Hello welcome back again. So, I will begin by summarizing what we did in lecture uh two. That is last time so one of the things that we did is to define drift velocity. So, this is the velocity which the charge carriers attain when subjected to an electric field.

Now what we did is to realize that the electrons in a conductor they are moving with great speed. Actually, it’s of the order of 106meter/second, but they move randomly now when they move randomly the net velocity of all the electrons taken together, because velocity is a vector and if I am summing over vectors in random directions, I get 0, but if I apply an electric field, then there would be a net drift or a velocity which they will pick up with respect to uh what it was in the absence of the electric field.

In other words, in that, in the absence of the electric field, the average velocity was zero, but in the presence of the electric field, the average electron velocity, which will be in a direction opposite to the direction in which the electric field has been applied will be the drift velocity or v d, we had obtained a relationship between the drift velocity and the current density. By saying that j is related to the drift velocity by minus n e velocity. Where n is the number density of electrons e is the charge, and this j and the drift velocity have a relative minus sign between them, and that is simply because we are talking about uh current the velocity of electrons, whereas the current positive current is defined as the current in which the direction at which the positive charges move, they we did do an estimate of the drift velocity for a typical conductor like copper, and we found that it is very small. Its typically vd magnitude is small. We found it to be a few millimeters per second, then we did a comparison of the drift velocities magnitude with other speeds, which are characteristic of the conductor. For example, we have already said that the random speed the thermal speed of electrons is of the order of 10 to the power 6 meter per second, which is, of course, a several orders of magnitude higher, and there is another scale there, which is when you switch on an electric field, what is the speed with which the electric field gets established and we found, since it is decided by the velocity of light?

The electric field is practically instantaneously established when we switch on an electric field, so drift velocity is very small. The having done that we found that, for a very large class of material, a simple relationship exists between the current density, j and the electric field e, and this is known as the ohm’s law. We found that we can write j equal to sigma e or, alternatively, the reverse relationship is called the conductivity and row the resistivity. Now, what we did say was rho and sigma are basically material property. That is the property which depends upon what material it is. We also said that it could depend upon things like temperature and pressure, but we have not said much about it today. We will try to talk about that also as well.

So, the point is: what is the resistance? So, what we said is this: that, while resistivity or conductivity are material properties, the resistance is a sample dependent property. Of course, it depends upon the conductivity or resistivity, but it also depends upon the sample geometry. So, we define the resistance, as I should more properly say that resistance, as seen between or as measured between two points across which we apply a potential difference, and the resistance r between such two points is defined to be delta v divided by so in other words, resistance is defined as the potential difference that must be applied across two points of a resistance in order to get a unit current and of course, we know, the unit of current is ampere, and this is volt. So therefore, resistance has the unit of ohms other than depending upon material property like conductivity and resistivity.

The resistance depends upon directly proportional to the length of the sample and has an inverse relationship with the area of cross section. We also provided an analogy between the conduction of electricity and heat conduction. We found that there is a similarity there having done that, we provided a microscopic view of ohm’s law and we basically pointed out that there is a characteristic time which is defined as the time that is taken or time that elapses between two successive collisions between the Electron and the ions or atoms in the medium, and that is called the relaxation time.

is called relaxation time and we showed that the conductivity is given as , and we also found out that the there is a relationship between the drift velocity and this relaxation time, which is simply . So not one thing that, in spite of the fact that in typical metal, is of the order of 1014 15 seconds e is a small quantity, but because of the fact that the number density is large and, of course, Because m appears here in this expression of the denominator. The number density in typical samples is of the order of 10 to the power 28 per meter cube, and so therefore, that explains why sigma is not very small, because well, v d is small, because n doesn’t appear there.

So, another thing which I have said, but I would like to emphasize, is that when we say resistance of a sample, it is actually a vague statement. It is a vague statement because we said that the resistance is proportional to length and inversely proportional to the area.

Now the question is: what is the length? Is that a standard thing? So, the point to realize is this: that very often, whenever we say that resistance of a sample is so much, we understand that the potential difference has been across the applied across the longer of the sides and that’s what we call typically as a length. But supporting is, supposing you applied the potential difference between the shorter side. Then, of course the resistance will change, so these are the things that we talked about last time and let us proceed with further data.

