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

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8.3 Selected examples of nanodevices and systems 303

Contact layer

Ferromagnetic layer

Contact layer

Insulating layer

Figure 8.32 The structure

of a magnetic tunnel

junction (MTJ).

magnetic read heads whose operation is based on the GMR effect with very high

sensitivity.

High-precision angle, position, and speed sensors based on the GMR are

widely used in automobiles – for example, in an anti-block braking system

(ABS), which provides straight-line motion of the vehicle while braking on a

slippery pavement. Modern computer, television, and video technologies are hard

to imagine without spintronic devices. Besides hard disks, spintronic devices can

be found in high-deﬁnition television equipment and in DVD drives. Apart from

in GMR devices, spin is used also in magnetic tunneling junctions (MTJs). The

structure of a valve built on the basis of an MTJ consisting of two ferromagnetic

layers separated by the insulator (for this purpose usually aluminum oxide is

used) is shown in Fig. 8.32.

It looks as if electric current, which ﬂows perpendicularly to the ferromagnetic

layers, will not ﬂow through such a structure at all. However, because of the

tunneling effect electrons are able to move from one ferromagnetic layer, through

the insulator, to another ferromagnetic layer. In MTJ valves, together with the

tunneling effect, the fact that electrons with different spins move differently

in a magnetic ﬁeld is used. If the magnetic ﬁelds in the two ferromagnetic

layers of the valve coincide directionally, then those electrons whose spin is

directed along the magnetic ﬁeld can easily tunnel from one ferromagnetic layer

to another. Therefore, the resistance of the valve is very low in this case. If

the magnetic ﬁelds in these layers are directed in opposite directions, then the

tunneling transmission would be more difﬁcult, since in one of the layers electrons

will be moving against the magnetic ﬁeld. Because of this, the resistance of the

valve increases by 20%–30%.

A prospective application of spintronics known as magnetic random-access

memory (MRAM) utilizes an MTJ. Its distinctive feature is that information is

saved neither in the form of electric charges on the capacitor plates, as is done in

dynamic memory DRAM, nor in the form of a trigger state as in static random-

access memory (SRAM), but in the form of the magnetization of a ferromagnetic

layer. The advantages of this type of memory are its low energy consumption, high

304 Nanostructures and their applications

speed of operation, and gigantic density of information storage. The maintenance

of the magnetization state does not require energy consumption, which is the case

with hard disks. The high density of information storage is deﬁned by the small

size of the memory cells, for which one requires neither a large number of

transistors, as for SRAM, nor separate circuits of regeneration, as for DRAM.

An MTJ can be used for MRAM to read information from a magnetic cell. The

latest models of MRAM have reached 256 kb. They use a power supply of 3 V

and provide a duration of the write/read cycle of 50 ns. These parameters allow

MRAM to compete directly with ﬂash-memory and offer the prospect of being

able to compete with conventional dynamic memory. Finally, MRAM will allow

the creation of computers whose dynamic memory data will not be lost when the

power supply is turned off.

The potential of spintronics is not exhausted by the technologies which have

already been developed. Despite the fact that research in this direction has been

going on for more than 10 years, there are many unsolved scientiﬁc and technical

problems. For example, to change the magnetization of some area of ferromag-

netic material, ﬁrst of all it is necessary to have a magnetic ﬁeld. Since we can

create a magnetic ﬁeld by the use of an electric current (we do not take into

account permanent magnets) there is a problem of localization of the magnetic

ﬁeld in the limited area. The smaller this area, the higher the density of infor-

mation storage on the magnetic carrier which could be achieved. Recently, a

particular experiment that showed the possibility of changing the magnetization

with the help of spin-polarized electrons was carried out. With the help of pho-

toemission from the semiconductor cathode, which is caused by polarized light, a

beam of spin-polarized electrons was produced, which in its turn was transmitted

through a magnetic ﬁlm of width several nanometers. When electrons passed

through the ﬁlm, their spins changed, together with the spins of electrons in the

magnetic ﬁlm, which means that the magnetization of the ﬁlm also changed. If

the number of electrons transmitted through the ﬁlm is comparable to the number

of atoms of the material, then the change of magnetization of the ﬁlm will be con-

siderable. This effect can be used for writing information as well as for reading

information with a lower intensity of the electron beam. Potentially this technol-

ogy can provide a remagnetization speed (i.e., the speed of reading/writing of

information) up to tens of GHz.

