 We had discussed in detail about the physical and mechanical properties of not only normal bulk materials, but also of nano structures and nano materials. The next topic we take up is the topic of electrical properties. And as before we will first describe some of the basic concepts involved in electrical properties and electronic properties, followed by electrical and electronic properties of nano structured materials and nano crystals and other nano entities. Further, we will take up part of this overall gamete of what you might call the electronic structure and electronic properties, magnetic properties and also optical properties that will be in the coming lectures. Now if I were to consider electrical properties, then before we had already noted that two important factors come into play when we want to understand the properties of materials. It is of course, at the heart of all the properties is what we may known as the electromagnetic structure. And crudely we would like to sort of divide this overall electromagnetic structure into what you may call the atomic structure, where we worried about the atomic entities and their positions. And also what you may call the electron distribution, which you might call the electromagnetic structure in this specific context. This electromagnetic structure can be thought of a in a simplified way as the special and energetic distribution of electrons. Take into account the charge and the spin of the electrons. In other words, when you are talking about electromagnetic structure typically unless we are really interested in very specific properties, typically the nucleus of the atom is usually ignored. So, we are worried about the electrons, we are worried about the electron density, we are worried about the spatial and energetic distribution of these electrons. And in this process, not only we are worried about the charge of the electrons in terms of its density, but also we worry about the spin of the electrons. And as we shall see later, the spin is the dominant force behind the magnetic properties, which we will take up. And usually in the common language, we call the language of bonding, this electron density distribution is interpreted as kind of a bonding. In other words, if the electron density distribution is high between the two atoms, we call it a covalent bond. If the electron density distribution is shifted towards one atom, we call it the ionic bond. And if the electron density distribution is delocalized, in other words, the electron belongs to the whole solid and not to a particular atom, we call it such a material or such a kind of a bonding as the metallic bonding. And just to reiterate, the nuclear aspects are usually ignored unless of course, we are worried about a certain kind of important phenomena, wherein the nucleus may play an important role. And in these set of lectures, we will not consider that further. And often when you are talking about electronic properties or electrical properties, we are interested in the response of a material to two kind of entities. One is the fields, the electromagnetic fields, which is essentially would be, which would mean that we are talking about optical kind of properties and other kind of external stimuli like heating. So, these two aspects we always keep in mind, when we are trying to understand for instance, any kind of properties. Suppose, I am talking about electrical conductivity, then I will worry about how electrons are accelerated, when you apply an electrical field and how this electrical conductivity is actually going to change with temperature. And as we shall see soon, many aspects of this response, that means the response of the material to external stimuli and these external stimuli could involve fields and heating and other kind of stimuli is governed by a quantity known as the band structure of the solid. In other words, we have individual atoms, which have discrete energy levels, but when these atoms come together, we have a concept of a band, wherein the electrons no longer belong to a single atom, but to belong to the entire solid as a whole. And this band structure is going to determine some of the important properties, which include the magnetic optical and electrical properties of the solid. Let us start with understanding, what you may call the resistivity and its variation across different kind of elements. You know, resistivity is the coefficient, when you are talking about the variation of resistance with geometrically related terms like the length of the conductor divided by the area of the conductor. Resistivity happens to be one of those kind of, one of the rare kind of material properties, which varies to about 25 orders of magnitude, when you go across from one element to the other. On one hand, we have extremely good conductors with very low resistivities like silver, which at 20 degree Celsius has resistivity of the order of about 10 power minus 8 ohm meters. On the other hand, we have extremely high insulators like few squads at 20 degree Celsius has resistivity of the order of about 10 power 17 ohm meter. So, this is one of those rare quantities, which varies orders of magnitude, which implies suppose I am using an aluminum as a conductor. And if on the surface of this aluminum conductor, you form an oxide, which is going to be Al 2 O 3, then you would notice that the resistivity is going to jump by orders of magnitude. And this implies that my electrical conduction is going to be very poor just by this mere process of oxidation at the surface of this aluminum conductor. There are materials with intermediate conductivity like silicon, which is in conductivity of the order of about 10 power 2 ohm meters. And if you look at how this resistivity changes with temperature, then the behavior 2 here too is very different. For instance, if you heat silver, then the resistivity would increase. On the other hand, when you heat silicon, actually the resistivity would drop. In other words, if I want to understand such phenomena, then as we pointed out in the previous slide, we will have to look at what is known as the band structure of the solid, which will tell us why such a behavior of resistivity is formed, when I go from one kind of a material like silver or copper or gold, which is a very good conductor to intermediate level conductor like germanium or silicon to a very poor conductor like diamond or PVC or fuse quads as the case may be. So, even for a simple property like electrical conduction and material property like resistivity, I have to look