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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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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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Appendix-C

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Appendix C

HAYNES – SHOCKLEY EXPERIMENT

Principle: Consider a uniformly doped n-type semiconductor bar in which N electron-hole pairs are generated instantaneously at x = 0 and t = 0. Assuming the semiconductor bar to be infinitely long in the x-direction

(Fig. C.2), the continuity equation for the excess holes under a constant applied electric field (1-dimensional equation) is given by

∂ (δp )

∂ 2 (δp )

∂ (δ p ) δ p

= Dp

−μp ε

2

∂t

∂x

τp

∂x

(C.1)

The solution of this equation is given by

Ne

−t / τ p

½

(4πD p t )

⎡ ( x − μ p εt )2 ⎤

⎥ exp ⎢ −

4D pt ⎥

This equation shows that the initial value of δp ( x, 0) is zero except at x = 0 where δp ( x, 0) approaches infinity i.e., the initial hole-concentration distribution corresponds to a Dirac delta function. For t > 0, the distribution has a

Gaussian shape. The half width of δp ( x, t ) will increase with time and its maximum amplitude will decrease with distance along the direction of the applied electric field with a drift velocity ν d = μ p ε . The total excess carrier density injected at time t into the semiconductor is obtained by integrating eqn C.2 with respect to x from − ∞ to + ∞ , which gives

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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 , − →

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