Another interesting effect is the formation of a pure spin ﬂux of electrons

without charge transport. In this experiment two colliding beams with oppositely

directed spins were formed. Thus, the transfer of spin charge was achieved

without an applied voltage. So far this phenomenon has been observed only at

distances of about several nanometers, nevertheless research in this direction

is continuing. One of the most important problems of spintronics is connected

with the materials that are used. In spintronics you have to use ferromagnetics,

whose magnetic properties give rise to various effects related to the spins of

electrons. But ferromagnetic materials are metals, while modern electronics

8.3 Selected examples of nanodevices and systems 305

is based on semiconductors. It is precisely the properties of semiconductors

which allow one to amplify current in transistors, whereas in metals this effect

is impossible. Therefore, in order to create a device that uses electron spin as

well as electron charge, it is necessary to have a ferromagnetic material that is

simultaneously a semiconductor. Research in this direction is very active and new

magnetic semiconductors that do not lose their magnetic properties even at room

temperatures have been fabricated. One such material is titanium dioxide (TiO

)

with cobalt (Co) as impurity, which has been grown in the form of nanometer

ﬁlms by molecular-beam epitaxy. Under high-vacuum conditions the beams of

atoms in certain proportions are directed onto a crystalline surface where they

form the desired crystalline structure. Despite some problems with the growth

of homogeneous ﬁlms, it is a very tempting material for the development of new

spintronic devices. Another similar material is an epitaxial ﬁlm of alternating

thin layers of GaSb and GaMn. The magnetic properties of this semiconductor

are preserved up to 130

◦

C, which is quite high enough for the needs of modern

electronics.

Using such materials, spintronics will allow us to process and store informa-

tion on the same wafer, which will lead to an increase in the speed of operations

and to a decrease in energy consumption. Integration of the achievements of elec-

tronics and spintronics will open new horizons in the development of traditional

computers as well as quantum computers and may extend the time of validity of

Moore’s law.

8.3.7 Graphene and carbon nanotubes

The research group at Georgia Institute of Technology announced in 2006 that

they had fabricated a ﬁeld-effect transistor based on graphene and a device that

uses the phenomenon of quantum interference. They stated that in the near future

a new class of graphene nanoelectronics with a characterististic thickness of

transistors equal to 10 nm will emerge. It is impossible to use graphene directly

to create a ﬁeld-effect transistor because graphene does not have a bandgap.

This means that we cannot achieve a substantial difference in resistivity at any

applied gate voltage, i.e., it is not possible to create two states for binary logic –

conducting and non-conducting states. Therefore, for device fabrication it is

necessary ﬁrst to create a gap of substantial width at room temperature in order

to reduce the input into the conductivity of the thermally excited electrons. One of

the possible ways to do this is by the formation of narrow graphene ribbons with

a width that provides quantum-dimensional effects in the plane of the sample.

At the same time, the width of the bandgap, E

, must be large enough to provide

the device with a non-conducting state (turn-off state) at room temperature (the

width of 20 nm possessed by the graphene nanoribbon corresponds to a bandgap

width of about 30 meV). Owing to the high mobility of electrons in graphene,

306 Nanostructures and their applications

the operating speed of such a transistor may be substantially higher than that of

the silicon transistors.

When depicting the band structure of graphene it is necessary to take into

account the fact that the outer shell of the carbon atom contains four electrons,

which form bonding with the neighboring atoms of the lattice by creating three

-hybridized orbitals. The fourth electron is in the 2p

-state. It is precisely

the 2p

-state which is responsible in graphite for the formation of interlayer

bondings, whereas in graphene the 2 p

-state is responsible for the formation

of energy bands. The bonding energy of atoms in a crystalline lattice rapidly

decreases with increasing interatomic separation. Therefore, the main input into

the bonding energy of an atom with other lattice atoms and the formation of

band structure arise from the atom’s interaction with its nearest neighbors. In

the electron tight-binding approximation, the energy spectrum of electrons in

graphene has the form

E =±





1 +4cos

(πk

a) + 4cos(π k

a)cos(πk

√

3a)



, (8.30)

where the + sign corresponds to electrons and the − sign to holes, and γ

is the

overlap integral. This expression is obtained by taking into account interactions

with the nearest neighboring atoms only.

