1

UNIT-IV

SEMICONDUCTOR & NANOMATERIALS

1. Free electron model

2. Kronig penny model

3. Effective mass

4. Fermi level for intrinsic and extrinsic semiconductor

5. P-N junction

6. Zener diode

7. photo diode

8. solar cell

9. Hall effect

10. elementary idea of nanostructures and nano materials

Free electron model: In order to explain the electrical and

thermal properties of metals in the year 1900, H. A. Lorentz and

Paul Drude propounded the free electron model. They made the

following postulates:

1. The outermost electrons (or the valence electrons) of the

constituent atoms of a metal are most weakly bound with

the atoms. Hence these electrons get separated from their

atoms and move freely inside the entire substance. These

electrons are called the free electrons or the conduction

electrons.

2. There is large no. of free electrons inside a metal and they

behave like the molecules of a gas enclosed in a vessel.

Hence they can also be referred as free electron gas. These

free electrons are responsible for the thermal and electrical

conduction of metal.

2

3. The free electrons in thermal equilibrium obey the Maxwell-

Boltzmann statistics. Each electron has 3 degrees of

freedom for translational motion.

4. According to Maxwell-Boltzmann statistics, mean energy

per electron at an absolute temperature T, is kT2

1, where k

is the Boltzmann’s constant.

5. Inside the metal, the free electrons move randomly with a

high speed and their speed depends on the temperature of

the metal. During motion, when they collide with the

positive ions their speed and direction of motion change

such that the rate of flow of electrons in a particular

direction is zero.

6. When the metal is kept in an external electric field, the free

electrons get accelerated in a direction opposite to the

external electric field but due to collisions with the positive

ions of the metal they begin to move with a constant

velocity which is called drift velocity. It is of the order of

10-4

m/s.

Success: On the basis of above model, the electrical and thermal

conductivity of a metal can be explained successfully. From this

model we find that electrical conductivity of metal decreases

with increase in their temperature and the ratio of thermal to

electrical conductivity of each metal at any temperature is

constant.

Failure: Although this theory explained some of the properties

of the metals yet it was unable to explain following features:

1. It could not explain the variation of electron specific heat

with temperature at low temperature. According to this

model the electron specific heat for metal is 3R which is

3

temperature independent but experimentally it was found

that specific heat of metals is temperature dependent.

2. According to free electron model the magnetic susceptibility

of a paramagnetic substance is inversely proportional to

temperature but experimentally it was found temperature

independent.

3. The mean free path calculated on the basis of free electron

model was ten times less than the experimentally calculated

value.

4. It was unable to explain the behaviour of semiconductors

and insulators.

5. It can not explain the origin of Pauli’s paramegnetism of

metals.

6. Monovalent metals (Cu, Ag) have been found to have

higher electrical conductivity than divalent (Cd, Zn) and

trivalent (Al, In) metals. If the conductivity is proportional

to the electron concentration than monovalent metals should

have lesser electrical conductivity compared to the divalent

and trivalent metals.

Sommerfeld’s Free Electron Theory: Sommerfeld modified

the Drude Lorentz free electron model on the basis of quantum

statistics. He made the following assumptions:

1. According to Sommerfeld, each free electron inside the

metal experiences an electrostatic attractive force due to all

the positive ions and an electrostatic repulsive force due to

the other electrons.

2. The force of repulsion due to mutual interactions of

electrons can be assumed to be negligible and the attractive

field due to positive ions can be considered to be uniform

4

everywhere inside the crystal. Thus each free electron inside

the metal is in an attractive potential field.

3. Since the crystal structure of solid is periodic i.e., in a solid

crystal, each positive ion is at a definite distance from each

other, therefore this potential field inside the metal must

also be periodic. But for convenience Sommerfeld assumed

that this potential inside the metal is constant.

4. Since no electron is emitted from the metal at an ordinary

temperature, therefore it can be assumed that the electron

inside the metal is more stable than outside the metal i.e.,

the potential energy of a stationary electron inside the metal

is less than the potential energy outside the metal.

5. Thus inside the metal electron can be assumed to be inside a

potential well of depth Es. Es is the difference in the energy

of the electron outside and inside the well.

