BAND THEORY OF

SOLID

Course: B.Tech

Subject: Engineering Physics

Unit: III

Chapter: 1

OBJECTIVES

• Effective Mass of electron

• Concept of Holes

• Energy Band Structure of Solids:

Conductors, Insulators and Semiconductors

• Semiconductors

Intrinsic and Extrinsic Semiconductors

• Type of diodes

Simple Diode

Zener Diode

EFFECTIVE MASS OF ELECTRON

An electron in crystal may behave as if it had a

mass different from the free electron mass m0.

There are crystals in which the effective mass

of the carriers is much larger or much

smaller than m0.

The effective mass may be anisotropic, and it

may even be negative.

The important point is that the electron in a

periodic potential is accelerated relative to

the lattice in an applied electric or magnetic

field as if its mass is equal to an effective

mass.

If the electron is free then E represents the kinetic energy only.

It is related to the wave vector k and momentum p by

Therefore, the quantum mechanical and classical free particles

exhibit precisely the same energy momentum relationship,

as shown below.

2 2 2

0 0

k pE

2m 2m

Effective Mass Expression

The equation is identical to Newton’s second law of

motion

except that the actual particle mass is replaced by an

effective mass m*.

g

2

2 2

dF m*.

dt

1m*

1 d E

dk

CONCEPT OF HOLES

Consider a semiconductor with a small

number of electrons excited from the valence

band into the conduction band.

If an electric field is applied,

• the conduction band electrons will

participate in the electrical current

• the valence band electrons can “move

into” the empty states, and thus can also

contribute to the current.

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CONCEPT OF HOLES

If we describe such changes via “movement” of the

“empty” states – the picture will be significantly

simplified. This “empty space” is called a Hole.

“Deficiency” of negative charge can be treated as a

positive charge.

Holes act as charge carriers in the sense that

electrons from nearby sites can “move” into the

hole.

Holes are usually heavier than electrons since they

depict collective behavior of many electrons.

Electrical current for holes and electrons in the same direction

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To understand hole motion, one requires

another view of the holes, which represent

them as electrons with negative effective mass

m*.

For example the movement of the hole think

of a row of chairs occupied by people with one

chair empty, and to move all people rise all

together and move in one direction, so the

empty spot moves in the same direction

Energy Band Structure of Solids

Conductor, Semiconductor and

Insulator

In isolated atoms the electrons are arranged in

energy levels.

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ENERGY BAND IN SOLID

The following are the important energy

band in solids:

Valence band

Conduction band

Forbidden energy gap or Forbidden band

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Valance band

The band of energy occupied by the valance

electrons is called valence band. The electrons in

the outermost orbit of an atom are known as

valance electrons. This band may be completely or

partial filled.

Electron can be move from one valance band

to the conduction band by the application of

external energy.

Conduction band

The band of energy occupied by the

conduction electrons is called conduction band.

This is the uppermost band and all electrons in the

conduction band are free electrons.

The conduction band is empty for insulator

and partially filled for conductors.

Forbidden Energy Gap or Forbidden band

The gap between the valance band and

conduction band on energy level diagram known as

forbidden band or energy gap.

Electron are never found in the gap. Electrons

may jump from back and forth from the bottom of

valance band to the top of the conduction band. But

they never come to rest in the forbidden band.

According to the classical free electron

theory, materials are classified in to three

types:

Conductors

Semiconductors

Insulators

Conductors

There is no forbidden gap and the conduction

band and valence band are overlapping each other

between and hence electrons are free to move about.

Examples are Ag, Cu, Fe, Al, Pb ….

Conductor are highly electrical conductivity.

So, in general electrical resistivity of conductor is

very low and it is of the order of 10-6 Ω cm.

Due to the absence of the forbidden gap, there is no

structure for holes.

The total current in conductor is simply a flow of

electrons.

For conductors, the energy gap is of the order of

0.01 eV.

