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ENGINEERING MATERIALSby

NareNAssistance Professor

Department of Physics

K L [email protected]@gmail.com

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mailto:[email protected]

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Electrical Properties

Electrical Conduction

Electron Band Structure of a materials

Dielectric characteristics

Materials

Metals Semiconductors Insulators

Electrical Properties of materials Their responses to an applied field

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• Ohm’s Law– At constant temperature the current flowing

through a conductor is directly proportional to the potential difference across the ends of the conductor.

• Macroscopic form

R

V = i

(ohms) wire,of resistance = R

(volts) V ,difference potential = V

(amperes) A current, electrical = i

:where

Electrical Conduction in Metals

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• The opposing force offered by the material to the flow of current.

• Depends on– Nature of the material (ρ).– Temperature.– Geometry/ dimensions (length L, area of

cross section A)

R = r (L/A)

Resistance Electrical Conduction in Metals

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• It is a material property.• It defines how difficult is it for current to flow.• Geometry independent.• Temperature dependent.

Resistivity

RA

ρ

surface area of current flow

current flow path length


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Examples of Resistivity (ρ)• Ag (Silver): 1.59×10-8 Ω·m• Cu (Copper): 1.68×10-8 Ω·m• Graphite (C): (3 to 60)×10-5 Ω·m• Diamond (C): ~1014 Ω·m• Glass: ~1010 - 1014 Ω·m• Pure Germanium: ~ 0.5 Ω·m• Pure Silicon: ~ 2300 Ω·m


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• It is the current flowing through unit area of cross section.

Current Density (J)

m section, cross of Area =

Amp conductor, he through tflowingcurrent = ImA / density,current = J

where

I = J

2

2

A

A


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)m (ty conductivielectrical =

m y,resistivit electrical =

m / V field, electric = EmA / density,current = J

where

E = E

= J

1-

2

Ohm's Law -- Microscopic FormElectrical Conduction in Metals

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Experimental verification of ohm’s lawElectrical Conduction in Metals

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Electrical conductivity varies between different materials by over 27 orders of magnitude, the greatest variation of any physical property

Metals: > 107 (.m)-1

Semiconductors: 10-6 < < 105 (.m)-1

Insulators: < 10-6 (.m)-1

Electrical Conduction in Materials

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Energy Band Structures in

Solids

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• In most of solids conduction is by electrons.

• σ depend on no. of electrons available

• The no. of electrons available for conduction depends on– Arrangement of electrons states or levels with respect to

energy.– The manner in which these states are occupied by

electrons.

Energy Band Structures in solids

Current carriers• electrons in most solids • ions can also carry (particularly in liquid solutions)

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IONIZATIONIf a sufficient amount of energy is absorbed by an electron, it is possible for that electron to be completely removed from the influence of the atom

An atom having more than its normal amount of electrons acquires a negative charge

The atom that gives up some of its normal electrons is left with fewer negative charges than positive charges

NEGATIVE ION

For ionization to take place, there must be a transfer of energy that results in a change in the internal energy of the atom.

POSITIVE ION

Isolated Atom


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Gas : Very little influence upon each other, and are very much like lone atoms.

Solid : The forces that bind these atoms together greatly modify the behavior of the other electrons.

One consequence of this close proximity of atoms is to cause the individual energy levels of an atom to break up and form bands of energy.

Discrete (separate and complete) energy levels still exist within these energy bands, but there are many more energy levels than there were with the isolated atom.

When atoms are spaced far enough apart


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The difference in the energy arrangement between an isolated atom and the atom in a solid.

Notice that the isolated atom (such as in gas) has energy levels, whereas the atom in a solid has energy levels grouped into ENERGY BANDS.


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WHY ENERGY BANDS ARE FORMED?

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• Electrons of one atom are perturbed by the electrons and nuclei of the adjacent atoms.

• Results in splitting of atomic states into a series of closely spaced electron states to from what are called ELECTRON ENERGY BAND.

• Extent of splitting depends on interatomic separation.

Electronic Band Structures

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Schematic plot of electron energy versus interatomic separation for an aggregate of 12 atomsUpon close approach, each of the 1s and 2s atomic states splits to form an electron energyband consisting of 12 states.


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• Valence band – filled – highest occupied energy levels• Conduction band – empty – lowest unoccupied energy levels

At Equilibrium spacing :Band formation may not occur


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• With in each band the energy states are discrete.

• No. of states with in each band will equal the total of all states contributed by the N atoms.– s band consists of N states– p band consists of 3N states

• Electrical properties of a solid depends on its electron band structure.


