Other than the mechanical properties of materials,

we have also discussed about the thermal properties of materials and the optical properties of

materials. Now today, we are going to look into another interesting set of properties,

which are known as the electrical properties of the material. As I told you that, materials

are becoming more and more multifunctional today. In fact, very shortly they are also

called functional materials, so functional materials actually solve more than one purpose,

they not only solve the mechanical purpose, but they also solve some electrical or may

be thermal or maybe optical or maybe magnetic functions they actually take care. So, in this particular talk we are going to

talk about the Ohm’s law which you all know, but for the sake of completeness we should

discuss it, the electrical conductivity, energy band structure of solid materials and then

conductivity in metals, semiconductors, ionic ceramics particularly that group of ceramics

which has the Ionic bonds and the polymers, so this is what we are going to talk about

in today’s lecture. So let us first start our life with the Ohm’s law. Now, this from your school days that the voltage

drop in a circuit, let us say this is our circuit here and we have a battery which is

the power source and then we have a resistor, it can be a variable resistor which means

you can change the resistance of a resistor, we have a ammeter which can actually measure

the current and then a part of, so this is the current that is flowing in the loop and

you need to complete the loop. Now there is one part here where you are actually putting

a new material let us say whose resistance is what you are interested to find out.

So what is the cross-sectional area of the material and what is the length of the material

and as the current I and you can also find out what is the voltage across it through

a voltmeter as you mean that the voltmeter is going to take minimum amount of resistance

so that it is going to actually measure the voltage drop between the 2 without adding

to the voltage drop and then the circuit is complete once it is coming back to the battery.

So with this kind of a simple circuit system if you actually note down that what is the

voltage drop and what is the current from the ammeter, then you will be able to find

out that what is the resistance of that particular length of the particular material. Now, this

resistance is an extrinsic property because it not only depends on the material property,

but also on the professional area on the length, etc. So the intrinsic material property that

you should go for is called resistivity which is also denoted by Rho, soRho for density

Rho for resistivity we should not actually confuse between the two because the unit here

is Ohm meter. Now with respect to the extrinsic of property

resistance, the relationship is simply given here as R equals Rho times L over A that is

the relationship. So once I know this material property of resistivity, once I know the length

of the material in the wire form and the cross-sectional area, I can find out what is the actual resistance

it will be or the it will be offering when you pass a current across it. Now then suppose I take 2 wires, one of diameter

D and length 2L and in the other case it is of diameter 2D and length L. So what is the

extrinsic resistance in the first case? Well, in the first case the length is 2L and the

resistivity is same Rho and the wire cross-sectional area is Pie D square by 4 so it will become

8 Rho L over Pie D square. On the other hand, if you consider the resistance of this one

R2, then resistance is same again Rho, length is now L and diameter is double now 2D, so

that you substitute and you will get it as Rho L over pie D square.

So what it means is that R2 is actually 1 by 8th of R1. So what it means is that if

I double the diameter, my resistance come down by 1 by 8th. On the other hand, if I

go for doubling the length, then my resistance increases. This is actually analogous to flow

of water in a pipe and this also demonstrates that the resistance depends on samples geometry

and size, but resistivity is an intrinsic material property. Now, similar to the resistivity there is another

very important term that we use that is called conductivity, which is the inverse of the

resistivity. So electrical conductivity is Sigma equals 1 over Rho and hence its unit

is Ohm meter inverseor sometimes people also call it as Siemens per meter. So this is indicative

of the case where a material is highly conductive and what is its level of conductivity corresponding

to electric current, we can measure it with respect to Sigma.

Now that we know that this Sigma is there with us, we can actually get a better form

of the Ohm’s law now in terms of the current density J, which is current per unit cross-sectional

area, J equals to I current per unit cross-sectional area. And voltage you already know that voltage

equals I R, so which means that I can write I equals V over R, so I can write this as

V over A R, right. Also, I can write R in terms of resistivity, which isRho times L

divided by cross-sectional area A. So if I substitute this expression here, I am going

to get V over A times Rho L over A. I can cross, area and area, so as a result

I am getting J equals V over Rho L. And also, I have already said by definition Sigma equals

1 over Rho, we already said that. So we can write that as Sigma and this voltage per unit

length is in meter, you can call it as the applied electric field E, so I can write it

as J equals Sigma E, where E equals voltage over the length. So thus, we can actually

get this beautiful relationship where the J the current density can be related to the

applied electric field with the help of electrical conductivity and that is a new form of the