From Eq. (8.30) it follows that near the points of intersection of the valence

band and the conduction band the dispersion law for the carriers in graphene can

be written in the following form:

E =±h

k, (8.31)

where v

is the Fermi velocity (its experimental value is v

= 10

−1

) and

k =



+ k

is the modulus of the wavevector in two-dimensional k-space. The

photon, whose mass at rest is equal to zero, has a similar spectrum. Therefore, we

can say that electrons and holes with energy deﬁned by Eq. (8.31) have effective

mass equal to zero too. The Fermi velocity, v

, plays the role of the effective

speed of light.

Transistors constructed on the basis of carbon nanostructures, as experts

expect to exist in the very near future, will exceed the parameters of the conven-

tional silicon devices. Up to now the efforts of researchers have been concen-

trated on the fabrication of single transistors on the basis of carbon nanotubes.

Researchers from IBM recently announced that they had fabricated the ﬁrst inte-

grated circuit (IC) constructed on the basis of a single carbon nanotube. The

importance of this achievement lies in the fact that the IC was fabricated using

standard semiconductor-technology processes. This is an important step towards

integration of the new technology with the existing electronic technologies. The

researchers from IBM developed a ring oscillator – an IC with the help of which

the developers can check the possibilities of new production processes or of new

materials.

8.3 Selected examples of nanodevices and systems 307

By fabricating an IC on a single carbon nanotube the researchers from IBM

have achieved a speed of operation a million times higher than that of numerous

ICs on nanotubes, which had been developed earlier. Despite the fact that this

speed is still lower than that which the latest silicon devices have, these researchers

from IBM are conﬁdent that new nanotechnologies will help to uncover the

enormous possibilities of carbon nanotube electronics.

Lately an active development of memory modules constructed on the basis of

carbon nanotubes has been undertaken. Such modules may become an alternative

to traditional ﬂash memory. Currently, energy-independent ﬂash memory is used

in various compact systems. In the main types of ﬂash memory double-gate MOS

transistors are used as elementary memory cells and it is expected that in future

memory cells will be based on carbon nanotubes.

8.3.8 Nanoelectromechanical systems

Nanoelectromechanical systems (NEMSs) are integrated nanodevices or sys-

tems that combine electrical and mechanical components, and are fabricated

using technologies that are compatible with the production of integrated circuits.

To realize advanced scientiﬁc ideas in this ﬁeld, it is necessary to utilize the

achievements of various branches of nanotechnology.

An NEMS is a miniature system that has sensory, computational, and produc-

tive functions. Sensory functionality concerns the ability to deﬁne the existence

and concentration of chemical or biological components in the environment.

Computational functionality means that the NEMS has a processor, which car-

ries out the necessary computations and prepares information for processing.

Productive functionality means that the NEMS has nanoscale actuating mecha-

nisms, so NEMSs can be used as controllers. Therefore, NEMSs usually contain a

combination of two or more devices operating on the basis of electrical, mechan-

ical, optical, chemical, biological, magnetic, and other properties, which are

integrated on a single chip or on a multichip board. Work on NEMSs has rapidly

become one of the leading ﬁelds of cutting-edge nanotechnology. Autonomous

NEMSs, e.g., nanorobots, are actively used in manufacturing, medicine, and

ecology.

Among the various elements of an NEMS, a very important role is played

by the load-bearing elements, which can be made of carbon nanotubes. The

controlled relative motion of layers in multiwalled nanotubes and their unique

elastic properties makes it possible to use them as moving elements of an NEMS.

Devices based on the relative motion of carbon nanotubes have recently been

suggested: rotational nanobearings, nanogears, nanoswitches, GHz oscillators,

Brownian nanoengines, nanorelays, and also a nanonut–nanobolt pair. Moreover,

there are experimentally developed nanomotors, which use as their axle and bush

the layers of multiwalled carbon nanotubes.

308 Nanostructures and their applications

(b)

(a)

Figure 8.33 Schematic

diagrams of

nanoactuators: (a) with an

inner layer as the stator

and (b) with an external

layer as the stator.

