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CHAPTER 6 Nanomaterials: Classes and Fundamentals182
Since we now have several working models for the categorization of
2-D nanomaterials, let’s move on to 3-D nanomaterials. Following
our previous definition, bulk nanomaterials are materials that do
not have any dimension at the nanoscale. However, bulk nano-
materials still exhibit features at the nanoscale. As previously
dis cussed, bulk nanomaterials with dimensions larger than the
nano scale can be composed of crystallites or grains at the nano-
scale, as shown in Figure 6.11. These materials are then called
nanocrystalline materials. Figure 6.12 summarizes 2-D and 3-D
crystalline structures.
Another group of 3-D nanomaterials are the so-called nanocompos-
ites. These materials are formed of two or more materials with very
distinctive properties that act synergistically to create properties
that cannot be achieved by each single material alone. The matrix of
the nanocomposite, which can be polymeric, metallic, or ceramic,
has dimensions larger than the nanoscale, whereas the reinforcing
phase is commonly at the nanoscale. Examples of this type of 3-D
nanomaterial are shown in Figures 6.13 and 6.14, where various
nanocomposites are shown. Distinctions are based on the types of
reinforcing nanomaterials added, such as nanoparticles, nanowires,
nanotubes, or nanolayers. Within the nanocomposite classifica-
tion, we should also consider materials with multinanolayers com-
posed of various materials or sandwiches of nanolayers bonded to
a matrix core.
Many applications, especially in nanoelectronics, require the use of
various kinds of physical features, such as channels, grooves, and
raised lines, that are at the nanoscale (see Figure 6.15). A typical
copper interconnect is shown in Figure 6.16. Nanofilms, nano-
coatings, and multilayer 2-D nanomaterials can be patterned with
various features at various scales. In the case of multilayered nano-
materials, the patterns can be made on any layer. These patterns can
have different geometries and dimensions at the nanoscale or at
larger scales. Most electronic materials fall into the category of pat-
terned 2-D nanomaterials. Figure 6.17 broadly summarizes types of
nanomaterials in relation to their dimensionalities.
6.2 SIZE EFFECTS
Surface-to-Volume Ratio Versus Shape
One of the most fundamental differences between nanomateri-
als and larger-scale materials is that nanoscale materials have an
extraordinary ratio of surface area to volume. Though the properties
of traditional large-scale materials are often determined entirely by
FIGURE 6.10
Two-dimensional nanocrystalline and
microcrystalline multilayered nanomaterials.
Nanocrystalline
multilayers
t ≤ 100 nm
Microcrystalline
multilayers
t ≤ 100 nm
FIGURE 6.11
Three-dimensional nanocrystalline nanomaterial in
bulk form.
d
Bulk
Nanoscale
FIGURE 6.12
Summary of 2-D and 3-D crystalline structures.
tn ≤ 100
nm
tn ≤ 100
nm
Ly
tn ≤ 100
nm
Lx
One layer
Substrate
Multiple layers
Substrate
Microcrystalline
structure
Substrate
Microcrystalline
layers
Substrate
Microcrystalline and
crystalline structures
Nanocrystalline structures
Large-Scale Forms
Nano-
crystalline
Micro-
crystalline
Crystalline
structure
(any
dimension)
Crystalline
structures
FIGURE 6.13
Matrix-reinforced and layered nanocomposites.