Ohm’s law. Now on the basis of this conductivity, solid

materials can be classified into 3 parts; conductors, where the conductivity is greater

than 10 to the power 7 per Ohm meter. Semiconductors, where it is in between 10 to the power – 6

and 10 to the power 4 per Ohm meter, and insulators, where it is actually less than 10 raise to

the power – 10 per Ohm meter, so let us look into each one of this now. Suppose, if I look into the conductivity of

the metals where we, well one of the highest conductivity is silver which is 6.8 into 10

to the power 7. And interestingly, all the almost all the generally the metals that we

use, all of them have 10 to the power 7 range of conductivity, but next to silver is copper

which is 6.0, gold 4.3 into 10 to the power 7 every case, aluminium 3.8, brass 1.6, iron

1.0, platinum 0.94, then carbon steel, further lower than iron because of the alloying 0.6,

stainless steel because of the addition of the carbon it is even lower 0.2, so that is

the level of electrical conductivity. Now, if you consider some of the materials

which are like semi conductive materials, silicon 4 into 10 to the power -4, germanium

2.2, gallium arsenide 10 to the power – 6 and InSb that is Indium based alloy, 2 into

10 to the power 4. So we can see here that really the semiconductors are between 10 raise

to the power – 6 to 10 raise to the power 4 per Ohm meter and conductors are greater

than 10 to the power 7 per Ohm meter. And if you consider the insulators in this

list, then graphite 3 into 10 to the power 4 to 2 into 10 to the power 5, then concrete

it is 10 to the power – 9 so it is so low. Except graphite, everything else graphite

is only having a good conductivity, not as well as metals but still quite a good conductivity,

soda lime glass 10 to the power – 10 to 10 to the power – 11, Porcelain – 10 to – 12,

borosilicate 10 to the power – 13, aluminium oxide less than 10 to the power – 13, fused

silica 10 to the power – 18. And if you consider the polymers where are

we like phenol formaldehyde 10 to the power – 9 to 10 to the power – 10, for PMMA 10

to the power – 12, nylon – 12 to – 13, polystyrene – 14, polyethylene – 15 to – 17 and Polytetrafluoroethylene

– 17. Having said that again there are some polymers like Polypyrrole, etc., which shows

much higher degree of conductivity, but we will come to it when we will discuss about

the polymers. Now, the next point is electronic and the

ionic conduction. How, basically the conduction takes place, there are 2 ways; one is that

conduction due to the flow of electrons and another is the ionic conduction, where current

is arising from net motion of the charged ions, so in order to explain that we have

to know little a bit about the energy band structures. Now, the farthest band from the nucleus is

filled with the valence electrons and that is known as the valence band, so this is our

valency point. Then there is an empty band which is called the conduction band, so this

is where our conduction is. The energy corresponding to the highest filled state at 0 degree Kelvin

is called the Fermi energy level, so this is our Fermi energy level that is the highest

energy state. Just keep in mind that your valency band is below this level in your conduction

band is above this level. There is a certain energy gap between the

conduction band and the valence gap and that is known as the band gap as you can see the

band gap here. Now, in metals the valence band is either partially filled like copper

or the valence and conduction bands, they overlap like in case of magnesium, so there

will be either an overlapping or they will be partially filled. On the other hand, the

insulators and semiconductors have completely filled valence band and empty conductive bands

like this that valence band completely filled and this is completely empty, this part is

completely empty. And the same thing is true for the insulators

that, this is empty and the only difference between semiconductor and the insulator is

that this gap is even more in the case of the insulators. So, the band gap is relatively

smaller in semiconductors, while it is very large in the insulators. So you need to supply

in case of metals only a little amount of energy so that the conduction band can then

start to flow the electrons and cause the generation of the current. So for metals, empty energy states are adjacent

to actually filled states, they are very close. And metals have high conductivity because

of the large number of free electrons that can be easily excited into empty space above

the Fermi energy level. Very low thermal energy excites electrons into this empty higher energy

states, so this is easy excitation through thermal energy is one of the tell-tale sign

of the metallic conduction, which is the conduction based on electron flow. Now, definitely then temperature should increase

this effect. However, when you increase the temperature it would actually cause greater

electron scattering due to the increased thermal vibration of the atoms. So hence, resistivity

of metals increases or conductivity decreases linearly with the temperature. So instead

of allowing the electrons to freely change this gap from the valence band to the conduction

band, what happens is that it increases the electron scattering and as a result of this

free flow actually gets hampered with the increase of the temperature.

And as you can see here this plot like pure copper, from – 200 onward how the resistivity

is continuously increasing that is for pure copper. Similarly, we will see copper with

nickel are smaller or bigger amount, each one of them has higher degree of resistivity,

but the effect is almost similar that it is increasing the resistivity with respect to

temperature. The total resistivity of a metal then is the sum of the contributions from

thermal vibration and impurities, both of them are important right.

