hhh(15)
Where are integers and are the confined sizes of material in z, y and x directions
respectively. From equations (14) and (15) it is clear that as the dimensions of confinement
increase the free space terms are replaced by the confinement terms in the energy states, which in
turn enhance the size dependence of the energy states. A similar size dependence of the energy
states can also be expected for holes in the valance band.
xyz nnn ,, cba ,,
2.3 Density of states
The density of states is defined as the number of energy states present in a unit energy interval
per unit volume of the material structure. The density of states gives an idea of the energies when
we combine the effect of confinement in one direction and unconfined in the remaining
directions. Now we will derive analytical expressions of density of states and see its effect on the
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electronic structure for bulk semiconductors (unconfined in all direction) and semiconductors
having confinement in one direction, two and three directions.
2.3.1 Density of states for bulk materials
One would like to know that how density of states depends on sizes of material in x, y and z
direction? For this let us consider the material having sizes in x, y and z directions
respectively as shown in figure 3. Let k be the wave vectors along x, y, z directions
respectively in k space.
zyx LLL ,,
zyx kk ,,
X
Y
Z
L Y
L X
L Z
Figure 3. Bulk material showing extension along all the three dimensions.
Under periodic boundary condition, the allowed values of wave vector along x, y and z
directions are
x
xL
k2
= (16a)
y
yL
k2
= (16b)
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z
zL
k2
= (16c)
The volume of a sphere of radius kin k space is given by
3*
3
4k=V (17)
Let n be the number of states/modes in the sphere, then
zyx kkk
Vn
*
=
or, zyx LLLk
n2
3
6= [Using equations (16a, b and c) and(17)]
When we consider the spins of electron, the total number of states in the sphere (say n ) can be
written as
=2n=n zyx LLLk
2
3
3(18)
Let be the number of states per unit volume of the bulk material. ThereforeVN3
=VN3zyx LLL
n = 2
3
3k (19)
The density of states for a three dimensional material defined asDN3E
NV
3 will therefore be,
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=DN3E
NV
3
From equation (19)
=DN3Ek
3
231
Or =DN3E
k
k
k
3
23
1
Or =DN3E
kk
2
2
(20)
But from equation (4)2
2 2
h
mE=k for free electron. Therefore equation (20) can be written as:
=DN3E
EmmE
21
2
1
222
221
hh
Or =DN3 Em 2
3
22
2
2
1
h(21)
This gives the density of states in three dimensions. It is obvious from this equation that
DN3 E (22)
This dependence of density of states for three dimensions is shown in figure 4.
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Density ofstates
Energy (E)
(N3D)
Figure 4. Variation of density of states with energy for bulk material i.e. a three dimensional
semiconductor.
2.3.2 Density of states for two dimensional materials
Now let us consider the density of states in a material structure having confinement in one
direction (say z) such as a quantum well. Earlier it was mentioned that the total energy of carriers
in such system is a sum of quantized energy along the quantized direction (Enz) plus the kinetic
energy of electron in other two free directions (Ex, y) i.e.
(23)yxzTotal EEnE ,+=
From equations (12) and (13), we can write
=TotalE ( 222
22
22yx
z
z kkmL
n
m++
hh ) (24)
Where Lz is the confinement width of the structure say the quantum well or width of
material in quantized direction and
n ......................3,2,1=z
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and k2=kx
2+ ky
2
For calculation of density of states in two dimensional materials we take area (A) instead of
volume (for the three dimensional case) in k space, which would be:
A= (25)2k
and the allowed values of wave vector along x and y directions would be:
kx=xL
2(26a)
ky=yL
2(26b)
The area of a given state/modes will be kx, ky.
