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

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7.8 A three-dimensional superlattice of quantum dots 253

The effective mass of an electron in the cubic three-dimensional superlattice is a

scalar, i.e., it does not depend on the direction of the K-vector:

∗

=±

. (7.259)

In the general case of a superlattice with different periods D

along the directions

of the main axes the inverse effective mass becomes a tensor:



∗



αβ

∂

∂k

. (7.260)

Example 7.10. Show that the electron motion in the three-dimensional superlat-

tice in an external electric ﬁeld, E

ext

, can be described by Newton’s second law

if the inverse mass of the electron is considered as a tensor with the components

deﬁned by Eq. (7.260).

Reasoning. The expression for the group electron velocity in the case of three-

dimensional motion can be written as follows:

= e

∂ω

∂k

+ e

∂ω

∂k

+ e

∂ω

∂k

= ∇

ω, (7.261)

where

∇

= e

∂

∂k

+ e

∂

∂k

+ e

∂

∂k

. (7.262)

Taking into account the relation of the electron energy to the frequency of

the electron wave, E(k) = h

ω, the group velocity of the wave packet which

represents the electron can be written through its energy as

∇

E(k). (7.263)

If the superlattice is in an external electric ﬁeld, E

ext

= e

+ e

,the

change in the electron energy, E, in unit time is deﬁned as

= F · v = eE

ext

· v = eE

ext



∇

E(k)



, (7.264)

where we considered the group velocity of the electron, v

, as the electron wave-

packet velocity, v. The electric force applied to the electron is equal to F = eE

ext

Using Eqs. (7.263) and (7.264), let us ﬁnd the acceleration, a, which the electron

gains in the external ﬁeld, E

ext

a =

(

∇

E(k)

)

∇



dE (k)



∇

ext

·∇

E(k))

∇



∂ E(k)

∂k

+ E

∂ E(k)

∂k

+ E

∂ E(k)

∂k



. (7.265)

254 Quantization in nanostructures

Let us write the x-projection of the acceleration, a:



∂

E(k)

∂k

+ E

∂

E(k)

∂k

+ E

∂

E(k)

∂k



. (7.266)

Analogous expressions can be written for the other two projections of the accel-

eration, a. These expressions, according to Newton’s second law in the general

form, can be presented as





∗



αβ

, (7.267)

where the summation is done over all three projections onto the coordinate axes

β = x, y, and z, and F

are the projections of the force F. On comparing the

expressions obtained, we ﬁnd that the magnitudes



∗



αβ

∂

∂k

(7.268)

are the components of the inverse effective mass tensor.

7.9 Summary

1. In a potential well of macroscopic size, the electron energy spectrum is quasicontin-

uous. With increasing energy the number of electron states in such a well increases

proportionally to E

3/2

and the density of states increases proportionally to

√

These dependences hold only in the case of a quadratic dependence of energy on

momentum.

2. A structure of size less than or about 100 nm is called a nanoobject or a nanostructure.

In a nanostructure the electron energy spectrum radically changes, which leads to

changes of the physical properties of an object. For a nanostructure that is formed

from a few atoms and placed in vacuum the potential barriers are high and the spacing

between levels is about several electron-volts.

3. In quantum wells (two-dimensional structures) the electron motion in one direction is

limited and the energy that corresponds to this motion is quantized. The motion in the

two other directions stays free and is characterized by a continuous energy spectrum.

Therefore, the electron spectrum in a quantum well is a set of two-dimensional

subbands, which have the form of paraboloids.

4. In quantum wires (one-dimensional structures) electron motion is limited in two

directions, with the corresponding energy quantization. The electron motion along a

wire (the third direction) is free and the energy spectrum is continuous. The electron

spectrum in a quantum wire is a set of one-dimensional subbands, which have the

form of parabolas.

5. In quantum dots (zero-dimensional structures) the electron motion is limited in all

three directions. The energy spectrum of such motion is completely quantized.

7.10 Problems 255

6. The density of states of a quantum well in any of its subbands is constant and does not

depend on energy. Each subband has the same input into the total density of states,

which is deﬁned by the electron effective mass and the width of the quantum well.

