Mitin V.V., Sementsov D.I., Vagidov N.Z. Quantum mechanics for nanostructures

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8.1 Methods of fabrication of nanostructures 263

GaAs

Substrate

Sources

Shutters

Molecular beams

Figure 8.2

Molecular-beam epitaxial

growth of

GaAs/Al

1−x

heterostructures.

Liquid Nitrogen

Titanium

Sublimation

Pump

Sample Mount

Main Flange

Knudsen Cells

Pyrometer

Actuator

Valve

As Cracker

Shutters

RHEED

Screen

Ionization

Gauge

Ion

Pump

Valve

Figure 8.3 A photograph

of a typical MBE chamber

(University at Buffalo).

GaAs (x = 0) to AlAs (x = 1). Additionally, the growth process can be controlled

by using shutters, which are positioned between the heaters and the substrate.

Using these shutters, we can abruptly interrupt or resume the deposition onto

the substrate of any molecular beam. For the discussed example of Al

1−x

As,

layers of materials with different x can be grown on top of each other. Figure 8.3

shows a picture of an actual MBE chamber used for research.

264 Nanostructures and their applications

One of the main disadvantages of MBE is its low rate of growth – approxi-

mately one monatomic layer per second or about 1 µm per hour. By controlling

the beams that are depositing material onto the substrate we can modulate layer by

layer the doping of the thin-ﬁlm structure with sufﬁcient accuracy. The dopants

can be of either p- or n-type. In such a way a superlattice with the desired number

of layers and order can be fabricated.

There are many other technologies that use different forms of epitaxy, which

can be realized from vapor, liquid or solid-state phases. Thus, for the growth of

epitaxial layers of A

compounds (compounds of elements from groups III and

V of the Periodic Table of the elements, for example, GaAs) vapor-phase epitaxy

from metallo-organic compounds is used. The methods of molecular-beam and

metallo-organic epitaxy are widely used for the fabrication of certain types of

structures in microelectronics (transistors, integrated circuits, and light-emitting

diodes), in quantum electronics (multilayer semiconductor heterostructures and

injection lasers), in the devices of integrated optics, in computer engineering,

and so on.

8.1.3 Self-organization and self-assembly

As we have already mentioned, the process of formation of complex ordered

structures from simple elements is called self-organization. Self-organization is

one of the most fascinating and interesting natural phenomena, being connected

with the formation of a more complex structure from an initial simpler one. In

physics and chemistry self-organization is the transition from chaotic distribu-

tions of atoms and molecules to ordered structures. The ﬁeld of science which

studies self-organized systems is called synergetics, which is derived from the

Greek word synergos, which means “working together.” The main idea of syner-

getics is the fundamental possibility of the spontaneous formation of order and

organization from disorder and chaos as a result of self-organization. There are

many types of self-organizing system in nature. A characteristic example is the

self-organization of the efforts of a large number of bees. Despite their apparently

chaotic movements during the formation of honeycombs, the honeycombs take

the form of a plane lattice consisting of practically perfect identical hexagons.

Among prospective approaches to the formation of nanostructures, those tech-

nologies which use the phenomena of self-organization to create nanostructures

from individual atoms (the so-called bottom-up technology) are becoming more

and more popular. One of the most difﬁcult problems that nanotechnology has

to resolve is how to make atoms and molecules group in a certain way, i.e., how

to self-organize them in the most stable structures that can be used for the fab-

rication of new materials and devices. In the near future the self-organization of

atoms and molecules will allow us to assemble materials from their constituents,

and it will not be necessary to assemble a nanostructure by putting it together

atom by atom manually. Manual assembly is not practical because it is a slow

8.1 Methods of fabrication of nanostructures 265

process and it requires disproportionate efforts. Therefore, nowadays the most

natural method of nanostructure fabrication is based on self-organization.

Many of us know how the billiard balls are organized by a rack before the

beginning of a game. In the closed space of the rack the balls spontaneously form

an equilateral triangle. If we were to scatter them into a big box and shake them

slightly, they would spontaneously take the form of a practically ideal box struc-

ture. In some cases atoms of the same type can be considered as homogeneous

spheres, which we can always order in the same way in a closed volume as bil-

liard balls do. In crystallography there is a term that deﬁnes such ordering – close

packing. Analogously to billiard balls, nanoparticles can spontaneously pack on

the surface of solid-state materials, forming regular geometrical structures. The

main reasons behind the formation of such orderings of nanoparticles are forces

that tend to decrease the overall surface area of nanoparticles and therefore their

surface energy (we talk about interaction forces between atoms in Appendix C).

