SEMICONDUCTOR PHYSICS

JNTU SNIST PPT ENGINEERING PHYSICS 2
SEMICONDUCTOR PHYSICS

1. Semiconducto
r Physics

2. Introduction
•Semiconductors are materials whose electronic
properties are intermediate between those of
Metals and Insulators.
•They have conductivities in the range of 10
-4
to
10
+4
S/m.
•The interesting feature about semiconductors is
that they are bipolar and current is transported
by two charge carriers of opposite sign.
•These intermediate properties are determined
by
1.Crystal Structure bonding Characteristics.
2.Electronic Energy bands.

3. •Silicon and Germanium are elemental
semiconductors and they have four valence electrons
which are distributed among the outermost S and p
orbital's.
•These outer most S and p orbital's of Semiconductors
involve in Sp
3
hybridanisation.
•These Sp
3
orbital's form four covalent bonds of
equal angular separation leading to a tetrahedral
arrangement of atoms in space results tetrahedron
shape, resulting crystal structure is known as Diamond
cubic crystal structure

4. Semiconductors are mainly two
types
1. Intrinsic (Pure)
Semiconductors
2. Extrinsic (Impure)
Semiconductors

5. Intrinsic Semiconductor
•A Semiconductor which does not have any
kind of impurities, behaves as an
Insulator at 0k and behaves as a
Conductor at higher temperature is known
as Intrinsic Semiconductor or Pure
Semiconductors.
•Germanium and Silicon (4
th
group
elements) are the best examples of
intrinsic semiconductors and they possess
diamond cubic crystalline structure.

6. Si
Si
SiSiSi
Valence Cell
Covalent bonds
Intrinsic Semiconductor

7. E
E
f
E
v
Valence band
E
c
Conduction band
E
c
Electron
energy
Distance
KE of
Electron
= E - E
c
KE of Hole
=
E
v
- E
Fermi energy level

8. Carrier Concentration in Intrinsic
Semiconductor
When a suitable form of Energy is supplied to a
Semiconductor then electrons take transition
from Valence band to Conduction band.
Hence a free electron in Conduction band and
simultaneously free hole in Valence band is
formed. This phenomenon is known as Electron -
Hole pair generation.
In Intrinsic Semiconductor the Number of
Conduction electrons will be equal to the
Number of Vacant sites or holes in the valence
band.

9. )1......(..........)()(
)()(
band theof top
ò
=
=
cE
dEEFEzn
EFdEEZdn
Calculation of Density of
Electrons
Let ‘dn’ be the Number of Electrons available between
energy interval ‘E and E+ dE’ in the Conduction band
Where Z(E) dE is the Density of states in the energy
interval E and E + dE and F(E) is the Probability of
Electron occupancy.

10. dEEEm
h
dEEZ
ce
2
1
2
3
3
)()2(
4
)( -=
*p
Since the E starts at the bottom of the Conduction band E
c
dEEm
h
dEEZ
e
2
1
2
3
3
)2(
4
)(
*
=
p
We know that the density of states i.e., the number of
energy states per unit volume within the energy interval
E and E + dE is given by
dEEm
h
dEEZ
2
1
2
3
3
)2(
4
)(
p
=

11. )exp()(exp)(
)exp(
1
)(
res temperatupossible allFor
)exp(1
1
)(
kT
EE
kT
EE
EF
kT
EE
EF
kTEE
kT
EE
EF
FF
f
F
f
-
=
-
-=
-
=
>>-
-
+
=
Probability of an Electron occupying an energy state E
is given by

12. )2.....()exp()()exp()2(
4
)exp()()2(
4
)exp()()2(
4
)()(
2
1
2
3
3
2
1
2
3
3
2
1
2
3
3
band theof top
ò
ò
ò
ò
¥
*
¥
*
¥
*
-
-=
-
-=
-
-=
=
c
c
c
c
E
c
F
e
E
F
ce
E
F
ce
E
dE
kT
E
EE
kT
E
m
h
n
dE
kT
EE
EEm
h
n
dE
kT
EE
EEm
h
n
dEEFEzn
p
p
p
Substitute Z(E) and F(E) values in Equation (1)

