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Solid State Physics: With An Introduction to Semiconductor Devices

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The branches of Solid State Physics are developing very fast. This new edition has been extensively enlarged by including numerous new topics and thorough revision. The following topics are covered in this book: Graphite structure, Fullerenes, Carbon clusters and nanotubes, Polymers, Solid-solutions, Origin of dislocations and interaction between dislocations, crystallographic zone-axis, Electron diffraction and diffraction patterns, scattering by a unit cell, Mott-transition, Gunn effect and Gunn diode, Cyclotron resonance for semiconductors, photodiode and phototransistor, characteristics of an ideal dc SQUID, the first cuprate superconductor La2=x Mx CuO4 (ground state and phase diagram), variation of Tc with x in superconductor Y1Ba2Cu3O6+x. This book has been written keeping in mind the UGC syllabus

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Solid-Ch-01

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1

Crystal Structure

1.1 CRYSTALLINE AND AMORPHOUS SOLIDS

On the basis of structure, solids may be divided into two broad categories – crystalline and amorphous. In crystalline solids, the atoms are stacked in a regular manner, forming a three-dimensional pattern which may be obtained by a three-dimensional repetition of a certain pattern unit. When the periodicity of the pattern extends throughout a certain piece of material, one speaks of a single crystal. In polycrystalline materials, the periodicity of structure is interrupted at the so-called grain boundaries such that the structure is periodic within a single grain and the size of the grains within which the structure is periodic may vary from macroscopic dimensions to several Angstroms*. When the size of the grains becomes comparable to the size of the pattern unit, one can no longer speak of crystallinity, rather one speaks of amorphous substances. Due to regularity in structure, there is a long-range order in single crystals. There remains no periodicity in structure in an amorphous substance and long-range order diminishes to a very short-range order.

 

Solid-Ch-02

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2

X-ray Diffraction and

Reciprocal Lattice

2.1

CHOICE OF X-RAYS, ELECTRONS AND NEUTRONS FOR

CRYSTAL STRUCTURE DETERMINATION

The interatomic spacing in crystals is of the order of 1 Å. To explore the structure of crystals, we require de Broglie waves, which when incident on solids, interact with its atoms. For this, we need wavelengths of the incident waves to be comparable with the interatomic spacing; visible or ultraviolet light (l ~ 5000 Å) cannot be used because it would not be able to provide the resolution necessary for this purpose. However,

X-rays or electrons or neutrons accelerated through suitable voltage may be used as an incident beam. We may have an idea of energies necessary for providing a resolution of 1 Å for the three cases as follows.

X-rays:

E = hv = c

=

h l

3 ¥ 1010 ¥ 6.62 ¥ 10 -27

(CGS units)

10 -8

= 19.86 ¥ 10–9 ergs = 1.24 ¥ 104 eV

2

Electrons: E =

=

2

p

(h / l )

=

(using de Broglie relation)

2m

2m h

2

-27 2

2

2m l

=

(6.62 ¥ 10 )

(CGS units)

-27

-16

2 ¥ 0.91 ¥ 10 ¥ 10

= 24.05 ¥ 10–11 ergs = 150.33 eV

 

Solid-Ch-03

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3

Crystal Imperfections

3.1 WHAT IS AN IMPERFECTION?

The concept of an ideal crystal with a perfect arrangement of atoms is, strictly speaking, valid only at the absolute zero temperature because then there is no entropy contribution. However, at a finite temperature, a certain native configurational disorder is introduced into the structure (a direct consequence of laws of thermodynamics) and solid becomes structurally imperfect. In fact, in almost all cases, a crystal just cannot be grown without an initial imperfection. Thus, imperfections are mistakes in the crystallographic structure of crystals.

3.2 IMPORTANCE OF LATTICE IMPERFECTIONS

Defects and imperfections affect especially the structure-sensitive properties of solids. As, for instance, it is well known that electrical and thermal conductivities are greatly reduced due to scattering of electrons and phonons by lattice defects. The semiconducting properties of solids are also considerably influenced by the presence of impurities in the lattice, which are responsible for creating localised (defect) levels in the energy gap between the valence band and conduction band. Dielectric behaviour of materials depends largely on the state of polarisation, temperature and perfection. Optical properties are also related to energy gaps (as in semiconductors) and different point defects (which is more apparent in ionic crystals), e.g. colour of some materials and sensitivity of photographic emulsions (to light) are due to imperfections.