Ordered arrays of gold nanoparticles of size 4 nm were obtained by self-

assembly for the ﬁrst time in 1995. The same year structures of CdSe nanopar-

ticles of size 5 nm were fabricated. Homogeneous initial nanoparticles result

in regular arrays of nanoparticles. The size-homogeneous nanoparticles may

be assembled in spatially ordered structures such as one-dimensional wires,

two-dimensional closely packed layers, and three-dimensional arrays or small

clusters. The type of organization of nanoparticles and the structure of the array

formed depend on the conditions of synthesis, the diameter of the particles,

and the nature of external forces acting on the structure. Two-dimensionally

and three-dimensionally ordered arrays of Pt, Pd, Ag, Au, Fe, Co, FePt, Fe

, CoO, CdS, CdSe, CdTe, and PbSe nanocrystals have been synthesized

to date. Orientationally ordered arrays have also successfully been synthesized

from anisotropic nanoparticles.

Nowadays we know methods of self-assembly that allow us to fabricate useful

ordered structures. To establish the special conditions required, the gravitational,

electric or magnetic ﬁelds, capillary forces, mutual wettability or non-wettability

of components of the system, and other factors must be taken into account.

Self-assembly processes are actively being used in manufacturing processes – in

particular, for the fabrication of a new generation of computer chips.

8.1.4 Fabrication of carbon nanostructures

One of the most studied types of carbon nanostructure is graphene, because

of its promising applications. Long before the fabrication of graphene it was

theoretically proved that it is impossible to obtain a free ideal two-dimensional

ﬁlm of monatomic thickness because of the instability caused by folding or

twisting. Moreover, thermal ﬂuctuations must lead to the melting of a two-

dimensional crystal at any non-zero temperature. The ﬁrst unsuccessful attempts

at making graphene, which was attached to another material, were undertaken

266 Nanostructures and their applications

using ordinary pencil lead and AFM to mechanically remove layers of graphite.

In 2004 graphene on a SiO

substrate was obtained, and stabilization of the

two-dimensional ﬁlm was achieved by bonding of the thin layer with SiO

The main method of obtaining graphene is either mechanical splitting off

from, or exfoliation of, graphite layers. The method of exfoliation is a relatively

simple and ﬂexible method, since it allows one to work with all crystals that

have weakly coupled layers. This method allows one to obtain graphene samples

of good enough quality, but it is not suitable for mass production of graphene

since it is a manual procedure. The mass production of high-quality samples of

graphene is still a very challenging problem. One of the prospective methods is

the procedure of placing graphene oxide in pure hydrazine, which is a chemical

compound of nitrogen and hydrogen. This procedure reduces graphene oxide to

a graphene single layer. Another method of obtaining graphene, which is more

suitable for mass production, is based on the thermal decomposition of SiC.

The ideal graphene consists exclusively of hexagons. The presence of pen-

tagons or heptagons will lead to the existence of various defects. The presence of

pentagons leads to the warping of the graphene plane into a cone. The presence

of heptagons leads to the formation of a saddle-shaped ﬂexing of the plane. The

combination of such defects and hexagons may lead to the formation of various

types of graphene surfaces. The presence of 12 hexagons leads to the formation

of fullerene.

From the moment of discovery of fullerenes and carbon nanotubes, numerous

methods for their fabrication were developed. The basis of some of these methods

is evaporation of graphite by arc discharge or by laser ablation. Other methods are

based on chemical transport reactions. An arc discharge is created in an evacuated

reactor. After this, its volume is ﬁlled by an inert gas, with the pressure being

about 0.5 bar. As a result of an arc discharge between two graphite electrodes

(an electric current of ∼50 A is maintained) carbon atoms leave the electrode

with the positive potential and are deposited onto the electrode with the negative

potential and onto the walls of the reactor. As a result, tiny particles of soot are

formed, which consist of arrays of carbon clusters differing in terms of size and

composition. Among these molecular clusters, fullerenes, C

, were discovered

for the ﬁrst time (see Fig. 1.9).