13. )3.....()(exp)()exp()2(
4
)(exp)()exp()2(
4
)exp()()exp()2(
4
0
2
1
2
3
3
0
2
1
2
3
3
0
2
1
2
3
3
ò
ò
ò
¥
*
¥
*
¥
*
-
-
=
+
-=
-
-=
=
+=
=-
dx
kT
x
x
kT
EE
m
h
n
dx
kT
xE
x
kT
E
m
h
n
dE
kT
E
EE
kT
E
m
h
n
dxdE
xEE
xEE
cF
e
cF
e
c
F
e
c
c
p
p
p
To solve equation 2, let us put

14. )exp()
2
(2
}
2
){()exp()2(
4
2
3
2
2
1
2
3
2
3
3
kT
EE
h
kTm
n
kT
kT
EE
m
h
n
cFe
cF
e
-
=
-
=
*
*
p
pp
)3(
2
)()exp()(
2
1
2
3
0
2
1
equationinsubstitute
kTdE
kT
x
xthatknowwe
p
=
-
ò
¥
The above equation represents
Number of electrons per unit volume of the Material

15. Calculation of density of
holes
)1......(..........)}(1){(
)}(1{)(
band theof bottom
ò
-=
-=
Ev
dEEFEzp
EFdEEZdp
Let ‘dp’ be the Number of holes or Vacancies in the
energy interval ‘E and E + dE’ in the valence band
Where Z(E) dE is the density of states in the energy interval
E and E + dE and
1-F(E) is the probability of existence of a hole.

16. dEEm
h
dEEZ
h
2
1
2
3
3
)2(
4
)(
*
=
p
Density of holes in the Valence band is
Since E
v
is the energy of the top of the
valence band
dEEEm
h
dEEZ
vh
2
1
2
3
3
)()2(
4
)( -=
*p

17. )exp()(1
exp
)}exp(1{1)(1
}
)exp(1
1
{1)(1
1
kT
EE
EF
valuesThigherfor
ansionaboveintermsorderhigherneglect
kT
EE
EF
kT
EE
EF
f
f
f
-
=-
-
+-=-
-
+
-=-
-
Probability of an Electron occupying an
energy state E is given by

18. )2....()exp()()exp()2(
4
)exp()()2(
4
)}(1){(
2
1
2
3
3
2
1
2
3
3
band theof bottom
ò
ò
ò
¥-
*
¥-
*
-
-
=
-
-=
-=
v
v
E
v
F
h
E
F
vh
Ev
dE
kT
E
EE
kT
E
m
h
p
dE
kT
EE
EEm
h
p
dEEFEzp
p
p
Substitute Z(E) and 1 - F(E) values in Equation (1)

19. ò
ò
ò
¥
*
¥
*
¥-
*
--
=
-
--
=
-
-
=
-=
-=
=-
0
2
1
2
3
3
0
2
1
2
3
3
2
1
2
3
3
)exp()()exp()2(
4
))(exp()()exp()2(
4
)exp()()exp()2(
4
dx
kT
x
x
kT
EE
m
h
p
dx
kT
xE
x
kT
E
m
h
p
dE
kT
E
EE
kT
E
m
h
p
dxdE
xEE
xEE
Fv
h
vF
h
E
v
F
h
v
v
v
p
p
p
To solve equation 2, let us put

20. )exp()
2
(2
2
))(exp()2(
4
2
3
2
2
1
2
3
2
3
3
kT
EE
h
kTm
p
kT
kT
EE
m
h
p
Fvh
Fv
h
-
=
-
=
*
*
p
pp
The above equation represents
Number of holes per unit volume of the
Material

21. Intrinsic Carrier Concentration
In intrinsic Semiconductors n = p
Hence n = p = n
i is called intrinsic Carrier Concentration
)
2
exp()()
2
(2
)
2
exp()()
2
(2
)}exp()
2
(2)}{exp()
2
(2{
4
3
2
3
2
4
3
2
3
2
2
3
2
2
3
2
2
kT
E
mm
h
kT
n
kT
EE
mm
h
kT
n
kT
EE
h
kTm
kT
EE
h
kTm
n
npn
npn
g
hei
cv
hei
FvhcFe
i
i
i
-
=
-
=
--
=
=
=
**
**
**
p
p
pp

