UNIT-IV

Formation of Energy bands in solids.

In an isolated atom the electrons are tightly bound and have discrete and sharp energy levels.
When two identical atoms brought together then outermost orbit of these electrons overlap and interact.
Therefore the energy of a particular electron is not unique rather two energies which are very close to
each other. Similarly when more atoms are close to each other, then a particular electron is having a
number of energies which are very close to each other. Hence instead of a sharp discrete energy this
electron is having a band of energy. Generally this happens in case of a solid which consists of a
number of atoms which are very close to each other.
The band corresponding to the outermost orbit is called conduction band. The next inner band
is known as valence band. The gap between two allowed bands is known as forbidden energy gap or
band energy.

Classification of solids based on Energy Bands

On the basis of energy bands solids are classified into Conductors,, Semiconductors, and Insulators.
Conductor

In case of conductor the conduction and valence bands overlap each other. Hence, electrons can easily
move from the valence to the conduction band even at room temperature and without application of
external energy. This makes availability of a large number of electrons for conduction. Also, the
resistivity of such solids is low or the conductivity is high.
Insulators

In case of insulator the energy gap (Eg) is very large

(>3 eV). Due to this large gap, electrons cannot be
excited to move from the valence to the conduction
band by thermal excitation. Hence, there are no free
electrons in the conduction band and no
conductivity.

Semiconductors:
In semiconductor the energy gap is small (< 3
eV). Since the gap is small, some electrons
acquire enough energy even at room temperature
and enter the conduction band. These electrons
can move in the conduction band increasing the
conductivity of the solid. The resistivity of
semiconductors is lower than that of insulators but
higher than that of conductors.
Fermi Level
According to fermi Dirac statistics the probability of an electron F(E) occupying an energy level E is
given by
1
F(E) = E  EF , Where EF is the fermi energy and F(E) is called fermi function.
1 e kT

It is observed that at 00K all the energy state below the Fermi level are fully occupied and all the energy
levels above the fermi level are empty.
So Fermi level can be defined as the level of maximum energy of the filled state at 00K
If E = EF then F(E) = ½
So the fermi level is defined as the energy level at which the probability of electron occupation is ½ at
any other temperature above 00K
Fermi Energy
Fermi energy is the energy of that state where the probability of electron occupation is ½ at any other
temperature above 00K. It can also be defined as the maximum energy of filled state at 00K
SEMICONDUCTORS

Those substances whose electrical conductivity lies between conductor and insulator are known as
semiconductors.e.g. germanium and silicon are important semiconductors which are vastly used in
the manufacturing of semiconductor devices, GaAs (gallium arsenide) and InSb (indium
antimonite), etc.

Properties:

 Semiconductors are tetravalent atoms.

 Conductivity increases with increasing temperature.
 Conductivity is due to holes and electrons.
 Forbidden energy gap is less than insulators.
 In a solid the bond between the atoms is covalent bond.

Types of semiconductors:

There are two types of semiconductors

1. Intrinsic semiconductors,
2. Extrinsic semiconductors.

Intrinsic semiconductors:

The semiconductor in purest form is known as intrinsic semiconductors.

Example: Pure germanium & silicon crystals.

Extrinsic semiconductors:

When a pure semiconductor is doped with suitable doping impurity is known as extrinsic
semiconductors.

Example: doped germanium and silicon.

Depending on the type of impurity added, the extrinsic semiconductors can be divided in to
two classes:
1. N-type Semiconductors
2. P-type Semiconductors
N-Type Semiconductor: When a group V dopants (pentavalent elements such as antimony, arsenic
or phosphorous) is doped with a intrinsic semiconductor, an extrinsic semiconductor is formed.
These pentavalent elements are called donor elements. A pentavalent atom is having five electron in
their outermost orbit. When this atom is doped to a pure semiconductor like Ge or Si which are
having four outermost orbit electron, then four electrons of the pentavalent atom formed four
covalent bond with four electrons of Ge or Si atom. The fifth electron is free which helps the
conduction of current. If a small amount of pentavalent substance is doped then a number of free
electrons will be available for conduction of current which increases the conductivity of the
semiconductor.
 In an n-type semiconductor, the majority carrier, or the more abundant charge carrier, is the
electron, and the minority carrier, or the less abundant charge carrier, is the hole.
Explanation
The increase in conductivity of an extrinsic semiconductor can be explained by energy band
diagram shown in figure. When donor impurities are added to an intrinsic semiconductor, allowable
energy levels (donor energy level) are introduced at a very small gap below the conduction band
where the donor free electrons accommodate as illustrated in figure. These donor levels are
essentially a discrete level because the added impurity atoms are far apart in the crystal structure
and hence their interaction is small. In the case of Silicon, the gap of the new discrete allowable
energy level is only 0.05 eV (0.01 eV for germanium) below the conduction band, and therefore at
room temperature almost all of the "fifth" electrons of the donor impurity are raised into the
conduction band and the conductivity of the material increases considerable.

