 Hello, I am S. A. Sivashankar from the Center of Nanoscience and Engineering at the Engineering Institute of Science. And I will begin today with the segment 3 of the course, Nanoelectronic Device Fabrication and Characterization, about which you have already heard from Professor Navkanth Bhatt and Professor K. N. Bhatt in earlier segments and those segments have dealt with device fabrication, scaling of devices, basically silicon devices, using essentially principles of scaling that are now well understood and even though scaling has taken devices now to the level of nanometer length scales that are the feature sizes of silicon circuits today in VLSI, they still do not make use of properties of materials at the nanoscale. So, what I propose to do in this segment of the course is to deal with materials and phenomena at the nanoscale, fabrication of such nanomaterials and nanostructures, characterization of these nanomaterials and structures and may be some devices and some selected applications. Therefore, this segment of the course actually deals with materials and their properties at the nanometer scale which I shall define shortly. I think it is appropriate now to say that you know a certain number of references are useful. Numerous books have been published on nanomaterials, nanostructures and devices and so forth, but I have just listed two of them here because they are generally well known and they have undergone a couple of editions. So, one is the book by Professor Kau and the other one is the book by Professor Poole and Owens and as we go along I will also refer to various publications in different lectures that are relevant to the material of discussion on hand. Now, what is nanomaterials, nanostructures and nanosystems? A broad definition is in order nanostructures and nanosystems are those whose extension is less than about 100 nanometers in at least one dimension. What it immediately implies is that nanosystems may be low dimensional that is they may have less than three dimensions with which you are normally familiar. So, there may be two dimensional materials one dimensional or even zero dimensional and there are already pretty familiar examples of these things and very interestingly all these examples that I have cited here are made of elemental graphene elemental carbon I am sorry. So, quantum dots or zero dimensional systems are exemplified by C 60 or the Bucky balls, nanotubes and nano rods are exemplified by carbon nanotubes which are now very well known and the most recent one is the two dimensional system namely the graphene sheet which is a single layer of carbon atoms that is of the graph graphite structure and of course, three dimensional particles and structures also in the nanometric size. Now, some history is in order while nanotechnology nanomaterials nanoscience and so on are very much in vogue today and a field of great deal of investigation it is useful to remember that nanoparticles were prepared nanoparticles of gold were prepared in the 1850s by Sir Michael Faraday by a technique that is pretty much similar to what people use today. And in industry catalysts made of finely divided particles in those days of course, the terminology of nanoparticles and nanosystems systems was not there. So, people call them finely divided particles. So, finely divided particles were used as industrial catalysts and certainly a spectacular use of nanotechnology of those days are stained glass art in churches and cathedrals in Europe which make use of metal nanoparticles. So, therefore, all these three are examples of nanotechnology of those days not necessarily nanoscience. Now, you probably have heard of a famous statement by Nobel laureate Professor Richard Feynman. He said in 1959 in a talk at the American Physical Society that there is plenty of room at the bottom. What he meant was that while macroscopic material was very interesting and under great investigation in those days in the period of solid state physics development, what he meant was that as one reduced dimensions of objects to the extremely small sizes beyond the visible region beyond the limits of what I can see then phenomena that occurred there were very interesting and what he actually anticipated in those days was scaling down of devices to such a degree that very very different phenomena would become possible. And I want to read here what he said was that you know it is staggeringly small world that is below in the year 2000 when they look back at this age they will wonder why it was not until the year 1960 that anybody began seriously to move in this direction. So, he anticipated that the world underneath so to speak was really going to be very very interesting and in fact defining. It is possible to say that the actual era of nanoscience and technology that we are in began in some way with the invention of the scanning tunneling microscope in about 1982 for which a Nobel prize was awarded where actually a quantum phenomenon was used as a way to look at materials on a very fine scale. So, in a scanning tunneling microscope a very fine tip made of a metal comes very close to a surface which it is investigating also of a metal in other words both are electrical conductors. And when even though they are not in touch because of the phenomenon of quantum tunneling a current flows between the sample and the tip. So, this called a tunneling current and this enables a close examination or taking a picture of the surface of the sample. These are actually made possible by the invention of p h o electric materials ceramic p h o electric materials actually in the bulk which made possible controlled motion on a nanometric scale. So, actually a ceramic technology of macroscopic dimensions enabled the invention of a device that would resolve objects into nanometric size. Soon after followed the atomic force microscope in 1986 while the limitation of the scanning tunneling microscope was that it could only look at electrically conducting or metallic objects. The atomic force microscope is different in the sense that it really measured the force between an atom on a sample and a tip and that atom at the tip of your