So let me define a new term, which is called mobility, dictionary wise when I say that something is mobile. Mobility is ability to move, but, of course, in physics we have to be lot more precise. I mean it’s not ability to move, but that’s where the name comes from. So let me say qualitatively: it is the ease with which a charge carrier moves inside a solid when an electric field is applied so notice that mobility depends upon how easily the charges move inside a conductor in an electric field. We will see that the mobility becomes actually lot more important in substances which are known as semiconductors, but at the moment we are talking about conductors. So therefore, we need a quantitative definition. Mobility, by definition, is a positive quantity and it is defined as the ratio of the drift velocity to the applied electric field notice that velocity meter per second electric field is, of course, volt per meter. So therefore, this is the unit of meter square by volt. Second, so this is the quantitative definition of mobility and let us try to see how it is connected to the characteristic times.

Remember that we had obtained this expression for the drift velocity, which is . So, if you substitute it for the electric field, this expression there, the drift velocity by this expression, you will find mu - will be given by e tau over m.

Now this enables you to determine the typical values of mu. Remember this is 10-19. I’m just doing an order. This is 1014or15, and mass of electron is (9 or 10)31. So, let’s take it as 10-30. So, this is typically of the order of 10-3to-4 meter square/four. Second, this is actually very small, so this is very important to realize that, though i said, mobility is the ease with which the electrons move in the presence of a semiconductor in the presence of an electric field. The value of mobility in case of conductors is actually not very much so usually.

This is measured in not in meter square by volt second, but in centimeter square per volt. Second, we’ll do some calculation we’ll see. This is not very large for a substance like copper, etcetera, where mobility becomes much more important or in this semiconductor devices, solid state devices. They require large mobility for their efficient working. For example, if you look at silicon at room temperature, this has a mobility. There are two types of charge: carriers in silicon or semiconductors in general. There is this electron mobility, so electron mobility is about 1400-centimeter square per volt.

Second, this is the electron mobility and there is a thing called a hole, mobility that is the mobility associated with the vacancies in a semiconductor, and that is for the case of silicon, is about a third of this value about 450 centimeters per volt. Second, now recall sigma expression, so sigma was n e square tau over mu. So, this is, if you pull out either so you find this sorry, this is any square tower mass, so this turns out to be e times n times mu, and that is just taking my expression from here that my mu is given by . So, notice that the conductivity has a simple relationship with the mobility, which is simply the electronic charge, multiplied by the number density times.

The mobility now in semiconductors, where both the electron and the holes contribute to conductivity. This takes an expression of this type. That is the charge n times the electron mobility plus the density of holes, which is usually represented by p times the whole mobility. We will be talking more about it, then in our discussion on semiconductors. So, let’s look at the copper which we have been talking about. Remember we calculated in one of the examples. The number density of copper was per meter cube, and we had seen that siemens per meter. So therefore, my mobility is, if you look at this expression, .

My mobility is simply sigma over any you substitute this, so this is . So, this is, you can calculate the number, but let us look at the order of magnitude in the denominator. You have got 109. So therefore, you take it up there, so you get 10-2 and there is a 5.8/8.5 and it works out to 0.0042, okay meter per meter square per volt second, which is 42-centimeter square per volt.

Second, I had already told you that the silicon, for example, has a fairly large electron mobility, which is 1400 also. Now I can look at this data and, in turn find out what is the drift speed like so look at? Let us look at v d expression. So, supposing I apply, let us say an electric field of 10 volts. Supposing I have e=volts. I just now calculated , so that gives you or in other words, 4.2 centimeter per second consistently giving us small number for drift velocity.

Let us return back to ohm’s law, which we talked about, so what we said is the ohm’s law is a linear relationship that exists between the applied voltage and the current. So, the typical iv relationship, if ohm’s law is valid, is given like this, and - and this thing slope here is tan inverse of the resistance V=IR. So that is typical relationship. Most of the time, this relationship will have some deviation from linearity, particularly in this region.

So this is, this is ohm’s law, and this is deviation from linearity now for a large range of current voltage relationship, the linearity is valid and in fact, most of the time in while we are at in our discussion of current electricity, we will be assuming ohm’s Law to be valid, but this is probably a good time to point out that this linearity is not true in many materials but, more importantly, another property in case of most of the conductors is that vi relations that I have given you? It is independent of the signature of v. What I mean is that the current that flows the magnitude of the current that flows. It does not depend upon the sign of v. Of course the direction will change, but it does not depend upon the sign of v, but so does not depend upon what it means is that if you have a resistance - and let’s suppose you apply potential difference between the two ends, with this side positive, this side, Negative, you get certain amount of current that if you change the polarity, that is the potential difference uh between the two ends, instead of this being positive, that being negative, if you apply make this negative, that positive the magnitude of the current for the same voltage, irrespective of its sign remains the same, but this is not true, particularly in when you go to semiconductors.