Matrix
reinforced with
nanoparticles
Sandwiches
Layered nanocompositesMatrix-reinforced nanocomposites
LaminatesMatrix reinforced with
nanowires/nanotubes
CHAPTER 6 Nanomaterials: Classes and Fundamentals184
FIGURE 6.14
Basic types of large-scale nanomaterials bulk
forms. The filler materials, whether 0-D, 1-D, or
2-D nanomaterials are used to make film and bulk
nanocomposites.
d 100 nm
d 100 nm
d 100 nm
Wires
Rods
Tubes
Basic
GeometryLarge Scale Forms
(dimensions at micro or macroscale)
Point
Line
SurfaceThin film
on substrate
0-D
1-D
2-D
Nanocomposite
thick film
Nanocomposite
thick film Bulk nanocomposites
Bulk nanocomposites
Bulk nanocomposites
the properties of their bulk, due to the relatively small contribution
of a small surface area, for nanomaterials this surface-to-volume
ratio is inverted, as we will see shortly. As a result, the larger surface
area of nanomaterials (compared to their volume) plays a larger
role in dictating these materials’ important properties. This inverted
ratio and its effects on nanomaterials properties is a key feature of
nanoscience and nanotechnology.
For these reasons, a nanomaterial’s shape is of great interest because
various shapes will produce distinct surface-to-volume ratios and
therefore different properties. The expressions that follow can be
used to calculate the surface-to-volume ratios in nanomaterials
with different shapes and to illustrate the effects of their diversity.
We start with a sphere of radius r. This is typically the shape of
nanoparticles used in many applications. In this case, the surface
area is given by
A r= 4 2π
(6.1)
whereas the volume of a sphere is given by
Vr
=
4
3
3π
(6.2)
185
FIGURE 6.15
Two-dimensional nanomaterials containing patterns of features (e.g., channels, holes).
FIGURE 6.16
Nanocopper interconnects used in electronic
devices. The copper lines were produced by
electrodeposition of copper on previously patterned
channels existent in the dielectric material.
(Courtesy of Jin An and P. J. Ferreira, University of
Texas at Austin.)
t ≤ 100
nm
t ≤ 100
nm
t ≤ 100
nm
t >100 nm
One layer
t ≤ 100
nm
Substrate
Multilayers
t ≤ 100 nm
Microsccale
structure
Substrate
Feature dimensions at
nanoscale, with layer
thickness often greater
Microscale and
macroscale features
Nanoscale Structures Large-Scale Forms
Microscale
Features
Nanoscale
Substrate
Features
Copper Dielectric
Size Effects
CHAPTER 6 Nanomaterials: Classes and Fundamentals186
Thus, the surface-to-volume ratio of a sphere is given by
A
V
r
r
r= =
4
4
3
3
2
3
π
π
(6.3)
On the basis of Equation 6.3, the results for various radii are shown
in Figure 6.18. Clearly, as the radius is decreased below a certain
value, there is a dramatic increase in surface-to-volume ratio.
Next, consider a cylinder of radius r and height H—for example,
a nanowire. In this case, the volume V = πr2H, whereas the surface
area A = 2πrH. Thus, the surface-to-volume ratio is given by
A
V
r H
rH r= =
π
π
2
2
2
(6.4)
The ratios of surface-to-volume as a function of critical dimension
for the cylinder case are shown in Figure 6.18. The trend is similar
to the sphere case, although the severe increase in surface-to-volume
ratio occurs at larger critical dimensions. Let’s now turn to a cube
of side L. In this case, the volume and surface area of the cube are
given by V = L3 and A = 6L2, respectively. Therefore the surface-to-
volume ratio of a cube is given by
A
V
L
L L= =
6 62
3
(6.5)
FIGURE 6.17
General characteristics of nanomaterial classes
and their dimensionality.
Classes
Dim
ensio
nalit
y
Class 1
Discrete nano-
objects
Class 2
Surface nano-
featured materials
Class 3
Bulk nano-
structured materials
Nanoparticles
(smoke, diesel fumes)Nanocrystalline films
Nanocrystalline materials
Nanoparticle composites
Nanorods and tubes
(carbon nano tubes)Nano interconnects
Nanotube-reinforced
composites
Nano surface layers Multilayer structures
0-D
All
3 d
ime
nsio
ns
on
na
no
sca
le
1-D
2 d
ime
nsio
ns
on
na
no
sca
le
2-D
1 d
ime
nsio
n
on
na
no
sca
le
Nanofilms, foils
(gilding foil)
FIGURE 6.18
Surface-to-volume ratios for a sphere, cube,
and cylinder as a function of critical dimensions.
