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Appendix-B

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Appendix B

CARRIER FLOW IN SEMICONDUCTORS

Current flow in semiconductors is chiefly by two processes, viz., (i) by drift and (ii) by diffusion

1. Carrier flow by drift

Consider a rectangular bar of a semiconductor of uniform cross section to whose opposite ends as applied a potential difference of V volts.

E

x=0

E

x=0

x=l

x=l

+e

e–

Jn

Jp

+

+

V

(a)

V

(b)

Fig. B.1 Drift currents in (a) n-type semiconductor and (b) p-type semiconductor

Under the action of the applied field E there will be directed motion of the carriers and the drift currents are given by

Jn

and

= neμ E n

(∵

J = σE , σn = neμn , σ p = peμ p

Jp = peμ p E where μn and μp are the electron and hole mobilities respectively.

2. Carrier flow by diffusion

Diffusion results from net flow of particles from a n(x) region of higher concentration to a region of lower concentration. Thus, diffusion flow is proportional to the concentration gradient.

Consider an n-type semiconductor bar having a x=0 decreasing electron concentration with distance x

(Fig. B.2).

)

(B.1)

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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.

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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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Untitled

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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.

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Appendix-A

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Appendix

619

Now, the decrease in the intensity of the transmitted beam due to scattering by the electrons in the slice is

dI = −dp = − I σe nρ 1× 1× dx

= − I σe nρdx

The minus sign indicates the decrease in the intensity. Therefore dI

∫ I = −∫ σe nρdx ln I = −σe nρc + constant. or

Now, we apply the boundary condition

I = I0 when x = x0

Then constant = ln I0 and

(A.2)

I = I0 exp (−σe )nρx

This eqn. can be written in terms of mass scattering coefficient ρm ( = nρe) :

I = I0 exp (−σm )ρx

Writing σ m p = μ, where μ, is the linear attenuation coefficient, we have

I = I 0 exp (−μx)

In terms of the linear attenuation coefficient μ, eqn (2) may be written as

μ=

− dI / I dx

(A.3)

Io

I

I = Ioe-µx

x

Fig. A.2 Intensity as a function of thickness of the absorbing material where μ is defined as the fractional fall in the intensity of the x-rays per unit thickness of the absorber. However, actual value of μ depends on the wavelength of x-rays and on the absorbing material.

The value of μ can be determined by finding the thickness of the absorbing material which reduces the intensity of the x-ray beam to half of its initial value. This thickness is known as half value layer xh.

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