So, if you look at the current voltage characteristic of a typical diode like a silicon diode, this is completely different from what you see in the case of a metal, so, for instance, for a silicon diode. When you apply a forward voltage that is positive v. The language that is used in diodes is if the diode is forward biased, then what you find is that for certain values of voltage, small values of voltage, the current essentially remains zero and then suddenly there is a threshold after which it rises sharply. This threshold for silicon is 0.7 volt, so that is the type of scale we are talking about, and this is this scale is about one or two volts in this case, the current I is in milliamps now something interesting happens when you apply voltage in the reverse direction.

The direction of current, of course, changes, but the current essentially remains zero for a very large value of voltage, even for 50, 60 volts or so, and then there is a particular value at which the what is known as a breakdown occurs. So, this is called breakdown. Voltage and this reverse breakdown voltage is greater than 50 volts.

Now, if you now look at a typical semiconductor, for example, if you look at gallium arsenide and look at its current voltage characteristic, there is something interesting you find so first thing that you notice is that the current voltage curve in here my current is in typically in milliamps reason volts, it starts with a linear, rather ohmic relation, and then there is a departure from the linearity and it sort of passes through a maximum and at some stage something interesting happens. It starts bending down. So, let’s look at that picture. A little more carefully, so I have three regions here. This is my region, one and this region, one is.

It follows Ohm’s law in radian 2 is a nonlinear region, and the last region that we have is a region where something interesting is happening that, as voltage, increases current instead of increasing, as is normally the case, it starts decreasing. So, in other words, this is actually a region which is showing negative resistance.

There is another thing that I would like to point out remember. I said that when I say that resistance of a sample is so much, you should be lot clearer and point out that the resistance, when I apply it across these points in a register, but on the other hand, we generally understand by length the longer side now in laboratory uses in many practical uses, we need resistances whose values are standard, and these are usually manufactured in bulk and supplied to the laboratories. There are typically two groups of resistances.

The first one is what is called a wire bound. These are made of alloys of material such as manganine Constantine. They are all alloys, necrom wires. The reason why these are used is because, as we will see later, the resistivity of a sample - or in this case resistance, because I am fixing the length and the cross section - it also has a dependence on temperature.

Now these are materials where the resistance is roughly independent of the temperature range. They are fairly tolerant towards a change in temperature and these are used when you want a typical use resistance, typically from fraction of ohms, to let us say several hundred ohms more common are what are called carbon resistances, which also has such study properties. Now, in carbon resistance, there is a color coding used to indicate what is the resistance? If you go to the lab and pick up a carbon resistance, you will find there are certain color bands. There I mean typically a carbon resistance would look like this. So let me suppose this is the resistance. There will be two lead wires across which you can apply the potential difference, but what you will find is there will be different colors here. I don’t have all the colors, but let me sort of draw some one or two that I have actually so this is color coding of resistances. So let me explain: how does this color-coding work most of the time, the resistances that you find in your labs have four bands, so there are four bands of colors and the way it works is this. The colors that are used here are black. I will explain some way of remembering the black brown red orange, yellow, green, blue, violet, gray and finally white. So let me explain how this works. Typically, these are four bands. I have just shown you three, but let me add another one out of this. The first two: they represent the significant figures, so let me explain what it means, so this is significant figure depending upon the color, we assign a value which is black, 0 brown, 1 red 2. 3. 4. 5. 6. 7. 8. 9.

So therefore, suppose you wanted to have the first two numbers as 23. Your first one band will be red, the next one will be orange or, for example, 47. The first one will be yellow the second one will be violet now. The third one is a multiplier. The multiplier is basically 10 to the power. Whatever is the digit, which represents this color I’ll, give an example to explain what happens so, for example, supposing I wanted to write 230?

How what will I do now this? I will write as . So, 23 is red orange, so it will be red first band red next band orange and one is brown. So next brand will be brown. There is a fourth band which tells you, what is the tolerance level, and this fourth band is either silver, which represents a ten percent tolerance or gold, which represents a five percent tolerance or no color. That is, the band is missing, actually missing band, which represents a pretty bad tolerance, which is 20%.

