 nanohub.org. You can follow along with this presentation using printed slides from the Nanohub. Visit www.nanohub.org and download the PDF file containing the slides for this presentation. Print them out and turn each page when you hear the following sound. Enjoy the show. So let me begin with lecture three. We are trying to understand some basics today, very basic, of quantum mechanics in order to understand what fraction of the total number of electrons available are available for conduction. Not all of them, as I mentioned in last class, that not all of them are available for electronic conduction. So let me explain a little bit about why do you need quantum physics and what the basic quantum concepts are. Now this will be very elementary, very practical, nothing complicated. And so we'll go through them real fast, because at the end we'll not really be using a lot of information from here. Just the basic few concepts, and that I will show. And let me, and then eventually we'll conclude. Now remember the original problem shown here in the left, a semiconductor in between two electrodes. And we are trying to find the number of electrons, free electrons available for conduction going from one contact to the other. Now if you could zoom in inside, which is the middle panel, I have shown here in yellow circles at the position of the atoms. And let's assume the red point is an electron. As you realize, when an electron is far from any of the atoms, the attractive potential it sees from the atoms is very small. As it goes closer to the atoms, then because the electron is negative and the core is positive, there will be a strong attraction. As it passes the atom, then again the potential will gradually go up. And so there is, as the electron goes through, there is this periodic potential going down, bobbing in and out. And so there will be potential going down and coming up, going down and coming up. And that's what I have shown on the right in an idealized manner by a series of sort of rectangular potential. In actual, it will be Coulombic potential, slightly more complicated, an idealized version. Now you can immediately see from here that not all electrons are equal. I mean, they are created equal, but they are not functionally equal because some electrons which are sort of sitting on the right, let's say, inside the potential, that means very close to the core of the atom, they will be just circling around the atom. They will not go anywhere. If you apply a field, they will not go anywhere. These are valence electrons inside the core shell. You remember 1s, 2s electrons, they don't go anywhere. So you cannot count them for electronic transport property. But the other one you see which is sort of above the potential, well, if you apply a field, they can go. So our whole purpose for next three lectures will be to calculate how many electrons do I have sort of are available for conduction, not tied down to individual atoms. You see? So we'll like to solve that problem. Now in solving that problem, this sort of the outline that what we'll first like to calculate that how many states do I have for electrons to sit, right? That will be chapter two and three in your book. And then how many, what fraction of them are occupied? Because if I multiply those two, then I will know how many are available for conduction. Now you see, number of states is like the number of chairs in this room. That means where electrons can sit, but doesn't necessarily mean that electron is there. And then the fraction means that the fraction that are actually occupied. For example, here you can see there are lots of chairs in this room, of course quantized, one chair for one person, but it's not all full. A fraction of them are full, some of them are empty. So when I want to know electronic transport, we are only interested in the fraction that are occupied. So those two information we'd like to calculate, but first the chairs or the places where the electrons can sit. And that's what quantum mechanics will give you. Quantum mechanics will not tell you where, what fraction of the states are occupied. It will simply tell you where electrons can sit and the rest will follow. Now this is a old version of quantum mechanics. This is how people teach things in high school and for us, or maybe in the college, this will be sufficient for our purposes. Now the quantum mechanics came about because a set of experiments could not be explained very simply. They tried, you know this history that in the beginning of the 20th century, there was a series of complicated experiment that could not be interpreted. And as a result by 1925, this quantum mechanics was born. And there are really four experiments or four ideas that leads to this. The first one is a blackbody radiation. And I'll explain what it means for electric effect. Einstein won the Nobel Prize for this work. Bohr-Atom, Niels Bohr work, and the wave particle duality. These four concepts will lead us to something called Schrodinger equation. And from Schrodinger equation, we'll be able to calculate how many places the electrons have for them to sit, like the counting of the number of chairs. So we'll see that. So the first experiment about blackbody radiation is the following. So if you take, let's say a sphere, a hollow sphere