Lectura Taller 4

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


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


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


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


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


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


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


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.

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