P-Type Semiconductor:
When a group III dopants (trivalent elements such as boron, aluminium or gallium) is doped with a
intrinsic semiconductor, P-type semiconductor is formed. These trivalent elements are called
acceptor elements. A trivalent atom is having three electron in their outermost orbit. When this
atom is doped to a pure semiconductor like Ge or Si which are having four outermost orbit electron,
then three electrons of the trivalent atom formed three covalent bond with three electrons of Ge or
Si atom. The fourth electron of the parent atom did not get any electron from doped atom to form a
covalent bond. This create a hole which is nothing but deficiency of an electron. If a small amount
of trivalent substance is doped then a number of holes will be available for conduction of current
which increases the conductivity of the semiconductor.
 In an P-type semiconductor, the majority carrier, or the more abundant charge carrier, is the
hole, and the minority carrier, or the less abundant charge carrier, is the electron.
Explanation
The increase in conductivity of an P-type semiconductor can be explained by energy band
diagram shown in figure. When acceptor impurities are added to an intrinsic semiconductor,
allowable discrete energy levels (acceptor energy level) are formed just above the valance band, as
shown in figure. Since a very small amount of energy (0.08 eV in case of Silicon and 0.01 eV in
case of Germanium) is required for an electron to leave the valence band and occupy the accepter
energy level, holes are created in the valence band by these electrons.

Difference Between Intrinsic and Extrinsic Semiconductor

Intrinsic Semiconductors Extrinsic Semiconductors
1 It is pure semi-conducting material and no 1. It is prepared by doping a small quantity of
impurity atoms are added to it. impurity atoms to the pure semi-conducting
material.
2 Examples: crystalline forms of pure silicon 2. Examples: silicon “Si” and germanium
and germanium. “Ge” crystals with impurity atoms of As, Sb,
P etc. or In B, Aℓ etc.
3 The number of free electrons in the 3. The number of free electrons and holes is
conduction band and the no. of holes in never equal. There is excess of electrons in n-
valence band is exactly equal and very small type semi-conductors and excess of holes in
indeed. p-type semi-conductors.
4 Its electrical conductivity is low. 4. Its electrical conductivity is high.
5 Its electrical conductivity is a function of 5. Its electrical conductivity depends upon
temperature alone. the temperature as well as on the quantity of
impurity atoms doped the structure.
Intrinsic carrier concentration
Expression for electron density

Let dne is the electron density in the conduction band in the energy interval E and E + dE
then,

dne  Z E F E dE ------------- (1)


Thus ne =  Z ( E ) f ( E )dE
Ec

Where F(E) is the probability of electrons occupying the energy state E.

and F E   1  expE  EF  / KT 
1

In conduction band E  EF  KT

So F E   e   E  EF  / KT  e  EF  E  / KT
Z (E) dE is the energy state density in the energy interval E and E + dE.
3 1
4
and is given by Z  E  dE 
h3
 e
2 m 2 E 2 dE

Since electron moves in a perpendicular potential hence, m is replaced by me* = effective mass and
since EC bottom level of conduction band and E be any arbitrary level in the conduction band
therefore E is replaced by E - EC
3 1
4
So, Z E dE 
h3

2me

 E  E  dE
2
C
2

Hence

3 1
4
dne  3 2me
h

  E  E  e 
2
C
2 E F  E  / KT
dE

 3 1
4
So n e   3 2 m e

  E  E 
2
C
2 e  E F  E  / KT dE
Ec h

3 
4 1
 3 2m e
h


e
2 E F / KT
 E  E C  2 e dE
 E / KT

Ec

Let E  EC  x  E  EC  x and dE  dx

When E  EC ; x  0

When E  ; x  

Hence
 3  1
4
n e  3 2m e
h

 e
2 E F / KT
x 2
e   EC  x  / KT dx
0

3  1 x
4
 3 2m e
h

e 2  E F  EC  / KT
x 2
e KT
dx
0

3
 1 x 2

Using gamma function, we have  x e dx   KT 
0
2 KT
2

3
4 3 
So, n e  3 2me e
h
 
 2  E F  EC  / KT
 KT  2
2
3

 2m KT   EF  EC  / KT
2
 ne  2 e 2  e
  N C e  EF  EC  / KT
 h 
3
*
 2m  KT  2

Where N C  2  e 2 
 h 
This is the expression for the electron density in conduction band.