22. Fermi level in intrinsic Semiconductors
sidesboth on logarithms taking
)exp()()
2
exp(
)exp()
2
()exp()
2
(
)exp()
2
(2)exp()
2
(2
pn torssemiconduc intrinsicIn
2
3
2
3
2
2
3
2
2
3
2
2
3
2
kT
EE
m
m
kT
E
kT
EE
h
kTm
kT
EE
h
kTm
kT
EE
h
kTm
kT
EE
h
kTm
cv
e
hF
FvhcFe
FvhcFe
+
=
-
=
-
-
=
-
=
*
*
**
**
pp
pp

23. E
E
f
E
v
Valence band
E
c
Conduction band
E
c
Electron
energy
Temperature
**
ehmm=

24. Thus the Fermi energy level E
F
is located in the
middle of the forbidden band.
)
2
(
that know tor wesemiconduc intrinsicIn
)
2
()log(
4
3
)()log(
2
32
cv
F
he
cv
e
h
F
cv
e
hF
EE
E
mm
EE
m
mkT
E
kT
EE
m
m
kT
E
+
=
=
+
+=
+
+=
**
*
*
*
*

25. Extrinsic Semiconductors
•The Extrinsic Semiconductors are those in which
impurities of large quantity are present. Usually,
the impurities can be either 3
rd
group elements or
5
th
group elements.
•Based on the impurities present in the Extrinsic
Semiconductors, they are classified into two
categories.
1. N-type semiconductors
2. P-type semiconductors

26. When any pentavalent element such as
Phosphorous,
Arsenic or Antimony is added to the intrinsic
Semiconductor , four electrons are involved in
covalent bonding with four neighboring pure
Semiconductor atoms.
The fifth electron is weakly bound to the
parent atom. And even for lesser thermal
energy it is released Leaving the parent atom
positively ionized.
N - type
Semiconductors

27. N-type
Semiconductor
Si
Si
Si
P
Si
Free electron
Impure atom
(Donor)

28. The Intrinsic Semiconductors doped with
pentavalent impurities are called N-type
Semiconductors.

The energy level of fifth electron is called donor
level.

The donor level is close to the bottom of the
conduction band most of the donor level
electrons are excited in to the conduction band at
room temperature and become the Majority
charge carriers.
Hence in N-type Semiconductors electrons are
Majority carriers and holes are Minority carriers.

29. E
E
d
E
v
Valence band
E
c
Conduction band
E
c
Electron
energy
Distance
Donor levels
E
g

30. Carrier Concentration in N-type
Semiconductor
•Consider N
d
is the donor Concentration i.e.,
the number of donor atoms per unit volume of
the material and E
d
is the donor energy level.
•At very low temperatures all donor levels are
filled with electrons.
•With increase of temperature more and more
donor atoms get ionized and the density of
electrons in the conduction band increases.

31. )exp()
2
(2
2
3
2
kT
EE
h
kTm
n
cFe
-
=
*
p
The density of Ionized donors is
given by
)exp(
)}(1{)(
kT
EE
N
EFdEEZ
Fd
d
d
-
=
-=
At very low temperatures, the Number of
electrons in the conduction band must be equal
to the Number of ionized donors.
)exp()exp()
2
(2
2
3
2
kT
EE
N
kT
EE
h
kTm
Fd
d
cFe -
=
-
*
p
Density of electrons in Conduction band
is given by

32. Taking logarithm and rearranging we get
2
)(
0.,
)
2
(2
log
22
)(
)
2
(2
log)(2
)
2
(2loglog)()(
2
3
2
2
3
2
2
3
2
cd
F
e
dcd
F
e
d
cdF
e
d
FdcF
EE
E
kat
h
kTm
NkTEE
E
h
kTm
N
kTEEE
h
kTm
N
kT
EE
kT
EE
+
=
+
+
=
=+-
-=
-
-
-
*
*
*
p
p
p
At 0k Fermi level lies exactly at the middle of the donor level
and the bottom of the Conduction band