6. The electrons present inside this well has so much energy

that it can move inside the well, but it cannot come out of

the metal surface.

7. Inside the well, all the energy states from zero to some

energy EF are filled up with electrons. The energy EF is

Metal Ø

EF

Es

Potential Well

5

called the Fermi energy. Thus Fermi energy is the

maximum kinetic energy of the electrons inside the metal.

8. The threshold energy or the work function for an electron

inside the well is Fs EE to cross this well.

9. He further specified the energy distribution function for an

electron viz. )/(1

1)(

kTEeEf

.Substituting the value of eα

askTEFe

distribution function becomes kTEE Fe

Ef/)(

1

1)(

.

10. The Fermi function predicts below EF all the energy levels

are completely occupied by the electrons.

Merits and demerits of Sommerfeld’s model: This model

successfully explained several properties of metals such as

electrical conductivity, thermal conductivity, specific heat and

magnetic susceptibility. But this model was unable to distinguish

between behaviour of metals, semiconductors and insulators.

Band Model: In order to remove the drawbacks of the

Sommerfeld’s free electron model Band Model was propounded.

Following features were discussed in this model:

1. According to this model the free electrons inside the metal

moves in the electric field of positive ions and of other free

electrons.

2. Since the crystal structure is periodic the potential energy of

free electrons also changes periodically with distance hence

the motion of the electron inside the metal is in the periodic

potential well.

6

3. The potential energy of a free electron at a distance x in the

potential field of an atom i is given as x

ZeU x

0

2

)(4

and the

graph plotted for it is a hyperbola.

U(x)x

4. Since inside the metal the atoms are arranged in a definite

order, therefore we obtain the influence of the other atoms

also on the potential energy curve and it’s combined effect

can be viewed as:

U(x) x

i j

7

5. All the electrons from the lowest energy to Eb are bound

with their atoms and can vibrate only with very small

amplitude.

6. The electrons with energy between Eb and EF move

anywhere within the metal, where EF is the Fermi energy

level. The work function for which can be given as

Fs EE where ES is the depth of the potential well.

U(x) x

0

Eb

EF

ES

Φ

i j k l m n

a

7. The electron is associated with the entire crystal, and not

only with an atom.

Kronig-Penny Model: To explain the behaviour of electrons in

the periodic potential, Kronig and Penny gave a simple one

dimensional model according to which the potential energy of an

electron can be represented by a periodic array of rectangular

potential well as shown in the fig. below:

8

i j

x

V(x)

V0

-b 0 a a+b 2a+b

Here the potential peaks obtained from the hyperbolic curves

have been assumed to be in form of rectangular peaks. Each

potential well represents the potential near an atom. If the time

period of potential is (a + b), then potential energy is zero in

ax 0 and potential energy is constant (= V0) in 0 xb ,

i.e.

In region 0,0 )( xVax

And in region 0)(,0 VVxb x (constant)

In both these regions, the Schrödinger wave equations for the

wave function ψn associated with nth

energy state of En electron

are,

In region ,0 ax 02

22

2

nn

n Em

dx

d

0)( xV

and in region ,0 xb 0)(2

022

2

nnn VE

m

dx

d

0)( VV x

……….(1)

Here the energy of electron En is very small in comparison to the

potential V0.

9

Assuming that as b tends to zero, V0 becomes infinite, Kronig

and Penny obtained the following condition for the allowed

wave function on solving the above equation:

kaaabmV coscossin)/( 2

0 …….(2) ; /2 nmE

and k is the wave vector.

If 2

0 /bamVP , which measures the area V0b of the potential

barrier, then increase in P means increase in the binding energy

of electron with its potential well.

Substituting the value of P in above eq. the condition for the

allowed wave function is

kaaa

aPcoscos

sin

…………..(3)

Hence for 2

3P the graph between the quantity on the left side

of above eq and a is shown below:

4 αa

a

a

aP

cos

sin

3

2

23

4

+1

-1

0

a b c dd’ a’b’c’

h’ hgg’ e’ f’ e f

p q srp’q’r’s’

The graph depicts the following facts:

10

1. Since the maximum and minimum possible values of the

term a cos are +1 and -1 hence two horizontal lines are

drawn on the Y-axis at y = +1and at y = -1.