7. Superlattices are structures with periodic positioning of quantum-dimensional objects

along one, two, or three directions. The electron wavefunction in a superlattice

must have the form of the wavefunction of a free electron modulated by a periodic

superlattice function.

8. The periodicity of the superlattice potential leads to the conservation not of electron

momentum, but of quasimomentum. For quasimomentum Newton’s second law in

the form dp/dt = F

ext

is valid. Here F

ext

is connected with the external ﬁelds and

does not include the periodic force of a superlattice, F

, which is related to the

superlattice potential by F

=−



∇U , with U being the potential of the superlattice.

9. The total external force is related to the electron acceleration by F

ext

= m

∗

a,where

the effective mass of the electron, m

∗

, reﬂects the character of the electron energy

dispersion in a superlattice. Internal forces (potentials) deﬁne the energy dispersion,

E( p), and, through the effective mass, m

∗

, deﬁne the electron motion under the

inﬂuence of external forces.

10. The interval of quasimomenta (−π h

/D ≤ p ≤ π h

/D) that contains all physically

equivalent electron states in a one-dimensional superlattice as in the case of oscil-

lations of a chain of atoms is called the ﬁrst Brillouin zone. The second Bril-

louin zone consists of two quasimomentum intervals, (−2π h

/D, −π h

/D)and

(π h

/D, 2π h

/D), and so on. The total length of each Brillouin-zone interval is equal

to 2π h

/D.

11. The boundaries of Brillouin zones correspond to wavevectors at which the electron

wave cannot propagate in the superlattice. At the boundaries of Brillouin zones the

electron energy in a superlattice has a discontinuity, which indicates the occurrence

in the electron energy spectrum of forbidden minibands, which alternate with allowed

minibands.

12. Each individual level of the identical quantum dots splits into the superlattice mini-

band if the quantum dots are assembled as a superlattice.

7.10 Problems

Problem 7.1. Find the expressions for the wavefunction and electron energy in a

symmetric rectangular quantum well with the following potential proﬁle:

U (r) =



, |x| > L

0, |x|≤L

(7.269)

The electron motion along the y- and z-directions is free.

Problem 7.2. Using the solution of Problem 7.1, consider the limiting cases of

very deep and very shallow quantum wells. Show that in a shallow quantum well

there is always at least one stationary state. Derive an approximate expression

256 Quantization in nanostructures

for the energy, E

, of this state and estimate it for U

= 2.5 × 10

−3

eV and

L = 2nm.(Answer:E

≈ 2.375 meV.)

Problem 7.3. In an asymmetric rectangular quantum well the motion of an

electron in the yz-plane is free and motion in the x-direction is limited by the

following potential:

U (x ) =







, x < 0,

0, 0 ≤ x ≤ L

, x > L

(7.270)

where U

< U

. Find the condition for which discrete quantum states with

energy E ≤ U

exist. Consider the limiting cases U

→∞and U

= U

Problem 7.4. Estimate the energy of the electron ground state in a quantum well,

where electron motion in the yz-plane is free but motion along the x-direction is

limited by the potential U(x):

U (x ) =



∞, x < 0,

bx, x ≥ 0.

(7.271)

Problem 7.5. Find the energy and the wavefunctions of the stationary states of

an electron in a quantum wire. The electron motion in the z-direction is free and

motion in the plane perpendicular to the z-direction is limited by the potential

U (ρ)(ρ

= x

+ y

U (ρ) =



0,ρ≤ a,

∞,ρ>a.

(7.272)

Problem 7.6. In a quantum wire the electron motion along the z-direction is free

and that in the plane perpendicular to the z-direction is limited by the potential

U (x, y):

U (x , y) =

β(x

+ y

)

. (7.273)

Find the energy, the wavefunctions, and the magnitudes of various projections of

the electron angular momentum onto the z-direction and occupation probabilities

for the states with quantum numbers n

= 1 and n

= 1.

Problem 7.7. The electron motion in a spherical quantum dot happens in the

following potential:

U (r) = U

−r/b

, U

< 0. (7.274)

Find the average electron energy of a ground state in the given quantum well.