Nanoscale materials have extremely high
surface-to-area ratios as compared to larger-scale
materials.
187
As shown in Figure 6.18, the overall trend remains for the case of
the cube, but the significant variation in surface-to-volume ratio is
observed at larger critical dimensions compared with the sphere
and cylinder cases.
After stressing the importance of the increase in surface area in
nanomaterials relative to traditional larger-scale materials, let’s put
this information into context. With the help of a few simple cal-
culations, we can determine how much of an increase in surface
area will result—for example, from a spherical particle of 10 µm
to be reduced to a group of particles with 10 nanometers, assum-
ing that the volume remains constant. To do this, first we calculate
the volume of a sphere with 10 microns. Following Equation 6.2
gives V (10 µm) = 5.23 × 1011 nm3. We then calculate the volume
of a sphere with 10 nm. Again, with the help of Equation 6.2, we
get V (10 nm) = 523 nm3. Because the mass of the 10 micron parti-
cle is converted to a group of nanosized particles, the total volume
remains the same. Therefore, to calculate the number of nanosized
particles generated by the 10 micron particle, we simply need to
divide V (10 µm) by V (10 nm)in the form:
NV m
V nm=
( )
( )=
×= ×
10
10
5 23 10
5231 10
119
µ .particles
(6.6)
Hence, so far we can conclude that one single particle with 10
microns can generate 1 billion nanosized particles with a diameter
of 10 nm, whereas the total volume remains the same.
We are thus left with the task of finding the increase in surface area
in going from one particle to 1 billion particles. This can be done by
first calculating the surface area of the 10 micron particle. Following
Equation 6.1 gives A (10 µm) = 3.14 × 108 nm2. On the other hand,
for the case of the 10 nm particle A (10 nm) = 314 nm2. However,
since we have 1 billion 10 nm particles, the surface area of all these
particles amounts to 3.14 × 1011 nm2. This means an increase in
surface area by a factor of 1000.
Magic Numbers
As discussed, for a decrease in particle radius, the surface-to-volume
ratio increases. Therefore the fraction of surface atoms increases as
the particle size goes down. In general, for a sphere, we can relate the
number of surface and bulk atoms according to the expressions
V r nA=
4
33π
(6.7)
Size Effects
CHAPTER 6 Nanomaterials: Classes and Fundamentals188
A r nA= 4 2 2 3π
(6.8)
where V is the volume of the nanoparticle, A is the surface area of
the nanoparticle, rA is the atomic radius, and n is the number of
atoms. On this basis, the fraction of atoms FA on the surface of a
spherical nanoparticle can be given by
Fr n
A
A
=3
1 3
(6.9)
We now consider a crystalline nanoparticle. In this case, in addi-
tion to the shape of the particle, we have to take into consid-
eration the crystal structure. For illustration purposes, we assume
a nanoparticle with a face-centered cubic (FCC) structure. This
crystal structure is of practical importance because nanoparticles
of gold (Au), silver (Ag), nickel (Ni), aluminum (Al), copper (Cu)
and platinum (Pt) exhibit such a structure. We start with the FCC
crystal structure shown in Figure 6.19. Clearly, the 14 atoms are
all surface atoms. If another layer of atoms is added so that the
crystal structure is maintained, a specific number of atoms must
be introduced. In general, for n layers of atoms added, the total
number of surface atoms can be given by
N nTotal
S = +12 22
(6.10)
On the other hand, the total number of bulk (interior) atoms can
be given by
N n n nTotalB = − + −4 6 3 13 2
(6.11)
Thus, Equations 6.10 and 6.11 relate the number of surface and
bulk atoms as a function of the number of layers. These numbers,
so-called structural magic numbers, are shown in Table 6.1.