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Energy Band Structure

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The concept of energy bands is particularly important in classifying materials as conductors, semiconductors, and insulators.

An electron can exist in either of two energy bands, the conduction band or the valence band.

All that is necessary to move an electron from the valence band to the conduction band so it can be used for electric current, is enough energy to carry the electron through the forbidden band.

The width of the forbidden band or the separation between the conduction and valence bands determines whether a substance is an insulator, semiconductor, or conductor.


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Energy level diagrams to show the difference between insulators, semiconductors, and conductors.


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The energy corresponding to the highest filled state at 0 K is called the Fermi energy


Cu-3d10 4s1- One 4s electron Mg-12-Two 3s electrons

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Conduction in SolidsPresence of an electric field :

Accelerated Electrons with energies greater than the Fermi energy ( Ef )

Conduction process

Free electrons Holes – SMC & Insulators

Electrical conductivity is a direct function

Distinction between conductors and nonconductors (insulators and semiconductors) -----

no: of free electrons and holes.

lies in the numbers of these free electron and hole charge carriers.

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Metals For an electron to become free, it must be excited or promoted into

one of the empty and available energy states above Ef .

Very little energy is required to promote electrons into the low-lying empty states.

Electrons are not locally bound to any particular atom

Conduction in Metals

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filled band

Energy

partly filled band

empty band

GAP

fille

d st

ates

Partially filled band• Metals (Conductors):

-- Thermal energy puts many electrons into a higher energy state.

Energy

filled band

filled band

empty band

fille

d st

ates

Overlapping bands

• Energy States:

-- for metals nearby energy states are accessible by thermal fluctuations.

Conduction in Metals

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To become free, therefore, electrons must be promoted across the energy band and into empty states at the bottom of the conduction band.

This is possible only by supplying to an electron the difference in energy between these two states, which is approximately equal to the band gap energy Eg

Excitation energy Nonelectrical source Heat or light

Number of electrons excited thermally (by heat energy) depends -->

1.Energy band gap width 2.Temperature

Conduction in Insulators & SMC Empty States adjacent to the top of the filled valence band are not available.

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At a given Temperature 1. Eg is small – conduction is high

2. Eg is large – conduction is low

Width of the band gapSMC – Narrow

Insulators -- Widely

Increasing the temperature Thermal energy increases

No of electrons excitation moreConductivity increases

Conduction in Insulators & SMC

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Semiconductors covalent valence electrons are not as strongly bound to the atoms

electrons are more easily removedby thermal excitation

Bonding

Insulating material ionic or strongly covalent

valence electrons are tightly bound

electrons are highly localized


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• Insulators: -- Higher energy states not accessible due to gap (> 2 eV).

Energy

filled band

filled valence band

empty band

fille

d st

ates

GAP

• Semiconductors: -- Higher energy states separated

by smaller gap (< 2 eV).

Energy

filled band

filled valence band

empty band

fille

d st

ates

GAP?

Conduction in Insulators & SMCWidth of the band gap

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Width of the band gap


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Electron MobilityAbsence of Electric field: (E=0)

1. Electrons move randomly through out the crystal in conductors

2. Net current – ZERO ( any direction)

3. No Drift velocity

Presence of Electric field:1.Electrons are accelerated in a single direction

2.This acceleration is opposed by the internal damping forces or frictional forces.

due to scattering of electrons by imperfections

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imperfection's

1. Impurity atoms

2. Vacancies

5.Interstitial atoms

3.Dislocations

4.Thermal vibration

Electron Mobility

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Each Scattering or Collision occurs:

A) e- losses kinetic energy Transfer to the lattice of metal

e- - lattice scattering

1. Existence of resistance 2. Warming up of the metalsEx: Electrical heaters

B) Changes its direction of motion

Electron Mobility

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• Electrons move randomly but with a net drift in the direction opposite to the electric field.

scattering phenomenon resistance

1. mobility of an electron2. drift velocity

vd = eEIt is represents the average electron velocity in the direction of the force imposed by the applied field. It is directly proportional to the electric field.

e – electron mobility [m2/V-s].