It has recently been shown that a two-walled carbon nanotube with non-

zero chiral angle can be used as a nanonut–nanobolt pair (as can be seen from

Fig. 8.5, the surface of a chiral nanotube is reminiscent of a thread on a bolt),

i.e., the motion along the nanotube axis of the stator is transformed into rotation

of the rotor (outer layers 2 and 4). This can be used for transformation of the

force directed along the nanotube axis into the mechanical torque, which causes

relative rotation of layers. Such a pair can be used in a so-called nanoactuator –

a device that makes an NEMS able to move. The operational principle of such a

device is the same as that of a humming-top toy. The principle of a nanoactuator

is shown schematically in Fig. 8.33.InFig.8.33(a), 1 denotes the inner layer of

a nanotube, whereas in Fig. 8.33(b) 1 denotes the outer layer of a nanotube. It

serves as the stator of the nanomotor and is ﬁxed. Other layers, labeled 2 and 3,

serve as the rotor. The relative position of these layers has to be ﬁxed. The stator

and rotor serve as a rotational nanobearing. We are considering now the motor

depicted in Fig. 8.33(a), but the operation of the motor shown in Fig. 8.33(b) is

analogous. The external layers serve for transformation of the force applied to

layer 4 and directed along the nanotube axis into torque, which rotates the rotor.

Such a transformation is possible if layers 2 and 4 form a nanonut–nanobolt

pair. The charges on the ends of layer 4 can be produced as a result of chemical

adsorption and can be used for the control of a nanoactuator by an external

electric ﬁeld.

We can force the internal part of a two-walled nanotube to oscillate. In order

to do this, the external layer has to be ﬁxed (see Fig. 8.34). The inner layer has

to be subjected to additional processing. At one end positive hydrogen ions are

absorbed and at the other end negative ﬂuorine ions. Under the inﬂuence of a

8.3 Selected examples of nanodevices and systems 309

(a)

(b)

(c)

Figure 8.34 Subsequent

positions of the two

layers of a two-walled

nanotube during the

half-period of oscillations:

(a) and (c) show the

maximum amplitude of

telescopic protrusion of

the inner layer of the

nanotube under the

action of the force F.

non-uniform electric ﬁeld the inner tube, which is a dipole, can move relative

to the external tube, as shown in Fig. 8.34. This oscillatory motion can also be

transformed into a rotational motion.

Currently there are numerous applications of nanotechnology. With its

help, new anticorrosive coatings are made, nanostructures allow a signiﬁcantly

increased lifetime of steel-work, provide reliable operation of electricity-supply

networks, reduce costs of equipment operation, and so on. According to some

scientists, in the future nanotechnology could save mankind. When we have

exhausted all the oil, coal, and gas on the Earth (which may be even sooner than

we expect), people will be using nanotailored solar batteries that will be able to

use solar energy for the production of hydrogen, oxygen, and electricity. In future

biology and medicine nanoparticles will deliver medicine to the required parts

of the human body. For example, nanoparticles mixed with blood will arrive at a

tumor and then, by exposure to infrared radiation, the tumor will be eradicated

because of overheating caused by absorption of radiation by the nanoparticles.

Nanotechnology for space exploration has great prospects. New nanomaterials

will make ﬂights not only to the Moon but also to Mars a reality. One more per-

spective is the utilization of powdered nanotechnology for the development of a

new generation of fuel cells that can produce electrical energy from any organic

fuel, with an efﬁciency no less than 60%–70%. It is impossible to mention or

foresee all the possible applications of nanotechnology. We can only mention

that, according to some experts, during the next decade the world market of

nanotechnology will reach the size of a trillion dollars and maybe will exceed

the value of all markets related to electronics, medicine, chemistry, and energy

altogether.

Appendix A

Classical dynamics of particles and waves

In this appendix we will study the basics of the classical mechanics of an individual

particle as well as a system of particles. We will treat, here, a particle as a mate-

rial point, i.e., an ideal object that has a mass and an inﬁnitesimally small size. The

importance of this chapter is deﬁned by the fact that any macroscopic structure or

medium can be considered as an array of material points interacting according to certain

laws.