So I increase the temperature, there is a thermal vibration contribution that is the

Rho t part of it and so that is the for example, the Rho t part of it. And also, there is the

effect of the impurities like copper I am writing 1.12percent nickel, so that is the

Rho i part of it, the impurity part of it. Also, plastic deformation Rho d sometimes,

so if you further deform the same material, if you further deform it, it will further

increase and that is the Rho d. So basically, total resistivity is actually

part of Rho t that is the resistivity due to thermal vibration, Rho i resistivity due

to the impurities andRho d that is resistivity due to the plastic deformation, so these are

the different things and they can act independently of one another that is very important thing

we have to keep in our mind. Now let us look into the energy band structures of insulators

and semiconductors. Insulators I told you that there will be a

wide band gap greater than 2 electron volt and very few electrons will be excited across

the band gap, so you can see here that there is a wide band gap here and the conduction

band remains empty. On the other hand, if you consider the semiconductor material then

there is a narrow band gap less than 2 electron volt and more electrons are excited across

the band gap, so more electrons can actually go from here to here, but if they go then

there will be a hole formation you know, so we will talk about it that is the good part

of the semiconductors. So there is a drift velocity Vd, which represents

the average electron velocity in the direction of the force imposed by the applied electric

field. So electrons, the moment you are applying the electric field then they will start to

flow and then the average velocity of these electrons is measured with the help of the

drift velocity Vd and this Vd is expressed as a function of Mu 0 times the electric field

E and where Mu 0 is the electron mobility that is the frequency of scattering events,

which is measured in terms of meter square per voltage second.

Now the conductivity Sigma of most materials may be expressed in terms of this Mu term,

which is the electron mobility and we can write it as the conductivity as n times magnitude

of the absolute electric charge on electron, which is 1.6 into 10 to the power – 19 C multiplied

by the Mu e, which is the electron mobility. So, if you have more free electrons per unit

volume, your conductivity will be more. If you have more electron mobility, because any

way absolute electric charge on electron is not changing. If you have more electron mobility,

then also you will get a higher conductivity in a metallic system. Now in the case of semiconductors, the conductivity

is lower than metals but higher than the insulators we have seen it just now in the table. Semiconductors

have relatively narrow band gap generally below 2 electron volts therefore, it is possible

to excite electrons from the valence band to the conduction band. Every electron that

is excited to the conduction band leaves behind a hole in the valence band.

So like 1 electron jumping from here to here, so there is a hole that is remaining here.

A hole is considered to have a charge that is of the same magnitude as that for an electron,

but of opposite sign that is what we call them and the electron deficiency in that region

and we call them as a hole. Now intrinsic semiconductors if you consider,

the electrical behavior here is based on the electronic structure inherent in the pure

material for example, silicon, and germanium. Now intrinsic semiconductors have 2 types

of charge carriers namely electrons and holes, so you can write it as n times, where n is

the number of free electrons per unit volume, so n times e times Mu e electron charge mobility

and p times e times magnitude of e times Mu h, where Mu h is the hole mobility.

So for intrinsic semiconductors, each electron excited to conduction band actually leaves

behind a whole in the balance band, so you may say n equals p equals n i that is the

intrinsic carrier concentration. In that case, you can simplify this expression further as

Sigma i equals n i times modulus of e times Mu e plus Mu h, this is the simplified relationship

of electric conductivity for the intrinsic semiconductors. What happens for the semiconductors if I increase

the temperature? Does it behave the same way as the metals? In intrinsic semiconductors,

the carrier concentration increases with temperature as more and more electrons are excited due

to the thermal energy and they are kind of encouraged to jump band gap and go to the

valence band. So conductivity increases here with temperature, which is opposite to the

metals so like silicon, the band gap is only 1.11 electron volt, germanium 0.67, GaP 2.25

eV, cadmium sulphide 2.40 eV. In fact, you can write actually that n i is

proportional to exponential e to the power – E gap over K T and where K is the Boltzmann

constant. So with respect to temperature, if you got this you will be able to see that

the intrinsic carrier concentration how sharply it is increasing for silicon and then it is

getting saturated with respect to temperature. So similar qualitative nature we will also

see for germanium, so this is opposite to the behavior that you will see in the metals. Now, very few semiconductors that we use today

are intrinsic semiconductors, mostly they are extrinsic semiconductors, which means

they are semiconductors because of the addition of impurity concentration, which is like something

like of one atom in 10 to the power 12 is sufficient to render silicon extrinsic at

room temperature. Now a higher valence dopant example, P 5 plus in Si 4 plus, that creates

an extra electron n-type, while a lower valence dopant or impurity like boron creates a hole

of p-type. So this increases the charge carrier concentration

and hence the enhancement in terms of the conductivity. So as you can see it here that

this is a silicon lattice structure and let us say you have actually added a phosphorus

here, so this is a higher valence dopant so as a result this will create an extra electron.