The assumption so far has been that a nanoparticle would exhibit a
cube-type shape. However, from a thermodynamic point of view, the
equilibrium shape of nanocrystalline particles is determined by
A i iγ∑ = minimum
(6.12)
where γi is the surface energy per unit area Ai of exposed surfaces, if
edge and curvature effects are negligible. For ideal FCC metals, the
surface energy of atomic planes with high symmetry should follow
the order γ111Pt < γ100Pt < γ110Pt due to surface atomic
density. On the basis of calculated surface energies, the equilibrium
crystal shape can be created. Among the possible shapes, the small-
est FCC nanoparticle that can exist is a cubo-octahedron, which is a
14-sided polyhedron (see Figure 6.20).
FIGURE 6.19
Face-centered cubic (FCC) structure. All 14 atoms
are on the surface.
FIGURE 6.20
The smallest FCC nanoparticle that can exist:
a cubo-octahedron. A bulk atom is at the center.
Others are surface atoms..
ba
c
189
This nanoparticle has 12 surface atoms and one bulk atom. If addi-
tional layers of atoms are added to the cubo-octahedral nano-
particle such that the shape and crystal structure of the particle are
maintained, a series of structural magic numbers can be found. In
particular, for n layers of atoms added, the total number of surface
atoms can be given by
N n nTotal
S = − +10 20 122
(6.13)
whereas the total number of bulk (interior) atoms can be given by
N n n nTotalB = − + −( )
1
310 15 11 33 2
(6.14)
Table 6.2 shows the number of surface and bulk atoms for each
value of n as well as the ratio of surface-to-volume atoms.
These structural magic numbers do not take into account the elec-
tronic structure of the atoms in the nanoparticle. However, some-
times the dominant factor in determining the minimum in energy
of nanoparticles is the interaction of the valence electrons of the
atoms with an averaged molecular potential. In this case, electronic
magic numbers, representing special electronic configuration may
Table 6.1 Structural Magic Numbers for a Cube-Type FCC
Nanoparticle
n Surface
Atoms
Bulk Atoms Surface/Bulk
Ratio
Surface
Atoms (%)
1 14 0 — 100
2 50 13 3.85 79.3
3 110 62 1.78 63.9
4 194 171 1.13 53.1
5 302 364 0.83 45.3
6 434 665 0.655 39.4
7 590 1098 0.535 34.9
8 770 1687 0.455 31.3
9 974 2456 0.395 28.3
10 1202 3429 0.350 25.9
11 1454 4630 0.314 23.8
12 1730 6083 0.284 22.1
100 120,002 3,940,299 0.0304 2.9
Size Effects
CHAPTER 6 Nanomaterials: Classes and Fundamentals190
occur for certain cluster sizes. For example, potassium clusters pro-
duced in a supersonic jet beam and composed of 8, 20, 40, 58, and
92 atoms occur frequently. This is because potassium has the 4s
orbital (outermost shell) occupied and thus clusters for which the
total number of valence electrons fill an electronic shell are espe-
cially stable. Thus, in general, electronic magic numbers correspond
to main electronic shell closings.
Surface Curvature
All solid materials have finite sizes. As a result, the atomic arrange-
ment at the surface is different from that within the bulk. As shown
in Figure 6.21, the surface atoms are not bonded in the direction
normal to the surface plane. Hence if the energy of each bond is
ε/2 (the energy is divided by 2 because each bond is shared by two
atoms), then for each surface atom not bonded there is an excess
internal energy of ε/2 over that of the atoms in the bulk. In addi-
tion, surface atoms will have more freedom to move and thus
higher entropy. These two conditions are the origin of the surface
free energy of materials. For a pure material, the surface free energy
γ can be expressed as
FIGURE 6.21
For each surface atom there is an excess internal
energy of ε/2 due to the absence of bonds.