Carbon nanotubes can be obtained using the same methods: laser heating

(or laser ablation), arc discharge, and chemical vapor deposition (in this case

a catalyst is added). The method of laser heating suggests the existence of a

graphite target containing particles of the catalyst. The graphite target has to

be inserted into the reactor. Particles of Fe, Ni, and Co, and atoms of rare-

earth elements, are used as catalysts. These particles become nucleation centers

of carbon nanostructures. When the high-intensity beam of the pulsed laser is

directed at the target, the carbon evaporates. The ﬂow of argon gas carries carbon

atoms from the high-temperature zone to a cooled collector, where the carbon

nanotubes are formed. In this way single-walled carbon nanotubes of diameter

8.1 Methods of fabrication of nanostructures 267

Figure 8.4 Armchair

carbon-nanotube

formation, showing

carbon dimer, C

,and

trimer, C

, units.

No. 10

No. 11

No. 7

No. 8

No. 1

1 nm

Chiral nanotube

nanotube

Armchair

Figure 8.5 Atomically

resolved STM images of

single-walled carbon

nanotubes with chirality.

Reprinted with

permission, from

P. Moriarty,

“Nanostructured

materials,” Rep. Prog.

Phys., 64, 369 (2001).

about 1 nm and length up to several millimeters are formed. At the point of

nucleation a carbon nanotube has a closed form, whereas from the side, where

the carbon atoms are added, it is open-ended.

Figure 8.4 shows how carbon atoms are added to a single-walled armchair

nanotube in the process of its growth. The sequential addition of C

dimers (clus-

ters of two carbon atoms) leads to the continuous growth of nanotubes. In order

to add hexagons and to eliminate pentagons, it is necessary also to use trimers C

(clusters of three carbon atoms). The inclusion of pentagons distorts the tube and

makes the nanotube close, thus the growth of the nanotube ends. The inclusion

of heptagons changes the size of the nanotube and its spatial orientation. More-

over, addition of “pentagon–heptagon” pairs brings about additional variations

of nanotubes. Many properties of single-walled nanotubes are deﬁned not only

by their diameter but also by the so-called chirality or torsion (see Fig. 8.5, where

the torsion (or chirality) of a carbon nanotube is shown).

To understand the notion of “chirality” or “chiral angle,” we will start with

a two-dimensional graphite sheet as shown in Fig. 8.6(a). The single sheet of

graphite called graphene has the form of a hexagonal lattice. Let a

and a

be the

graphene lattice vectors, and let n and m be integers. The diameter and helicity of

the nanotube are uniquely characterized by the vector C = na

+ ma

≡ (n, m)

that connects crystallographically equivalent sites on a two-dimensional graphene

sheet. Here a

1,2

are in units of a

√

3, with a

being the carbon–carbon distance.

268 Nanostructures and their applications

(0,0)

(11,7)

(0,7)

zigzag

(a)

zigzag

armchair

(b)

(11,0)

Figure 8.6 (a) The relation

between the hexagonal

carbon lattice and the

chirality of carbon

nanotubes. A carbon

nanotube can be

constructed from a single

graphite sheet by rolling

up the sheet along the

wrapping vector C.(b)

Fragments of “armchair”

and “zigzag” carbon

nanotubes.

By using the vector C, a carbon nanotube can be constructed by wrapping a

graphene sheet in such a way that two equivalent sites of the hexagonal lattice

coincide. The wrapping vector C deﬁnes the relative location of these two sites.

In Fig. 8.6(a), the wrapping vector C connects the origin (0, 0) and the point with

coordinates (11, 7). Thus, a nanotube denoted by indices (11, 7) is formed. A tube

is called an armchair tube if n equals m, and a zigzag tube in the case of m = 0

(see the fragments of “zigzag” and “armchair” carbon nanotubes in Fig. 8.6(b)).

Having wrapping vectors along the dotted lines leads to tubes that are of zigzag

or armchair form. All other wrapping angles lead to chiral tubes whose wrapping

angle is speciﬁed relative to either the zigzag direction θ or the armchair direction

ϕ = 30

◦

− θ .Bothθ and the wrapping angle (chiral angle) ϕ are in the range

(0, 30

◦

) as a result of the hexagonal character of the two-dimensional lattice of

the graphene. Dashed lines are perpendicular to C and run in the direction of the

tube axis indicated by the vector T. The solid vector H is perpendicular to the

armchair direction and speciﬁes the direction of nearest-neighbor hexagon rows.