Now let is the number of states in k space area. Thereforen
n =yxkk
A
Or using equation (25),
n =yxkk
k
2
(27)
From equations (26a and b) and (27), we can write
n =4
2
yxLLk
If we consider the spins of electron, the total number of states in area (let n ) can be written as:
n =2 =2 n4
2
yxLLk
Or =n2
2
yxLLk(28)
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Let is the number of states per unit area in real space of the two dimensional material, in
that case,
VN2
=VN2 yxLL
n
Therefore, using equation (28),
=VN22
2
yxLLk
yxLL
1
Or =VN22
2k(29)
Let be the density of states for two dimensional materials, which as per the above definition
is
DN2
E
N V
2 . Then,
=DN2E
N V
2 =
2
2k
E[From equation (29)]
Or =D
N2 E
k
2
2
1
Or =DN2E
k
k
k
2
2
1
Or =DN2E
kk
(30)
From the solution of Schrdinger equation for free electron motion as from equation (4), we
know that2
2 2hmEk = , therefore equation (30) reduces to
=DN22
1
2
2
1
2
221
hh
mE
E
mE
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Or
22h
mND = (31)
This shows that the density of states for a two dimensional structure is constant and does not
depend on the energy due to free motion of electron. It may be noted here that this is the energy
density of the sub-band for a given kz (orEnz). There will be an additional
2h
mterm for each
successive kzorEnz. Therefore the comprehensive density of states can be written as
(=
=...2,1
22
zn
zD EnEm
Nh
) (32)
Where
is a step function and,
when zEnE= , =1
and for zEnE , =0
A schematic dependence of density of states on energy for a two dimensional material structure
such as quantum well is shown in figure 5.
Density ofstates
Energy (E)
(N'2D)
Figure 5. Variation of density of states with energy for quantum wells i.e. a two dimensional
semiconductor.
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2.3.3 Density of states for one dimensional materials
The one dimensional material structure is defined as that which has two dimensions confined
(say z and y directions) and one dimension unconfined (say x direction). The total energy of the
system in this case can be written as
xyzTotal EEnEnE ++=
Or =TotalE ( 22
22
22
222
x
y
y
z
z k
mL
n
mL
n
m
hhh+
+
) (33)
Where Lz and Ly are widths of material in quantized directions and n are integers.yz n,
And k2= kx
2.
In k space total length will be 2k because we consider both positive and negative directions.
Therefore the number of states/modes ( n ) along this length is
xk
k2
=n
Or kLx
=
n (34)
where the allowed value of kx isxL
2
After considering the two spins of electrons, the number of states per unit length (let n ) can be
written as
kL
n x
==
22n
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when E=Enz,,ny , =1
and forE En z,ny, =0
The behavior of density of states for a one dimensional system is shown in figure 6.
Density ofstates
(N'1D)
Energy (E)
Figure 6. Variation of density of states with energy for quantum wire i.e. a one dimensional
semiconductor
2.3.4 Density of states for zero dimensional materials
A material having all dimensions confined (say z, y and x), is known as zero dimensional
material. In this case there are no dimensions in which free motion of charge carriers is possible.
The total energy of this system will be
xyzTotal EnEnEnE ++=
=TotalE
22
22
22
222
+
+
x
x
y
y
z
z
L
n
mL
n
mL
n
m
hhh(38)
Where are integers.xyz nnn ,,
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In this case there are no allowed values of wave vector along x, y, and z directions, therefore the
density of states can be written as
xyzD nnEnEN ,,0 = (39)
A schematic of the functional dependence of density of states on energy for a zero dimensional
system is shown in figure 7.
Density ofstates
Energy (E)
(N'0D)
Figure 7. Variations of density of states with energy for quantum doti.e. a zero dimensional
semiconductor.
2.4 Applications of Nanostructured Semiconductors
In the last decade nanostructured materials have attracted much attention from physicists,
chemists and engineers due to their intriguing electrical and optical properties, which are found
to be different from those of bulk materials. As seen above the electrical and optical properties of
nanostructured semiconductors are highly size dependent and thus can be modified by varying
the size alone, keeping the chemical composition in tact [1-23]. These size variations and
confinement dimensions only categorize the nanostructured materials as quantum wells, quantum
wires, and quantum dots. Even today Nanoscience and nanotechnology continue to be an active
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and challenging subject in material science and other fields. Nanostructured materials have
variety of applications in the areas of biotechnology and medicine [24-29, 37-41], semiconductor
quantum devices [73-90], solar cells [63-72], quantum computations [91-94], semiconductor
lasers [42-62] etc. To discuss all these applications in detail here is beyond the scope of this
chapter. Therefore we will touch upon some of the applications, especially that of nanostructured
semiconductors.
Fluorescence microscopy is one of the finest methods for biological research and has emerged as
an inescapable tool for cell biologists. It provides real-time imaging of living cells with high
resolution. But for fluorescence microscopy one needs Biological labeling. Conventionally
biological labeling is carried out by using organic dyes such as tetramethylrhodamine (TMR).