Electron Mobility

used to describe

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Electrical conductivity proportional to number of free electrons per unit volume, Ne, and electron mobility, e

metal >> semimetal > semi

Nmetal >> Nsemi

Electron Mobility

= Nee e

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Electrical resistivity of metals

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Metals are extremely good conductors of electricity

Discuss conduction in metals in terms of the resistivity

large numbers of free electrons

excited into empty states above the Fermi energy

crystalline defects serve as scattering centers for conduction electrons

concentration of these imperfections temperature, composition


raises the resistivityincreasing their number

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experimentally total = thermal+impurity+deformation


The electrical resistivity versus temperature for copper and three copper–nickel alloys, one of which has been deformed.

Matthiessen’s rule

Influence of Temperature : pure metal and all the copper–

nickel alloys

Resistivity rises linearly with temperature *

T Vibrations and lattice defects electron scattering increases

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Impurity resistivity ρi is related to the impurity concentration Ci in terms of the atom fraction (at%100)

The influence of nickel impurity additions on the room temperature resistivity of copper

Influence of Impurities

where A is a composition-independent constant (impurity and host metals)

Two-phase alloy consisting of α and β phases

where the V’s and ρ’s represent volume fractions and individual resistivities for the respective phases.

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Result of increased numbers of electron-scattering dislocations.

Influence of Plastic Deformation

Raises the electrical resistivity

Its influence is much weaker than that of increasing temperature or the presence of impurities.

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SilverBest material for electrical conductionVery expensive

CopperTo achieve low resistance

Remove gases included in metal

Soft material

Mechanical strength CuBe is used

AluminiumWeightMore resistant to corrosion

Electrical characteristics of commercial alloys

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Furnace heating elements

high electrical resistivity

energy loss by electrons

dissipated as heat energy

Electrical characteristics of commercial alloys

Ex: Nichrome, a nickel–chromium alloy

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Semiconductors have a resistivity/resistance between that of conductors and insulators

Their electrons are not free to move but a little energy will free them for conduction

Their resistance decreases with increase in temperature

Electrical properties are extremely sensitive to the presence of impurities.

Semiconductors

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INTRINIC SEMICONDUCTORS

The Silicon, Si, AtomSilicon has a valency of 4 i.e. 4 electrons in its outer shell

Each silicon atom shares its 4 outer electrons with 4 neighbouring atoms

These shared electrons – bonds – are shown as horizontal and vertical lines between the atoms

This picture shows the shared electrons

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Silicon – the crystal latticeIf we extend this arrangement throughout a piece of silicon…

We have the crystal lattice of silicon

This is how silicon looks when it is 0 K

It has no free electrons – it cannot conduct electricity – therefore it behaves like an insulator

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Electron Movement in SiliconAt room temperature

An electron may gain enough energy to break free of its bond…

It is then available for conduction and is free to travel throughout the material

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Hole Movement in Silicon

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Hole Movement in SiliconThis hole can also move…

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Heating Silicon

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Intrinsic ConductionTake a piece of silicon…

This sets up an electric field throughout the silicon – seen here as dashed lines

And apply a potential difference across it…

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Intrinsic Conduction

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Intrinsic Conduction

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Intrinsic Semiconductors• Consider nominally pure

semiconductor at T = 0 K• There is no electrons in

the conduction band

• At T > 0 K a small fraction of electrons is thermally excited into the conduction band, “leaving” the same number of holes in the valence band

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• This hole is positive, and so can attract nearby electrons which then move out of their bond etc.

• Thus, as electrons move in one direction, holes effectively move in the other direction

Electron moves to fill hole

As electron moves in one direction hole effectively moves in other

EXTRINIC SEMICONDUCTORS

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• Prepared by adding (doping) impurities to intrinic semiconductors

• Doping is the incorporation of [substitutional] impurities (trivalent or pentavalent) into a semiconductor according to our requirements• In other words, impurities are introduced in a controlled manner

• Electrical Properties of Semiconductors can be altered drastically by adding minute amounts of suitable impurities to the pure crystals

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Doping

• Pentavalent • Group VA elements

– Phosphorous– Arsenic– Antimony

• Trivalent • Group III A elements

– Boron – Gallium – Indium

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The Phosphorus AtomPhosphorus is number 15 in the periodic table

It has 15 protons and 15 electrons – 5 of these electrons are in its outer shell

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Doping – Making n-type Silicon

We now have an electron that is not bonded – it is thus free for conduction

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Doping – Making n-type Silicon

As more electrons are available for conduction we have increased the conductivity of the material

If we now apply a potential difference across the silicon…

Phosphorus is called the dopant

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Extrinsic Conduction – n-type Silicon

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The free electrons in n type silicon support the flow of current.

This crystal has been doped with a pentavalent impurity.