And on the other hand, if I add something like boron, so this is a lower valence dopant

so electrons will travel from silicon towards this system. So thus, one is where it is actually

giving the free electron then it is n-type and when it is contributing to the making

of holes, then it is actually p-type, so this is the extrinsic semi conduction system. So in n-type, for each impurity atom one energy

state known as the donor state is introduced in the band gap just below the conduction

band. Whereas, in the p- type for each impurity atom one energy state known as the acceptor

state is introduced in the band gap just above the valence band, so like in the n-type if

you look at it that as the electron has travelled up, so you are actually getting here the donor

state that is introduced in the band gap. And on the other hand, in the p-type impurity

you are actually getting an acceptor state as the band gap, so this is what the extrinsic

semi conduction system is. Now large number of electron can thus be excited

from the donor state by thermal energy in n-type extrinsic semiconductors. So further

to this if you apply this actually, if you go to the last slide that if you increase

the temperature, then you can actually create large number of these donor states. So, number

of electrons in the conduction band is far greater than number of holes in the valence

band and that can be written in terms of n the magnitude of e times the Mu e.

In the case of p-type conductors on the other hand, number of holes is much greater than

electrons, p much greater than n due to the presence of the acceptance states and hence

the conductivity for such material is p times mod of e times Mu h. So one case it is Mu

e and other case it is Mu h, so this point we have to keep in our mind. Now what is the

effect of temperature in the case of extrinsic semiconductors? Here it is different from the intrinsic for

example; an n-type conductor will exhibit 3 regions. In the low-temperature region,

known as the Freeze-out region you will see that the charge carriers cannot be excited

from the donor level to conduction band due to insufficient thermal energy. In the intermediate

temperature range, what you are going to see, so this is the intermediate temperature range.

Almost all the donor atoms are ionized and electron concentration is approximately equal

to donor content, this region is also known as the extrinsic region.

In the high-temperature region, sufficient thermal energy is available for electrons

to get excited from the valence to the conduction band and hence behaves like an intrinsic semiconductor.

So with respect to temperature, you have these 3 regions, the Freeze-out region, extrinsic

region and intrinsic region. In the extrinsic region as you can see that electron concentration

is remaining almost constant and then as you are increasing temperature, the intrinsic

region is getting dominated in the system. Now, in terms of the Ionic ceramics, it possesses

a large band gap usually it is greater than 2 electron volts like the semiconductor materials.

Thus, at normal temperature only very few electrons may be excited across the band gap

by the available thermal energy hence it has a very small value of conductivity. However,

for the ceramics I told you this point earlier that it can also be conducted by ions called

ionic conduction. This may either occur in conjunction with or separately from electronic

conduction. There is a new thing that you have, which is ionic conduction.

Both cations and anions in ionic materials possess an electric charge which is capable

of migration under the electric field. Thus, for ionic material conductivity equal to the

sum of electronic and ionic contribution, so they have conductivity as an electronic

contribution and an ionic contribution. Now, this contribution will predominate depending

on the material, its purity and temperature. But still, most ionic materials remain insulative

even at elevated temperature. So, can you think of any scope of applying this particular

property? For example, we can tell you that one such

interesting thing that by properly engineering the point defects, it is possible to convert

ceramics into semiconductors. Example, if you make Indium tin oxide ITO, which has 74percent

Indium, 8percent stannum and 18percent oxygen by weight. ITO is often used to make transparent

conductive coatings for display such as your LCD displays, flat panel displays, plasma,

touch panels and electronic ink applications. In fact, if you look at the aircraft, then

on that window you use ITO films deposited on windshields which are used for defrosting

aircraft windshields, the heat is generated by applying voltage across the film. So thus,

by properly engineering point defects, you can actually convert the ceramics at least

to the semiconductor level. And ITO defrosting coating on an Airbus cockpit window is what

is can be taken as an example of the whole thing. Now, in terms of the polymers, polymers are

in general insulators, they are much more insulators than the ceramics. But some polymers

can be made conductive in two ways; one is introducing an additive to the polymer to

improve conductivity such as carbon black, graphite plate or even silver flex you can

use. The other is, creating polymers with inherent conductivity by doping. Examples

of conducting polymers are like Polyparaphynylene, Polypyrrole, Polyaniline, Acetolpolymers,

so these polymers become conductive when doped with appropriate impurities such as AsF5,

SbF5 or Iodine. Polypyrrole has been tested as microwave absorbing,

so “stealth” radar visible screen coatings. It is interesting to know that in 2000, a

Noble prize was awarded for the development of electrically conductive polymers to Professor

Alan Heeger, Alan MacDiarmid and Hideki Shirakawa. So thus, electrically conductive polymer is

a very interesting application of the polymers. This is where we will come to an end, in the

next lecture we will learn about the dielectric concept and we will solve a problem based

on the Ashby approach, thank you. Keywords- Ohm’s Law, Electrical Conductivity,

Energy band structure, Conductivity in metals, semiconductors, ionic ceramics and polymers.

thanks sir

thanks sir,nicely explained