However the organic dyes have certain limitations such as sensitivity to pH changes,
susceptibility to photo bleaching, fixed fluorescence spectra and separated absorption and
fluorescence maxima etc. In recent years, semiconductor quantum dots have been applied
biological labeling, which overcome the above mentioned drawbacks [25]. It provides the option
of continuously tuning the emission wavelength of the semiconductor by changing the size of its
quantum dots. For example it is possible to tune the emission of quantum dots of CdSe over the
entire visible spectrum by changing the radius of its dots. Quantum dots have narrow emission
spectra (for example that of CdSe have a bandwidth of about 30nm) therefore a single light
source can be used for excitation of multiple semiconductor quantum dots to emit different
colors. Each of these different - colored quantum dots could be functionalized with a unique
ligand-biomolecule pair, which in turn facilitates binding it to a particular biological structure.
This makes multicolor experiments more convenient. In addition to this, semiconductor quantum
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dots are inorganic and more robust for photobleaching than organic dyes; it is also insensitive to
pH changes of the biological environments. These optical properties of semiconductor quantum
dots make them ideal biological labeling materials which have applications in bioengineering as
a well as in cell and molecular biology.
The applications and uses of semiconductor lasers are well known to scientific as well medical
communities and industries. Conventional sources of lasers use gases, organic dyes and
semiconductors etc. as their active mediums. The semiconductor lasers are widely used for
applications in compact disc players, laser printers etc. Those are also well known to play critical
and important role in optical communication, spectroscopic studies, defense and numerous
industries etc. In the case of nanostructured semiconductor lasers, the carrier confinement and
nature of electronic density of states of the nanostructures make it more efficient for devices
operating at lower threshold currents than lasers with bulk materials. The size dependent
emission spectra of quantum wells, quantum wires and quantum dots make them attractive lasing
media. The performance of quantum dot lasers is less temperature dependent than conventional
semiconductor lasers [45]. Due to size dependent emission of nanostructured semiconductors, in
principle, a single device can work for a blue, green or a red laser.
Nature has provided us abundant energy through sun light, which is commonly known as solar
energy. The physicists, chemists and engineers are trying to develop devices called Solar cell to
harness this energy with higher efficiencies. Due to size dependent optical and electrical
properties, the nanostructured materials can play an important role for increasing the efficiencies
of solar cell. The conventional solar cells are made by semiconductor material like silicon and
organic dyes or combination of both. But the efficiencies of these solar cells are in range of 15%
to 30%. Due to this lower efficiency, it requires large solar panel which increases the cost, size as
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well as the maintenance related problems. A wide gap semiconductor such as Zinc Oxide (ZnO)
grown in a nanostructured form is more efficient and convenient for photovoltaic applications as
a part of a solar cell [67]. The nanostructured surfaces coated with extremely thin absorber lead
to low optical losses and increases carrier transfer between active region and contact. The
semiconductor quantum dots have tunable size dependent absorption and emission characteristics
so it can be made to absorb the sunlight through out from UV to IR region. This is not possible
with organic dyes. The quantum dots are more photo-stable and the absorption cross section of
quantum dots is an order of magnitude higher than organic dyes. These properties make the
nanostructured semiconductors as ideal and efficient elements for solar cells. The nanostructured
thin film solar cells are more advantageous over the conventional solar cells because those have
less material utilization, enhanced efficiency and in some cases low processing costs etc [71].
In addition to the aforesaid applications, there are many other applications of nanostructured
semiconductors like light emitting diodes, quantum dots as memory elements, quantum
computation, sensors for toxic gases and chemicals, quantum well modulators, nano-scale
semiconductor quantum device for information and telecommunications and electroluminescent
devices etc [73-94]. It is possible to make small and efficient optical modulators using
semiconductor quantum wells [10, 88, and 89]. It is known that the optical absorption of the
quantum wells can be changed by applying an electric field perpendicular to its surface. Also in
case of quantum wells the change in absorption is relatively large which makes it suitable for
optical modulators. In the field of quantum information processing, using multi-color laser pulses
from a semiconductor quantum dot array, it is possible to perform single as well as two qubit
gate operations. By these means one can implement basic quantum computation operations on
time scale much shorter than the exciton dephasing time [93].
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References
P.Harrison, Quantum Wells, Wires and Dots, John Wiely, New York (2000)1.
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3.
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5.
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7.
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