2. Solution of the eq.(3) can be given by the intersection points

a, b, c,….and a’, b’, c’…….which means that the solution of

the eq.(3) can be possible only in some specific regions.

3. The ranges π to q, 2π to r,……represents the forbidden

energy gap.

Thus the Kronig –Penny model, we get the following

conclusions:

1. In the energy spectrum of metals there exist several bands

separated by the forbidden energy region. The energy band

completely filled with electrons are called the valence band

and the energy band which is either completely empty or is

partially filled is called the conduction band.

Ele

ctr

on

en

erg

y in

cry

sta

l

Conduction band

Valence band

Forbidden energy gap

2. As the value of αa increases, the width of the allowed

energy bands increases.

3. With increase in the binding energy V0 of electrons or with

increase in the value of P, the width of a particular allowed

energy band decreases and when the binding energy

becomes infinite, the allowed energy becomes very narrow

i.e. the energy spectrum becomes the line spectra. In other

11

words at P = ∞ energy levels become discrete while at P = 0

the energy levels become continuous.

E

E2

E1

0 010.4

4π/PP/4π

pq

r

s

t

u

Forbidden

energy

gap

4. At the wave vector ank / the energy is discontinuous

and these values of k correspond to the boundaries of

Brillouin Zones. For 1n we get the first Brillouin zone.

5. The energy in an energy band is a periodic function of k.

6. The no. of total possible wave functions in an energy band

is equal to the no. of unit cells.

7. The velocity of free electron is zero at the top and bottom of

an energy band and it is maximum at the point of inflexion

of energy band.

8. At kT 0 , the effective no. of electrons in a completely

filled band is zero while the effective no. of electron is

max. in a band filled upto the point of inflexion. At absolute

12

zero the energy level completely filled by the electron is

called the Fermi level

Effective mass:

1. In solid state physics, a particle's effective mass is the mass

it seems to carry in the semi classical model of transport in a

crystal.

2. It can be shown that, under most conditions, electrons and

holes in a crystal respond to electric and magnetic fields

almost as if they were free particles in a vacuum, but with a

different mass.

3. This mass is usually stated in units of the ordinary mass of

an electron me (9.11×10-31

kg).This experimentally

determined electron mass is called the effective mass m*

4. The cause for deviation of the effective mass from the free

electron mass is due to the interactions between the drifting

electrons and the atoms in a solid.

En

Es

EFP P

0

F

a

a

+k1-k1

13

Expression for effective mass: According to wave mechanics,

the velocity of electron corresponding to the wave vector k is

equal to the group velocity of waves representing it i.e.

dk

d , where ω is the angular frequency of the de-broglie

waves.

If E is the energy of electron, then E or

dEd hence,

dk

dE

1 ……….(1)

Let there be only one electron initially in the k state in the first

Brillouin zone. Now if an external electric field ε is applied on

the electron for a very short duration dt, then displacement of

electron in time dt will be dt and force on electron due to the

electric field will be e . Hence the gain in the energy of

electron

dE force x displacement

or dt

dk

dEedtedE

or dtdk

dEedk

dk

dE

or rate of change in wave vector

e

dt

dk ……………(2)

from (1),acceleration of electron dt

dk

dk

Ed

dt

da .

12

2

Substituting the value of dk/dt from (2) we get ,

14

2

2

dk

Edea

……….(3)

comparing the above equation with Newton’s second law we

conclude that the proportionality factor may be regarded as mass

and is known as effective mass m* .Thus the effective mass of

the electron m* is 2

2*

dk

Edm

………..(4)

The effective mass of the electron can be determined with the

help of graph plotted between the energy E and wave vector k.

We find that up to E < EF, i.e. in the lower half part of the energy

band the value of m* is positive and in the upper half of part of

the band E > EF the value of m* is negative. At the points of

inflexion i.e. at E=EF the value of m* becomes infinite. An

electron with negative effective mass is called an electron hole

and electron-hole pair is called an exiton.

E

kx

E

a

a

Allowed Bands

15

a

a

mdk

dE

a

a

kx

Semiconductors: Semiconductors are the materials whose

conductivity lies between conductors and insulators. According

to Band Theory they are characterized by a narrow energy band

gap(Eg~1eV)

16

1. The Fermi energy EF is midway between the valence band

and the conduction band.

