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

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B.4 Light and its wave properties 373

into four regions. In region 1 there is no absorption because 



is practically equal to zero.

This region is called the optical-transparency window. In this region 



> 1 and thus the

wave speed in the medium is less than the speed of light in vacuum. Region 2 is called

the optical-absorption window because in this region 



has sufﬁciently large magnitude.

In this region the medium becomes non-transparent and waves with the corresponding

frequencies quickly fade in the medium. Note that any dielectric medium, even if it does

not have free carriers of charge, always has a frequency interval within which you can

see strong absorption. This is caused by quantum transitions in atoms. Region 3 is called

the metallic-reﬂection window. It begins from frequencies at which there is practically no

absorption, and it ends at frequencies for which 



→ 0. In this region 



< 0and



= 0.

Thus, the coefﬁcient of wave reﬂection from the interface is practically equal to unity,

which is characteristic of metals. Region 4 also is a region with optical transparency, since

at these frequencies 



= 0. In this region 0 <



< 1, thus the wave’s phase velocity is

greater than the speed of light in vacuum.

As we mentioned before, monochromatic waves are only an idealization and real waves

propagate in the form of wave packets transferring energy with the group velocity, v

dω

. (B.195)

There is a relation between phase and group velocities:

= v

− λ

dλ

. (B.196)

The region where dv

/dλ>0 is the region of so-called normal dispersion and in this

region v

. In the region of so-called anomalous dispersion we have dv

/dλ<0

and v

. In the absence of dispersion, when the phase velocity does not depend on

the wavelength, v

= v

and the wave packet propagates without spreading. Vacuum is

a medium of this kind.

Example B.6. Estimate the real and imaginary parts of the dielectric constant, ,for

graphite at the X-ray wavelength of λ = 0.05 nm. The density of graphite is ρ = 1.6 ×

kg m

−3

Reasoning. Note that the carbon atom has four electrons in its outer shell (the electron

conﬁguration of carbon is 1s

, see Table 6.7). Hard X-rays ionize these electrons

and make them free, i.e., uncouple them from their nuclei. Thus, the returning force from

the core of the atom in Eq. (B.180) is equal to zero, i.e., κ = 0, so the frequency ω

can

also be considered equal to zero. As a result Eqs. (B.193)and(B.194)taketheforms



(ω) = 1 −



+ 4β

= 1 −

+ 4β

, (B.197)





(ω) =



2β

ω(ω

+ 4β

)

2ω

ω(ω

+ 4β

)

, (B.198)

where ω



/(

)



1/2

is the so-called plasma frequency. For its calculation, let

us estimate the concentration of free electrons created by X-rays. The number of carbon

374 Appendix B. Electromagnetic ﬁelds and waves

atoms in the unit volume, n

,is

ρ N

, (B.199)

where A = 12 × 10

−3

kg mol

−1

and N

= 6.02 × 10

mol

−1

. After carrying out the

requisite calculations, we obtain

n = 4n

= 3.2 × 10

−3

and the plasma frequency is ω

≈ 3.2 × 10

−1

. The wavelength λ = 0.05 nm corre-

sponds to the frequency ω = 2πc/λ = 3.8 ×10

−1

, which is three orders of magnitude

higher than ω

. The parameter β plays the role of the scattering frequency and it has the

following order: β ∼ 10

−10

−1

. Therefore, 



is equal to zero with high precision,

which indicates that graphite is transparent for X-rays. The magnitude of the real part of

the dielectric constant is practically equal to unity because 1 − 



∼ 10

−6

. In accordance

with Eqs. (B.137)and(B.138), the coefﬁcients of reﬂection, R, and transmission, T ,of

X-rays incident from vacuum on graphite are equal to

R =

1 −

√



1 +

√



≈ 0, (B.200)

T =

1 +

√



≈ 1. (B.201)

B.5 Summary

1. Maxwell’s equations are the main equations of classical electrodynamics, which suf-

ﬁciently correctly describe the processes of propagation and interaction of electro-

magnetic waves with various media up to frequencies of 10

−10

Hz. At higher

frequencies the quantum character of the interaction of electromagnetic radiation with

a substance comes to dominate.

2. Maxwell’s equations are a system of linear partial differential equations of ﬁrst order,

whose unknowns are the ﬁeld intensities, E and H, and ﬂux densities D and B of

the electric and magnetic ﬁelds, respectively. For these magnitudes the superposition

principle is valid.