ε/2
Table 6.2 Structural Magic Numbers for a Cubo-Octahedral
FCC Nanoparticle
n Surface
Atoms
Bulk Atoms Surface/Bulk
Ratio
Surface
Atoms (%)
2 12 1 12 92.3
3 42 13 3.2 76.4
4 92 55 1.6 62.6
5 162 147 1.1 52.4
6 252 309 0.8 44.9
7 362 561 0.6 39.2
8 492 923 0.5 34.8
9 642 1415 0.4 31.2
10 812 2057 0.39 28.3
11 1002 2869 0.34 25.9
12 1212 3871 0.31 23.8
100 98,000 3,280,000 0.029 3.0
191
γ = −E TSS S
(6.15)
where ES is the internal energy, T is the temperature, and SS is the
surface thermal entropy. Equally important is the fact that the
geometry of the surface, specifically its local curvature, will cause
a change in the system’s pressure. These effects are normally called
capillarity effects due to the fact that the initial studies were done in
fine glass tubes called capillaries. To introduce the concept of surface
curvature, consider the 2-D curve shown in Figure 6.22. A circle of
radius r just touches the curve at point C. The radius r is called the
radius of curvature at C, whereas the reciprocal of the radius
k r= 1
(6.16)
is called the local curvature of the curve at C. As shown in Figure
6.22, the local curvature may vary along the curve. By convention,
the local curvature is defined as positive if the surface is convex and
negative if concave (see Figure 6.23). As the total energy (Gibbs free
energy) of a system is affected by changes in pressure, variations
in surface curvature will result in changes in the Gibbs free energy
given by
∆ ∆G PVV
r= =
2γ
(6.17)
On the basis of Equation 6.17, the magnitude of the pressure dif-
ference increases as the particle size decreases, that is, as the local
curvature increases. Therefore, at the nanoscale, this effect is very
significant. In addition, because the sign for the local curvature
depends on whether the surface is convex or concave, the pressure
inside the particle can be higher or lower than outside. For example,
if a nanoparticle is under atmospheric pressure, it will be subject
to an extra pressure ∆P due to the positive curvature of the nano-
particle’s surface, described in Equation 6.17.
Another important property that is significantly altered by the cur-
vature effect is the equilibrium number of vacancies (see the section
on crystalline defects in Chapter 4). In general, the total Gibbs free
energy change for the formation of vacancies in a nanoparticle can
be expressed by
∆ ∆ ∆G G GvTotal
vbulk
vexcess
= +
(6.18)
where ∆Gvbulk is the equilibrium Gibbs free energy change for the
formation of vacancies in the bulk and ∆Gvexcess is the excess Gibbs
free energy change for vacancy formation due to curvature effects.
Assuming no surface stress, ∆Ω
Gr
vexcess =
γ, where Ω is the atomic
FIGURE 6.22
Surface curvature in two dimensions.
r
C
FIGURE 6.23
Concave and convex surface curvatures.
Convex Concave
Size Effects
CHAPTER 6 Nanomaterials: Classes and Fundamentals192
volume, γ the surface energy, and r the radius of curvature, Equa-
tion 6.18 can be rewritten as:
∆ ∆Ω
G Gr
vTotal
vbulk= +
γ
(6.19)
Therefore, the total equilibrium vacancy concentration in a nano-
particle can be given by
XG
k TvTotal v
Total
B
= −
exp∆
(6.20)
where kB is the Boltzmann constant and T the temperature. Insert-
ing Equation 6.19 into Equation 6.20 yields
XG
k T rk TvTotal v
bulk
B B
= −
−
exp exp∆ Ωγ
(6.21)
For the bulk case, where curvature effects can be neglected, the con-
centration of vacancies can be expressed as
XG
k Tvbulk v
bulk
B
= −
exp∆
(6.22)
However for a nanoparticle, the concentration of vacancies can be
written as
X Xrk T
vTotal
vbulk
B
= −
expΩγ
(6.23)
As discussed, by convention, the local curvature is defined as posi-
tive if the surface is convex and negative if concave. Therefore, for a
convex surface, Equation 6.23. can be rewritten as
X Xrk T
vTotal
vBulk
B
= −
1Ωγ
(6.24)
On the other hand, for concave surfaces, the mean curvature is given
by −1/r, and thus Equation 6.23 becomes
X Xrk T
vTotal
vBulk
B
= +
1Ωγ
(6.25)
This means that the vacancy concentration under a concave surface
is greater than under a flat surface, which in turn is greater than under
a convex surface. This result has important implications for nano-
particles due to their small radius of curvature, playing a significant
role in a variety of properties such as heat capacity, diffusion, catalytic
activity, and electrical resistance, thereby controlling several process-
ing methods such as alloying and sintering. Figure 6.24 shows the
FIGURE 6.24
Diffusivity at 900° in silver, gold, and platinum
nanoparticles of different sizes normalized with
respect to bulk diffusivities.