2. At T=0, the valence band is filled and the conduction band

is empty

3. However for semiconductors the band gap energy is

relatively small (1-2eV) so appreciable numbers of electrons

can be thermally excited into the conduction band

4. Hence the electrical conductivity of semiconductors is poor

at low T but increases rapidly with temperature.

Semiconductors can be classified into two categories:

1. Elemental Semiconductors

2. Compound Semiconductors

Elemental Semiconductors: Chemically pure semiconductors

are known as elemental or intrinsic semiconductors. Pure Ge and

Si are well known examples of elemental semiconductors. They

are tetravalent and have four electrons in the outermost orbit of

the atom.

17

There are two types of charge carriers in semiconductors:

electrons in conduction band and holes and valence band. All

charge carriers, electrons and holes are thermally generated.

Electrons and holes are equal in numbers because they are

always formed as electron hole pairs and they are evenly

distributed throughout the crystal. The behaviour of intrinsic

semiconductors with temperature can be studied:

1. at 0 k:

At 0 k, all valence electrons are strongly bounded to their

atoms and they spend most of the time between

neighbouring atoms

It takes large energy to force an electron out of the bond.

Therefore there are no free electrons drifting about within

the material at a temperature of absolute zero.

Because of this semiconductors at 0 k cannot conduct

electricity.

Conduction Band

Valence Band

Ec

Ev

EFEg

E

Distance

Si Si Si

Si Si Si

Si Si Si

18

2. at room temperature:

Thermal energy can dislodge some electrons from their

bonds.

Whenever a covalent bond is ruptured by thermal energy, a

valence electron becomes free.

These electrons make transitions from valence band to

conduction band after acquiring thermal energy.

Simultaneous to the generation a free electron, an empty

space known as hole arises in the valence band.

These thermally generated electron-hole pair causes

electrical conduction in intrinsic semiconductors.

Si Si Si

Si Si Si

Si Si Si

Broken

Covalent

Bond

Vacancy

free

electron

T>0 k

Ec

Ev

EFEg

E

Distance

Compound Semiconductors: Two group IV elements, III-V

group elements or II-VI group elements have average of four;

hence they also show the semiconductor properties and are better

known as the compound semiconductors, e.g.

IV-IV elements: SiC

III-V elements: GaP, GaAs, InAs

19

II-VI elements: ZnS, CdS, CdSe, CdTe

Extrinsic Semiconductors: If a small amount of pentavalent or

trivalent impurity is added into a pure semiconductor crystal,

then the conductivity of the crystal increases appreciably and the

crystal is known as extrinsic semiconductor. The process of

adding impurity is known as doping and the impurity element is

known as dopant. They can be classified into two categories:

1. P- type semiconductors

2. N-type semiconductors

P- type semiconductors:

1. when a trivalent ( boron, aluminium, gallium or indium)

atom replaces a Ge (or Si) atom in a crystal lattice only

three valence electrons are available to form covalent bonds

with the neighbouring Ge (or Si) atoms.

2. This results into an empty space known as hole.

3. When a voltage is applied this vacancy is filled by the

electron bound to the neighbouring Ge (or Si) atom thereby

creating new vacancy there.

4. This process continues and hole moves in the crystal lattice.

5. The conduction mechanism in these semiconductors with

acceptor impurities is predominated by positive carriers

which are introduced into valence band. This type of

semiconductors is known as p-type semiconductors.

20

6. In p-type semiconductor the holes are the ‘majority carriers’

and the few electrons thermally excited from the valence

band into the conduction band are ‘minority carriers’.

N- Type semiconductors: 1. When a small amount of pentavalent (antimony,

phosphorous or arsenic) atom is added to Ge (or Si) four of

these valence electrons form bonds with the neighbouring

Ge(or Si) atoms.

2. The fifth electron is loosely bound.

3. At room temperature this extra electron becomes

disassociated from its atom and move through the crystal as

a conduction electron when a voltage is applied to the

crystal.

4. This extra electron is called ‘donor’ and the crystal is known

as n-type semiconductor.

5. The impurity atoms introduce discrete energy levels for the

electrons just below the conduction band, called the donor

levels.