at some of the details of the band structure or the electronic structure of the material. The simplest kind of a theory, which has been proposed to understand what you might call a phenomenon like conductivity or the variation of conductivity temperature or the what you might call the origin of resistivity, when you actually apply an electric field is known as the free electron theory. In the free electron theory, the outermost electrons of the atoms are only considered to be taking part in conduction. In other words, these electrons belong to the whole solid and not to a particular atoms and these electrons are assumed to be free moving through the whole solid. So, they are delocalized, they do not belong to a single atom and this electron gas is often called a Fermi gas or a free electron cloud. And it is assumed that the potential field due to the ion course is constant and the potential energy of electrons is not a function of position. Though it is moving around in a potential of these ion course, it is assumed to be a constant that means, there is no special variation of these what you might call the potential in which these electrons are moving. And it is to be noted that the kinetic energy of the free electron is much lower than the bound electrons in an isolated atom. So, now we have a simplified version of what you might call the real picture, which is called a free electron theory. But interestingly, this free electron theory is able to explain many concepts, many simple concepts and we will take them one by one. And we will slowly build towards what you might call the region where we need to transcend this free electron theory and go into a what you might call a band structure, which will be needed for the explaining the conductivity of semiconductors and insulators. The starting point for all this is that we have something known as the wave particle duality of electrons. In other words, every moving particle has certain wave nature. This wave nature becomes more and more important when the mass of the particle becomes small. So, for a very lightly massed particle like electrons, the wave nature becomes extremely important. And we will see later on that this wave nature is that the heart of what you might call quantization of energy levels, when we are talking about a nano crystalline material or a nano structure, wherein we want to understand the conductivity. So, and this wave length is given by h by m b and typically in a free electron theory, you would plot what is known as the energy of the various levels, the orbital levels as a function of k, which is the defined as 2 pi by lambda, which is goes as the inverse of the lambda. So, this wave number or the wave vector is what we plot against energy and in a free electron theory, the energy is a function of k square with other constants add around including the mass of the electron. In other words, you have what you might call a continuum of energy levels, which increases parabolicly and electrons can occupy these continuum of energy levels. And this implies that if suppose an electron is sitting at the top most energy level, there are energy levels are free above that to which it can be excited. So, they are in proximity to this energy level and if you are looking at an increase in the wave de Broglie wave length, then you would notice that the k decreases and the energy decreases. So, in this, but of course, if you look little closely, you will have to note that these energy levels though have to be as if from this kind of an equation looks continuous, they are actually slightly discrete because of the Pauli's exclusion principle, which tells that no 2 electrons can actually have the same set of quantum numbers. And therefore, if you look closely, they may be a closely spaced electron levels, but nevertheless for now we will assume they are continuous and therefore, there is a series of levels, which can be available for the electron to occupy. Another kind of a diagram, which we will encounter, wherein we do not plot energy versus the wave number or wave vector and which is where when we try to notice a different kind of plot a different kind of quantity, which is called the density of states, which is also encountered more frequently than an e k kind of a diagram. Now, we have to of course, note that when you write down the energy as half m b square, if you are ignoring the relativistic corrections. Now, we noted that suppose I fill start filling electrons and I know that, therefore, there are n number of orbitals, which need to be filled with these electrons, there will be obviously an highest energy level, which is filled. So, all the electrons in the solid might fill up and still you will only reach a certain energy level, which means all the states below this energy are filled up by electrons and this is the top most energy level, which is filled. This top most energy level, which is filled is called the Fermi level and typically the concept of Fermi level is defined as its highest energy level filled, but at 0 Kelvin. Now, if in other words, if I look at my energy versus the probability of finding an electron, I would note that at 0 Kelvin, the curve, which is the green curve is the one, which is valid. In other words, up to the highest level of Fermi level, all the energy levels are filled. That means, the probability of finding an electron is 1, but beyond the Fermi level, no energy levels are filled at 0 Kelvin, which means the probability of finding an electron above the Fermi level is 0. So, it is 0 here. Now, suppose you and this Fermi level can be given by a formula like this as E f is h cross by 2 m into k f square, where and which is related to n, which is the total number of orbitals, which are available. In other words, in the k x, k y, k z space, you would notice that the energy of the highest electron, which is a constant energy surface is a sphere and this sphere, the electrons lying on that sphere have an energy, which is equal to the E f. Now, if I look at the total number of orbitals with energy below E and of course, I can replace E by E f at 0 Kelvin is given by the function n, which is v by 3 pi square v is the volume of the specimen and it goes as 2 m e by h cross square and the important thing to note is this exponent 3 by 2. The importance of this exponent will come, when we do not track actually the total number of orbitals, but we go to the what I mentioned as the density of states. Now, suppose I heat this specimen above 0 Kelvin, then what will happen progressively with