Expression for hole density

If nh be the hole density then in a similar procedure by substitution of E = EV - E in the expression

of Z(E) and integrating from -  to Ev we can derive
3
*
 2m  KT   E E 2
 / KT
nh  2  h 2  e
V F
 NV e  E  E V F  / KT

 h 
3
*
 2mh KT  2
Where NV  2  
 h2 
Where mh* is the effective mass of the hole.

Expression for Intrinsic Carrier concentration

In case of intrinsic semiconductor the electron concentration and hole concentration are same.

So, ne = np = ni = intrinsic carrier concentration

3 3
* *
 2m  KT  2  E  E  / KT  2me  KT  2  E  EC  / KT
Now ni2  ne nh = 2 h 2  e
V F
x 2 2  e
F

 h   h 
 3
 2 KT  3
 E  E  / KT
  me mh  e
* * 2
= 4 2
V C

 h 
3
 2 KT  3 E g

  me mh  e
* * 2
= 4 2
KT
(Since Eg = Ec – Ev)
 h 
3

 2 KT  3 E 2 g

  me mh  e
* * 4
Hence ni = 2  2
2 KT

 h 
1  Eg

or n i   N C NV  e 2 kT 2

Variation of Intrinsic carrier concentration with temperature

The equation of intrinsic carrier concentration can be re written as:
3 3

 2 KT  2 * * 34 2KTE g
 2 K  2 * * 34 32 2KTE g

ni = 2  2   me mh  e = 2 2   me mh  T e
 h   h 
The above relation indicates that the carrier concentration varies with temperature. It may be
approximated to
3 2500 E g
21.7
ni  10 T  10 2 T

The following important points may be inferred from the above relation:
1) The intrinsic concentration is independent of the Fermi level
2) The intrinsic concentration depends exponentially on the band gap value.
3) The intrinsic concentration also strongly depends on the temperature
4) The factor 2 in the exponent indicates that two charge carriers are created when one covalent
bond is broken.
Fermi Level

In case of intrinsic semiconductor the electron concentration and hole concentration are same.

i,e , ne = np
3 3
* *
 2m  KT   E 2
 EC  / KT  2m  KT   E  E 2
 / KT
2 e 2  e 2 h 2  e
F V F
So, =
 h   h 
3 EF  EC 3 EV  E F

 m  *
e
2
e KT
= m  *
h
2
e KT

3
*
2 EF
 mh  E KT E 2 V C

 e KT
 *  e
 me 
Taking logarithms on both sides, we have
2 EF 3 mh*  EV  EC   E  EC  3kT mh
*

 ln   So, EF   V  ln *
KT 2 me*  KT   2  4 me

EV  EC EC  EV
Now   EV
2 2
EC  EV 3KT mh* Eg 3KT mh*
So, EF   EV  ln * =  EV  ln *
2 4 me 2 4 me
If we denote the top of the Valance band Ev as zero level then EV = 0

Eg 3kT mh*
So EF   ln *
2 4 me

EV  EC
If we assume me*  mh* then EF 
2
If we denote the top of the Valance band Ev as zero level then EV = 0

EC Eg
Then EF = 
2 2
Fermi Level and its variation with Temperature in an Intrinsic semiconductor

Fermi level is located half way between the valence and conduction band if me*  mh* .
If me*  mh* fermi level is just above the middle.
If me*  mh* fermi level is just below the middle.
With an increase in temperature the Fermi level gets displaced up word slightly towards the
* * * *
conduction band if me  mh and down word towards the valence band if me  mh .
* *
In most of the materials the shift of fermi level on account of me  mh is insignificant. Therefore
the fermi level on an intrinsic semiconductor may be considered as independent of temperature and
at middle of the band gap.