3. Electric and magnetic ﬁelds possess energy. The volume energy density at each point

of an electromagnetic ﬁeld is equal to

w =

E · D + H · B



+ µµ

4. The vectors E, H,andk of an electromagnetic wave in an isotropic medium are

perpendicular to each other and in this order form a right-handed system of vectors.

For a monochromatic wave the oscillations of E and H are harmonic.

5. The Poynting vector, P = E ×H, deﬁnes the ﬂux energy density which is carried by an

electromagnetic wave, which has the dimensionality [W m

−2

]. In an isotropic medium

this vector is parallel to the wavevector k.

B.6 Problems 375

6. In absorbing media the amplitude of an electromagnetic wave exponentially decreases

during its propagation. The effect of localization of electromagnetic wave energy in

the subsurface layer of an absorbing sample is called the skin effect and it can most

clearly be observed in metals.

7. At the interface of two media, which is not charged, and there is no current ﬂowing

along this interface, the tangential components of the ﬁeld intensity and normal com-

ponents of ﬂux density of electric and magnetic ﬁelds must be continuous. To ﬁnd the

amplitudes of transmitted and reﬂected waves we have to solve a boundary problem,

which requires sewing ﬁelds at the interface taking into account the above-mentioned

boundary conditions.

8. Interference, polarization, diffraction, and dispersion, which occur widely in nature

and are observed throughout the entire electromagnetic spectrum, are explained as

arising solely because of the wave nature of electromagnetic radiation.

B.6 Problems

Problem B.6.1. Maxwell’s equation (B.16) can be rewritten in the following form:

∇ × H = j

dis

+ j, (B.202)

where we introduced the density, j

dis

, of the displacement current, which is measured in

[A m

−2

dis

∂D

∂t

. (B.203)

We see from Eq. (B.202) that it is precisely the displacement currents which are

responsible for the appearance of vortex magnetic ﬁelds in the case when there is no

current, j, of external charges. Let a monochromatic plane wave described by Eq. (B.56),

with an amplitude of the electric ﬁeld of E

= 10

−1

and frequency ω = 3 × 10

−1

(red light), propagate in vacuum. Find the amplitudes of the displacement current,

dis,0

, and magnetic ﬁeld intensity, H

,ofthewave.

Problem B.6.2. The point of interference, SP, of two monochromatic plane waves,

propagating in vacuum, is deﬁned by the radius vector r, which forms angles ϕ

and ϕ

with the wavevectors, k

and k

, of the two waves (Fig. B.13). Find the frequency of the

wave at the point SP which corresponds to the intensity maximum of the wave ﬁeld.

Problem B.6.3. Derive the equation for the location of maxima and minima in the

interference pattern. Assume that the setup of Young’s scheme (the two-slit experi-

ment; see Fig. B.9) is placed in water to perform the same experiment as was done in the air.

Problem B.6.4. The ﬁeld intensities of electric and magnetic ﬁelds in a non-absorbing

medium oscillate in phase. Show that, for a wave in an absorbing medium, the oscillations

of the vectors E and H do not coincide in phase. Find the relation of the phase shift of

these two vectors to the conductivity of the medium, σ .

Problem B.6.5. The conductivity of sea water is σ

s.w.

≈ 5 

−1

and that of copper

is σ

≈ 6 × 10



−1

. Compare the values of skin depth for these two media at

376 Appendix B. Electromagnetic ﬁelds and waves

Table B.6.1. The skin depth, δ, of sea water and copper

Frequency (Hz)

Skin depth (m) 10 10

Sea water 71.4 2.26 7.1 ×10

−2

2.3 ×10

−3

Copper 2.1 ×10

−2

6.6 ×10

−4

2.1 ×10

−5

6.6 ×10

−7

Figure B.13 The

interference of two plane

waves with wavevectors

and k

. Here

= r cos ϕ

and

= r cos ϕ

frequencies f = 10, 10

, 10

, and 10

Hz, considering for simplicity that the dielectric

constant, , and magnetic permeability, µ, equal unity for both media. (Answer:

Table B.6.1.)

Problem B.6.6. Let us consider an optical micro-resonator that consists of two parallel-

plate mirrors applied from the opposite sides of a narrow layer of optical glass with width

L = 10 µm. Find the number, N , of possible types of electromagnetic ﬁeld oscillations

in the region between the mirrors (i.e., standing waves with different wavelengths) that

can be excited in the visible range of wavelengths. Take the dielectric constant of glass

as  = 1.55 and its magnetic permeability as µ = 1. (Answer: N = 27.)