 

Solid-Ch-04

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4

Bonding in Solids

4.1 INTRODUCTION

In any solid, the mutual interatomic forces are basically electrostatic in nature, and the primary differences among different types of solids depend on the ways in which the valence electrons of the constituent elements are distributed, i.e. it depends on the type of bonding. Thus, on the basis of bonding type, we have the following five categories of solids:

(i) Ionic solids (e.g. alkali halides, alkaline oxides etc.)

(ii) Covalent solids (diamond, silicon etc.)

(iii) vander Waal bonded molecules (O2, H2, solid-He, Kr, Xe)

(iv) Hydrogen-bonded solids (ice, some fluorides and compounds having water of crystallisation)

(v) Metallic solids (various metals and alloys). Different physical, chemical and electrical properties of a material is determined by particular type of bonding present in it.

Before discussing various types of binding, we will consider general nature of atomic bond between two atoms in a solid.

4.2 GENERAL NATURE OF COHESION BETWEEN TWO ATOMS IN A SOLID

 

Solid-Ch-05

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5

Lattice Vibrations and

Thermal Properties

5.1

PHONONS

As a result of their thermal motion, the lattice ions in a solid vibrate about their equilibrium positions. All the elastic properties, compressibility and propagation of acoustic waves in solids are then described in terms of a continuum theory disregarding the atomic structure of the lattice.

When we supply thermal energy to a lattice-ion, it will be rapidly distributed throughout the entire lattice by the mutual interaction between the ions. As a result, the ions vibrate about their mean positions at all temperatures above absolute zero, the restoring force being provided by the force between the ions due to the chemical bonds. Thus, local excitations lead to collective-vibrations of the whole ion system. Collectivecoordinates (normal coordinates) are generally used for the mathematical description for describing oscillations in continuum theory. Similar to light quanta (photons), the lattice vibrations can be quantized and the associated quanta are elementary excitations termed as phonons. Phonons are Bosons and are, therefore, described by statistics different from that of electrons. Electrons follow Fermi-Dirac statistics whereas phonons follow the

 

Solid-Ch-06

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6

Transport Properties

6.1 DRUDE’S MODEL

During the nineteenth century, there was no precise concept known for atomic structure. The electron was discovered by Thomson in 1897, and this discovery had a vast and immediate impact on theories of structure of matter. Three years later (i.e. in 1900) Drude gave his theory of electrical and thermal conduction by considering the metals to be containing free electrons, and thereby applying the kinetic theory of gases to metals.

Accordingly, it is assumed that when atoms of a metallic element are brought together to form a metal, the valence electrons get detached and wander freely within the metal, whereas ions remain intact and play the role of immobile particles. Surrounding the nucleus are Za electrons of total charge –eZa. Out of these, there are Z relatively weakly bound valence electrons and remaining (Za – Z) are relatively tightly bound and are known as core electrons. When atoms condense to form the material, core electrons remain bound to the nucleus, but valence electrons detach from their parent atoms and are called conduction electrons. They are akin to atoms in a gas moving against a background of heavy immobile ions (see Fig. 6.1).

 

Solid-Ch-07

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246 Solid State Physics

7

Band Theory

There are two approaches to discuss formation of energy bonds in solids: (i) atomistic approach and

(ii) one-electron approach.

In the atomistic approach, electrons are assumed

2

E to be tightly bound to individual atoms. As atoms are brought together to form a crystal, interaction between n=3

2 the neighbouring atoms causes the electron energy

1 levels of individual atoms to spread into bands of n=2 energies.

1

In the one-electron approximation, we study the

2 behaviour of a single electron in the potential field n=1 established by the lattice atom cores and modified by

1 the presence of all the other free electrons. The various r0 permissible energy values (levels) obtained for this r electron represent the allowed energy levels of all the

Fig. 7.1 Energy states for electron in diatomic electrons.