probe. So, that it did not depend on the electrical conduction that would be required in an STM. So, this force actually exists between any two atoms and therefore, it makes no distinction between a conducting surface and a non conducting surface. As a result of it was possible to probe the surfaces of any kind of a material whether conducting or non conducting and in fact, the atomic force microscope has really become the principle on which a more general technology known as scanning probe microscopy through which one can measure electrostatic forces, magnetic forces and so forth p h o forces. So, in all these technologies that are based on these AFM it has been come possible to look at objects on a very fine scale. Equally importantly the atomic force microscope and the scanning tunneling microscope of today can look at objects in the ambient while objects had been looked at in electron microscopes those generally require a vacuum and pretty elaborate systems. The atomic force microscope really enabled the birth of nano technology of today because it is possible through an AFM or a similar device to look at objects in the ambient on a table top so to speak. So, you could say that the current era of nano science and technology began with the atomic force microscope. Now, some idea of the length scales involved so what I have shown here is the scale of things human scale as opposed to the microscopic scale. We are interested at the bottom of the scale on the view graph where we are talking about somewhere between 1 and 100 nanometers where different kinds of nanostructures and quantum devices and of course, natural objects like proteins and DNA and so on belong. Now, these are examples where we have a molecule of a DNA which can contain millions of atoms nanowires and carbon nanotubes which can contain hundreds to millions of atoms depending on the length and so forth. So, these are all objects of interest in the current nanoscience nanotechnology context. It is always useful to have an idea of actual dimensions and numbers. The Bohr radius which is the radius of the first orbit in the hydrogen atom measures 0.053 nanometers or half an angstrom. The van der Waals radius of a carbon atom is about 0.17 nanometer 1.7 angstroms and it is possible to line up 3 carbon atoms along a 1 nanometer line. So, if you had a cube of a nanometer measuring 1 nanometer by 1 nanometer by 1 nanometer of carbon atoms it would contain 27 carbon atoms or in the current graphene sheet which is a two dimensional object measuring 1 nanometer and 1 nanometer then you would have 9 carbon atoms and in a cube of 10 nanometers on the edge there will be 27000 carbon atoms. Now, you could say that a typical nanosystem would be made of hundreds to tens of thousands of atoms. By comparison if you had a carbon cube measuring 1 micrometer on the edge or 1000 nanometers on the edge it would contain something like 27 billion carbon atoms that is already a large collection it is no longer in the nano regime and the behavior of such a an object or a particle would be the same as the behavior of a macroscopic object. So, it is no longer in the nano regime. Now, atoms we all know for example, the Bohr radius the treatment of the atom and the hydrogen atom and so on requires a proper understanding of the behavior of atoms requires quantum mechanics. So, it is necessary to apply quantum mechanics to understand the behavior of atoms when there is the in other words quantum mechanics is required when systems are sufficiently small. How small that is depends on the strength of interactions involved we will come back to that. So, for example, clusters of atoms quantum dots electronic properties of nano wires and thin films and so forth would require quantum mechanical description to be able to understand their behavior fully. An essential aspect of such an approach is the concept of the density of states that is the number of energy states per unit energy interval typically 1 electron volt. This density of states determines many of the processes in the world of the small. It depends on the it is important to note that the density of states depends on the dimensionality of the system that is whether it is a one dimensional two dimensional three dimensional system or even zero dimensional. Therefore, the very functional form of the dependence of the density of states on energy depends on the dimensionality. So, these factors that require quantum mechanics to describe nano systems and the importance of the density of states together can and lead can and do lead to the behavior of nano systems that is substantially different from the behavior observed in the macro world. As I just said quantum mechanics dominates or even dictates the behavior of atoms whereas, a different discipline statistical mechanics is actually pertinent to the understanding of the behavior of ensembles of atoms that is large collections macroscopic collections of atoms and molecules are described with the help of statistical mechanics because it is not possible to follow the motion of individual atoms in a large collection. Therefore, they can such a large collection can be described by statistics. Now, on the one hand you have nano systems where you are atomic systems where you require quantum mechanics on the other hand you have nano systems where you have hundreds or thousands of atoms that are involved. So, actually the science and technology of nano systems is where the quantum behavior the atomic behavior of matter intersects with the behavior of complex systems which requires statistical mechanics. So, you have an intersection now of quantum mechanics and statistical mechanics that is appropriate to understand nano systems. As I said nano systems may consist of tens of thousands of atoms and their behavior is necessarily quantum mechanical and as I said how quantum mechanical how microscopic depends on the strength of interactions of a particular system we will get to that later. Now, one of the necessary aspects of statistical mechanics is that fluctuations are important when you have large ensembles as I said you cannot describe every particle in an ensemble. So, while