Problem B.6.7. The density of charge distribution in the electron cloud of a hydrogen

atom in the ground state is deﬁned by the following expression:

ρ(r) =−

πa

−2r/r

, (B.204)

where r

is the radius of the ﬁrst Bohr orbit and e is the electron charge. Find the energy

of the electrostatic Coulomb interaction of the electron cloud with the nucleus of the

hydrogen atom (i.e., with the proton).

Problem B.6.8. Find the screening radius (Debye radius), r

, and potential, ϕ,ofa

point defect with charge q in an n-type semiconductor at room temperature, T = 300 K.

B.6 Problems 377

Assume that positively charged ions of univalent impurities are located at the points of

the lattice and that the free electrons are in thermal equilibrium. The energy distribution

of electrons is described by Boltzmann’s classical function. The density of impurities is

= 10

−3

and the dielectric constant of the crystal is  = 10. (Answer: r

≈ 1.2 nm.)

Problem B.6.9. A plane electromagnetic wave from a medium with index of refraction

is obliquely incident on an interface with a medium with index of refraction n

. Find

the condition of total internal reﬂection, the wave ﬁeld in the second medium, and the

depth of penetration into this medium.

Problem B.6.10. Find the energy, W , which is transmitted in vacuum by a plane

electromagnetic wave propagating in the x-direction through a plane contour with area

S = 1m

perpendicular to this direction during one second. Assume that the amplitude

of the electric ﬁeld intensity is equal to E

= 10

−4

−1

and that the frequency is equal

to f = 10

Hz. (Answer: W ≈ 1.3 ×10

−15

J.)

Problem B.6.11. A linearly polarized wave is normally incident on a slab of thickness

L. The dielectric constant of the slab is , its conductivity is σ = 0, and its permeability

is µ = 1. Find the amplitudes of the waves reﬂected from and transmitted through the

slab and of the wave inside of the slab. Find the energy coefﬁcients of reﬂection and

transmission. Find the angle of minimal wave reﬂection from the slab.

Appendix C

Crystals as atomic lattices

Three-dimensional quantum-dot superlattices can be considered as nanocrystals. Spherical

nanoparticles consisting of a big enough number (from 10 to 1000) of atoms or ions, which

are connected with each other and are ordered in a certain fashion, can be considered as the

structural units of such nanocrystals. Examples of nanocrystals that are of natural origin

are the crystalline modiﬁcations of boron and carbon which have as their structural units

the molecules B

and C

. The boron molecule B

consists of 12 boron atoms, and the

carbon molecule C

, which is called fullerene, consists of 60 carbon atoms. The fullerene

molecule resembles a soccer ball, i.e., it consists of 12 pentagons and 20 hexagons, with

carbon atoms at their corners. These nanoparticles form face-centered superlattices with

a period of about 1–10 nm. At these distances between molecules of C

weak molecular

forces, which provide the crystalline state of fullerene, act.

In addition to nanocrystals of natural origin, numerous artiﬁcial three-dimensional

superlattices consisting of various types of nanoparticles have been fabricated. The vari-

ety of nanocrystalline structures as well as of conventional crystals is deﬁned by the

differences in the distribution of electrons over the quantum states of atoms. The most

signiﬁcant role in the formation of individual nanoparticles as well as of crystals is played

by the electrons in the outer shells of atoms. First of all, the energy minimum of a nanopar-

ticle itself is predominantly deﬁned by the interaction of valence electrons. Conﬁgurations

of nanoparticles in which electrons completely ﬁll the energy shells have stable structures

consisting of a certain number of atoms. For this type of structure there exist so-called

magic numbers, which deﬁne the total number of atoms or ions in a nanoparticle. For

one group of metals (Ag, Au, Cu, and others) nanoparticles are formed on the basis of a

face-centered cubic elementary cell built around the central atom and have the following

series of magic numbers: N =1, 13, 55, 147, 309, 561, . . . For another group of metals

(Mg, Zn, and others) nanoparticles form on the basis of a hexagonal densely packed

elementary cell and have the following magic numbers: N =1, 13, 57, 163, 321, 581, . . .

Owing to the exchange of valence electrons or their collectivization, and due to the

spatial distribution of their wavefunctions, arrays of atoms, ions, or nanoparticles will,

under certain conditions, form regular crystalline structures of various conﬁgurations.

