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Solid-Ch-10a

Ajay Kumar Saxena Laxmi Publications PDF

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.

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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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Ajay Kumar Saxena Laxmi Publications PDF

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.

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Solid-Ch-03

Ajay Kumar Saxena Laxmi Publications PDF

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.

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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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