and a little opening from one side and increase its temperature, then and notice the light that is coming out of that little opening on the side, shown here, for example, in the red. Now these colors don't take it seriously. It is just for illustrative purposes, but essentially some wavelength is coming out at some wavelength you may or may not be able to see. Some let's say in the infrared, some in the ultraviolet, but the point is the wavelength or the corresponding color is shifting as you increase the temperature. And what people have noticed for a long time that as you increase the temperature, the wavelength of the light that is coming out of that box or that sphere gradually decreases. So the x-axis is wavelength, but you can see this is in micrometer and log scale and the y-axis is the amount of light that is coming out per unit wavelength at a given temperature. So let's say you have a vertical thing at 10 micrometer, that is how much light is coming out. If you raise the temperature to a thousand degree, well, the wavelength shifts to a lower value, the color changes, but the intensity might also go up. 2000 degrees, very hot, most silicon and other materials will melt at that point, but the wavelength has shifted downward and the intensity has gone up. Intensity going up, we sort of understand, right? It's getting heated. Of course, the intensity will go up, but the wavelength shifting down is not yet clear, but that will come a little bit later. Now, although we saw in the last slide that one wavelength was associated with every temperature, but if you looked a little bit carefully, you would see that that wavelength, that single wavelength I showed at a given temperature, let's say at 2000 degree, that single wavelength is not really one wavelength. If you look at finally, you see it's primarily looking great, let's say, but if you looked carefully, you would see a little bit of the infrared and a little bit on the other side also, and this is the spectrum, this almost like a parabolic structure, that is what you will get. That is that, yes, this is a log plot on the y-axis. So yes, there will be one dominant color, right? That's dominant one, but there is also a little bit of other colors mixed in. And one of the challenges in the beginning of the 20th century was that people couldn't explain how this shape comes about. They could explain almost everything, but they couldn't explain the shape, and that is when people began to realize that there is something wrong in the theory. So they could do almost, but not quite. So one group of people could explain why the higher wavelength part behaves in this particular way. So for example, the higher wavelength part, they said they made a calculation, very simple calculation, and they said that it should be going as KT, T is the temperature, K is the Boltzmann constant, KB is the Boltzmann constant divided by lambda over four. That's what they said, and you can see that that's about right because if you take a log on both sides, it says the log of U, which is the energy that is coming out part unit wavelength, that should have a slope of minus four on a log-log plot. And you can easily check that for one order of magnitude change in the lambda axis, you indeed have four orders of magnitude change in the y-axis. So this equation is indeed right. But what is it that it cannot explain? It cannot explain the turning over. It says as the wavelength is getting smaller and smaller, your energy will keep getting bigger and bigger. That's obviously not true. So that's wrong. And so there's another experiment or another theory, which is in fact a little bit earlier in 1888. That was this Wayne's formula. He said, well, the lower part, you can explain by this formula. And again from pure thermodynamics, you could get this formula. Now, you don't have to really know the derivation of it. You will learn it in other courses. That's not the point. But the point is that again, you could explain the lower part in this particular fashion and what Planck did. Planck was a young person at that time, relatively young person. What he simply did? He said, well, I know the high energy part or high wavelength part. I know the answer for the low wavelength part. Well, I'll make up a formula. No theory, nothing. I'll make up a formula that sort of interpolates between the two. And that's exactly what he did. He was at a conference and one morning in the conference, he simply proposes this formula. It's a pure fitting formula, no physical content in it when he proposed. That sort of goes in the right limit to either of these things. And I will ask you to check. For example, when lambda is very large, when lambda is very large, you can see that beta to the power lambda t, beta to the power lambda t, will it be? It'll be a small quantity, right? Small quantity. And when you have a small quantity, then you can expand it as one plus lambda, beta divided by lambda t. What is beta? Beta is, by the way, is for the time being, it's just a constant. We're just assuming it as a constant. And from that, you can easily get the religion's formula in one end. And similarly, you can get the vanes formula at the other end if lambda is small. You see, if the lambda is small, then the exponential term in the bottom is very large, much