Semiconductor
Physics
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 +4S/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.
• 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 Sp3 hybridanisation.
• These Sp3 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
Semiconductors are mainly two types
1. Intrinsic (Pure) Semiconductors
2. Extrinsic (Impure) Semiconductors
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 (4th group elements) are
the best examples of intrinsic semiconductors and
they possess diamond cubic crystalline structure.
Intrinsic Semiconductor
Valence Cell
Covalent bonds
Si
Si
Si
Si
Si
Conduction band
Ec
KE of
Electron
= E - Ec
Ec
E
Electron
energy
Ef
Fermi energy level
Ev
Valence band
Distance
KE of Hole
=
Ev - E
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.
Calculation of Density of Electrons
Let ‘dn’ be the Number of Electrons available between
energy interval ‘E and E+ dE’ in the Conduction band
dn  Z ( E )dE F ( E )
top of the band
n
 z ( E ) F ( E )dE................(1)
Ec
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.
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
4
Z ( E )dE  3 (2m) 2 E 2 dE
h
3
1
4
Z ( E )dE  3 (2me ) 2 E 2 dE
h
3
1
Since the E starts at the bottom of the Conduction band E c
4
Z ( E )dE  3 (2me ) 2 ( E  Ec ) 2 dE
h
3
1
Probability of an Electron occupying an energy state E is
given by
1
F (E) 
E  Ef
1  exp(
)
kT
For all possible t emperat res
u E  EF  kT
1
F (E) 
E  Ef
exp(
)
kT
E  EF
EF  E
F ( E )  exp (
)  exp(
)
kT
kT
Substitute Z(E) and F(E) values in Equation (1)
top of the band
 z ( E ) F ( E )dE
n
Ec

4
E E
n   3 (2me ) 2 ( E  Ec ) 2 exp( F
)dE
h
kT
Ec
3
1
3 
1
4
 2
n  3 (2me )  ( E  Ec ) 2 exp(EF  E )dE
h
kT
Ec

4
EF
E
 2
2
n  3 (2me ) exp( )  ( E  Ec ) exp( )dE.....(2)
h
kT Ec
kT
3
1
To solve equation 2, let us put
E  Ec  x
E  Ec  x
dE  dx

3
1
4
E
E
 2
F
2
n  3 (2me ) exp( )  ( E  Ec ) exp(
)dE
h
kT 0
kT

3
1
E x
4
E
n  3 (2me ) 2 exp( F )  ( x) 2 exp ( c
)dx
h
kT 0
kT

3
1
E

E
4
x
 2
F
c
2
n  3 (2me ) exp(
)  ( x) exp (
)dx.....(3)
h
kT
kT
0

1
2
3
x

2
we know that  ( x) exp(
)dE  (kT )
kT
2
0
1
2
substitute in equation (3)
1
2
3
3
E

E
4

 2
F
c
2
n  3 (2me ) exp(
) {( kT )
}
h
kT
2
2me kT 32
EF  Ec
n2 (
) exp(
)
2
h
kT
The above equation represents
Number of electrons per unit volume of the Material
Calculation of density of holes
Let ‘dp’ be the Number of holes or Vacancies in the
energy interval ‘E and E + dE’ in the valence band
dp  Z ( E )dE {1  F ( E )}
Ev
p
 z ( E ){1  F ( E )}dE................(1)
bottom of the 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.
Density of holes in the Valence band is
4

Z ( E )dE  3 (2mh ) E dE
h
3
2
1
2
Since Ev is the energy of the top of the valence band
4
Z ( E )dE  3 (2m ) ( Ev  E ) dE
h
3
 2
h
1
2
Probability of an Electron occupying an energy state E is
given by
1
1  F (E)  1  {
}
E  Ef
1  exp(
)
kT
E  E f 1
1  F ( E )  1  {1  exp(
)}
kT
neglect higherorder term s in above exp ansion
for higherT values
E  Ef
1  F ( E )  exp(
)
kT
Substitute Z(E) and 1 - F(E) values in Equation (1)
Ev
 z( E ){1  F ( E )}dE
p
bottom of the band
p
Ev


4
E  EF
(2m ) ( Ev  E ) exp(
)dE
3
h
kT
3
 2
h
1
2
4
 EF
E
p  3 (2m ) exp(
)  ( Ev  E ) exp( )dE....(2)
h
kT 
kT
3
 2
h
Ev
1
2
To solve equation 2, let us put
Ev  E  x
E  Ev  x
dE  dx
4
 EF
E
p  3 (2m ) exp(
)  ( Ev  E ) exp( )dE
h
kT 
kT
3
 2
h
Ev
1
2
Ev  x
4
 EF
p  3 (2m ) exp(
)  ( x) exp(
)(dx)
h
kT 
kT
3
 2
h
0
1
2