The famous three laws of classical dynamics formulated by Isaac Newton more than

three centuries ago were for a long time considered as the basis for the description of

motion of all objects – ranging from nanoparticles to cosmic objects. With the development

of physics, it became clear that the region of validity of the classical description of a

particle’s motion is limited. Newton’s mechanics cannot be applied for the description of

motion with velocities close to the speed of light. For the description of such a motion,

relativistic mechanics was introduced at the beginning of the twentieth century. The typical

formulae of relativistic theory, in contrast to the corresponding formulae of classical

mechanics, contain the radical



1 −(v/c)

, which contains the square of the ratio of the

particle’s velocity, v, to the speed of light in vacuum, c. For a wide group of problems

the ratio v/c  1; therefore, usage of the classical description at velocities that are lower

than the speed of light by two orders of magnitude leads to an error that is signiﬁcantly

smaller than the error in measurements of the corresponding physical magnitude. However,

Newton’s mechanics cannot be applied for the description of elementary particles. The

motion of electrons in atoms, molecules, nanostructures, semiconductors, and metals is

not governed by the laws of classical mechanics. Nevertheless, the role of the laws of

classical mechanics in the modern physics of the nanoworld is still signiﬁcant. These laws

deﬁne the motion of heavy particles, for example the motion of atoms or ions, and the

wave motion of atoms or ions in crystalline structures.

In the current appendix we will also study free harmonic and damped oscillations

of an individual particle, discuss their governing equations, and ﬁnd their solutions. We

will consider small-amplitude oscillations of a particle, with harmonic and anharmonic

approximations for the description of oscillations for an arbitrary proﬁle of the potential

well.

310

A.1 Classical dynamics of particles 311

r(t)

(t)

y(t)

z(t)

r = xi + yj + z

Figure A.1 The Cartesian

system of coordinates and

a particle’s trajectory, r(t).

A.1 Classical dynamics of particles

A.1.1 Newton’s second law of motion

According to the classical description of particle motion, a particle’s location in space

at any instant of time, t , is deﬁned by three spatial coordinates, which depend on the

choice of the system of coordinates. One such system – the Cartesian coordinate system –

consists of three mutually orthogonal axes x , y,andz that intersect at the origin of

coordinates (x = y = z = 0, with respect to which x(t), y(t), and z(t) deﬁne the location

of the particle at any given time t (see Fig. A.1)). For convenience the position of a particle

can be written in the form of its radius vector, r(t):

r(t) = ix(t) + jy(t) +kz(t), (A.1)

where i, j,andk are unit vectors directed along their respective coordinate axes. Since

these vectors are perpendicular to each other, their scalar products are:

i · i = j · j = k · k = 1andi ·j = i ·k = j ·k = 0. (A.2)

The array of points deﬁned by the vector r(t) along a particle’s motion deﬁnes the trajectory

of the particle (Fig. A.1).

The change of a particle’s state is described by Newton’s second law, which in its

general form can be written through the momentum of a particle, p:

p = mv, (A.3)

where

v =

. (A.4)

312 Appendix A. Classical dynamics of particles and waves

Here, m is the mass and v is the velocity of the particle. According to Newton’s second

law of motion, the derivative of the momentum p with respect to time t is equal to the

sum of all forces acting on the particle:

= F, (A.5)

where the resultant force, F, is deﬁned by a vector sum of all forces F

actingonthe

particle:

F = F

+ F

+···+F



k=1

, (A.6)

where n is the total number of forces acting on the particle.

Now, let us discuss some of the forces existing in nature that can act on a particle,

which has mass, m, and charge, q.

(a) Gravitational force, F

= mg, (A.7)

where g, called the acceleration due to gravity, denotes the free-fall acceleration

caused by the gravitational ﬁeld at a point close to the Earth’s surface where the

particle is located. The acceleration due to gravity, g =|g|, is approximately 9.8

−2

and is directed towards the center of the Earth.

(b) In electric and magnetic ﬁelds, the Lorentz force, F

, is a sum of electric and magnetic

forces applied to a particle of charge q:

= q

(

E + v × B

)

, (A.8)

where E is the electric ﬁeld strength and B is the magnetic ﬂux density. The vector

product, v ×B (which is part of Eq. (A.8)), is given, according to the deﬁnition of

the vector-product operation, by the third-order determinant:

v ×B =



ijk



= i



− v



+ j

(

− v

)

+ k



− v



(A.9)

where we used the general rule for ﬁnding determinants. Let us note that because of

the small masses of the electron and proton the gravitational force is several orders

of magnitude smaller than the force of the electric ﬁeld under laboratory conditions,

even if the strength of the electric ﬁeld is weak. For an electron these forces become

of the same order of magnitude, i.e., eE ≈ F

, if the magnitude of the electric ﬁeld,

E, is about

E =

≈ 6 × 10

−11

−1

, (A.10)