Problem 7.8. Electron motion takes place in an inﬁnitely deep double-quantum-

dot structure and is limited by the potential U(x, y, z):

U (x , y, z) = U (x)U(y)U(z), (7.275)

7.10 Problems 257

where

U (x ) =







∞, x < 0, x > L

0, 0 < x < b, L

− b < x < L

, b ≤ x ≤ L

− b,

(7.276)

U (y) =



∞, y < 0, y > L

0, 0 ≤ y ≤ L

(7.277)

U (z) =



∞, z < 0, z > L

0, 0 ≤ z ≤ L

(7.278)

Find using ﬁrst-order perturbation theory the expression for the electron energy

states in a double-quantum-dot structure.

Problem 7.9. Prove that in isotropic three-dimensional systems the density

of states per unit volume of momentum space (the number of states per unit

interval of momentum modulus) is proportional to the square of momentum.

Problem 7.10. The electron motion in a quantum wire is free along the z-direction

and that in the xy-plane is limited in the region 0 < x < L

and 0 < y < L

barriers of inﬁnite height. Find the number and density of states in the lowest-

energy subband of the quantum wire.

Chapter 8

Nanostructures and their applications

Nanotechnology is based on the ability to manipulate individual atoms and

molecules in order to assemble them into bigger structures. Such artiﬁcial

nanoscale structures, usually fabricated using self-assembly phenomena, possess

new physical, chemical, and biological properties. The fabrication of various

types of nanostructures and study of their properties require new technological

means and new principles.

Nanotechnology has initiated a new so-called bottom-up technology. The

bottom-up technology is based on the self-assembly phenomenon, i.e., the process

of formation of complex ordered structures from simpler ones. The main idea

of this technology is in the development of the controlled self-assembly of the

atoms, molecules, and molecular chains into nanoscale objects. The bottom-up

technology allows the fabrication of nanoobjects, such as quantum dots, quantum

wires, and superlattices.

The bottom-up approach is opposite in principle to the traditional approach,

which may be called the top-down approach, which is based on the sequen-

tial decrease of the object’s size by means of mechanical or chemical pro-

cessing for the fabrication of objects of nanoscale size (nanoobjects). Thus,

for example, some of the nanoparticles can be obtained by grinding mate-

rial consisting of particles of micrometer or larger size in a special grinder.

The traditional technologies include laser methods for the processing of semi-

conductor surfaces and making masks of various conﬁgurations and sizes for

photolithography.

In this chapter we will brieﬂy consider the main fabrication and characteriza-

tion techniques for nanostructures and will give some examples of applications

of nanostructures in modern nanoelectronics.

8.1 Methods of fabrication of nanostructures

In this section we consider the main techniques of fabrication of nanostructures.

258

8.1 Methods of fabrication of nanostructures 259

Illuminating radiation

Mask

Photoresist

Film to be patterne

Substrate

Develop

Etch pattern

Imaging system

Illuminated areas

Ion-oxide layer

Quartz plate

egative resist:

exposed rendered

insoluble

Positive resist:

exposed is

removed

Figure 8.1 Standard steps

of a photolithographic

process.

8.1.1 Photolithography

Lithographic methods are the main representatives of the top-down techniques

in the fabrication of devices and integrated circuits (ICs). Under the term

“lithography” one usually understands a set of physicochemical processes of

layer-by-layer formation of the patterns of ICs and of elements of nanostruc-

tures. The decrease of microchip sizes can be achieved by decreasing the pattern

size which is formed with the help of lithography on the surface of solid-state

materials.

From optics it is known that the resolution of an optical system increases with

decreasing wavelength of the light. Therefore, to fabricate smaller elements by

lithographic methods it is necessary to switch to shorter wavelengths. Depending

on the wavelength and the methods of generating radiation, one can distinguish

among four types of photolithography that are in use: (1) optical lithography, (2)

X-ray lithography, (3) ion-projection lithography, and (4) electron-beam lithog-

raphy. Optical lithography, which is often also called photolithography,isaset

of methods of formation of the given relief on the sample surface with the help

of a focussed light beam.