Ec

Ev

EA

Eg

Acceptor

Level

Conduction Band

Valence Band

21

6. In n-type semiconductor majority carriers are electrons

while the minority carriers are holes formed due to

thermally ruptured covalent bonds.

Conduction Band

Valence Band

Ec

Ev

EDEg

Donor Level

Fermi Level:

1. "Fermi level" is the term used to describe the top of the

collection of electron energy levels at absolute zero

temperature.

2. This concept comes from Fermi-Dirac statistics. Electrons

are fermions and by the Pauli exclusion principle cannot

exist in identical energy states.

3. So at absolute zero they pack into the lowest available

energy states and build up a "Fermi sea" of electron energy

states.

4. The Fermi level is the surface of that sea at absolute zero

where no electrons will have enough energy to rise above

the surface.

22

5. The concept of the Fermi energy is a crucially important

concept for the understanding of the electrical and thermal

properties of solids.

6. Both ordinary electrical and thermal processes involve

energies of a small fraction of an electron volt. But the

Fermi energies of metals are of the order of electron volts.

7. This implies that the vast majority of the electrons cannot

receive energy from those processes because there are no

available energy states for them to go to within a fraction of

an electron volt of their present energy.

8. Limited to a tiny depth of energy, these interactions are

limited to "ripples on the Fermi Sea. At higher temperatures

a certain fraction, characterized by the Fermi function, will

exist above the Fermi level.

9. The Fermi level plays an important role in the band theory

of solids. In doped semiconductors, p-type and n-type, the

Fermi level is shifted by the impurities, illustrated by their

band gaps. The Fermi level is referred to as the electron

chemical potential in other contexts.

23

Fermi Dirac Distribution Function: The Fermi function f(E)

gives the probability that a given available electron energy state

will be occupied at a given temperature. The Fermi function

comes from Fermi-Dirac statistics and has the form

The basic nature of this function dictates that at ordinary

temperatures, most of the levels up to the Fermi level EF are

filled, and relatively few electrons have energies above the

Fermi level. The illustration below shows the implications of the

Fermi function for the electrical conductivity of a

semiconductor.

The band theory of solids gives the picture that there is a sizable

gap between the Fermi level and the conduction band of the

semiconductor. At higher temperatures, a larger fraction of the

24

electrons can bridge this gap and participate in electrical

conduction.

Although the Fermi function has a finite value in the gap, there

is no electron population at those energies. The population

depends upon the product of the Fermi function and the electron

density of states. So in the gap there are no electrons because the

density of states is zero. In the conduction band at 0K, there are

no electrons even though there are plenty of available states, but

the Fermi function is zero. At high temperatures, both the

density of states and the Fermi function have finite values in the

conduction band, so there is a finite conducting population.

Fermi Level in Intrinsic semiconductors:

Concentration of electron in Conduction- Band:

1. In conduction band the electrons are free to move anywhere

as a free particle of effective mass me*.

Conduction Band

Valence Band

Ec

Ev

EFEg

E

Distance

25

2. The electrons in conduction band per unit volume lying

between energy E and E+dE can be calculated as:

CE

dEEFEZn )()(

Where EC is the energy at the bottom of the conduction band

and Z(E) is the density of states at the bottom of the

conduction band and have the value,

2

12

3*

3)()2(

4)( Ce EEm

hEZ

, for E > EC

F(E) is the Fermi-Dirac function and is given by,

kTEE FeEF

/)(1

1)(

C

F

E

kTEECe dEe

EEmh

n/)(

21

23

*

3 1

1)()2(

4

if kTEE F then 1 in the denominator of F(E) is

negligible, i.e.

C

F

E

kTEECe dEe

EEmh

n/)(

21

23

*

3

1)()2(

4

C

F

E

kTEE

Ce dEeEEmh

n/)(2

12

3*

3)()2(

4

C

FCC

E

kTEEEE

Ce dEeEEmh

n/)(2

12

3*

3)()2(

4

C

CFC

E

kTEE

C

kTEE

e dEeEEemh

n/)(2

1/)(2

3*

3)()2(

4

The integral in above eq. is of the standard form which has a

solution of the following form,

26

2/3

0

2/1

2adxex ax

where kT

a1

2/3/)(2/3*

3)(

2)2(

4kTem

hn

kTEE

eFC

or,kTEEe FCe

h

kTmn

/)(

2/3

2

*22

,

putting

2/3

2

*22

h

kTmN e

C

we get

kTEE

CFCeNn

/)( , where NC is known as effective

density in conduction band.