increasing temperature is that more and more electrons will be excited to these higher energy levels and there is no barrier to this excitation, because these energy levels are closely spaced and are just above the Fermi level, which implies that now my probability of finding an electron above E f increases. That means, some electrons, which are originally here are actually excited to this level and that means that then the probability of finding an electron below the Fermi level slightly reduces and above the Fermi level increases. This probability of finding an electron at any finite Kelvin temperature is given by this Fermi Dirac function. In other words, I multiply my probability of finding an electron at 0 Kelvin by this Fermi Dirac function and therefore, I find the probability of finding an electron at any finite temperature using this Fermi Dirac function. Therefore, with increasing temperature what happens is that more and more electrons are promoted to higher and higher energy levels, which are now accessible to the system, because energy levels are continuous and parabolic and therefore, the tail of the Fermi Dirac function starts to grow. This implies that I can actually increase the energy of the electrons. Of course, at the expense of some of the electrons, which had a lower energy level by merely heating the solid. Now, this process can also be done by an alternate method and that is by applying an electric field and this application of an electric field would lead to what we know as the conduction of these free electrons. Since there are empty energy levels above the Fermi level, then in the presence of the electric field, there is a redistribution of the electron occupation energy levels. In other words, in the absence of an electric field, you notice that all the electrons are filled up to the top most level, which is now my Fermi level. The green line is shown here and all the levels above the Fermi level are completely vacant. But what happens when you actually apply an electric field that there is an asymmetry between the positive k x direction and the negative k x direction. So, this is my positive k x direction and this is my negative k x direction. And therefore, more electron levels are occupied in the positive k x direction. That means electrons are promoted, the electrons which originally were occupying this level are actually now promoted to this higher energy levels. And since like before for the thermal excitation, there were free energy levels in the what you might call in the e k diagram. Therefore, these electrons which gain momentum can actually occupy these levels and therefore, you will have electrical conduction. And of course, you can write down the force experienced by the electron in the presence of the electric field e as f equal to e into e, where e is the electron charge and m is the mass of the electron, if you want to write down the acceleration as m a. So, I have two ways of actually exciting electrons to higher energy levels. So, point number one of course, is that there are continuous set of energy levels and at 0 Kelvin it is filled to a certain energy level known as e f beyond e f there are no energy levels filled, but there are continuous free energy levels available above the e f also. And I can actually excite electrons to these energy levels by either heating the solid or by applying electric field. In the case of heating of the solid, there is actually no net flow of electrons in any given direction, but when you apply an electric field electrons tend to flow down the electric field gradient. And therefore, because of the electric field they do flow and because of that you have actually electrical conduction, but if you if you just notice this part that f is equal to m a that means the electrons will be accelerated by the presence of this electric field. And what you would expect with time is that the electrons keep on gaining velocity and soon they will obtain a very high velocities and perhaps they will get to a close to a velocity of light, but then that is not what is found this is because no material is perfect. And what really happens is that this when the electrons are accelerated or constant acceleration they gain velocity. When they approach a velocity known as the v d they get scattered they suffer collisions and when an electron suffers a collision its velocity actually falls down to 0. Now, because of the presence of electric field again the electron is accelerated at constant acceleration we assume it to velocity v d, but then it again suffers a collision and the velocity comes down to 0. This of course, is an idealistic picture in reality of course, you might find that one electron accelerates a little more compared to another electron before it suffers collision, but what we are talking about is average quantities here. Now, so what we are saying is that in the presence of the electric field the electron velocity increases by an amount about its usual velocity which may be a diffusive velocity to an amount called the drift velocity. So, the drift velocity is the velocity acquired in the presence of the electric field and this velocity is lost on collision with the obstacles. We will of course, soon note that what kind of obstacles are these which are giving rise to these collision which is what is leading to what you might call loss in acceleration of the electron or in other word this is the at the heart of the quantity called resistivity or the origin of resistance, because if there were no collisions then the material will be will be conducting smoothly and what you may have is very good conduction, but because of these collisions in the at these obstacles you are actually having the concept of resistivity. The average time between collisions which can be seen as the quantity here here between. So, you have this average time between collisions is given by the quantity tau and the time actually the distance actually travel during this period is called a mean free path. The word mean is because as I told you each electron is following different kind of a time before it suffers a collision and what is drawn here is for an average kind of a number here over when you when you statistically average over a large number of electrons. And therefore, we have a mean path which an electron travels before it gets scattered and the time in which during the period in which it actually experiences a uniform acceleration is given by this tau which is called the average collision time or sometime you can even call it more