7.1

molecule (1 and 2 refer to two states created when stabilising the molecule)

DEVELOPMENT OF ENERGY BANDS

IN SOLIDS (ATOMISTIC APPROACH)

We have seen that when two H atoms are brought together, the original 1s wave functions of the two

 

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250 Solid State Physics

Æ Æ

Æ Æ

TR yn ( k , r ) = exp (i k ⋅ Rl ) yn ( k , r )

(∵ TR yn = ll yn and |l l|2 = 1)

(7.4)

Then Bloch’s theorem states that eigenfunctions yn satisfy

→ →

Æ Æ

yn ( k , r + Rl ) = exp (i k ⋅ Rl ) yn ( k , r )

(7.5)

If we substitute

Æ Æ

Æ Æ

Æ Æ

yn ( k , r ) = e i k ◊ r un ( k , r )

(7.6)

in Eq. (7.5), we have

exp [ i k ⋅ ( r + Rl )] u n ( k , r + R l )

→ →

= ei k

⋅ Rl

→ →

ei k

⋅r

un ( k , r )

Therefore

Æ Æ

un ( k , r + Rl ) = un ( k , r )

(7.7)

Æ Æ

that is, un ( k , r ) have the periodicity of the lattice. The functions given by Eq. (7.6) are called Bloch functions.*

The crystal electrons which these functions describe are correspondingly called Bloch electron. The form of

Æ

the eigenfunctions (7.6) give the first due to the physical meaning of the k . If we put u = constant, y is given

Æ Æ

by y = C e i k ◊ r. The electron in this case behaves as a free particle and is represented by a plane wave of wave

Æ

vector k . If transferred to the crystal case, then functions of Eq. (7.6) correspond to the situation that Bloch electron is represented by a plane wave, which is modulated by periodicity of the lattice. This is the physical interpretation of the Bloch theorem.

 

Solid-Ch-08

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8

Fermi Surfaces

8.1 FREE-ELECTRONS

In case of one-electron model, we may ignore the interactions between the valence electrons and the (Hartree

Æ or Hartree–Fock) potential V( r ) seen by each valence electron, as it moves through the crystal may be taken as constant. The electrons are then described by plane-wave states

Æ Æ

Æ

i k.r

| k = y Æk = e

(8.1)

having energies

2

Æ

E( k ) = k

(8.2)

2m

Æ

We know that these states satisfy the Bloch theorem. The surfaces of constant energy are spheres in kÆ

Æ space. The allowed values of k are distributed with density V/8p 3 in this space. But for each value of k , there are two electron states of opposite spin. If there are Z electrons/atom, 1 atom/cell and N unit cells/unit volume, then there will be ZN electrons per unit volume in real space and to satisfy the Pauli’s principle, we need to fill the states upto a wavevector (number) kF given by

4 3 2 p kF

= ZN, i.e. kF = (3p 2ZN )1/3

3

8p 3

It is then said that the Fermi surface is a sphere of radius kF.

Thus, Fermi surface is a surface of constant energy

 

Solid-Ch-09

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9

Semiconductors and

Semiconductor Devices

9.1 CONDUCTOR, INSULATOR AND SEMICONDUCTOR

A conductor is any material that will support a generous flow of charge when a voltage source of limited magnitude is applied across its terminals. An insulator is a material that offers a very very low level of conductivity (under pressure) from an applied voltage source. A semiconductor is a material that has a conductivity level somewhere between the extremes of an insulator and a conductor.

8

–1 –1

–20

–1 –1

A pure metal at 1 K may have a conductivity ~

- 10 ohm m against a low of 10 ohm m for an

–7 extreme insulator. Semiconductors have typical conductivity values in the range 10 to 1 ohm–1 m–1. In

Table 9.1, typical resistivity values are provided for the aforementioned three broad categories of materials.

Typical resistivity values

Table 9.1

Conductor

–6

10 W-cm

(copper)

Semiconductor

Insulator

50 W-cm (Ge)

50 ¥ 103 W-cm (Si)

1012 W-cm

(mica)

The most useful feature of semiconductors is that their electrical conductivity generally decreases with increasing purification in contrast to metals (where conductivity always increases with purification).