Concentration of holes in Valence-Band: If F(E) is the

probability for the occupancy of an energy state E by an

electron, then the probability that the energy state is vacant is

given by [1- F(E)]. Since a hole represents a vacant state in

valence band, the probability for occupancy of a state at E by a

hole is equal to the probability of absence of electron at that

state. The hole density in the valence band is therefore given by,

VE

dEEFEZp )](1)[(

kTEEkTEE FF eeEF

/)(/)(1

1

1

11)](1[

if kTEEF

then the exponential factor in the denominator of above eq. will

be much grater than unity hence for all values of E in the valence

band, kTEEFeEF

/)()](1[

27

hence,

V

F

E

kTEEdEeEZp

/)()(

21

23

*

3)()2(

4)( EEm

hEZ Vh

V

F

E

kTEE

Vh dEeEEmh

p/)(2

12

3*

3)()2(

4

V

VVF

E

kTEEEE

Vh dEeEEmh

p/)(2

12

3*

3)()2(

4

V

VVF

E

kTEE

V

kTEE

h dEeEEemh

p/)(2

1/)(2

3*

3)()2(

4

kTEEh FVeh

kTmp

/)(

2/3

2

*22

or, kTEE

VFVeNp

/)( where

2/3

2

*22

h

kTmN h

V

is known as

effective density of states in the valence band.

Fermi level in intrinsic semiconductor: In an intrinsic

semiconductor, the free electron and hole concentrations are

equal, i.e. p n

kTEE

V

kTEE

CFVFC eNeN

/)(/)(

taking log on both sides and rearranging the terms, we get

kT

EE

N

N

kT

EE VF

C

VFC )(ln

)(

)(ln)( VF

C

VFC EE

N

NkTEE

VC

C

VF EE

N

NkTE ln2

28

2

)(ln

2

1 VC

C

VF

EE

N

NkTE

substituting the values of NV and NC we get,

2

)(ln

4

3*

*

VC

e

hF

EE

m

mkTE

if

**

eh mm then 2

)( VCF

EEE

i.e. Fermi level is mid-way between the valence band and

conduction band ,in intrinsic semiconductor.

Fermi level in extrinsic semiconductor: The Fermi level of an

extrinsic semiconductor is determined by the intrinsic properties

of the material and the dopant levels. This includes a function

with three terms.

EF = EF (Intrinsic)+EF (Impurities)

i

AD

n

pVC

n

NNkT

m

mkT

EE

2sinhln

4

3

2

1

*

*

Term (1) is the centre of the band gap.

Term (2) is the effect of unequal hole and electron effective

masses.

Term (3) is the effect of the donor and acceptor atoms.

At low, but not too low, temperature the extrinsic term

dominates. At higher temperature the concentration of intrinsic

carrier’s increases and the intrinsic terms dominate. That is, the

sinh-1

term tends to zero because ni gets large.

29

With this approximations for n type semiconductors we have

i

ADF

n

NNkTE ln)(IntrinsicE F

and for p type semiconductors

i

DAF

n

NNkTE ln)(IntrinsicE F

P-N junctions: A semiconductor device can be defined as a unit

which consists, partly or wholly, of semiconductor materials and

which can perform useful functions in electronic apparatus, e.g.

p-n junction diode, transistor etc. P-N junction is a system of two

semiconductors in physical contact, one with excess of electrons

(n- type) and the other with excess of holes (p- type). There are

three methods for preparing p-n junction:

30

1. The grown junction method

2. The alloying method

3. The diffusion method.

In the grown junction method, a crystal of semiconductor is

grown, for example from an initially n-type melt, which while

the crystal is being formed, is counter doped by adding enough

acceptor impurities so that the subsequently grown portion of the

crystal is p- type. But this technique involves the difficulty of

locating the junction in the grown crystal and the difficulty of

attaching leads to the narrow grown junction regions and thus it

is not adopted frequently. This process generally produces

graded p-n junctions.