Fermi level in Intrinsic semiconductor Variation of Fermi level with temperature

Electrical Conductivity of intrinsic semoconductor

Electrical conductivity = σ = σe + σh = (neeμe + nheμh)

Where μe and μh are the mobility of electron and hole respectively, e is the electronic charge

Since in case of intrinsic semiconductor ne = nh == ni

So, σ =nie(μe + μh)

3

 2 KT  2 3 E g

  me mh  e
* * 4
But ni = 2  2
2 KT

 h 
3

 2 KT  2 * * 43 2KTE g

So σ = 2e  e   h   2   me mh  e
 h 
 Eg

 σ=A e 2 KT ------------------------(1)
3

 2 KT  2 * * 43
 Where A is a constant and A = 2e  e   h   2   me mh 
 h 
(in this case μe and μh both are proportional to
3

T2
Now taking logarithm of both the sides of equation (1)

We have, ln σ = lnA – Eg/ 2KT

If we draw a graph by taking 1/T along X-axis and lnσ

along Y- axis then the nature of the graph will be like
the figure which shows that conductivity increases
with increase in temperature.

Carrier concentration in N type semiconductor:

Density of electrons in a conduction band is

3

 2me KT   E F  E C  KT
2
ne  2  e

 h2 
If Nd is the density of donor atom and Ed is the donor energy level. At the low temperature donor
levels are filled with the electron. When temperature increases donor atoms are ionized and density
in conduction band increases.

Now density of ionized donor atom is

 
 1 
N d 1  F E d   N d 1   Ed  E F  
 
 1  e KT 
In this case we assume that the Fermi level lies just above the donor level such that EF  Ed  KT

1
Ed  EF  Ed  E F 
 
Hence 1  e KT   1  e KT
 
 

 Ed  E F 
So, N d 1  F E d   N d e KT

At low temperature concentration of electron in conduction band is equal to the concentration of

ionized atoms in donor level

So, ne  N d 1  F E d 
3
 2m e  KT  2  E F  EC  Ed  EF 
2  e KT
 N d e KT
 h 2 
 

 E  E C  E d  E F   Nd
 exp F  3
 KT  
 2me KT  2
2 
 h 2 
 

2E F EC  E d Nd
   log 3
KT KT 
 2m e KT  2
2 
 h 2 
 

E C  E d KT Nd
 EF   log 3
2 2
 2m e  KT 2
2 
 h2 
 
1
E  Ed N d  2
 EF  C  KT log 1
2 3
   2
  2m e KT  2

2
  
h2 
   

EC  Ed
At 0o K,  EF
2

I.e. at 0o K Fermi level lies exactly at the middle of the donor level and the bottom of the conduction
band.

So,
 1
EC  E d  N d 2
 KT log 1
 EC
2 3
 2
  2me  KT  2
 2   

3   h2  
  
 2me KT 2
ne  2  e

KT

 h2 
3 1

 2me KT
 2

 
2
N d 2 e
E d  EC
2 KT
h2  1
   3
 2

2 2me KT  
2

  h2  
   

3

1  2me KT  E2d KT
4  EC
 ne  2N d  2 

 e

 h2 
This is the expression for the density of electrons in conduction band at low temperature
which is directly proportional to the square root of the donor concentration.

Variation of fermi level

Variation with concentration
(a) N-type semiconductor
When a small pentavalent impurity is added (low concentration) to a pure semiconductor then donor
levels formed just below the conduction band and the fermi level lies in between donor level
(ED)and bottom of conduction band (EC). In this case the donor level is discrete due to large
separation between the atoms and no interaction at all.
When donor concentration is increased (medium doping) then separation between impurity atom
reduces and overlapping of orbits takes place by which donor level form an energy band and the
fermi level is pushed further towards conduction band and decreases forbidden energy gap..
When donor concentration is further increased (heavy doping) donor level broaden and decreases
the energy gap further and the donor level overlap with the conduction band and fermi level further
pushed towards conduction band and lies with in conduction band.
Thus with increase in concentration of donor impurity fermi level shift towards conduction band
and finally moves to conduction band when the donor band overlaps on the conduction band.
 Light doped Medium doped Heavy doped

(b) P-type semiconductor

When a small trivalent impurity is added (low concentration) to a pure semiconductor then acceptor
level formed just above the valance band and the fermi level (EF)lies in between acceptor level
(EA)and top of valence band (EV). In this case the acceptor level is discrete due to large separation
between the atoms and no interaction at all.
When acceptor concentration is increased (medium doping) then separation between impurity atom
reduces and overlapping of orbits takes place by which acceptor level form an energy band and the
fermi level is pushed further towards valence band and decreases forbidden energy gap..
When acceptor concentration is further increased (heavy doping) acceptor level broaden and
decreases the energy gap further and the acceptor level overlap with the valence band and fermi
level further pushed towards valence band and lies with in valence band.
Thus with increase in concentration of acceptor impurity fermi level shift towards valence band and
finally moves to valence band when the acceptor band overlaps on the conduction band.