casually as a mean free time. So, now I can write my f is equal to m a as f is equal to m v d by tau which we have noted is equivalently written as e where e is the electric field and not the energy. So, therefore, I can write down my v d which is now my average or the peak velocity gain as e tau by m. Now, what is the importance of these two quantities because there are two important quantities we have introduced in this slide. One is the what you might call the mean free time or the average collision time and other is what you might call the m f p which is otherwise known as the mean free path. So, this slide involves two concepts one concept is the concept of acceleration of electron to a peak velocity v d and which we call the drift velocity. And the second important concept is that we cannot keep on accelerating electrons because they suffer collisions which is the origin of resistance. And this time average time over which the electron is actually accelerated is called the average collision time or the mean free time and the path equivalent to that is the mean free path. Now, I can relate this mean free path to the conductivity of the material by writing noting that the flux of electrons and now flux is nothing but the flux flow of electrons is what you might call the current density which is given as J e and the conductivity can be written as flux per unit potential gradient. In other words J e the flux current density is nothing but conductivity into e the electric field or the potential gradient and this function is very similar to our ohms law and this can actually be verified by using by comparing the dimensional quantities because on the left hand side you have ampere per meter square which is now my flux and on the right hand side I have conductivity which is 1 by ohm meter into volt per meter. And on the other hand if I track my v equal to i r that is volt per ohm and on the right hand side is of course, I take the ohm from here to the bottom side therefore, the ohm has come here and this is equal to ampere. And therefore, if you compare the two equations you can see that both sides if you divide by meter square you can see that volt per ohm per meter square you take the volt above is similar to this kind of a quantity. In other words this is some form of the ohms law which you are very familiar with now this current density J can be written as number of electrons into the electronic charge into the drift velocity. And going back to the previous slide we know that v d can be written as e tau by m and substituting that I can write it as n e square tau e by m. In other words and combining this equation for J with the equation for the flux I can get the conductivity as n e square tau by m. In other words my conductivity is directly related to my what you may call the average collision time. If I have a longer average collision time that means I would have a higher conductivity for the material. And if I have a material where because we will see the origin of all these collisions if therefore, if the electrons suppose more and more collisions per unit time that means that such a material is going to be a poor conductor. And we should note of course that we are still talking in the regime of what you might call a free electron kind of a picture. Now, we have a few more things to say about the mean free path which we had pointed out before the mean free path obviously can be written down as the drift velocity into the average collision time. And this is the mean distance travel by the electron between successive collisions. And for an ideal crystal with no imperfections the mean free path we expect at 0 Kelvin to be about infinity. That means in an ideal crystal there are no collisions and the conductivity is infinite. Because we know now that the mean free path or the average collision time is directly proportional to the conductivity. And if the mean free time goes to infinity the conductivity goes to infinity. But as I pointed out there are scattering centers in the material which reduce my mean free path. And therefore, increase the resistivity of the material. Now what are these what is the origin of these kind of a scattering centers one is obviously thermal vibration. That means if there are atomic vibrations and collective quantized modes of these vibrations are called phonons. And therefore, if an electron suffers a collision with a phonon its velocity can be reduced to 0. The second kind of an origin to these kind of scattering centers is solute or impurity atoms. That means and of course when I say impurity or solute I mean it could be a substitutional element intentionally added or it could be an unintentional element present in any material. And this kind of impurity atoms actually distort the lattice. And therefore, cause an imperfection in the lattice. And therefore, can act like a scattering center for these conduction of electrons. Further we also know that a crystal could have additional kind of defects like dislocations, green boundaries, stacking faults and other kind of defects which also can act like scattering centers for these electrons. And therefore, instead of the mean free path being infinity the mean free path typically reduces to a very small number. And you would notice that the mean free path typically for a good conductor like gold or silver is about 15 nanometers. On the other hand even for a slightly less conductor less conducting material like aluminum you notice that the mean free path has already reduced to about 15 nanometers. So, and for mean free path for copper is about 39 nanometers. Clearly you see that the mean free path is of the nanoscale. So, automatically the question will which will come to our mind is that suppose I have a nano crystal. And if this nano crystal is of the size of the mean free path then what happens to its conductivity. So, this is the question which we will try to understand very soon. But before that we will say a few more things about the quantity which is the thermal vibration or phononic scattering. Now, at any temperature bureau above 0 Kelvin the atomic vibration which leads to the phononic scattering is going to lead to an increase in the resistivity of the material. Now, the mean free path does not have a constant kind of a function. The mean free path is a function of the temperature at which you are making your measurement. And it goes as 1 by T cube approximately at very low temperatures. And at slightly higher temperatures it goes as 1 by T. And the other factor which we saw was the impurity