bigger than minus one. So get rid of minus one. The exponential goes up and you are set. You find the vanes formula. But this was a fitting formula. He didn't really understand that what it means. But within a few months, he was able to come up with an explanation. And the explanation was the following. What I'm doing that is easier to work often in terms of frequency rather than the wavelength. So you can simply do a conversion between frequency and the wavelength because these are related, right? These are related by the fact that the velocity of light, remember we are talking about light, right? Light emission that is given by lambda, the wavelength is inversely proportional to the frequency because lambda multiplied by F is the velocity of light. You can simply do a conversion and you can find that how per unit frequency rather than per unit wavelength, how the energy changes. Okay. Now then that formula that you have over there, he interpreted it in the following way. He said, in order to understand what that formula means, that why the light is coming out with this strength at this wavelength, let's interpret it in the following way. He said there is three pieces. One piece is the number of modes you have in that cavity. Remember that was a spherical box. So number of standing waves that you can have in the cavity. That's number one. Goes as F square. The second piece was that of all those modes, one-year fraction are occupied. So for example, the mode, let's say the red one on the left, that has the lowest energy, right? Because that's the simplest mode and that will have a certain occupation probability because this is the lowest energy on the right side. The blue one is again, there is a corresponding occupation probability for that one because it's a little bit higher energy because it has more modes, more oscillations. And the final he said, the remaining piece that remains, which is HF, H is Planck's constant because he's doing the derivation. So it's better associate his name. So H is Planck's constant and F, so he says that piece must be because this is the total energy. If I know the number, F square is the number of modes, the fraction occupied, so that gives me the total number of modes that are available. If I wanted to get energy, then the remaining piece must be energy per unit, per unit of those photons. So that is what he said, that energy per mode. So what this derivation at the end, you don't really have to remember any of this. The bottom line is this. At the end of the day, what he concluded, that the light that is coming out of the tiny hole from that black body, that must be coming out with a quantized energy equal to HF. If I see a particular frequency coming out, there must be a photon correspondingly with energy, HF. That's it. It cannot come in, let's say, 2 third HF. It cannot do that. It has to be HF or 2 HF for 2 photons, 5 HF for 5 photons. Light coming out in a quantized form. It is a completely different idea compared to before. But then came along Einstein and he said something which is the complementary part of it. That not only light comes out when you hit something up, not only light comes out as a quanta, but light also gets absorbed as a quanta. That is the second experiment. Oh, by the way, so before I get there, this information is, although it's old derivation, is often used. For example, people often say that, you know, this big bank theory, how to prove it, and then the background temperature of the universe is some three degrees or so. How do they know those things? Like how do they know the temperature of the sun? I cannot go there and actually put a thermometer in the thasan, obviously. How do they know that? They exactly use this formulation. So they take the spectra. You can see the intensity as a function of wavelength here shown in millimeter. You can immediately see that the same bell shaped curve or similarly parabolic curve, you fit this curve and then once you fit this curve, you backstrike the temperature. And once you backstrike the temperature, you actually know what the temperature of the original object was. That is how people still use it all the time. It's a very recent. This is a Kobe satellite data, very recent, but you see that that formula is used all the time. By the way, this you cannot see because this is in microwave range. Now the second experiment was this photoelectric effect. What is photoelectric effect? If you take a semiconductor and then shine a flashlight on it, you'll see that electrons like popcorns is getting out of that material. So these are electrons getting out in response of photon, photoelectric effect. Now, what was the amazing thing is that these electrons that come out have very specific energy. Given that you shine a light of a given frequency or given energy, if you shine it, it will come out always the very specific energy. And this is how it works. Assume this is a material, the top surface is the vacuum level, meaning that's where an electron can become completely free. And let's say all the electrons are pulled down somewhere minus W, my W is a work function. Because of course, if the electrons didn't have lower energy compared to vacuum, all the electrons will simply go out, dissolve out. So they are being confined in sort of a container with