The angle between T and H is the chiral angle ϕ. The unit cell of a nanotube can

be constructed by ﬁnding the smallest lattice vector T which connects equivalent

points of the lattice.

Depending on the magnitude of the diameter and the chiral angle (or chi-

rality), nanotubes may have either metallic or semiconductor properties. In the

process of fabrication two thirds of all nanotubes have semiconductor proper-

ties and the rest have metallic properties. Generally, nanotubes free of defects

and without chirality have metallic properties. Such nanotubes consist of rows of

hexagons parallel to the nanotube axis. If catalyst particles are not used, so-called

multiwalled nanotubes are formed.

8.2 Characterization with nanoscale resolution 269

Example 8.1. A carbon nanotube of radius r = 5 nm is placed in a magnetic

ﬁeld with magnetic ﬂux density B = 1 T, oriented in the direction of the nanotube

axis, which is parallel to the surface of the nanotube. Find the orbital magnetic

moment of an electron rotating around the nanotube axis and moving along the

nanotube surface. Compare this magnitude with the magnetic moment of the

electron in the hydrogen atom.

Reasoning. When an electron moves with the velocity v perpendicular to the

magnetic ﬁeld it experiences a magnetic force, F

, which is equal to

= ev B.

The electron undergoes centripetal acceleration and Newton’s second law may

be written as

= ev B. (8.1)

Finding from the last equation the orbital velocity v, we can write the period of

rotation, T ,as

T =

2πr

2πm

. (8.2)

The rotating electron creates a current, I , equal to

I =

. (8.3)

Owing to the rotation, the electron acquires the following magnetic moment:

µ = SI = πr

I =

. (8.4)

For the electron motion around the nanotube axis the magnetic moment is equal

to µ ≈ 3.5 ×10

−25

. The electron magnetic moment in the hydrogen atom

is equal to the Bohr magneton: µ

= 9.27 × 10

−24

. Therefore,

≈ 32. (8.5)

8.2 Tools for characterization with nanoscale resolution

The minimum size of objects that the human eye can discriminate at the distance

of their best visual acuity (about 25 cm) is about 0.1 mm. To study smaller

objects, optical devices are used – from the simplest ones such as a magnifying

glass to more complex ones consisting of several lenses, such as an optical

microscope. Modern optical microscopes have magniﬁcations up to 1500. This

means that, with the help of these devices, we can discern objects of size about

−7

m, i.e., objects of size about 100 nm. The increase of the resolution of

optical microscopes is hindered by diffraction. If the objects are separated by

a distance less than d ≈ λ/n, then they are indistinguishable. Here, λ is the

wavelength of the light and n is the index of refraction of a medium. The visible

270 Nanostructures and their applications

Incident light

Wavelength of light

Near field

Far field

Sample surface

Apertured tip

5 nm

Figure 8.7 Near-ﬁeld

optical microscopy

(schematic, not to scale).

part of the electromagnetic spectrum ranges from 400 to 800 nm. Therefore, even

theoretically with the help of the most powerful optical microscope possible we

cannot discern objects of size less than 200 nm. Some biological objects such

as cells, whose size is of the order of hundreds and thousands of nanometers,

can be studied using optical systems. Atoms, whose size is about one tenth of a

nanometer, cannot be seen using optical microscopes. The simplest solution for

this problem is to decrease the wavelength of the light. To discern atoms with the

help of such a microscope, we have to decrease the wavelength of the radiation

by a factor of 1000.

8.2.1 Near-ﬁeld scanning optical microscopy

The near-ﬁeld scanning optical microscope (NSOM) was introduced in 1972.

This microscope is a special form of scanning-probe microscope, which utilizes

visible light as its probe. The resolution of conventional optical microscopes,

as we have noted, is limited by the diffraction limit of the light, and therefore

it is limited by the wavelength of the light, which is about 0.5 µm. In the

NSOM this limitation is overcome and the resolution is increased by one order

of magnitude. Figure 8.7 illustrates how the diffraction limit of the light could be

overcome in NSOM. The scanning tip used in an NSOM is called an apertured tip.