900°C
Size (nm)
0
0.0
0.5
1.0
2.0
1.5
5 10 2015
Concave surfaceSilverGoldPlatinum
Convex surface
Norm
aliz
ed d
iffu
siv
ity
193
effect of curvature on the diffusivity of nanoparticles of silver, gold,
and platinum.
Clearly, for nanoparticle sizes below 10 nm, the effect is quite sig-
nificant. This behavior has profound consequences on the sinter-
ing of nanoparticles. In fact, when two nanoparticles are in contact
with each other (see Figures 6.25 and 6.26), the neck region
between the nanoparticles has a concave surface, which results in
reduced pressure. As a consequence, atoms readily migrate from
convex surfaces with positive curvature (high positive energy) to
concave surfaces with negative curvature (high negative energy),
leading to the coalescence of nanoparticles and elimination of
the neck region. In other words, nanoparticles exhibit a high ten-
dency to sintering, even at room temperature, due to the curvature
effect.
One other important physical property of a material is its lattice
parameter. Because this parameter represents the dimensions of the
simplest unit of a crystal that is propagated in 3-D, it has significant
impact on a variety of properties. To understand the effects of scale
on the lattice parameter, we consider the Gauss-Laplace formula
given by
∆Pd
=4γ
(6.26)
where ∆P is the difference in pressure between the interior of a
liquid droplet and its outside environment, γ is the surface energy,
and d is the diameter of the droplet. If the droplet is now solid
and crystalline with a cubic structure and lattice parameter a (the
droplet is now a nanoparticle), we can write for the compressibility
of the nanoparticle:
KV
V
PO T
=∂
∂
1
(6.27)
which measures the volume change of the material as the pres-
sure applied increases, for a constant temperature. It is normalized
with respect to Vo to represent the fractional change in volume with
increasing pressure. In this case, Vo = a3. Equation 6.26 can then be
inserted in Equation 6.27, giving
γ
d
K a
a=
3
4
∆
(6.28)
Since the surface energy increases as the particle decreases, because
the radius of curvature decreases, Equation 6.28 reveals that the
FIGURE 6.25
Aberration-corrected STEM image of two
nanoparticles sintering at room temperature.
(Courtesy of Michael Asoro, University of Texas
at Austin; Larry F. Allard, Oak Ridge National
Laboratory; and P. J. Ferreira, University of Texas
at Austin.)
FIGURE 6.26
Schematic showing the sintering process of two
nanoparticles. R is the radius of the convex surface
and r is the radius of the concave surface.
r
R
Size Effects
CHAPTER 6 Nanomaterials: Classes and Fundamentals194
lattice parameter is reduced for a decrease in particle size (Figure
(6.27)).