this potential of minus W. Now when light comes in, light of 8F when that comes in, then electrons are pumped out of the material. This whole region shown here in blue are full of electrons. So electron is pumped out. Now what would be the energy of those electrons? You see, it came in with HF, that was the total amount of energy, in order to jump to the top floor, or so from the basement to the ground floor, you lost minus W or W. So HF minus W would be the available energy for the electrons that are coming out. Now how would you measure that velocity, half V square? So people had this ingenious idea. So what they did was they put a cathode nearby, and when the electrons came, it's like a break. So if you apply a negative voltage, and the electrons is coming with a certain velocity V, for right voltage, so it is going to break the electrons. Because it's moving, coming close with a velocity V, and you are putting a negative potential for the electrons, which is negatively charged. So it's going to break it, and if the potential is just right, the retarding potential is just right, it will just come and stop. And at that voltage, you can say what the original velocity was, right? So that is how they measured for every photon coming in at a certain frequency F, what the corresponding velocity V is, by measuring the retarding potential that is required to slow them down to velocity zero. And this is the expression you can easily see, that this is how it should work out. And experimentally, this is indeed how it worked out. That is that if your wavelength of the light is less than W, nothing is going to come out. It's going to jump up, but this doesn't have enough energy to be free, it will then go down. So you do not get anything below W, and above W, then it's going to come out linearly. And so this experiment proved conclusively that indeed light is absorbed as a quanta also. Last experiment, blackbody, it says light is coming out of a quanta, this one is saying light is being absorbed by a quanta, right? Now, if something is being emitted as a quanta and something is being absorbed as a quanta, there must be something inside that material that is saying that why it is quantized, always is quantized. And so the idea is the following, that that must mean that in the surface of that volume, there must be the atoms must be doing something. It must be absorbing things in quanta and emitting things in quanta. And the only way it can do so, if the individual levels in the atom itself are somehow quantized. Remember, that's what we're trying to get, right? Not all electrons are the same, they sit at a particular place. So we are getting in that direction because now we are seeing that electrons cannot sit arbitrarily at any place. There are specific chairs for the electrons to sit and that is what we are going to discuss now. So the solution came not from the atoms of the black body, which is too complicated, but from the analyzing the spectra of hydrogen, which is a simple atom and that is where we'll pick up the story. So what people often saw, that in the hydrogen spectra, then certain lines are missing. If you look at the transmission, certain lines are missing and they could fit it with this formula, you can see em sub n is em comma n proportional to a constant and that constant is called a Riedberg constant because here's the first person to actually fully explain this. One divided by m square minus one over n squared, m and n are two integers. It cannot be 3.2. It has to be one, two, three, four, whatever that number is. And that is explained by the following relationships. So you have let's say e one, if you have e one and if you have e two, then e two one. So m equals one and n equals two. If you insert it in this formula, you can say explain why the light that is coming out has this quantized energy. And similarly when light is coming in, it will promote a carrier, let's say for electron from e one to e two. So this is explaining why things get absorbed in a quanta and also emitted as a quanta. Now one thing I want you to remember here, for hydrogen, look at this number minus 13.6 and in a few minutes, we'll see why the constant is also minus 13.6. So let's explain this one, how it comes about. So this was explained by Bohr, you know. This is a very simple old quantum mechanics. He simply said the following. He just made up this assumption that if something is going around a atom, then only orbits that are allowed are quanta that has quantized angular momentum. So angular momentum is ln and angular momentum is given by m naught v and rn. Now hopefully you haven't forgotten about angular momentum but you can always look it up in a high school or college physics book. And he says that it has to be quantized, n h bar. And n can only be one, two, three and those values, it cannot be a fraction. And from that, you can easily calculate from that relationship that the velocity at which an electron goes around the core is inversely proportional to r. That farther out you are, it will go with a smaller velocity. Let's say. And m naught is the mass of the free electron. Now from that, you can also have a second relationship. No, not from that, you can also have a second relationship that the velocity at which the energy at which electrons are trying to