 

Solid-Ch-09a

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Semiconductors and Semiconductor Devices

Fig. 9.23

357

Qualitative plots of the space charge, electric field and barrier potential near an abrupt pn-junction. In this case the donor concentration is greater than the acceptor concentration, but charge neutrality requires that the areas in curve (a) be equal about x = 0

9.16.3 Width of the Depletion Region

The width of the depletion region tells us how wide the region is where there is an upset in the bulk equilibrium conditions. We will now derive an expression for the depletion region width d ∫ (dp + dn). This width is determined by the impurity concentrations and the voltage.

We are considering here the junction where the impurity concentration changes abruptly (from acceptors to donors) at x = 0. The space charge density in the p-region is – qNA and in the n-region is qND. Since the number of negative charges must be equal to the sum of the positive charges, we have

(qA) ND dn = (qA) NA dp

[9.89(i)]

ND dn = NA dp

[9.89(ii)]

or

(A is the junction area).

 

Solid-Ch-10

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10

Magnetic Properties

10.1 ORIGIN OF MAGNETIC PROPERTIES OF MATERIALS

We know that a small circular current-carrying loop has a magnetic moment associated with it. So motion of electrons in atoms is responsible for magnetism and quantised nature of electronic motion gives rise to the fundamental unit of magnetic dipole moment, the Bohr magneton mB.

For a circular loop of area A and carrying a current I, the dipole moment is IA. For an electron of charge e and mass m rotating in a circular orbit of radius r at an angular velocity w, the magnetic dipole moment is w

1 m = IA = - e p r2 = - ewr2

(10.1)

2p

2

F I

H K

The angular momentum

Æ

| J | = m | r × v | = mwr2

(10.2)

therefore,

Æ

m =−

FH e IK J

2m

(10.3)

The angular momentum is quantised in units of h/(2p) where h is Planck’s constant. Therefore, the

Æ

lowest non-zero value for | m | is mB = eh/4p m = 9.2741 ¥ 10–24 J T –1 (SI units) or 9.2741 ¥ 10–21 erg Oe–1 (CGS units).

No electron can have a magnetic moment below mB.

Thus, orbiting/spinning of electrons on the whole may impart a permanent magnetic moment to atoms.

 

Solid-Ch-10a

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Magnetic Properties

487

This gives the increase in exchange energy when two equal spins are rotated from exact parallelism to make some small angle f with each other.

Now, let f0 denote the total change of angle between two domains and the change occurs in N equal steps, so that the change of angle between neighbouring spins is f0 /N. Then, the exchange energy between each pair of neighbouring atoms is

FG φ IJ

H NK

0

(DEex)pair = JeS 2

2

(10.137)

and the total energy of the array of atoms

J S 2 f 20

(DEex)total = e

(10.138)

N

This shows that the exchange energy decreases when N increases. Now, one may argue that why does not the wall become infinitely thick (to increase N)? This is explained by the concept of anisotropy energy.

Since the spins within the wall are nearly all directed away from the easy direction, an anisotropy energy is associated with the wall which is roughly proportional to the thickness of the wall. This energy also needs to be minimum. Thus, the actual thickness and the energy of the wall is, therefore, the result of compromise between the two energies, viz., the exchange energy and the anisotropy energy.

 

Solid-Ch-11

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Dielectrics, Plasmons, Polarons and Polaritons 505

11

Dielectrics, Plasmons,

Polarons and polaritons

11.1 INTRODUCTION

In insulating materials (dielectrics), electrons are very tightly bound to the atoms (and forbidden gap in the energy band picture is comparatively larger). Consequently, electrons cannot be made free, however they can only be displaced a bit within the molecule (under the application of an external electric field) and their cumulative effect accounts for the characteristic behaviour of dielectric materials. Thus, under the action of a strong electric field, the centres of the positive charges are displaced slightly in one direction (i.e. the direction of the field) and that of negative charges in the opposite direction. This gives rise to a local electric dipole moment throughout the crystalrelative to lattice deformation. The internal distortion in the lattice can be studied in detail if one has the knowledge of the dielectric constant.