Of course, you would sort of wonder how does one ever remember such things when we were in school? We were given a mnemonic to remember this, so I will repeat that you might have your own, but the one that I learnt is a phrase like this bb Roy of Great Britain has a very good wife good to remember it. So, you realize all that happens is black blue brown, red green orange, great green blue, then of course violet, grey and white, so you could have your own. There are if you look up internet, you will find several, but let me explain this by the following: supposing you have a color combination of this type, supposing you have a yellow, you have a violet, you have a red and a silvery. These are the four bands. Then, if you look at my table there, yellow was four violet was seven red was 2 and silver.

Of course, I told you a tolerance, so we’ll come to silver tolerance, which is a 10 tolerance. So, what it says is this: this 2 represents 47. Our third one represents 102, So . This is what is meant by tolerance, so this nothing but 4.7 kilo, ohms ± 10 percent. Occasionally, but not in your labs, you might find a band with five of them, in which case what happens is the same principle is true, but the first three figures then represents the significant figure. So, you realize this will be useful if you want to represent bigger or higher values of resistance. Having said that, I did mention that the resistance depends upon. Resistance of a sample depends upon temperature.

Let us see why and how? Now the typical variation of resistivity of a sample with temperature has been found to be roughly linear. Now, since is a linear curve, you can take any point as your reference now, if you take any point as your reference, let me call this sum: temperature and let’s say the corresponding resistance is . Then I can represent over this entire length. Rho minus rho, 0 equal to rho 0 times some constant . Alternatively, is given by . So, therefore look at this relationship. This is your resistance. At some temperature t reference alpha is called temperature coefficient of resistivity and, of course, t is the temperature at which you want to find out what is the resistance like? So, basically, what we are trying to tell you is this that the like you know when you apply a temperature, you have thermal expansion. You have, for example, the length changes, delta, l, okay, so in thermal expansion, what we do is to say that the length changes a very similar relationship here. So and if you realize that this is the change in the resistivity when the temperature changes by t minus t0. So, this quantity can be written as . Now this is the definition of the temperature coefficient of resistivity and sometimes in some material depending upon the temperature range. This relationship may not remain valid, in which case you should probably add corrections like etcetera.

So then, for a material such as copper, for instance, versus temperature, if you do it, their variation is like this, so there is a wide length range in which the linearity is valid. But of course, there are some corrections here, so this is typically copper. Necrom is a much better. This is copper. If you look at necrom, this is actually much better is almost linear, but if you look at some semiconductors, the behavior is different. Basically, it goes like this now. Let us look at.

Why is this happening so forgetting about whether it is actually linear or not? I understand here that the resistance or resistivity increases with increase in temperature. Why does it happen in order to understand remember how did resistance arise? We said, as temperature increases my charge. Carriers have a higher velocity, because of thermal velocity is becoming more importantly, the ions in the solid they also start vibrating.

So, as a result, the frequency of collision increases - and this is very similar to the example that I gave you - that if you are randomly moving around in a room where there are chairs now, as long as the chairs are static, you would be still moving randomly. But suppose in the process chairs also started moving randomly. Then, of course your probability of collision becomes much more, and it is because of that that the resistance increases, because, as the probability of collision increases, the relaxation time decreases further.

Now, what happens in a semiconductor? Once again, I must tell you, I occasionally bring in semiconductors so that you can relate to such things when a complete discussion of semiconductor is taken up in later lectures, so in semiconductors. This is not the primary mechanism. What happens in semiconductors is the number density of charge carriers to begin with is low.

Now, as you increase temperatures, the number of charge carriers increase, and that is the predominant contribution to increased conductivity in case of semiconductors, which means the resistivity decreases. In fact, this is the best way in which you can distinguish a conductor from a semiconductor. So, the reason is this: supposing we say we ask the question: what is a good cut now you would say well good conductors are those whose conductivity value is high, but then that’s a loose definition, because how high is high?