Semiconductors

Semiconductors are materials whose electrical

resistivity lies between insulator and conductor.

Examples are silicon (Si), germanium (Ge) ….

The resistivity of semiconductors lie between 10-4 Ω

cm to 103 Ω cm at room temperature.

At low temperature, the valence band is all most full

and conduction band is almost empty. The forbidden

gap is very small equal to 1 eV.

Semiconductor behaves like an insulator at low

temperature. The most commonly used

semiconductor is silicon and its band gap is 1.21 eV

and germanium band gap is 0.785 eV.

When a conductor is heated its

resistance increases; The atoms vibrate more

and the electrons find it more difficult to move

through the conductor but, in a

semiconductor the resistance decreases with

an increase in temperature. Electrons can be

excited up to the conduction band and

Conductivity increases.

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InsulatorsIn insulator, the valence band is full but the

conduction band is totally empty. So, free electronsfrom conduction band is not available.

In insulator the energy gap between the valence andconduction band is very large and its approximatelyequal to 5 eV or more.

Hence electrons cannot jump from valence band tothe conduction band. So, a very high energy isrequired to push the electrons to the conductionband.

The electrical conductivity is extremely small.

The resistivity of insulator lie between 103 to 1017

Ωm, at the room temperature

Examples are plastics, paper …..

TYPES OF SEMICONDUCTORS

Semiconductors

Intrinsic Semiconductor Extrinsic Semiconductor

p - type n - type

INTRINSIC SEMICONDUCTOR

The intrinsic semiconductor are pure semiconductormaterials.

These semiconductors posses poor conductivity.

The elemental and compound semiconductor can beintrinsic type.

The energy gap in semiconductor is very small.

So even at the room temperature, some of electronsfrom valance band can jump to the conduction bandby thermal energy.

The jump of electron in conduction band adds oneconduction electron in conduction band and creates ahole in the valence band. The process is called as“generation of an electron–hole pair”.

In pure semiconductor the no. of electrons inconduction band and holes in holes in valence bandsare equal.

EXTRINSIC SEMICONDUCTOR

Extrinsic semiconductor is an impure semiconductor

formed from an intrinsic semiconductor by adding a

small quantity of impurity atoms called dopants.

The process of adding impurities to the

semiconductor crystal is known as doping.

This added impurity is very small of the order of one

atom per million atoms of pure semiconductor.

Depending upon the type of impurity added the

extrinsic semiconductors are classified as:

(1) p – type semiconductor

(2) n – type semiconductor

The application of band theory to n-

type and p-type semiconductors shows that extra levels

have been added by the impurities.

In n-type material there are electron energy

levels near the top of the band gap so that they can be

easily excited into the conduction band.

In p-type material, extra holes in the band gap

allow excitation of valence band electrons, leaving

mobile holes in the valence band.

P – TYPE SEMICONDUCTOR

The addition of trivalent impurities such as

boron, aluminum or gallium to an intrinsic

semiconductor creates deficiencies of valence electrons,

called "holes". It is typical to use B2H6 diborane gas to

diffuse boron into the silicon material.

6

N – TYPE SEMICONDUCTOR

The addition of pentavalent impurities such as

antimony, arsenic or phosphorous contributes free

electrons, greatly increasing the conductivity of the

intrinsic semiconductor. Phosphorous may be added

by diffusion of phosphine gas (PH3).

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SIMPLE DIODE (P N- JUNCTION

DIODE) The two terminals are called Anode and Cathode.

At the instant the two materials are “joined”, electrons

and holes near the junction cross over and combine with

each other.

Holes cross from P-side to N-side and Free electrons

cross from N-side to P-side.

At P-side of junction, negative ions are formed.

At N-side of junction, positive ions are formed.

8

Depletion region is the region having no free carriers.

Further movement of electrons and holes across the junction

stops due to formation of depletion region.

9

Depletion region acts as barrier opposing further

diffusion of charge carriers. So diffusion stops within

no time.