Strain Confinement
Planar defects, such as dislocations are also affected when present
in a nanoparticle. As discussed in Chapter 4, dislocations play a
crucial role in plastic deformation, thereby controlling the behavior
of materials when subjected to a stress above the yield stress. In the
case of an infinite crystal, the strain energy of a perfect edge disloca-
tion loop is given by
Wb r
cS ≅ µ
π
2
4ln
(6.29)
where µ is the shear modulus, b is the Burgers vector, r is the radius
of the dislocation stress field, and c is the core cutoff parameter.
If the crystal size is reduced to the nanometer scale, the disloca-
tion will be increasingly affected by the presence of nearby sur-
faces. As a consequence, the assumption associated with an infinite
crystal size becomes increasingly invalid. Therefore, in the nano-
scale regime, it is vital to take into account the effect posed by the
nearby free surfaces. In other words, there are image forces acting
on the dislocation half-loop. As a consequence, the strain energy
of a perfect edge dislocation loop contained in a nanoparticle of
size R is given by
Wb R r
RS
d≅−
µ
π
2
4ln
(6.30)
where rd is the distance between the dislocation line and the surface
of the particle and the other symbols have the same meaning as
before. A comparison of Equations 6.29 and 6.30 reveals that for
small particle sizes, the stress field of the dislocations is reduced.
In addition, the presence of the nearby surfaces will impose a force
on the dislocations, causing dislocation ejection toward the nano-
particle’s surface. The direct consequence of this behavior is that
nanoparticles below a critical size are self-healing as defects
generated by any particular process are unstable and ejected.
Quantum Effects
In bulk crystalline materials, the atomic energy levels spread out
into energy bands (see Figure 6.28). The valence band, which is
filled with electrons, might or might not be separated from an
FIGURE 6.27
Lattice parameter of Al (aluminum) as a function
of particle size. (Adapted from J. Woltersdorf, A.S.
Nepijko, and E. Pippel, Surface Science, 106, pp.
64–69, 1981.)
Particle diameter (nm)
Lattic
e p
ara
mete
r (n
m)
195
FIGURE 6.28
Energy bands in bulk conductors, insulators, and
semiconductors.
empty conduction band by an energy gap. For conductor mate-
rials such as metals, there is typically no band gap (Figure 6.28a).
Therefore, very little energy is required to bring electrons from the
valence band to the conduction band, where electrons are free to
flow. For insulator materials such as ceramics, the energy band gap
is quite significant (Figure 6.28b), and thus transferring electrons
from the valence band to the conduction band is difficult. In the
case of semiconductor materials such as silicon, the band gap is
not as wide, and thus it is possible to excite the electrons from the
valence band to the conduction band with some amount of energy.
This overall behavior of bulk crystalline materials changes when the
dimensions are reduced to the nanoscale. For 0-D nanomaterials,
where all the dimensions are at the nanoscale, an electron is con-
fined in 3-D space. Therefore, no electron delocalization (freedom
to move) occurs. For 1-D nanomaterials, electron confinement
occurs in 2-D, whereas delocalization takes place along the long
axis of the nanowire/rod/tube. In the case of 2-D nanomaterials,
the conduction electrons will be confined across the thickness but
delocalized in the plane of the sheet.
Therefore, for 0-D nanomaterials the electrons are fully confined.
On the other hand, for 3-D nanomaterials the electrons are fully
delocalized. In 1-D and 2-D nanomaterials, electron confinement
and delocalization coexist.