The stable state in such structures is almost completely provided by the forces of electro-

static attraction involving the subsystem of valence electrons in ion shells. In addition to

attraction forces at large distances, signiﬁcant forces of repulsion can occur at small dis-

tances between atoms and ions because of overlap of electron shells. There are signiﬁcant

differences in behavior of electrons among metals, dielectrics, and semiconductors that

378

C.1 Crystalline structures 379

are deﬁned mainly by their different band-energy spectra. We will consider some of the

features of the formation of the above-mentioned structures and their energy spectra.

C.1 Crystalline structures

C.1.1 Spatial periodicity

All objects consist of a collection of atoms, molecules, or ions. The average distance

between them is deﬁned by their interaction with each other and by temperature. In the

crystalline state the particles of a medium are arranged in an ordered fashion, and the type

of ordering depends on the chemical species as well as on the details of the interaction

among the particles.

In 1912 the German physicist Max von Laue discovered the diffraction of X-rays

from crystals. Since X-rays are electromagnetic waves, diffraction can take place only

in spatially-periodic structures consisting of atoms or ions with interatomic distances

comparable to the wavelength of X-rays. Modern experimental methods such as scanning

tunneling microscopy (STM) allow us to see directly the regular spatial distribution of

atoms in a crystal.

The main property of crystalline structures is their so-called translational symmetry.

The crystal can be thought of as being generated by the repetition of a basic atomic group

over a ﬁxed vector, which is called the period of a crystalline lattice.Insuchacase

it is said that there is long-range order in a crystalline structure, i.e., the particles are

distributed in an orderly fashion throughout the entire crystal.

In addition to the translational symmetry in ideal crystals, there is so-called point

symmetry. A crystal that has point symmetry coincides with itself after an operation of

rotation and/or reﬂection has been carried out. The symmetry of a crystalline lattice

depends on the distribution of its structural elements in space. Symmetry deﬁnes the

physical properties of a crystal and the directions along which these properties become

apparent. For example, after a rotation of a cube around one of its three axes drawn through

the centers of the opposite cube plane, the cube will coincide with itself after each 90

◦

rotation. If we divide a cube by a plane going through the diagonals of the opposite bases

and then reﬂect it with respect to this plane, then the cube will coincide with itself. The

existence of such elements of symmetry can be demonstrated on the crystalline structure

of sodium chloride (NaCl). It is represented by a cubic lattice with alternating ions of

sodium (Na) and chlorine (Cl) in the sites of the lattice. Each Na

ion is surrounded by

six Cl

−

ions, and in its turn each Cl

−

ion is surrounded by six Na

ions (see Fig. C.1).

The minimal distance between ions is 2.81

A. Let us note that on a macroscopic scale we

are dealing with a very large number of crystalline elements. For example, if the linear

dimensions of a crystal are about 1 cm, the crystal contains of the order of 10

ions.

How can one build an inﬁnite lattice of a simple (consisting of only one type of atoms

or ions) crystal? Let us choose the origin of coordinates in any arbitrary place, and let us

put an atom (or ion) at this point. Let us draw three mutually-orthogonal vectors a

, a

and a

(in general they are not required to be mutually orthogonal), whose length is equal

to the distance between the nearest atoms (or ions). The parallelepiped whose edges are

the vectors a

, a

,anda

is called a unit cell. The three vectors a

, a

,anda

are also

called basis vectors, because, using them, one can build an inﬁnite spatial lattice. In order

380 Appendix C. Crystals as atomic lattices

−

Figure C.1 The unit cell of

the NaCl crystalline

structure.

to do this it is necessary to move all atoms (or ions) of a lattice from their initial positions

by distances equal to the translational vectors a

= n

+ n

, (C.1)

where n

, n

,andn

are arbitrary integer numbers. We can choose unit cells with other

basis vectors and, by translation of these unit cells, we may obtain the same spatial

crystalline lattice. A unit cell of minimal volume is called a primitive cell (see Fig. C.2

for examples of translational symmetry).

There are many physical phenomena in which the atomic structure of the material is

not manifested directly. While studying these phenomena the material can be considered

as a continuum, disregarding its internal structure. Such phenomena include, for example,

thermal expansion of bodies, their deformation under the inﬂuence of external forces,

dielectric permeability, and many optical phenomena. These material properties are char-

acteristic for a continuum and are called macroscopic properties. The indicated properties

signiﬁcantly depend for most crystals on the direction along which the measurements are

carried out. The origin of this dependence is connected with the structure of the crystal and

its symmetry. For example, stretching a cubic crystal in the direction parallel to the edges

of cubic cells of a lattice will not be the same as stretching along the diagonals of the cells,

because the coupling energy between atoms depends on the distance between them. The

dependence of physical properties on the direction is called anisotropy. Anisotropy is the

characteristic property of crystals and in this sense they are fundamentally different from

isotropic media such as liquids and gases, whose properties are the same in all directions.