In the simplest scenario the photolithography process consists of several steps

(see Fig. 8.1). In the ﬁrst step the surface of the solid-state material (substrate)

is covered with a layer of photosensitive material. The structure of the photosen-

sitive layer changes under the inﬂuence of optical radiation. This photosensitive

260 Nanostructures and their applications

layer is called the photoresist.Aphotomask is placed on the surface of the pho-

toresist or somewhere between the photoresist and source of light. The photomask

consists of transparent and non-transparent regions. The next step of the pho-

tolithographic process is called exposure. The surface of the solid-state material

covered by the photoresist and by the photomask is subjected to radiation from

the optical or X-ray source. As a result of photochemical reactions underneath

the transparent regions of the photomask, the photoresist changes its properties.

In the case of a positive-resist process the exposed photoresist is removed during

the so-called etching process. In the case of a negative-resist process the exposed

region becomes resistant to the special chemical agents, called etchants, and

the unexposed parts of the photoresist are removed by the etching. In both cases

we will end up with a certain image on the surface of the substrate, such that some

parts of the substrate are covered by the photoresist and some parts are open. The

next step is chemical etching, which is based on the dissolving by etchants of the

part of the sample area which is not protected by the photoresistive layer. Thus,

patterns of complex conﬁgurations can be developed on the substrate.

As we have already mentioned, the radiation wavelength is one of the most

important characteristics of the source of optical radiation which is used for

the illumination of the photoresist. Because of the diffraction phenomenon the

wavelength used cannot be larger than the feature size that we would like to obtain

by photolithography. If we use a source of radiation with wavelength 1 µm, then

the minimum feature size in the pattern will be of the same order of magnitude.

To obtain a pattern with nanometer feature size we have to use so-called deep-

ultraviolet light with a wavelength of the order of tens or hundreds of nanometers.

In modern deep-ultraviolet photolithography a wavelength of λ = 200–400 nm

is used. The sources of such a radiation are powerful Hg–Xe gas-discharge

lamps and excimer lasers whose output power reaches hundreds of watts. The

radiation wavelength of the excimer lasers depends on the composition of the

active medium of the laser, for example, KrF (λ = 248 nm), ArF (λ = 193 nm),

and F

(λ = 157 nm) are widely used. These lasers have a pulse duration of

5–20 ns with a repetition frequency of about 4 kHz and output power up to 50 W.

Further increase of the photolithographic resolution can be achieved by using as

a source of optical radiation excimer lasers with a wavelength of λ = 13.5 nm.

In X-ray lithography soft (or low-energy) X-ray radiation with a wavelength of

0.5–5 nm is used for illumination.

In conclusion, let us note that the huge success of microelectronics in the last

quarter of the twentieth century was due to the fast advances in the development

of these processes. Many technological methods of microelectronics have the

potential to be used also in nanoelectronics, with reduction of the characteristic

feature size. In the transition from microelectronics to nanoelectronics special

attention is paid to the self-organization processes which take place during epi-

taxial growth of nanostructures, atomic-force epitaxy in colloid solutions, etc.

In particular, the rate of development of nanoelectronics in general depends

8.1 Methods of fabrication of nanostructures 261

entirely on the rate of development of industrial technologies for nanomaterials

and nanostructures.

8.1.2 Epitaxy

Epitaxy is one of the methods of nanostructure fabrication using a bottom-up

approach. Most of the fabrication methods using the assembly of nanoobjects

from individual atoms are based on the phenomenon of condensation. By con-

densation we usually understand the transition of the substance from the gaseous

state to a liquid or solid state as a result of its cooling or compression. Rain, snow,

and dew all result from natural phenomena of condensation of water vapor from

the atmosphere. The condensation of a vapor is possible only at temperatures

below a critical temperature for a given substance. In a similar fashion we can

condense atoms and molecules of other chemical elements. Condensation and

the opposite process – evaporation – are examples of phase transitions of the sub-

stance. The process of phase transition of a gas into a liquid or of a liquid into a

solid substance occurs during a certain time. During the initial stage nanoparticles

are formed, with their further transformation into macroscopic objects occurring

subsequently. We can fabricate nanoparticles by “freezing” phase transition at

an earlier stage. As a result of condensation we can obtain fullerenes, carbon

nanotubes, nanoclusters, and nanoparticles of various sizes.