scattering. And we said that the material with an alloying element or with an impurity is going to have a high resistivity as compared to a completely pure metal. Now, the increase in resistivity is approximately proportional to the amount of alloying element added. And therefore, if I have a very pure material like copper and with a very low defects densities. And I plot its resistivity as a function of temperature. Then I would notice suppose I plot the resistivity of pure copper. And now I am talking about a pure copper with low density of imperfections. That means I am ignoring my stacking faults and dislocations. And I track its resistivity as a function of temperature. I would note that with decreasing temperature my phononic scattering would reduce. And the resistivity keep on decreasing. In other words my conductivity is going to continuously increase. And what you would expect at 0 Kelvin or as you tend to 0 Kelvin is that the material will tend to some kind of a superconductor. Because this is now pure copper. And there are no phononic scattering and there are no other kind of lattice defects which can give rise to scattering. And therefore, you expected to become a superconductor. And on the other hand suppose I have a copper nickel alloy. This alloy even though the resistivity decreases with temperature. Which is what we expect because the phononic contribution is going to decrease. But then there is always a residual resistivity even as we tend to what 0 Kelvin. And this is coming from the fact that now you have an alloying element like nickel which is going to give rise to what you might call the impurity scattering or the alloying element scattering. And as you increase the amount of alloying element the resistivity say now suppose I am at a vertical temperature like about 100 Kelvin. You would notice that at this 100 Kelvin you would notice that my resistivity increases as you add the alloying element. And you add little more of the alloying element resistivity further increases. That means with more and more alloying element you are going to have higher and higher residual resistivity as you tend towards 0 Kelvin. So, we know now that if I want to make a perfect conductor then I have to avoid all these scattering entities which includes presence of solute elements and the presence of defects. And if I reduce my what you may call the phononic scattering contribution by reducing the temperature then I would have an increase in conductivity. So, to reiterate the message of all these slides so far in the free electron model there are free energy states available where electrons can be excited. And if you applying a electric field then it is going to be a directed motion of electrons in the presence of a what you may call temperature. This is going to be random motion of electrons in the presence of an electric field electron is accelerated but suffers collisions because of which the velocity comes down to 0. And this time the path it travels freely before a collision is suffered is called a mean free path. And the mean free path or the equivalent mean free time can actually related to the conductivity. That means you increase the mean free path then the conductivity increases. Equivalently suppose if I note that the mean free path of most materials good conductors is of the order of about tens of nanometers. If I make a nano material or a nano crystal which is of with whose size of this order then I would expect some drastic changes in the properties. I had pointed out that the free electron theory cannot explain all kind of concepts. And it is a sort of a simplified theory the more or the more regress way of understanding the conduction properties and the overall electronic structure is what is known as the band structure. Now in a simple physical way of understanding the band structure is to assume that you have these individual atomic levels when the atoms are far apart. So, you have an infinite separation here atomic separation on the x axis and the energy levels on the y axis. And when the electrons when the atoms are very far apart here at infinity then these atoms do not talk to each other the electronic energy levels do not talk to each other. And essentially you have discrete energy levels. But as the electrons come the atoms come closer the outermost energy levels start to overlap and given the polys exclusion principle they spread into a band. In other words now my 3 D electrons would split into a band like this and the 4 s electrons would form a band like this. And now these 3 D and 4 s bands belong to the entire solid and not to the individual atomic levels. And now therefore there is an equilibrium atomic separation at which you can see that there are energy bands. The for instance the 4 s band starts here and ends here and the 3 D band starts here and ends here. There is some more detail about these 3 D and 4 s bands and their overlaps which does not become obvious when you look at a diagram like an energy atomic separation diagram. And therefore we will take up something known as the density of state diagram very soon. So, the message of the slide is that at equilibrium atomic separation the outer electronic energy level give rise to bands. The core levels continue to remain discrete that means they have the atomic character still. Bands may overlap and fill in parallel over a range of energy values as shown in the figure here. That means that once I have when once I am in the energy region between here and here that implies that I am going to be filling my 3 D and 4 s band parallely. As a range of energy values allowed in a band and of course we know from polys exclusion principle that they are discrete, but they are closely spaced. Any radiative transition from these outer levels to a core level has a bronze range of wavelengths. Now, because if you look at the atomic emission spectra or an absorption spectra that will tend to be very very sharp. But suppose you have a radiative transition from these bands they will have a continuous energy value because you these band themselves have a continuous range of energy levels. Now, as I pointed out that certain important details are hidden away when you plot a diagram like this wherein you are showing an overlap between the 3 D and 4 s bands and this kind of a diagram would be for a typical transition metal. So, we have to invoke the concept known as the density of states. Density of states is defined as the number of available states in a given interval of energy. Previously we had noted the value n