go away or tear away from this attractive potential is m naught v squared divided by rn. You have seen this expression before. And that must be equal to the Coulomb potential between the two. Now why Q squared? Because proton has a Q, electron has a Q. Because it is hydrogen, one proton. So therefore you have Q squared. If it was something else, let's say helium, then it would be two Q squared, right? Because two Q from the proton and one Q for the electron. And four pi epsilon rn squared. This is Coulomb's law. Now this relationship, if you combine, insert the velocity v from the previous relationship in here, you immediately get an amazing relationship. Which says that as soon as angle is quantized and energy is conserved, then the orbit around which electrons can go, r sub n, is quantized as well. Look at this formula. Except for n, everything is a constant. Epsilon naught is a dielectric constant. H bar is a Planck's constant. Q and m naught, of course, all constants. So rn, for n equals one, rn will have a value. n equals two, rn will have four times the value, right? Because it goes as n squared and so on and so forth. But we are not interested in rn. We want to know what the energies are. So we'll have to work a little bit more. So that was the rn from the previous one. Now you know the kinetic energy is half mv squared and from the last relationship of energy conversion, you can always express it in terms of the Coulomb energy as well. You can do it just two lines. I didn't repeat it here. And you also know the potential for an electron bringing from the infinity to any this point r away from the center is given by q squared divided by four pi epsilon rn, right? This is something we also know. What is the total energy? Well, it's the sum of the potential and the kinetic energy. And if we sum them, we'll get this explicit relationship. And then finally, you have the rn you can see. Let me go back. The rn you see in this expression, that is the only unknown, but we have already calculated rn on the top. You put it in and you get an expression for energy which has a bunch of constant but goes as one over n squared. And do you see the 13.6? The 13.6 was the first level. So that is what this energy level is. And if you wanted to know how much energy difference there is between two steps, two layers, you will take this one en and you will take em, subtract them and that will give you this formula, right? Now I want you to remember one thing from here that we'll use later on. Most of this information we'll not use later on is that this formula had an m naught free electron mass. Later on we'll see that this formula will be modified for some other cases. And I also want you to see the presence of epsilon naught because this is sort of in vacuum. Electrons going around in vacuum. But we'll later see when electrons go around in a solid, of course you cannot take epsilon naught, this is a solid. And so these two numbers will change later on and without these two numbers changing, most of the computers that you use today wouldn't be usable. And so we'll see how these two numbers are changed and how they affect device performance. Okay, so you know why then electrons are absorbed and emitted as quanta. The last thing before we get to Schrodinger equation and that's where we'll end is this notion of a wave particle duality. Now the first expression I have written is something E equals square root of this entire expression you may or may not have seen this before. But you actually know this one very well. This is simply and this I haven't written is E equals MC squared. You know the Einstein's famous relationship, the atom bomb and everything. That relationship written out in a long hand because you can see that this M that we have for electrons is not really the rest mass. This is the mass the electron has when it's moving at a velocity. If you wanted to relate that electron mass with the rest mass, then you should have use this formula. Do you remember that when electron moves close to the velocity of light, then its mass begins to be very high. And so you can see from this expression that the denominator will actually blow up if you get close to zero when V approaches C. This is that expression you should check it out. So it's E equals MC squared written in a slightly different form. Now let's first think about photons. So let's think about photons. Now photons you remember is emitted as HF, right? It's quantized, emitted as HF. Free F is a frequency. So the left hand side I should replace it at HF. On the right hand side, photon doesn't have any mass, rest mass. It doesn't have any rest mass because it's always moving at a velocity of C. So it's M naught is zero. And so if you take under the square root, you'll see HF is equal to P. P is the momentum, C is the velocity of light. And then you can easily express P by using this relationship by noting that that F divided by C is the wavelength because that's the velocity of light is constant. And this lambda, one over lambda, you can also express it in terms of wave vector K. Wave vector K, right? It's two pi over lambda. These are some, no physics here. Just some redefinition. But what it is saying that photons move, the photon that moves with a momentum P is related to its wave vector because we