In the alloying method an alloy pellet or foil is melted upon a

semiconductor base crystal, which contains impurities that give

rise to the conductivity type opposite to that of the original

crystal. After melting it the temperature is raised to high value

such that the molten alloy dissolves away the underlying

semiconductor to some extent. The liquid phase which we get

then is cooled slowly. Then some of the dissolved

semiconductor atoms re-crystallize at the liquid solid interface.

Since the liquid phase contains impurities associated with the

conductivity type opposite to that of the original crystal, the re-

grown material will be of the opposite conductivity and there

will be a p-n junction at the interface between original and re-

grown material. But this process generally produces abrupt p-n

junctions. But because of ease and simplicity this method is

quite popular.

On the other hand in diffusion method, an impurity atom is

diffused at an elevated temperature into a base crystal whose

31

conductivity type is opposite to that which is produced by the

presence of the diffusing impurity in the crystal lattice. Diffused

junctions can be either graded or abrupt depending upon the

diffusion time. This method is also popular because of its

simplicity.

p-n junction characteristics: Following features can be

observed in a p-n junction:

-

-

-

+

+

+

+

+

+

+

-

-

-

-

-

-

--

-

-

--

+

+

+

+

++

-X1 X20

V1

V2

VB

Negative space

charge

Positive space

charge

Depletion region

vo

lta

ge

Ch

arg

e d

en

sit

y

_

+

Distance

Concentration of

holes

Concentration of

acceptors

p-type n-type

+

+

32

1. On each side of the junction there exist free carriers which

were shown in the fig. by un-circled charges. These free

charges, holes on the p-side and electrons on the n-side, are

present because of the thermal ionisation of the donors and

acceptors.

2. Along with the free charges there exist ionised impurities,

which can not move shown encircled in the fig. The acceptors

impurities on the p-side are negatively charged. On the other

hand the donor impurities on the n-side are positively charged

as they have supplied electrons to the conduction band.

3. Due to thermal agitation the holes from the p-side diffuse over

to the n-side and electrons from the n-side diffuse into the p-

region.

4. After the formation of junction the charges on both side

penetrates the boundary region to get recombined.

5. Away from the boundary charged impurity atoms get

neutralised by the space charge of the free carriers present in

both the regions by recombination.

6. Near the junction region, the impurities have no neutralizing

space charge and thus near the junction on the n-side there

exist ionised donors and on the p-side there exist ionised

acceptors.

7. Because of such uncovered ions a potential barrier or junction

barrier is developed.

8. The region across the p-n junction in which the potential

changes from positive to negative is called depletion region

which consist immobile charges and also known as space

charge region.

33

V-I characteristic of P-N junction: When a dc voltage is

applied to a device it is said to be biased. A p-n junction can be

biased in two ways:

1. Forward Biased

2. Reverse Biased

P-N junction under forward biased: To apply forward bias

positive terminal of battery is connected to the p-side and

negative terminal of battery is connected to the n-side.

-

-

-

- -

-

-

- +

+

+

+ +

+

+

+

Depletion region

p n

VB

34

The applied forward potential establishes an electric field which

acts against the potential barrier field and reduces the barrier

height. It increases the probability of majority carriers crossing

the junction and thus diffusion current density Jdiff increases.

Since it does not affect the influence of minority carriers hence

drift current density Jdrift remains unaffected.

Here the negative terminal of external causes an increase in

electron energy and an upward shift of all energy levels on the n-

side. Similarly, the positive terminal causes an increase in hole

energy and hence a lowering of all levels on p-side.

As the displacement of energy levels occur in opposite direction,

the Fermi levels EFn and EFp get separated by an amount of

energy eV added by the voltage source and thus the height of the

potential barrier is reduced by an amount of energy e(VB-V).