Under these conditions of confinement, the conduction band
suffers profound alterations. The effect of confinement on the
resulting energy states can be calculated by quantum mechanics,
as the “particle in the box” problem. In this treatment, an elec-
tron is considered to exist inside of an infinitely deep potential
well (region of negative energies), from which it cannot escape and
is confined by the dimensions of the nanostructure. In 0-D, 1-D,
En
erg
y
En
erg
y
En
erg
y
Conductor
Insulator
Semiconductor
Gap
Vacant state Vacant state Vacant state
Occupied state
Occupied state
Occupied state
Occupied state
Occupied state
Occupied state
Valence band
Conduction band
Valence band
Conduction band
Valence band
Conduction band
Size Effects
CHAPTER 6 Nanomaterials: Classes and Fundamentals196
and 2-D, the effects of confinement on the energy state can be
written respectively as
(0-D) EmL
n n nn x y z=
+ +( )π 2 2
2
2 2 2
2
ℏ
(6.31a)
(1-D) EmL
n nn x y=
+( )π 2 2
2
2 2
2
ℏ
(6.31b)
(2-D) EmL
nn x=
( )π 2 2
2
2
2
ℏ
(6.31c)
where h ≡ h/2π, h is Planck’s constant, m is the mass of the elec-
tron, L is the width (confinement) of the infinitely deep potential
well, and nx, ny, and nz are the principal quantum numbers in the
three dimensions x, y, and z. As shown in Equations 6.31a–c, the
smaller the dimensions of the nanostructure (smaller L), the wider
is the separation between the energy levels, leading to a spectrum of
discreet energies. In this fashion, the band gap of a material can be
shifted toward higher energies by spatially confining the electronic
carriers.
Another important feature of an energy state En is the number
of conduction electrons, N (En), that exist in a particular state.
As En is dependent on the dimensionality of the system (Equa-
tions 6.31a–c), so is the number of conduction electrons. This also
means that the number of electrons dN within a narrow energy
range dE, which represent the density of states D(E), i.e., D(E) =
dN/dE, is also strongly dependent on the dimensionality of the
structure. Therefore the density of states as a function of the energy
E for conduction electrons will be very different for a quantum
dot (confinement in three dimensions), quantum wire (confine-
ment in two dimensions and delocalization in one dimension),
quantum well (confinement in one dimension and delocalization
in one dimension), and bulk material (delocalization in three-
dimensions; see Figure 6.29).
Because the density of states determines various properties, the use
of nanostructures provides the possibility for tuning these pro-
perties. For example, photoemission spectroscopy, specific heat,
the thermopower effect, excitons in semiconductors, and the super-
conducting energy gap are all influenced by the density of states.
Overall, the ability to control the density of states is crucial for
applications such as infrared detectors, lasers, superconductors,
single-photon sources, biological tagging, optical memories, and
photonic structures.
FIGURE 6.29
Density of states in a bulk material, a quantum
well (2-D nanomaterial), a quantum wire
(1-D nanomaterial), and a quantum dot
(0-D nanomaterial).
E
E
E
E
D(E
)D
(E)
D(E
)D
(E)
3-D bulk
2-D
quantum
well
1-D
quantum
wire
1-D
quantum
dot
197
FURTHER READING
C. P. Poole, Jr. and F. J. Owens, Introduction to nanotechnology, Wiley-Interscience, 2003. ISBN 0-471-07935-9.
A. S. Edelstein, and R. C. Cammarata (eds.), Nano materials: Synthesis, properties, and applications, Institute of Physics, 1996, ISBN 0-7503-0578-9.
R. T. De Hoff, Thermodynamics in materials science, McGraw-Hill, 1993, ISBN 0-07-016313-8
M. Muller, and K. Albe, Concentration of thermal vacancies in metallic nanoparticles, Acta Materialia, 55, pp. 3237–3244, 2007.
J. Woltersdorf, A. S. Nepijko, and E. Pippel, Dependence of lattice para-meters of small particles on the size of nuclei, Surface Science, 106, pp. 64–69, 1981.
R. Lamber, S. Wetjen, and I. Jaeger, Size dependence of the lattice param-eter of small particles, Physical Review B, 51, pp. 10968–10971, 1995.
C. E. Carlton, L. Rabenberg, and P. J. Ferreira, On the nucleation of partial dislocations in nanoparticles, Philosophical Magazine Letters, 88, pp. 715–724, 2008.
C. Kittel, Introduction to solid-state physics, John Wiley & Sons, Inc. 6th ed., 1986.
Further Reading