33. Density of electrons in the Conduction band
kT
EE
h
kTm
N
kT
EE
h
kTm
N
kT
EE
kT
EE
kT
E
h
kTm
N
kT
EE
kT
EE
kT
E
h
kTm
NkTEE
kT
EE
kT
EE
h
kTm
n
cd
e
dcF
e
dcdcF
c
e
dcdcF
c
e
dcd
cF
cFe
2
)(
exp
])
2
(2[
)(
)exp(
}
])
2
(2[
)(
log
2
)(
exp{)exp(
}
])
2
(2[
)(
log
2
)(
exp{)exp(
}
}
)
2
(2
log
22
)(
{
exp{)exp(
)exp()
2
(2
2
1
2
1
2
1
2
3
2
2
1
2
3
2
2
1
2
3
2
2
1
2
3
2
2
3
2
-
=
-
+
-
=
-
-+
+
=
-
-+
+
=
-
-
=
*
*
*
*
*
p
p
p
p
p

34. kT
EE
h
kTm
Nn
kT
EE
h
kTm
N
h
kTm
n
kT
EE
h
kTm
n
cde
d
cd
e
de
cFe
2
)(
exp)
2
()2(
}
2
)(
exp
])
2
(2[
)(
{)
2
(2
)exp()
2
(2
4
3
2
2
1
2
3
2
2
1
2
3
2
2
3
2
2
1
-
=
-
=
-
=
*
*
*
*
p
p
p
p
Thus we find that the density of electrons in the
conduction band is proportional to the square
root of the donor concentration at moderately low
temperatures.

35. Variation of Fermi level with temperature
To start with ,with increase of temperature E
f
increases slightly.
As the temperature is increased more and more
donor atoms are ionized.
Further increase in temperature results in
generation of
Electron - hole pairs due to breading of covalent
bonds and the material tends to behave in
intrinsic manner.
The Fermi level gradually moves towards the
intrinsic Fermi level E
i.

36. P - type Semiconductors
•When a trivalent elements such as Al, Ga or Indium
have three electrons, added to the Intrinsic
Semiconductor all the three electrons of these are
involved in Covalent bonding with the three neighboring
Si atoms.
•These like compound accepts one extra electron, the
energy level of this impurity atom is called Acceptor
level and this acceptor level lies just above the valence
band.
These type of trivalent impurities are called acceptor
impurities and the semiconductors doped with the
acceptor impurities are called P-type Semiconductors.

37. Si
Si
Si
In
Si
Hole
Co-Valent
bonds
Impure atom
(acceptor)

38. E
E
a
E
v
Valence band
E
c
Conduction band
E
c
Electron
energy
distance
Acceptor levels
E
g

39. •Even at relatively low temperatures,
these acceptor atoms get ionized taking
electrons from valence band and thus
giving rise to holes in valence band for
conduction.

•Due to ionization of acceptor atoms only
holes and no electrons are created.
•Thus holes are more in number than
electrons and hence holes are majority
carriers and electros are minority
carriers in P-type semiconductors.

40. Then current
density
Then conductivity
)1.........(
E
nev
E
J
EJ
d
=
=
=
s
s
s
)2........(Ev
E
v
nd
d
m
m
=
=As we know that mobility of
electrons.
Drift Current
The moment of electron in the presence of
electric field.

41. Substitute the drift velocity value in
equation 1
EnedriftJ
ne
nn
n
m
ms
=
=
)(

42. In case of semiconductor, the drift current density
due to holes is given by
eEpdriftJ
pP
m=)(
Then the total drift current density
)()()( driftJdriftJdriftJ
pn
+=

43. eEpeEn
pn
mm+=
pn
pn
epen
E
driftJ
drift
pneEdriftJ
mms
mm
+==
+=
)(
)(
)()(
For an intrinsic Semiconductor, n = p = n
i
, then
)()(
pnii
endrift mms +=

44. Diffusion:
Due to non-uniform carrier concentration in a
semiconductor, the charge carriers moves from a
region of higher concentration to a region of
lower concentration. This process is known as
diffusion of charge carriers.