-

-

-

- +

+

+

+

VB-V

p n

free electron

flow

35

V

R

mA

+

-

+ -

Forward voltage

Fo

rwa

rd c

urr

en

t

Depletion region

eVF

Forward bias

p-Type n-Type

EFp

EFn

e(VB -V)

Jhp

Jhn

Jen

Jep

E

Energy band diagram of a p-n junction under

forward bias P-N junction under reverse biased: To apply reverse bias

positive terminal of battery is connected to the n-side and

36

negative terminal of battery is connected to the p-side. The

applied reverse potential establishes an electric field which acts

in the same direction as that of the potential barrier field and

thus increases the barrier height by an amount of energy of

e(VB+V). In this case the energy levels of n-side are displaced

downwards and energy levels of p-side are displaced upwards.

Due to the large height of the barrier diffusion of majority

carriers totally stops while drift of minority carriers is not

affected and thus we get only drift current in this case.

V

R

μA

+

-

+ -

Re

ve

rse

cu

rren

t

Reverse voltageVBR

37

-

-

-

- +

+

+

+

p n

VB+V

-

-

-

-

+

+

+

+

Depletion region

eVR

Reverse bias

p-Type n-Type

EFp

EFn

e(VB +V)

Jhn

Jep

E

Energy band diagram of a p-n junction under

reverse bias

Electron drift

hole drift

38

PHPTODIODE

Its working is based on photo conduction from light.

A photo diode is a semiconductor made of photosensitive

semiconductor material.

In such a diode, a provision is made to allow the light of

suitable frequency to fall on it.

The conductivity of p-n photodiode increases with the

increase in intensity of light falling on it.

Symbolically, a photodiode is shown in the fig. below:

Fig shows an experimental arrangement in which the

photodiode is reverse biased but the voltage applied is less

than the break down voltage.

When visible light of energy greater than forbidden energy

gap (i.e. h>Eg)is incident on a reverse biased p-n junction

photodiode additional electron-hole pairs are created in the

depletion layer (or near the junction).

These charge carriers will be separated by the junction field

and made to flow across the junction.

The value of reverse saturation current increases with the

increase in the intensity of incident light as shown in the fig.

below:

It is found that the reverse saturation current through the

photodiode varies almost linearly with the light flux.

The photodiodes are used preferably in reverse bias

condition because the change in reverse current through the

photodiode due to change in light flux can be measured

directly proportional to the light flux. But it is not so when

the photodiode is forward biased.

39

When the photodiode is reverse biased, then a certain

current exists in the circuit even when no light is incident on

the p-n junction of photodiode.

This current is called dark current. A photodiode can turn its

current ON and OFF in nanoseconds. Hence photodiode is

one of the fastest photodetector.

Photodiodes are used for following purposes:

1. In photodetection for optical signals.

2. In demodulation for optical signals.

3. In switching the light on and off.

4. In optical communication equipments.

5. In logic circuits that require stability and high speed.

6. In reading of computers, punched cards and tapes etc.

Solar cell: Solar cell is basically a solar energy converter.

It is a p-n junction device which converts solar energy into

electric energy.

A solar cell is symbolically shown in fig. (a) and in

construction along with circuit in fig. (b)

+

-

a

40

A solar cell consists of silicon or gallium-arsenide p-n

junction diode packed in a can with glass window on top.

The upper layer is of p-type semiconductor.

It is very thin so that the incident light photons may easily

reach the p-n junction.

On the top face of p-layer, the metal finger electrodes are

prepared in order to have enough spacing between the

fingers for the light to reach the p-n junction through p-layer.

When photons of light (of energy h>Eg) fall at the junction,

electron-hole pairs are generated in the depletion layer (or

near the junction) which move in opposite directions due to

junction field.

The photo generated electrons move towards n-side of p-n

junction.

The photo generated holes move towards p-side of p-n

junction.

R

GLASS

LIGHT

METAL FINGER

ELECTRODES

METAL CONTACT

N

P

+

-b

41

They will be collected at the tow sides of the junction,

giving rise to a photo voltage between the top and bottom

metal electrons.

The top metal contact acts as positive electrode and bottom

metal contact acts as negative electrode.

When an external load is connected across metal electrodes

a photo current flows.

Applications:

1. Solar cells are used for charging storage batteries in say

time, which can supply the power during night times.

2. The solar cells are also used in artificial satellite to operate

the various electrical instruments kept inside the satellite.

3. They are used for generating electrical energy in cooking

food.

4. Solar cells are used in calculators, wrist watches and light

meters (in photography).