in the free electron model. We have noted the n goes as e power 3 by 2 and which is scaled by the volume of the material. Now, the density of states is nothing but the number of energy states available in a given energy interval of energy and this available implies we are talking about available to be filled by electrons. So, I can calculate my density of states as d n by d e the n formula as I showed you before. And therefore, if I do this differentiation I would find that the density of state goes as e power half. In other words if I plot my density of states with energy in the free electron model I would note that it goes like an e power half kind of a relationship. And at 0 Kelvin you know that the energy is only up to Fermi level or field which we pointed out. In other words all the energy levels below the Fermi level are occupied which is given by the blue region in this plot and all the energy levels above the above E f level are all vacant. But we had also pointed out when I want to track the number of energy levels at any finite Kelvin temperature then I need to multiply my function by what is known as the Fermi Dirac function. And I would do the same thing here to find the density of states at any finite Kelvin temperature and that means at t greater than 0 Kelvin to find the density of states I will multiply the density of states at 0 Kelvin which is of course the region blue shaded region in the curve and multiplied by the p of e which is given by the Fermi Dirac function. And therefore, this function close to the E f will start to develop this kind of a curved region. In other words there will be a small tail you know the energy there will be electrons states which will be go vacant here and there will be electron levels which will be occupied here at higher energy level. So, this is at a finite Kelvin temperature in other words I have taken my density of states is 0 Kelvin and multiplied by the Fermi Dirac function. Now, this kind of a picture is not actually valid when you are close to the what you might call the band edge this kind of a picture is only valid when you are in the completely free electron domain that means the electron is not filling any effect of the potential of the ion course. And once it starts to feel that potential this kind of a function which is given by e power half has to be modified. And the way to do it is by invoking the effective mass of the electron and the new density of states can be plotted instead of e power half. Then I have the e v minus e power half where e v is the energy corresponding to the band edge. In other words close to the what you might call when the Fermi surface actually starts to talk to the Brule zone then what happened you actually modify the function and the function starts to look something like this. In other words this is the regime where I can talk about absolutely a free electron. And this is the regime where I need to modify this e power half kind of a functionality. Later on when we talk about nanostructured materials we will see how this e power half kind of a function is actually modified in the case of a nanostructured material. And how we actually even a very good conductor can actually start to become an insulator because of the limitation or quantizations in one or more dimensions. If you look at we had also talked about in the previous slide that actually bands can fill parallely. That means the 3D and 4S bands can fill parallely and in that case the density of state function needs to be even further modified. And it looks a little more complicated as in the case here you can see that it starts to look a little more complicated when more than one band is overlapping. And this is typical of divalent metals and this diagram here corresponds more closely to the transition metals like iron. In the case of transition metals you would notice that the 4S band which is a broad band in the density of states is parallely filling compared to the also with the 3D band which has a high density of states localized to a small region in energy. This has important implications because now we are not just merely talking about a free electron picture we are talking about a picture where I am taking into account what you might call the density of states. In other words it is a number of states available in a small interval of energy DE and this is going to determine much of my important properties. For instance now the fact that this 3D band is highly localized as a narrow bandwidth tells me that the electrons here the 3D electrons are not fully itinerant. That means they have a partial kind of a localization character and later on we will see this plays a very important role in the magnetism of iron. So, to summarize this slide we note that we have normally what we plot in the case of materials is the what you call the density of states which is the number of energy levels available in a small interval of energy and this density of states function has to be initially of course looks like an e per half function in the free electron regime. But has to be modified when I am talking about overlapping of bands and close to a bandage the function becomes little more complicated. The density of states function is highly altered with respect to a normal material. So, now what happens in the case of this kind of wire if I keep on exciting the system that means that originally in a free electron picture I could keep on exciting the system and I would find always there are energy levels to be occupied that means the material will behave like a conductor. But now you see that up to a certain extent of course I have a density of states which is now falling with energy not increasing with energy and beyond this kind of a critical energy there are no density of states available for the electron to be promoted. Till of course the next quantum level is occupied that implies that this material now starts to behave in some sense like a semiconductor not it is not like a metallic conductor which we expect even though we started off with a metal. But just by reducing dimensions the density of states has been altered and I have developed a band gap in my density of states. And we had originally said that this band gap is characteristic of a insulator or a semiconductor. So, now this is a pure metal starting to look like a semi metal or to look like a semiconductor or an insulator purely by the quantum confinement effect which is coming from the fact now that the density of state functionality has been altered in the case of a what you