45. Diffusion of charge carriers
Drifting of
charge
carriers
x
Diffusion of charge carriers in a Semiconductor

46. Let Δn be the excess of electron concentration.
Then according to Fick’s law, the rate of
diffusion of electrons
x
n
D
x
n
n
¶
D¶
-=
¶
D¶-
µ
)(
)(

47. )(
)]([
n
x
eD
n
x
De
n
n
D
¶
¶
=
D
¶
¶
--=
The diffusion current density due to holes
)]([)( p
x
DediffusionJ
pP
D
¶
¶
-+=
)(p
x
eD
pD
¶
¶
-=
Where D
n
is the diffusion of electrons, the
diffusion current density due to electrons is
given by J
n(diffusion)

48. The total current density due to electrons is
the sum of the current densities due to drift
and diffusion of electrons
)()( diffusionJdriftJJ
nnn +=
)(
)(
p
x
eDEpeJ
Similarly
n
x
eDEneJ
ppp
nnn
D
¶
¶
-=
D
¶
¶
+=
m
m

49. Direct band gap and indirect band gap
Semiconductors
•We known that the energy spectrum of an electron
moving in the presence of periodic potential field is
divided into allowed and forbidden zones.
•In Crystals the inter atomic distances and the internal
potential energy distribution vary with direction of the
crystal.
• Hence the E-k relationship and hence energy band
formation depends on the orientation of the electron
wave vector to the Crystallographic axes.
•In few crystals like gallium arsenide, the maximum of the
valence band occurs at the same value of k as the
minimum of the conduction band as shown in below, this
is called direct band gap semiconductor.

50. Valence band
Conduction
band
gE
k
E
k
E
g
E
Valence
band
Conduction
band

51. •In few semiconductors like Silicon the
maximum of the valence band does not always
occur at the same k value as the Minimum of
the conduction band as shown in figure. This
we call indirect band gap semiconductor.
•In direct band gap semiconductors the
direction of motion of an electron during a
transition across the energy gap remains
unchanged.
•Hence the efficiency of transition of charge
carriers across the band gap is more in direct
band gap than in indirect band gap
semiconductors.

52. Hall Effect
When a Magnetic field is applied perpendicular to a
current Carrying Conductor or Semiconductor, Voltage is
developed across the specimen in a direction perpendicular
to both the current and the Magnetic field.

This phenomenon is called the Hall effect and voltage so
developed is called the Hall voltage.

Let us consider, a thin rectangular slab carrying Current in
the X-direction. If we place it in a Magnetic field B which is in
the y-direction.
Potential difference Vpq will develop between the faces p and
q which are perpendicular to the z-direction.

53. i
B
X
Y
Z
V
H
+
-
_
_
_
_
_
_
_
_
_
__
_
_
_
_
_
_ _ P
Q
N – type Semiconductor

54. Magnetic deflecting force
citydrift velo is vWhere
)(
)(
d
BvE
qEBvq
dH
Hd
´=
=´
Hall eclectic deflecting force
H
qEF=
When an equilibrium is reached, the Magnetic deflecting
force on the charge carriers are balanced by the electric
forces due to electric Field.
)(BvqF
d´=

55. ne
J
v
d=
The relation between current density and drift velocity is
Where n is the number of charge carriers per unit volume.
BJ
E
ne
tcoefficienHallR
BJRE
BJ
ne
E
B
ne
J
E
BvE
H
H
HH
H
H
dH
´
Þ=
´=
´=
´=
´=
1
),.(
)(
)
1
(
)(
)(

56. If V
H
be the Hall Voltage in equilibrium , the Hall Electric field.
IB
LV
R
Bd
A
I
RV
A
I
J
JBdRV
d
V
JB
R
JB
E
R
d
V
E
H
H
HH
HH
H
H
H
H
H
H
=
=
=
=
´=
=
=
sample, theof thickness theis L If
)(
density current area sectional cross isA If
1
slab. theof width theis d Where

57. •Since all the three quantities E
H
, J and B are
Measurable, the Hall coefficient R
H and hence
the carrier density can be find out.
•Generally for N-type material since the Hall field
is developed in negative direction compared to
the field developed for a P-type material,
negative sign is used while denoting hall
coefficient R
H .