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The Boron AtomBoron is number 5 in the periodic table

It has 5 protons and 5 electrons – 3 of these electrons are in its outer shell

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Doping – Making p-type Silicon

Notice we have a hole in a bond – this hole is thus free for conduction

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Doping – Making p-type Silicon

If we now apply a potential difference across the silicon…

Boron is the dopant in this case

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Extrinsic Conduction – p-type silicon

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This crystal has been doped with a trivalent impurity.

The holes in p type silicon contribute to the current.

Note that the hole current direction is opposite to electron current so the electrical current is in the same directionNareN @KL U

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Extrinsic conductivity—p type• Every acceptor generates excess mobile holes

(p=Na).• Now holes totally outnumber electrons, so

conductivity equation switches to p domination.

hahhe eNepepen

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Ef=Edonor= Ec-0.05eV

Ef=Eacceptor= Ev+0.05eV

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• Intrinsic: # electrons = # holes (n = p) --case for pure Si

• Extrinsic: --n ≠ p --occurs when DOPANTS are added with a different # valence electrons than the host (e.g., Si atoms)

nee peh

• N-type Extrinsic: (n >> p)• P-type Extrinsic: (p >> n)

no applied electric field

5+

4+ 4+ 4+ 4+

4+

4+4+4+4+

4+ 4+

Phosphorus atom

no applied electric field

Boron atom

valence electron

Si atom

conduction electron

hole

3+

4+ 4+ 4+ 4+

4+

4+4+4+4+

4+ 4+

hei en

310*, inpn

he epen

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Variation of carrier concentration with temperature in intrinsic semiconductors

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Variation of carrier concentration with temperature in extrinsic semiconductors

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+

• The converse transition can also happen.

• An electron in CB recombines with a hole in VB and generate a photon.

• The energy of the photon will be in the order of Eg.

• If this happens in a direct band-gap s/c, it forms the basis of LED’s and LASERS.

e-

photon

Valance Band

Conduction Band

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• The magnitude of the band gap determines the differences between insulators, s/c‘s and metals.

• The excitation mechanism of thermal is not a useful way to promote an electron to CB even the melting temperature is reached in an insulator.

• Even very high electric fields is also unable to promote electrons across the band gap in an insulator.

Insulators :

CB (completely empty)

VB (completely full)

Eg~several electron volts

Wide band gaps between VB and CB

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Metals :

CB

VB

CB

VB

• No gap between valance band and conduction band

Touching VB and CB Overlapping VB and CB

These two bands looks like as if partly filled bands and it is known that partly filled bands conducts well.

This is the reason why metals have high conductivity.

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The Concept of Effective Mass :

Comparing

Free e- in vacuum

An e- in a crystal

In an electric field mo =9.1 x 10-31

Free electron mass

In an electric field

In a crystal m = ?

m* effective mass

• If the same magnitude of electric field is applied to both electrons in vacuum and inside the crystal, the electrons will accelerate at a different rate from each other due to the existence of different potentials inside the crystal.

• The electron inside the crystal has to try to make its own way.

• So the electrons inside the crystal will have a different mass than that of the electron in vacuum.

• This altered mass is called as an effective-mass.

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Direct an indirect-band gap materials :

• For a direct-band gap material, the minimum of the conduction band and maximum of the valance band lies at the same momentum, k, values.

• When an electron sitting at the bottom of the CB recombines with a hole sitting at the top of the VB, there will be no change in momentum values.

• Energy is conserved by means of emitting a photon, such transitions are called as radiative transitions.

Direct-band gap s/c’s (e.g. GaAs, InP, AlGaAs)

+

e-

VB

CBE

k

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• For an indirect-band gap material; the minimum of the CB and maximum of the VB lie at different k-values.

• When an e- and hole recombine in an indirect-band gap s/c, phonons must be involved to conserve momentum.

Indirect-band gap s/c’s (e.g. Si and Ge)

+

VB

CB

E

k

e-

Phonon Atoms vibrate about their mean

position at a finite temperature.These vibrations produce vibrational waves inside the crystal.

Phonons are the quanta of these vibrational waves. Phonons travel with a velocity of sound .

Their wavelength is determined by the crystal lattice constant. Phonons can only exist inside the crystal.

Eg

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• The transition that involves phonons without producing photons are called nonradiative (radiationless) transitions.

• These transitions are observed in an indirect band gap s/c and result in inefficient photon producing.

• So in order to have efficient LED’s and LASER’s, one should choose materials having direct band gaps such as compound s/c’s of GaAs, AlGaAs, etc…

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