you can describe an average there can also be fluctuations. Therefore, fluctuations are an inescapable aspect of any ensemble atoms and this fluctuations play an especially significant role in small or nano systems because fluctuations are relatively large in small systems that is because fluctuations scale as square root of n by n or 1 over square root of n with respect to the mean energy of the system. So, this comes out of statistics or statistical mechanics in small systems this square root of n by n approaches unity although of course, it is less than unity is of the order of unity whereas, this ratio is much smaller of the order of 10 to the minus 11 in a macroscopic ensemble of let us say an avagadro number of atoms. Therefore, fluctuations are hardly important in macroscopic systems whereas, they become very important in microscopic systems or nano systems. On the other hand complexity of the systems that is the complexity of the behavior of these systems is an exponential function you can say it varies as p to the n divided by n. Therefore, on the one hand you have a set of fluctuations that are important in nano systems. On the other hand you have complexity that increases with n. One way to look at these things is to recognize for example, that at the nanometer scale systems are small enough for fluctuations to be significant whereas, the systems are also complex enough therefore, at the nano scale you have a convergence of fluctuations and complexity. Now, this can be looked at in the following fashion. So, one can draw a graph that describes on the one hand fluctuations, fluctuations come down as the size increases whereas, complexity goes up as size increases. So, this is complexity and this is fluctuation. So, you really have a region here where in the nano systems you have the simultaneous importance of fluctuations and complexity. So, that is what actually signifies nano systems and as I said will come to this a bit later and try to elaborate on these things. Now, fluctuations have become important in a way that is understandable from the context of VLSI integration. I am not really going to talk about the scaling and so on, but what is happening now is that because of the extremely small dimensions the feature sizes in today's integrated circuits in production, there is actually a factor that comes into play that is due to fluctuation. That is you have millions and millions even billions of circuits in a single wafer and you are subjecting them to various kinds of processes and these processes have to take place on a very fine scale, nanometer scale on a wafer for example. So, therefore, there can be variations from let us say spot to spot so to speak on the wafer and therefore, there can be an influence of these fluctuations that can affect the yield of a semiconductor manufacturing process. This is not really to do with nano materials as such, but I wanted to point out how fluctuations can come into play in the actual fabrication of devices as the dimensions really come down to very small sizes. Now, why do we want to study nanomaterials and nanostructures? There are very interesting phenomena that arise which is actually the subject of this course. There are chemical aspects taking advantage for example, of the large surface to volume ratio, interfacial surface chemistry become very important and these can be exploited to advantage. There are very significant electronic and optical effects due to something known as quantum confinement will come to change in the density of states, electron tunneling and so forth. Some of these things are already evident in today's circuits because after all when the gate oxide thickness comes down very much then tunneling which is the quantum mechanical effect is already beginning to be important in real estate circuits and then there is this so called giant magneto resistive effects which is evident from nano scaled multi layered thin film structures which was actually invented around the same time as atomic force microscope and today already it is part of the magnetic storage devices. In other words, it is nano technology actually implemented in practice that is a part of the nano effects in magnetism. There are also interesting mechanical effects for example, improved strength of materials, optical effects which is really the other side of the electronic effects, single photon phenomena and so forth and photonic band gap engineering which is already a well developed technology, nano fluidics and thermal effects such as increase the thermoelectric performance in nano scale materials and so on. So, these are all examples of the motivation for studying nano materials for the expansion of nano science and the development of nano technology. A couple of vivid examples of this what is shown here is the progressive change in the appearance of nanoparticles of cadmium selenide which is a direct band gap semiconductor as the size of the particles is increased from 2 nanometers to 8 nanometers or reduced from 8 nanometers to 2 nanometers. What you see here is the same material under fluorescence and the fluorescence from the same material under UV illumination. You can see that in fact, because of the change in the size of the particles progressively from 8 nanometers to 2 nanometers, you have a complete change in the appearance from red all the way to blue as the size is reduced and this in fact is a quantum mechanical effect something known as the blue shift which arises from quantum confinement that we will refer to as we go along that we will learn about soon. So, this is in semiconductors now gold I mentioned nanoparticles of gold actually Michael Faraday made them 150 years ago. Now, we all think of gold as having well golden appearance yellowish appearance, but when you size it down as you can see here in this view graph when you bring down the size of gold in nanoparticles or in nano rods then the color of gold changes it can even be green as opposed to being yellow. So, that depends on the size of the nanoparticles in the top view graph or in this on the aspect ratio of nano rods in the bottom one where the aspect ratio changes