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Solid-Ch-12a

Ajay Kumar Saxena Laxmi Publications PDF

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 , − →

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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

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Solid-Ch-07

Ajay Kumar Saxena Laxmi Publications PDF

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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Solid-Ch-14

Ajay Kumar Saxena Laxmi Publications PDF

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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Solid-Ch-09

Ajay Kumar Saxena Laxmi Publications PDF

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).

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