Current through the diode under no-bias condition is

zero.

REVERSE BIAS

Positive of battery connected to n-type material

(cathode).

Negative of battery connected to p-type material

(anode).

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REVERSE BIAS…..

Free electrons in n-region are drawn towards

positive of battery, Holes in p-region are drawn

towards negative of battery.

Depletion region widens, barrier increases for the

flow of majority carriers.

Majority charge carrier flow reduces to zero.

Minority charge carriers generated thermally can

cross the junction – results in a current called

“reverse saturation current” Is , Is is in micro or

nano amperes or less. Is does not increase

“significantly” with increase in the reverse bias

voltage

FORWARD BIAS

Positive of battery connected to p-type (anode)

Negative of battery connected to n-type (cathode)

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FORWARD BIAS…

Electrons in n-type are forced to recombine with

positive ions near the boundary, similarly holes in

p-type are forced to recombine with negative ions.

Depletion region width reduces.

An electron in n-region “sees” a reduced barrier at

the junction and strong attraction for positive

potential.

As forward bias is increased, depletion region

narrows down and finally disappears – leads to

exponential rise in current.

Forward current is measured in milli amperes

12

ZENER DIODE

A diode which is heavily doped and whichoperates in the reverse breakdown region with asharp breakdown voltage is called a Zener diode.

This is similar to the normal diode exceptthat the line (bar) representing the cathode isbent at both side ends like the letter Z for Zenerdiode.

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In simple diode the doping is light; as aresult, the breakdown voltage is high and notsharp. But if doping is made heavy, then thedepletion layers becomes very narrow and eventhe breakdown voltage gets reduced to a sharpvalue.

Working Principle

The reverse breakdown of a Zener diodemay occur either due to Zener effect oravalanche effect. But the Zener diode isprimarily depends on Zener effect for itsworking.

When the electrical field across the

junction is high due to the applied voltage, the

Zener breakdown occurs because of breaking

of covalent bonds and produces a large number

of electrons and holes which constitute a steep

rise in the reverse saturation current (Zener

current IZ). This effect is called as Zener effect.

Zener current IZ is independent of the

applied voltage and depends only on the

external resistance.

I-V CHARACTERISTIC OF A ZENER

DIODE

The forward characteristic is simply that

of an ordinary forward biased junction diode.

Under the reverse bias condition, the

breakdown of a junction occurs.

Its depends upon amount of doping. It can

be seen from above figure as the reverse voltage

is increased the reverse current remains

negligibly small up to the knee point (K) of the

curve.

At point K, the effect of breakdown

process beings. The voltage corresponding to

the point K in figure is called the Zener

breakdown voltage or simply Zener voltage

(VZ), which is very sharp compared to a

simple p-n junction diode. Beyond this voltage

the reverse current (IZ) increases sharply to a

high value.

The Zener diode is not immediately burnt

just because it has entered the breakdown

region.

The Zener voltage VZ remains constant

even when Zener current IZ increases greatly.

The maximum value of current is denoted by

IZ max and the minimum current to sustain

breakdown is denoted by IZ min. By two points A

and B on the reverse VI characteristic, the Zener

resistance is given by the relation,

rz = ( Δ VZ / Δ IZ) -----(1)

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ZENER DIODE APPLICATIONS:

I. Zener diodes are used as a voltage

regulator.

II. They are used in shaping circuits as peak

limiters or clippers.

III. They are used as a fixed reference voltage

in transistor biasing and for comparison

purpose.

IV. They are used for meter protection against

damage from accidental application of

excessive voltage.

REFERENCE BOOKS AUTHOR/PUBLICATION

ENGINEERING PHYSICS S S PATEL (ATUL PRAKASHAN)

MODERN ENGINEERING

PHYSICSA S VASUDEVA

ENGINEERING PHYSICS K. RAJGOPALAN

IMAGE REFERENCE LINKS

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