might call a quantum buyer as you would like it. Now what happens when I before of course I take up zero nano crystals it is important to see that there are actual examples of this kind of 1 d nano materials which are now like for instance my carbon nano tubes. And in this carbon nano tubes we will take up an important phenomena which is known as the ballistic transport of electrons which is coming from the second effect which we talked about here. We had said that there are two effects which are very very important and when you are talking about the conduction of electrons or conductivity one is the quantum confinement of which we have been talking about so far which determines what you may call the an altered density of states. But additionally we also have this issue which we talked about the mean free path becoming comparable to the size of the system. And when there are no scattering events within the system then the electrons can actually just be accelerated or can keep on gaining velocity and that implies that you can have what is known as ballistic transport of electrons. Scattering does not lead to a loss of kinetic energy and electrons can move unimpeded to the system. The ballistic transport is observed when the mean free path of the electron is bigger than the size of the material. And in these cases the electron transport mechanism changes from diffusive to ballistic and ballistic transport is coherent in the what you might call the terms of wave mechanics or quantum mechanics terms. Now the optical analogy of a ballistic transport is light transmission through a waveguide where there is no loss. In other words now there is a lossless transmission of in the course in the waveguide there is lossless transmission of light in this case there is lossless transmission of electrons. And there are actual structures which are metallic carbon nanotubes which is a 1 d nano structure where in transport along the length can become ballistic. And when this happens you can actually have a current density of the order of about 10 power 9 ampere per meter square. And if you compare it with a metal like copper which has a current density of 10 power 6 you can clearly see that this now this carbon nanotube has become really a very good conductor along the length. And we are talking about those kind of keralities where you actually m and n numbers where in you actually have what you call a metallic carbon nanotube. It is also seen that ballistic conduction may be observed in silicon nanowires at very low temperatures like 2 to 3 Kelvin where foronic scattering becomes a small contributor. So, to summarize this slide we have noted that a system size can become so small that normal drift transport or diffusive transport switches to what you might call ballistic transport where there is no scattering of the electron. And when this happens you can obtain very high current densities as in the case of metallic carbon nanotubes along the length of the nanotube. And this kind of a current density is much higher than that what you typically obtain even in a good conductor like copper. And there are other systems like silicon nanowires too wherein we would expect some kind of a ballistic transport at very low temperatures. Of course, we have to note that this metallic carbon nanotube has to be defect free because all these if there are any defects in the crystal then or in the nanotube then that would lead to scattering which would lead to further which will not give us this kind of high current density. So, it is indeed surprising that in nanotubes you can get very high conductivities and very high current densities which is coming from the fact that now the length scale of the problem has gone to nano size. So, after having talked about what you might call bulk materials and then we have talked about 2D nanomaterials then we went on to talk about 1D nanomaterials wherein we saw that even metallic materials change their behavior. We can now talk about what you might call the 0D nanomaterials or quantum dots. And here all the energy levels are discrete because then now the system is confined in all three dimensions. So, such a system is now a quantum dot and it is confined in x, y and z all three dimensions. And therefore, this implies that there are no free electrons in the system. There are no free dimensions along which the electron can freely move about and there is quantization or confinement in all three directions. And therefore, now such a system is described by three quantum numbers the energy levels n x, n y and n z. And if you look at the density of states of such a system this becomes what you might call delta function in the ideal limit. In other words the density of states for us quantum dot becomes similar to that of an atom and metals start to behave like insulators. You can see that if I plot my density of states in of course, the ideal situation they start to behave like delta functions in other words they are discrete. And if electrons have to be promoted they have to be promoted from one energy level to the other by giving a discrete amount of energy. More realistic of course, if you have the size of the system little larger then you would notice that there will be little broadening of these energy levels. Nevertheless they remain discrete and such a system you know the conductivity would be limited and metals would start to behave like insulators at very very small length scales. So far we have been talking about conduction which is called the normal conduction wherein you have a free electron or an electron which is travelling down the potential. But there is another kind of a quantum effect which is called tunneling in which case an electron actually has a finite probability of being found beyond a barrier. So, suppose this is my conductor and this is a barrier in classical terms the electron cannot be found beyond this barrier this potential barrier. So, but in quantum mechanics actually there is a finite probability of actually finding this electron outside the barrier. So, this implies that if I have a small resistor of a thickness smaller than this what you might call the decay length for this tunneling. Then actually such if this region is conducting and this region is conducting I am actually transport electrons purely by the tunneling mechanism. So, this tunneling has no classic analog and it is purely a quantum effect and when you are talking about tunneling there is a very closely interesting kind of an effect which is called the Coulomb blockade which is