Optical property of materials

1. Electronic and Optical Properties of Materials
MS-380

6. Relationship between Band Structure and Atomic bonding
for Conduction of free electrons

7. Electron Mobility

8. Electrical resistivity of metal

9. Influence of Temperature
Influence of Impurities
Influence of Plastic Deformation

10. Free Electron Theory
 Outermost electrons of the atoms take part in
conduction.
 These electrons are assumed to be free to move through
the whole solid.
Free electron cloud / gas, Fermi gas.
 Potential field due to ion-cores is assumed constant.
Potential energy of electrons is not a function
of the position (constant negative potential) .
 The kinetic energy of the ‘free’ electron is much lower
than that of bound electrons in an isolated atom

12. K =π/L, 2π/L…….
K can takes both positive and negative values
K =±π/L, ±2π/L…….
In general, K =±nπ/L,
where the quantum number n takes on
successively increasing integer values
𝐴 = 𝜋𝑟2
𝐸 =
ℎ2𝑛2
8𝑚𝐿2

13. For three-dimensional solid
𝐸 =
ℎ2[𝑛𝑥2 + 𝑛𝑦
2 +
𝑛𝑧
2]
8𝑚𝐿2

14. Probability of finding an electron:
P α A2 (square of the amplitude of the wave).
Therefore P is maximum at midpoint for λ=2L
Contradictory
Waves are considered to be travelling waves with a constant velocity, so that the time
averaged probability of finding an electron is constant throughout the solid.

16.  At zero K the highest filled energy level (EF) is called the Fermi level
 If E F is independent of temperature (valid for usual temperatures)
► Fermi level is that level which has 50% probability of occupation
by an electron (finite T)
Fermi level

19. n= 1028 m-3, e = 1.6 x 10-19C
Ʈ = 10-14 s, m = 9.1 x 10-31 kg

21. Diffraction Phenomenon

22. Energy Gap in Solids
 In the model of free electrons, the potential field is assumed to be
constant.
 In reality, potential energy of an electron will be undoubtedly a
function of its position.
 This fact combined with the diffracting properties of electrons has
an important effect on the electron energy calculations.
 Consider the motion of electrons along only one axis.
 Electrons moving to the right or to the left are scattered by the
atoms in the path

23.  If, the spacing d between neighboring atoms along the axis is such
that the Bragg equation nλ = 2d sin Ф
 is exactly satisfied, the electrons are totally reflected.
 For the unidirectional motion, the atoms along the axis can be
visualized as successive ‘planes’ perpendicular to the direction of
electron motion so that sin Ф = 1.
 Then, from above Eq. , the wavelength λ of the electron takes on
critical values of 2d, d, 2d/3, and
 If Expressed in terms of the wave number k (k = 2Π/ λ), we get the
 critical conditions at k = ± Π /d, ± 2Π /d, 3 ± Π /d,…..

24.  Wave form of the electrons under such critical conditions comes out
to be two standing waves rather than the travelling wave.
Standing waves can be obtained from linear combinations of two
travelling waves moving in opposite directions.
 Two possible combinations (the sum and the difference) of the
travelling waves give respectively a cosine wave and a sine wave.
These waves have the maximum
amplitude at fixed points in space.
 Probability of finding an electron
at a point is proportional to the
square of the amplitude of the
wave at that point.
 Electron is localized here.

25.  The E–k relationship is replotted in, taking into account the
deviations from the average potential energy
 . At a critical condition, electron is described by a
standing wave and is therefore localized
 Deviations from the average potential energy appear as a break in
the E–k curve
 Slope is zero at critical values

26.  In the three-dimensional crystal, the electrons can move in any
direction.
 The Bragg angle Ф can have any value from 0 to 90°.
 Critical values of k will then correspond to n k = ±n Π /d sin Ф and not
±n Π /d.

27.  Ranges of k that lie between the forbidden gaps are called Brillouin
zones.
The first Brillouin zone contains all the energy levels up to the first
E–k discontinuity,
Second zone contains all levels between the first and the second
discontinuities and so on

28. Intrinsic Semiconductors
 A very large field is required for electronic transition
 The other possibility is by thermal excitation.
 Conduction is due to the intrinsic processes.
 A pure crystal of silicon or germanium is an intrinsic
semiconductor

29.  The number of electrons excited across the gap can be calculated
from the Fermi–Dirac probability distribution.

30.  Probability P(E) of an electron occupying energy level E
 Number n of electrons promoted across the gap
 Number holes and electrons are equal. Ie

31.  Under an external applied field both electrons and holes
move in the opposite directions.
 Conductivity depends on the concentration of these charge
carriers

32. Energy gap can be computed from the slope of straight line
between log of conductivity against the reciprocal of temperature

33. Extrinsic Semiconductors
 Doped pure semiconductor.
 Addition of impurities markedly increases the conductivity of a
semiconductor.
 Large orbit means that the fifth electron is almost free and is at an
energy level close to the conduction band.

34.  Phosphorus atom donate its electron to the semiconductor.
 The energy level of the fifth electron is called the donor level
 n –type semiconductors.
 Energy required to excite the fifth electron into the conduction band
is known as the ionization energy.
 Ionization energy of an impurity atom is very small.
 So, at room temperature, a large fraction of the donor level
electrons are excited into the conduction band.
 According to the law of mass action, the product of the number of
electrons in the conduction band and the number of holes in the
valence band must be constant

35.  The law of mass action also valid here.
 Positive holes become the majority charge carriers.
 Such an extrinsic semiconductor is called a p-type semiconductor.