Is it ? Is it ? Is there a sharp number? The answer is no, but this is a clear-cut distribution. If you look at the way the resistance of a sample increases when you increase the temperature, if the substance happens to be a conductor, then the resistance will rise with the increase in temperature, in other words, the conductivity, the conductance will decrease. But if you have a semiconductor as you increase the temperature, the conductivity increases the resistance decreases, so this is a much better way of distinguishing. So let me then take an example and work out a few things with which takes care of some of these explain. Some of these things in detail earlier I had talked about drift velocity of copper. I will just change because copper, aluminum, etcetera are typical good conductors. Actually, silver also is, but then one doesn’t play around with silver that much because this of its expense. So let me take aluminum now an aluminum has three valence electrons and at zero degrees centigrade. It has a resistivity of meter. It’s temperature coefficient, which we represented by alpha, is per degree, kelvin or per degree centigrade doesn’t make any difference because, as you know, I am talking about units of temperature, so it doesn’t matter one degree. Kelvin difference is also 1° centigrade difference. So, first thing that we do is we want to calculate the resistivity at room temperature? Let me take the room temperature. This is winter season to be, let us say, 25° centigrade. I told you anything you can take as a reference. So therefore, 25° centigrade is rho at 0 degree into 1 plus alpha times, delta, t and delta t is the change in temperature, which is . Now you can see what this is rho 0, and this is 25 into 4, roughly, so it’s 100 into 10 to the power minus 1. So, it is 1. one roughly one point one, and there is a little bit: zero. Seven five etcetera.

So, therefore, if you look at the resistivity at 25°, it will simply 1.1 times that which simply makes if this was 2.7 - and you add another 0.2, it’s about 2, so 2.7*1.1, so about 2.9 into of course, per meter, return back to properties of aluminum has an atomic mass of 27 and it has a mass density of about 2700. This makes our calculations little easier.

We do the same calculation as we did in case of copper, so we find out how many numbers of atoms are there in aluminum, and that is clear because I have a mass density, which is the mass of 1 meter cube. Then I divide it by the atomic mass, but I take care to see write it in kg number of atoms is . The Avogadro’s number. So, this is roughly. This is per meter. Cube. Now, if I assume that aluminum contributes 3 all three of its valence electrons to the electron gas, then my n will be three times that which is per meter cube.

So, this is this: is your electron density? You should be always careful. What we need for calculation of conductivity is the electron density. Here we are talking about mass density. That is what is its weight per unit volume or mass per unit volume. So, this is what we have got, and so, if you look at sigma, I use my usual n e square tau over m formula and substitute the conductivity values, and you find that this is of the order of the tau is of the order of seconds, you have calculated the mobility, which is sigma over n e. I will not repeat this calculation because we have calculated the sigma and then n we have got, and then e of course is given. If you do this, it works out to 12 cm square per volt.

Second, the corresponding mean free path is obtained by multiplying this number, with the typical value of the electron velocity thermal velocity, which is . This is about 14.4 nanometers or so basically, what has happened is this. So, as temperature t increases, we have the following relationship for conductors the resistivity road increases. In other words, if you take a sample of a fixed dimension, of course, then the resistance r also will increase sigma naturally decreases now the collision time or relaxation time power decreases, because there are more kinetic energy of thermal kinetic energy and the mean free path.

Lambda Also decreases all this is of course applicable for conductors. Let me give you an example of how this temperature dependence of resistance or resistivity can be used to determine the temperature of an unknown heat bath. For instance, we have a platinum resistance thermometer whose thermal element has the following: values of resistance at t=0°. The resistance r of the sample is 5 ohms and at t=100° centigrade. Resistance is 5.4 ohms. This is the property. The calibrated values and when the same thermometer is put in a heat bath of unknown temperature, the resistance becomes 6 ohms.

The question is: what is the temperature of this heat path now? The first thing is, we know that the resistivity rho is related to at any temperature. It is related to the resistivity at certain reference temperature by where alpha is the temperature coefficient of resistivity and t is the change in temperature from this reference temperature. In this case, we take the reference temperature to be 0° centigrade and .

Now, since we are talking about a particular sample, the resistance obviously follows the same rule, because the dimensions have to be multiplied on both sides, so resistance r also follows . So, if you substitute the given values, 5-point r, 4 ohms is equal to 5 ohms here into and if you solve this equation, you find out the value of alpha to be given by per degree centigrade.

Now I substitute this equation: and take if alpha r is taken to be 6 ohms. So, I have . That’s alpha times delta t. This is the new delta t, and if you solve for this, I get delta t is equal to 250 degrees centigrade. Since my reference temperature, with respect to which I had my 5-ohm resistance, was 0 degrees, so the temperature of the heat bath is 250 degrees. According to this method, one thing that we observed is that the reference point can be anything, and that is because of the linearity of this relationship. Let me conclude this lecture by saying that copper at 0 degrees has a resistivity of meter. I am asking what should be the temperature at which its resistivity will double look at this I have row well alpha.

We have just now found out from the previous example we did find out for platinum, but the value of alpha is known, and so therefore, I can substitute it here. All that I am asking is what should be the temperature at which my rho t will be equal to two times zero. You could work this out and we will continue with this next time.

Thank You