11.2 MACROSCOPIC DIELECTRIC CONSTANT

 

Solid-Ch-12

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12

Superconductivity

12.1 THE DISCOVERY

Dutch physicist H. Karnerlingh Onnes was successful in liquifying helium in 1908 (nitrogen liquifies at 77.4

K). The attainment of liquid He temperatures opened a new regime of low temperatures and it was discovered by him in 1911 (while investigating the electrical properties of frozen mercury) that the electrical resistance of

Hg completely disappeared on approaching 4.2 K. It was discovered that the disappearance did not take place gradually, but abruptly. Thus, the mercury at 4.2 K enters a new state, which owing to its particular electrical properties is called the state of superconductivity. Superconductivity was thus born in 1911.

The phenomenon of superconductivity is manifested in the ‘complete’ vanishing of electrical resistance at a finite (low) temperature called the critical temperature (denoted by Tc).

The variation of resistance with temperature for mercury in the Onnes experiment is depicted in

Fig. 12.1. The latest data show that the resistivity of a superconductor is below 10 –27 W-cm in contrast with the resistivity of copper (an excellent conductor), which is 10–9 W cm. Thus, there is no doubt here that we are dealing with ideal conductivity (total vanishing of electrical resistance) in superconducting state (in contrast Fig. 12.1 Temperature dependence of resistance of a with perfect conductivity of a good conductor). normal metal and a superconductor

 

Solid-Ch-12a

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568 Solid State Physics

12.18 THE BCS GROUND STATE

A weak attractive interaction, resulting from electron–phonon interaction leads to the formation of Cooper pairs. A single pair formation leads to an energy reduction of the Fermi sea. The new ground state of the Fermi sea (after Coopper-pair formation) is achieved through a complicated interaction between the electrons. The total energy reduction is not given by simply summing the contributions of single pairs because the effect of each single pair depends on those already present. Thus, we require the minimum total energy of the whole system for all possible pair configurations taking into account the kinetic one-electron component and the energy reduction due to ‘pair-collisions’, i.e. the electron–phonon interaction. The kinetic component is given by

Ekin = 2

2 2

∑w ξ

where

k k

x=

k

k

- E F0

2m

(12.75)

0 wk is the probability that the pair state ( k→

A , − kB ) is occupied and EF = EF (T = 0ºK).

Æ

Æ

The total energy reduction due to the pair collisions ( k→A , − →

 

Solid-Ch-13

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Elastic Constants of Crystals

591

13

Elastic Constants of Crystals

13.1 INTRODUCTION

Solids are not absolutely rigid: by application of suitable forces, they can be made to change in both size and shape. However, when these changes are not too great, they return to their original size and shape when forces are removed. This property of solids is termed elasticity and is common to all classes of solids.

Here, we shall discuss the elasticity of single crystals and find their elastic constants. While dealing with this, we treat the crystal as a homogeneous continuous medium rather than a periodic array of atoms. This continuum approximation is usually valid for elastic waves of wavelengths longer than 10–6 cm, i.e. for frequencies below 1012 Hz.

We now analyze the forces acting on a crystal and the consequent deformation which may be a change in shape or size or in both. This is done in terms of stress and strain respectively. Next we shall relate stress and strain linearly in accordance with Hooke’s law to find the elastic constants.

 

Solid-Ch-14

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608 Solid State Physics

14

Fabrication of Integrated Circuits

The physical realization of large number of electronic elements (or semiconductor devices) separately integrated on a single semi-conductor layer so as to perform the functions of a complicated circuit is an integrated circuit.

Monolithic integrated circuit is one in which all circuit components are fabricated on to a tiny silicon dice. The following are the various steps in the fabrication of semiconductor integrated circuits:

14.1 CRYSTAL GROWTH

The process of the preparation of single crystals of Si :

Crystals are grown using SiO 2 Czokralski method. The crystal grown is in the form of a cylinder of 1 to

4 inches diameter depending on the pulling rate, melt temperature and other external factors.

Silicon is an abundantly available material as sand (SiO2). This is chemically treated to obtain highly purified polycrystalline Si which is then used for single crystal growth. The basic arrangement of the method is shown in Fig. 14.1. gas inlet

 

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