The apertured tip is actually a “light funnel.” The visible light emanates from

the narrow side of the light funnel, which has a diameter of 10–30 nm. The light

is registered by the detector either after its reﬂection from the sample or after

transmission through the sample. The intensity of the optical signal is registered

by the detector and the set of data from the scanned surface forms the NSOM

image of the surface. With the help of an NSOM we can form the image of

8.2 Characterization with nanoscale resolution 271

the scanned surface in visible light with a resolution of about 15 nm if the

following condition is satisﬁed: the distance between the light source and the

sample must be very small – about 5 nm. By making the scanning tip approach

so close to the sample surface, we make the light interact with the surface

before its diffraction, thus increasing the resolution of optical microscopy (see

Fig. 8.7). The systems for placing the apertured tip and maintaining a constant

distance between the sample and the light source are the most complicated parts

of an NSOM. Usually, the apertured tip is made by heating an optical ﬁber and

stretching it until its end has a small diameter, whereupon the other end of the

ﬁber is chopped off. After this the optical ﬁber, except the end of the apertured tip,

is covered with a metallic layer for better optical transmission. Another method

for the fabrication of apertured tips is the following: a small hole is drilled

in the sharp end of an atomic-force-microscope (AFM) cantilever and light is

directed into this hole (we will consider AFM techniques in Section 8.2.3). The

development of effective apertured tips is nowadays a ﬁeld of intensive research

efforts.

An NSOM must keep the distance between the sharp end of the scanning

tip and the sample constant in order to obtain an optical image of the surface

being studied. To do this, the traditional AFM methods for maintaining constant

deﬂection of an AFM cantilever may be used. The distinctive feature of the NSOM

compared with other scanning methods is that the image is formed directly in

the optical range including visible light, but its resolution is much higher than

the resolution of traditional optical microscopes.

8.2.2 Scanning-tunneling microscopy

As we have shown previously, the behavior of an electron in an atom is described

by its wavefunction, the square of whose modulus deﬁnes the probability density

of the electron being at a given point of space at a given time. The wavefunc-

tions of the electrons which belong to an atom have non-zero values outside of

the atom, i.e., at distances greater than the average size of the atom. Therefore,

when we bring atoms that belong to different objects closer to each other, the

wavefunctions of the valence electrons overlap before the forces of interatomic

repulsion begin to manifest themselves. As a result there is a possibility of

electron transfer from one atom to another, i.e., the exchange of electrons that

belong to different objects is possible without mechanical contact. This effect

may be realized if one of the objects has freely moving electrons, the other has

vacant electron states, and the applied voltage between those objects is equal to

or greater than the breakdown voltage of the air gap. If the voltage is much less

than the breakdown voltage, the electric current occurring under these condi-

tions is due to the tunneling effect and the current itself is called the tunneling

current.

272 Nanostructures and their applications

~0.5 nm

Sample

Tip

Figure 8.8 The scanning

process in a

scanning-tunneling

microscope.

In tunneling microscopy, the tunneling current between the probe (or tip)

and the conducting surface is measured at each point of the scanned sur-

face (see Fig. 8.8). If we move the tip step by step (with a step of about the

size of the atomic radius) while measuring the tunneling current, we will “feel”

the individual atoms of the scanned surface. By scanning we mean the motion of

the tip along the surface from one point to another. The picture of the distribution

of the current measured over the sample surface is visualized on a computer dis-

play and allows us to study the surface topology on the atomic scale. The tunneling

phenomenon was utilized in 1981 as the basic principle of the scanning-tunneling

microscope (STM). One of the conductors had the form of a sharp needle, which

was called the tip, and the other conductor was the surface of the object being

studied.

In order to move electrons from the end of the tip to the studied surface,

the conduction electrons must gain a certain energy. The energy depends on

the distance, L, between the tip and the surface of the object, the voltage, U ,

between them, and the work functions, A

and A

, of the tip and the sample,

respectively. On bringing the tip to the surface at a separation of L ≈ 0.5 nm and

applying a voltage U ≈ 0.1−1 V a current, I

, is produced due to the tunneling

effect:

≈ en

vS. (8.6)

Here, n

and v are the concentration and velocity of electrons, and S is the

cross-section of the electron beam. The concentration of electrons in the beam is

related to the concentration of electrons in the tip, n

,by

= Dn, (8.7)