Light doped Medium doped Heavy doped

Variation with Temperature
(a) N-type semiconductor
At 00K the fermi level EFn lies
between the donor level and the
bottom of the conduction band.
EC  ED
i,e EFn  at T = 0K
2
When temperature increases the
donor level depleted and the fermi
level moves downwards. At the
temperature of complete depletion
of donor levels (Td) the fermi level
coincides with the donor level (ED)
Thus EFn = ED at T = Td
As the temperature goes further
above Td Fermi level shifts down
words linearly and at a temperature Ti the N-type semiconductor losses its extrinsic character and
fermi level approaches the intrinsic value
Eg
Thus EFn = EFi = at T  Ti
2
(a) P-type semiconductor
At 00K the fermi level EFn lies
between the acceptor level and the
top of the valence band.
EV  E A
i,e EFn  at T = 0K
2
When temperature increases the
acceptor level is gradually filled
and the fermi level moves upwards.
At the temperature of saturation
(Ts) the fermi level coincides with
the acceptor level level (EA)
Thus EFn = EA at T = Ts
As the temperature goes further
above Ts Fermi level shifts up
words linearly and at a temperature
Ti the N-type semiconductor losses
its extrinsic character and fermi
level approaches the intrinsic value
Eg
Thus EFn = EFi = at T  Ti
2
Hall Effect :- If a metal or a semiconductor carrying a current I is subjected to a transverse
magnetic field B, a potential difference VH is produced in a
direction normal to both the magnetic field and current
directions. This is called Hall effect.

Explanation

Any plane perpendicular to the current flow direction is an

equipotential surface. Therefore, the potential difference
between front and rear faces of such surfaces is zero. Now if
a magnetic field is applied normal to such a surface and
hence to the direction of current flow in it , then a transverse potential difference is produced
between the two faces.

Derivation of Hall Co-efficient

Let us consider a P-type semiconductor slab

ABCDEFGH. Let DH= length of the slab
AD = width ‘w’, (distance between the surface
where Hall voltage is developed)
DC = thickness ‘t’ (distance between the end
along the direction of current)
Area of the side ABCD = A= wt
Let ‘p’ be the hole density
Let a current I passed through the slab along X-
axis. (along the thickness)
Then I = peAVd , Where Vd is the drift velocity
and ‘e’ is the electrical charge associated with hole.

I
Then the current density J= = peVd -------(1)
A
Let a transverse magnetic field B is applied along HD direction. Due to this magnetic field a
Lorentz or magnetic force FL = eVdB will be developed by which the holes will be drifted towards
the surface ABDF. (by Flemings left hand rule magnetic force will act along Z-axis). Also
equivalent negative charges will be drifted towards CDGH. These opposite charges produce an
electric field call hall field EH. Due to this Hall field an electric force (FE = eEH) will experience and
under equilibrium position electric force is equal to Magnetic force. i,e FE =FL. => eEH = eVdB.

VH
If VH is the Hall voltage, then EH = -------------(2)
w
VH
So, =BVd ----------(3)
w
J VH BJ BI BI
From equation (1) Vd = , So = = =
pe w pe peA pewt
BI
So, VH = ------------------(4)
pet
This is the expression for Hall Voltage
Hall Coefficient
Hall coefficient RH is defined as Hall field per unit current density per unit magnetic field
VH
EH
Thus RH   w ----------(5)
JB JB
 BI BJ 1
Using equation (4) RH = = =
petwJB peJB pe
1
Thus RH = ----------(6)
pe
This is expression for Hall coefficient
From equation (5)
VH VH
RH = w = w = VH t
JB IB IB
wt
Vt
Thus RH = H This is another expression for Hall coefficient.
IB
# Hall coefficient is negative for n-type semiconductor (because e is negative) while the same is
positive in the case of p-type semiconductor (because e is positive)
# Unit of RH is m3/C or m3/A-s.
Applications of Hall Effect:
Determination of Carrier Concentration
The expressions for the carrier concentrations of electrons (n) and holes (p) in terms of Hall
coefficient are given by

1 1
n and p respectively
RH e RH e
Determine the Mobility (Hall Mobility)
Mobility expression for the electrons (μn) and the holes (μp), expressed in terms of Hall coefficient
is given by,

μ n =σ n R H and μh = σh R H respectively

Where, σn and σp represent the conductivity due to the electrons and the holes, respectively.
Determination Magnetic Flux Density
Magnetic flux density can be determined by using the formula

VH t
B=
RHI
Determination of drift velocity

VH
Drift velocity can be determined by using the formula Vd 
Bw