Ev  E F
4
x
p  3 (2m ) exp(
)  ( x) exp( )dx
h
kT
kT
0
3
 2
h
1
2
1
2
Ev  E F
4

p  3 (2m ) exp(
)(kT )
h
kT
2
3
 2
h
3
2
2m kT
Ev  EF
p  2(
) exp(
)
h
kT

h
2
3
2
The above equation represents
Number of holes per unit volume of the Material
Intrinsic Carrier Concentration
In intrinsic Semiconductors n = p
Hence n = p = n i is called intrinsic Carrier Concentration
ni2  np
ni  np
Ev  E F
2mhkT 32
E F  Ec
2mekT 32
)}
) exp(
)}{2(
) exp(
ni  {2 (
2
2
kT
h
kT
h
Ev  Ec
2kT 32   34
)
ni  2( 2 ) (me mh ) exp(
2kT
h
 Eg
2kT 32   34
)
ni  2( 2 ) (me mh ) exp(
2kT
h
Fermi level in intrinsic Semiconductors
In int rinsicsemiconduct orsn  p
2me kT 32
E F  Ec
2mh kT 32
Ev  E F
2(
) exp(
)  2(
) exp(
)
2
2
h
kT
h
kT
2me kT 32
E F  Ec
2mh kT 32
Ev  E F
(
) exp(
)(
) exp(
)
2
2
h
kT
h
kT
mh 32
Ev  Ec
2 EF
exp(
)  (  ) exp(
)
kT
me
kT
t akinglogarit hmson bot h sides
Conduction band
Ec
Ec
E
Electron
energy
mh*  me*
Ef
Ev
Valence band
Temperature
mh
Ev  Ec
2 EF 3
 log(  )  (
)
kT
2
me
kT
mh 32
E  Ec
3kT
EF 
log(  )  ( v
)
4
me
2
In int rinsicsemiconduct or weknow t hatme  mh
Ev  Ec
EF  (
)
2
Thus the Fermi energy level EF is located in the
middle of the forbidden band.
Extrinsic Semiconductors
• The Extrinsic Semiconductors are those in which
impurities of large quantity are present. Usually,
the impurities can be either 3rd group elements or
5th 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
N - type Semiconductors
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 Semiconductor
Free electron
Si
Si
P
Si
Si
Impure atom
(Donor)
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.
Conduction band
Ec
Ec
E
Ed
Donor levels
Electron
energy
Ev
Valence band
Distance
Eg
Carrier Concentration in N-type Semiconductor
• Consider Nd is the donor Concentration i.e., the number
of donor atoms per unit volume of the material and Ed 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.
Density of electrons in conduction band is given by
2mekT 32
EF  Ec
n  2(
) exp(
)
2
h
kT
The density of Ionized donors is given by
Ed  E F
N d {1  F ( Ed )}  N d exp(
)
kT
At very low temperatures, the Number of electrons in the
conduction band must be equal to the Number of ionized
donors.
2mekT 32
EF  Ec
Ed  EF
2(
) exp(
)  N d exp(
)
2
h
kT
kT
Taking logarithm and rearranging we get
E F  Ec
Ed  E F
2me kT 32
(
)(
)  log N d  log 2(
)
2
kT
kT
h
Nd
2 E F  ( Ed  Ec )  kT log
3

2me kT 2
2(
)
2
h
( Ed  Ec ) kT
Nd
EF 

log
3
2
2
2me kT 2
2(
)
2
h
at.,0k
EF 
( Ed  Ec )
2
At 0k Fermi level lies exactly at the middle of the donor level
and the bottom of the Conduction band
Density of electrons in the conduction band
2me kT 32
E F  Ec
n  2(
)
exp(
)
2
h
kT
( E  Ec ) kT
Nd
{ d

log
}  Ec
3

2
2
2me kT 2
2(
)
2
E F  Ec
h
exp(
)  exp{
}
kT
kT
E  Ec
( E  Ec )
exp( F
)  exp{ d
 log
kT
2kT
E  Ec
( E  Ec )
exp( F
)  exp{ d
 log
kT
2kT
E  Ec
exp( F
)
kT
1
2
1
2
(Nd )
Ec