know about the wave vectors of a wave. We generally don't talk about the momentum. Of a wave. But these two are nice related. P equals H bar K. Let's say that's one relationship we will use in the next few minutes. So let's try to derive, Quoton could derive the Schrodinger equation. This is how it works. Remember that original relationship is equal to MC squared written in this strange form. Now we are not talking about photons. Two slides ago, we had been talking about photons. Now we are talking about electrons. Now the electrons, the momentum is very small. Now in that case, an M naught is not equal to zero. MC squared is a very big quantity. So if you expand under this square root, let's say, this you should check that you can expand it in this particular form. M naught C squared, you can see the first term and the rest of the terms can be expanded in this particular form. Check it out when you go home. I can take that M naught C squared multiplied by one to the left hand side and that gives me E minus M naught C squared. And on the right hand side, I have that M naught C squared multiplied by P squared C squared, the second term. And that's the second term you can see P squared divided by two M naught. And I have added a V, arbitrarily added a V. But apart from that, I haven't done anything from the first line to the second line. Now that extra energy beyond the rest mass will say that that's equal to HF to the left hand side. And to the right hand side, do you see? YP should be replaced by H bar K, right? That was the last slide. So I replace that with H bar K squared divided by two M. And that's what I have. This is a relationship I have for electrons. Previous one is for photons, this for electrons. So that's it. One more slide and we'll be done. We'll be in good shape. Now, the first thing is that, remember, I have just copied the last equation from last slide, H omega is H squared K squared divided by two M naught plus V. Now let's assume, and this is out of the blue, no reason for this particular assumption. But let's assume that there is a function called psi and whose name is a wave function. That is given by, that's like a wave, electron wave, is given by e to the power i omega t minus Kx. This is the equation of a plane wave. When a wave goes from one point to the next for electromagnetic wave, this is what we write generally. But we are writing it this time for electrons, which is supposed to be not a wave, but we are writing it anyway. Now you can see that if I take a first derivative of that function, then, sorry, then I will have, I will pick up a i omega, because with respect to time, I will pick up i omega in front. If I take derivative with respect to x, what should I pick up? Not i omega, I will pick up a i K. And if I do it twice, then I will do i K squared. And so you can see that minus sign in the second derivative for the wave function. Now look at this. On the left-hand side, I have an expression for omega. And on the right-hand side, I have an expression for K squared. If I plug it back in in the top equation, lo and behold, this is an equation you get, which is, do you see this? You simply have to multiply the top equation with psi throughout, left and right, and then replace these identity identities. And you get a Schrodinger equation out of relativity and out of this wave particle duality. It's a very simple way to get the full Schrodinger equation. And this equation is going to tell us where the electrons sit and how many chairs I have in a semiconductor. And then I will fill it up later on in chapter four. So I will end here. So what I've tried to do is this is, I've tried to mention that this is a part of three lectures on trying to find where the electrons sit. And it came about because, because when you try to multiply the total number of electrons with the density of the particles or density of the atoms we have per centimeter cube, the result didn't make any sense. So we are looking for the fraction of electrons that actually contribute to the transport. And we are beginning, this is the first class for Schrodinger equation you can already see that electrons don't sit everywhere. They have a specific place to sit. And this is solution of the Schrodinger equation will tell us where they sit. And so in the next class, we will see how to solve Schrodinger equation for toy problems in a small potential. But the real interesting thing, as I mentioned, and that's how we'll conclude, is this Schrodinger equation, if we have to solve for the real crystal with 10 to the power 20, 10 to the power 21 atoms, no computer in the whole world will be able to solve that problem. Impossible, no computer. And not only now, I cannot even imagine that any time, even in the next 200 years, there'll be a computer so we can solve it. But because of the periodicity of the crystal, you see, what we'll be able to do, we'll be able to solve that problem in essentially two slides. The solution of all the electrons in 10 to the power 20 electrons, where they sit. And that will be something very interesting. And once we get that conclusion, then we'll be able to apply it to all the materials that are of interest, crystalline, amorphous, polycrystalline, all those materials, we should be able to use them for. Okay, all right. Thank you.