observed. And we will take this example because such an kind of effect can actually put to some kind of an application. There is something called a tunnel junction and this tunnel junction behaves like a resistor. Now, I can take a metal which is a conductor where an electrons can normally flow and on in between this metal and another metal I put an insulator. And as I just pointed out if this insulator is thin enough then actually I can flow tunneling current from one metal layer to another if I apply a potential. This tunneling current is obviously going to be smaller than the normal current you would observe when you have a flow through the conductor. And this resistance in the increases exponentially with the barrier thickness and now I am talking about barrier thickness which is very small of the order of nanometer. If this insulating barrier becomes large then I would find that there is no current which is transported from this green metallic region which is shaded green to the next metallic region which is also shaded green. So, for this kind of a tunneling current to take place the thickness of this barrier the insulating layer has to be small. But addition to the fact that this layer this layer introduce a resistance to this flow of current through this kind of a set up it additionally also has capacitance. Because this insulator can is also a dielectric and as you know dielectrics can store charges and therefore, it can also behave like a capacitor. The important point to note is that the current through this junction this tunnel junction passes as one electron at a time. Sometimes of course, co-tunneling can occur where two electrons can also pass through, but typically it is one electron at a time which actually tunnels through this tunnel barrier. Now, this implies that the tunnel junction capacitor is charged with one tunneling electron once this tunnel barrier has been charged with one tunneling electron. This implies that if I want to push the second electron into through this system across this voltage. The first electron which is already built a charge because of the capacitance of this insulating layer work has to be done by the external voltage to push the second electron. That implies for an additional charge q to be introduced into the conductor work has to be done against the electric field of the pre existing charges residing in the insulating layer of course, it should this should be the insulating layer. Charging an island with capacitor c that island of course, insulating island we are talking about with an electron of charge e requires e square by 2 c. That implies that if I do not supply this much of energy I cannot push a second electron in an insulating through the insulating layer which is already been charged with the first electron. That implies if I now plot my voltage current characteristics this becomes very very different from that what we normally observe for normal materials which is like a normal conductor we know the v equal to i r kind of a relationship. That means it goes the current goes linearly with voltage, but you can see that in such a system where in I have a metal insulator metal configuration you can see that the current voltage characteristics has a staircase kind of a structure. Now, so at low voltages first of all no there is current itself is suppressed that means if you have very low voltages you see that no current passes through the system and this is called the Coulomb blockade. That means that I have no current if I have low voltages in a normal conductor if you have low voltages you will still have some current the current value will be small. Typically for a Coulomb blockade to be observed you have to work at low temperatures because otherwise at higher temperatures thermal excitation can transport the electron instead of tunneling. That means purely by thermal excitation this barrier may be broken and therefore, you will have some conduction we could thermal excitation and therefore, you will not you will be not observing the Coulomb blockade phenomena. So, what is the essence of this Coulomb blockade phenomena you see that initially of course at low voltages there is no current at all and when you put sufficient voltage you can see that the system charges to one electron, but further increase in voltage does not in this regime you can see that in this regime between from here to here further increase in voltage is not leading to an increase in current because the pre existing charge in the capacitor is effectively stopping a second electron from entering the capacitive system. That implies that I have to again break down this capacitor for which I have to employ a higher voltage and once I have charge the system at two electrons again you find that there is a plateau in the current voltage characteristics. Therefore this is a very interesting system which has been put to use in developing devices like single electron transistors and Coulomb blockade thermometers, but you can clearly see that there is no classical analog of such a behavior in the voltage current characteristics space which is coming from a thin insulator. And the fact that this insulator is now in the nano scale and it is what you might call you at the heart of this whole process is what you might call tunneling current which only happens in the nano scale. And if I make up thick material then the tunneling current will fall down to 0 because the resistance increases exponentially with the barrier thickness. So we can see that there are very very interesting phenomena and which are very very different from bulk scale materials which you observe regarding the electrical conductivity in nano scale materials. And some of this can actually be put to very good use in the form of devices and various kind of counter connects. Additionally we have to remember that when you are making a we are trying to reduce the dimension of because when you are trying to do very large scale integration for devices and trying to reduce the dimensions of various wires and interconnects. We have to note that some of these quantum effects may come in some of these confinement effects may come in and therefore the materials will start to behave very differently than the bulk. And therefore newer and newer technologies have to be discovered before the size can be reduced to very small scale. But addition with those problems we can see that there are newer possibilities which are opening up because of newer phenomena which kick in when you actually go down to the very small scale.