}
3 1

kT
2me kT 2 2
[ 2(
)
]
h2
1
2
(Nd )
}
3 1

2me kT 2 2
[ 2(
) ]
h2
(Nd )
( Ed  Ec )
exp
3 1
2kT
2me kT 2 2
[ 2(
)
]
h2
2me kT 32
E F  Ec
n  2(
) exp(
)
2
h
kT
1
2
2me kT 32
(Nd )
( E d  Ec )
n  2(
) {
exp
}
3 1
2

h
2kT
2me kT 2 2
[ 2(
) ]
2
h
1
3

2

m
kT
( Ed  Ec )
e
2
4
n  2( N d ) (
) exp
2
h
2kT
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.
Variation of Fermi level with temperature
To start with ,with increase of temperature Ef 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 Ei.
P-type semiconductors
• When a trivalent elements such as Al, Ga or Indium have
three electrons in their outer most orbits , added to the
intrinsic semiconductor all the three electrons of Indium are
engaged in covalent bonding with the three neighboring Si
atoms.
• Indium needs one more electron to complete its bond. this
electron maybe supplied by Silicon , there by creating a
vacant electron site or hole on the semiconductor atom.
• Indium 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 the acceptor
impurities are called P-type semiconductors.
Hole
Co-Valent
bonds
Si
Si
In
Si
Impure atom
(acceptor)
Si
Conduction band
Ec
Ec
E
Eg
Electron
energy
Acceptor levels
Ev
Valence band
temperature
Ea
• 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.
• Equation of continuity:
• As we have seen already, when a bar of n-type
germanium is illuminated on its one face, excess charge
carriers are generated at the exposed surface.
• These charge carriers diffuse through out the material.
Hence the carrier concentration in the body of the
sample is a function of both time and distance.
• Let us now derive the differential equation which governs
this fundamental relationship.
• Let us consider the infinitesimal volume element of area
A and length dx as shown in figure.
• If tp is the mean lifetime of the holes, the holes lost
per sec per unit volume by recombination is p/tp .
• The rate of loss of charge within the volume under
consideration
p
 eAdx
tp
If g is the thermal rte of generation of hole-electron
pairs per unit volume, rate of increase of charge wthin
the volume under consideration
 eAdxg
• If i is the current entering
the volume at x and i + di
the current leaving the
volume at x + dx, then
decrease of charge per
second from the volume
under consideration = di
• Because of the above
stated three effects the
hole density changes with
time.
• Increase in the number of
charges per second
within the volume
Increase = generation - loss
eAdx
dp
p
 eAdxg eAdx  dI
dt p
tp
 eAdx
dp
dtp
Since the hole current is the sum of the diffusion current and the drift current
I   AeD p
dp
 Ape  h E
dx
Where E is the electric field intensity within the volume. when no external
field is applied, under thermal equilibrium condition, the hole density
attains a constant value p0.
dp
under these condit ionsdi  0 and
0
dt
p0
g
tp
thisequationindicatesthat therateof generationof
holesis equal to therateof loss due to recombination
under equilibrium condit ions.
com bain,.eq s ...3,4 & 5.
( p  p0 )
dp
2 p
d ( pE)

 Dp 2  h
dt
tp
x
dx
This is called equation of conservation of charge or the continuity equation.
if p is a funct ionof bot h tand x,
part ialderivat ives should be used.
( p  p0 )
p
2 p
 ( pE)

 Dp


h
t
tp
x 2
x
if we are considering holesin t he n - t ypemat erial
pn
( p n  p0 n )
 2 pn
 ( pn E )

 Dp


h
t
tp
x 2
x
if we are considering elect ronsin t he p - t ypemat erial
n p
t

( n p  n0 p )
te
 Dn
 2n p
x 2
 e
 (n p E )
x
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.
E
Conduction
band
E
Conduction
band
Eg
Eg
k
Valence band
k
Valence
band
• 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.
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 (i) in the xdirection.
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.
Z
+
VH
-
+
+
+ P
+
+
Y
+ + + + + + + +
+
+ ++ + + + + Q + + +
X
i
B
P – type semiconductor
Z
_ _
VH
_
_ P_
+
_
Y
_
_ _ _
_
X
i
_
_ _Q_ _ __
B
N – type semiconductor
Magnetic deflecting force
F  q(vd  B)
Hall eclectic deflecting force
F  qEH
When an equilibrium is reached, the magnetic deflecting force on
the charge carriers are balanced by the electric forces due to
electric Field.
q(vd  B)  qEH
EH  (vd  B)
Wherevd is drift velocity
The relation between current density and drift velocity is
J
vd 
ne
Where n is the number of charge carriers per unit volume.
E H  ( vd  B )
J
EH  (  B)
ne
1
E H  (  JB)
ne
E H  RH  JB
1
EH
RH ( Hall,.coefficient ) 

ne JB
If VH be the Hall voltage in equilibrium ,the Hall electric field.
VH
d
Whered is the width of the slab.
EH 
EH
JB
1 VH
RH 

JB d
If t is the thicknessof thesample,
T henits cross sect ionis dt and currentdensity
I
J
dt
VH  RH JBd
RH 
I
VH  RH ( ) B
t
V t
RH  H
IB
• Since all the three quantities EH , J and B
are measurable, the Hall coefficient RH and
hence the carrier density can be found out.
• Generally for N-type material since the Hall
field is developed in negative direction
compared to the field developed for a Ptype material, negative sign is used while
denoting hall coefficient RH.