Using the condensation method of fabrication of nanoparticles, we evapo-

rate from macroscopic objects atoms that are assembled into nanoobjects. The

evaporation can be done by thermal heating or by laser heating of the macro-

scopic material. The evaporated atoms have to be transferred into regions with

lower temperatures, where the condensation of atoms into nanoparticles takes

place. The controlled condensation of atoms on a crystal surface (substrate)is

the foundation of epitaxial technology.

Epitaxy (the term is derived from the Greek words epi, which means on, and

axis, which means arrangement,ororder) is an arranged growth of a substance

during the process of condensation onto a substrate. Epitaxy of atoms on a

crystalline surface can be done from the liquid phase as well as from the vapor

phase. The process of epitaxy usually begins with the nucleation on the substrate

of islands, which coalesce with each other, forming a continuous ﬁlm. Modern

epitaxial techniques allow one to control the growth with resolution up to a

single atomic layer and to alternate layers with different physical and chemical

properties.

One of the modern methods of epitaxial growth of nanostructures is molecular-

beam epitaxy (MBE), which is based on the interaction of several molecular

beams with a heated monocrystalline substrate. Molecular-beam epitaxy is an

improved version of the technique of thermal sputtering under conditions of ultra-

high vacuum. The pressure of the residual gases in the vacuum chamber of MBE

is maintained at a level lower than 10

−8

Pa. The ﬂuxes of atoms (or very rarely

262 Nanostructures and their applications

entire molecules) are formed by the evaporation of liquids or sublimation of solid-

state materials, which are located in the source – so-called Knudsen effusion cells

or simply effusion cells. The effusion cell is a crucible of cylindrical or conical

form of diameter 1–2 cm and length 5–10 cm. The outlet of the effusion cell is a

round opening called a diaphragm of diameter 5–8 mm. The crucible is usually

made of high-purity pyrolytic graphite or boron nitride (BN).

The ﬂuxes of atoms (or molecules) of the necessary chemical elements are

directed towards and subsequently deposited onto a substrate to form a substance

of the desired composition. The number of effusion cells depends on the com-

position of the ﬁlm and the dopants. To grow elementary semiconductors such

as Si or Ge, we need only one source of main material and sources of dopants

of n- and p-types. In the case of compound semiconductors (binary or ternary

alloys) we need a separate source for each component of the material which

is to be grown. The temperature of the effusion cells deﬁnes the magnitude of

the ﬂux of the particles deposited onto the substrate and it is strictly controlled.

The control of ﬂux is provided by so-called shutters, which shut off the various

ﬂuxes. The homogeneity of the grown material over the surface and its crystalline

structure are deﬁned by the homogeneity of the molecular beams. In some cases,

to increase the homogeneity of the ﬁlm the substrate with the forming ﬁlm is

constantly made to spin.

The epitaxial growth of semiconductor compounds involves a series of steps.

The most important steps are (1) adsorption of atoms and molecules by the

substrate, which leads to the nucleation and growth of the layer, and (2) migration

and dissociation of the adsorbed particles. The growing material establishes a

crystalline structure, which is deﬁned by the crystalline properties of both the

substrate and the deposited material. The atoms that are deposited onto the

substrate are adsorbed by the surface. During the ﬁrst stage, called physisorption,

the physical adsorption is due to weak van der Waals and (or) electrostatic

forces. During the second stage, called chemisorption (chemical adsorption), the

molecules of the substance undergo a transition to a chemisorbed state during

which electron transfer takes place, i.e., a chemical reaction between the surface

atoms and the newly arrived atoms occurs. The binding energy of chemical

adsorption is higher than of physical adsorption.

Figure 8.2 shows the main elements of the apparatus for the fabrication of

semiconductor ﬁlms of Al

1−x

As on GaAs substrate using MBE. Each heater

has a crucible, which serves as a source of one of the components of the compound

materials. The evaporated material is deposited onto the substrate with a relatively

slow deposition rate under conditions of ultra-high vacuum. The heaters are

arranged in such a way that the maxima of the distribution intensities of the beams

are on the substrate. By selecting the temperatures of heaters and of a substrate

we can fabricate structures and ﬁlms with complex chemical compositions. For

example, in the case of Al

1−x

As, by controlling x, i.e., by controlling the

fractions of Al and Ga, it is possible to grow a wide spectrum of materials, from