C.1 Crystalline structures 381

Figure C.2 Translational

symmetry of a

three-dimensional lattice.

Planes of

symmetry

Axes of

symmetry

Center of

symmetry

Figure C.3 Axes, planes,

and the center of

symmetry in a simple

cubic lattice.

C.1.2 Symmetry elements and types of crystalline lattices

What are the main symmetry elements of an object? Let us brieﬂy describe them.

Plane of symmetry. The presence of a plane of symmetry means that one part of

the ﬁgure coincides with the other if we move all its points equal distances to the

other side of a plane in a direction perpendicular to it. Thus the plane of symmetry

can be treated as a mirror, through which one part of the ﬁgure is reﬂected (see

Fig. C.3).

Axis of symmetry. An axis of symmetry is a straight line around which rotation of

the ﬁgure by a certain angle makes the ﬁgure coincide with itself. The order, n,

382 Appendix C. Crystals as atomic lattices

of the axis of symmetry is deﬁned by the equation

n =

360

, (C.2)

where α is the minimal angle of rotation at which the ﬁgure coincides with itself.

Only ﬁve axes of symmetry can exist in crystals: of ﬁrst, second, third, fourth,

and sixth orders (see Fig. C.3).

Center of symmetry. A crystal has a center of symmetry if any straight line drawn

through it at the opposite sides of a crystal goes through identical points. Thus,

there are equal planes, edges, and angles at the opposite sides of the crystal with

respect to the center of symmetry (see Fig. C.3).

There are 32 possible combinations of planes, axes, and centers of symmetry in

crystals. In general, a crystal does not possess only one element of symmetry. The full

set of symmetry elements is called the symmetry group. Why is the symmetry group so

important for the physics of crystals? It turns out that the symmetry group of the crystal

very often deﬁnes the physical properties of the crystal. Depending on the relation of

magnitudes and on the self-orientation of edges of the unit cell, there are 14 types of

crystalline lattices, which are called Bravais lattices (see Fig. C.4). The above-mentioned

14 types of lattices constitute 7 different systems: triclinic, monoclinic, orthorhombic,

tetragonal, rhombohedral, hexagonal,andcubic. Each of the systems is characterized

by the ratio of sides of the unit cells and the angles α, β,andγ between them. Many

important materials have simple (or primitive), base-centered, body-centered,andface-

centered Bravais lattices. If the sites of a crystalline lattice are located only at the vertices

of a parallelepiped, which represents a unit cell, then this lattice is called primitive or

simple. If there are additional sites at the center of the parallelepiped’s base, then this

lattice is called base-centered. If there is a site at the center of the intersection of the

spatial diagonals, then this lattice is called body-centered. Finally, if there are sites at the

centers of all of the lateral faces, then the lattice is called face-centered.

Almost half of the elements from the Periodic Table of the elements form crystals of

cubic or hexagonal symmetry, which we will study now in more detail.

Three lattices are possible for the crystals of cubic systems: simple, body-centered,

and face-centered (see Fig. C.5). In the cubic system all the angles of the unit cell are

equal to 90

◦

and all the edges are equal to each other. A right prism with a rhombus of

angles 60

◦

and 120

◦

as the base represents the unit cell of the hexagonal system. The two

angles between the unit cell’s axes are right angles and one of them is equal to 120

◦

(see

Fig. C.4).

In most cases we consider a crystal as a system of hard spheres that are in contact with

each other. A structure in which the spheres are as densely packed as possible corresponds

to the minimal energy of the structure. Let us compare three possible cubic structures in

such a model. In a simple cubic structure atoms are present only at the sites of a cube. In

this case one atom corresponds to one primitive cell. In a face-centered cubic lattice atoms

are present not only at the sites of a lattice, but also at the centers of six faces (NaCl has

such a structure). In a body-centered cubic lattice the atoms are at the sites of the cube and

in addition there is one at the center of the cube. The least densely packed is the simple

cubic structure, and the chemical elements prefer not to crystallize in such structures,