changing the color of gold as we see it. So, these are vivid examples of how the properties of materials can be altered by controlled nanostructuring of materials. I already said that nanotechnology is not new we had the industrial catalyst and the stained glass windows one thing I forgot about is the silver halide photograph photography the wet photography which is of course, more or less vanished now, but all these are examples of what may be called old technology. The new technology is the new technology is a list that I have shown here which is just a partial list designer drugs and so forth. The difference between the old technology and the new technology is that many times because of the lack of ability to explore and understand the behavior of particles on a very fine scale there was no clear understanding of the principles of work working of for example, let us say the stained glass windows it was a sort of a black magic as to why spectacular colors were achieved in those context whereas, we now are in a position to be able to understand these things in a systematic fashion and therefore, develop them in a systematic fashion. So, a matrix of sorts can be defined where we obtain fundamental understanding that comes through characterization and experimentation we know the loss of physics that apply and therefore, modeling and simulation are possible and therefore, it is possible to integrate this knowledge into useful devices and actually employ them. So, this is the development of nanotechnology of the current day having given this somewhat lengthy introduction we are probably in a position to proceed further what we need to understand more formally is that there are size dependent effects in materials that is as you size it down from macroscopic scale to nanometric scale of the way of the kind that we defined there are two different kinds of size dependent effects that one can observe the first kind is those that scales smoothly with size and these are related to the fraction of atoms at the surface of a particle a surface of a nano system. The other kind is those kinds of effects which show discontinuous behavior because of quantum effects and one example is in the cluster of atoms which will describe as we go along. So, first we come to the kind that very smoothly with size and we ask what happens when you scale down the obvious thing that happens when you scale down is there is a smaller number of atoms in a grain of material in a particle of material automatically it also means that therefore, a larger fraction of the atoms within the grain are at the surface they lie on the surface of the grain. Let us look at a simple example and then go to an analytical description of what is meant by this. Let us take silicon the prototypical semiconductor we are all familiar with silicon has the diamond structure with a lattice constant of 5.43 angstroms and with the knowledge of these dimensions it is possible to is simple arithmetic to show that a silicon nano cube 1 10 nano meters on the edge is made of about 6250 unit cells or about 50,000 silicon atoms. Now, each of these nano cubes measuring 10 nano meters on the edge is composed of I mean it has 6 phases of course and each of these phases is made of 340 unit cells per phase or 680 surface atoms per phase there are 6 phases therefore, there are about 4000 atoms of silicon on the surface of this 10 nano meter cube which has a total of about 50,000 atoms. Therefore, something like 10 percent of the atoms in this nano cube lie on its surface whereas, if you go to a bulk silicon film which measures about 1 micrometer thick I already said that a micrometer is already a very large dimension in this world because it comprises millions of atoms or billions of atoms. Therefore, we are in the macroscopic world when you come to a micrometer thickness as opposed to nano meters. So, if you had a silicon film 1 micrometer thick on a 10 centimeter square wafer it is easy to show that there are about 5 into 10 to the power of 20 atoms in this one and also to show that about 1.4 into 10 to the 17 of these are on the surface that is only about 0.03 percent as opposed to 10 percent in the earlier case or on the surface. Therefore, the surface to volume ratio increases progressively as you reduce the particle size. So, this is just a numerical example and this indicates in a rather vivid fashion that in nanoscale materials the surface or the boundary or the interface is very important. Now, there is an analytical way to describe this progression and that quantity is called dispersion. Dispersion is the fraction of atoms of a solid at its surface. So, dispersion is denoted by f and it is a fraction of atoms of a solid at its surface. If you consider a sphere the total number of atoms n scales with the volume which varies as r cubed 4 by 3 pi r cubed as you know. Therefore, n goes as r cubed or r goes as n to the power of one third. The surface area varies as r squared meaning that the number of atoms at the surface varies as r squared. Thus for a sphere dispersion f varies as r squared by r cubed or dispersion f goes as r to the power of minus 1 which means that dispersion f goes as n to the power of minus one third. So that is scaling law for a spherical object. But one can show readily that this relationship this functional relationship is valid for a long cylinder or for a thick film or a rectangular plate. So, this is a general result that the dispersion goes as n to the power of minus one third. So, this is a scaling law. Now, one can actually consider a different kind of a geometric shape to look at dispersion and that is a cube. Now, consider a cube with n atoms along each edge. Let us try to draw this. So, we have a cube let us say a cube with 3 atoms on the edge. So, you can imagine that this goes this way. So, it is a cube. So, such a cube would have as you know 27 atoms. Now, what we will do is to consider the number of atoms in such a cube on lying on the surface. If you had only a cube with 2 atoms on the edge then every one of them all the 8 atoms would lie at the surface. Whereas, if you had a cube with 3 atoms on the edge with a total of 27 you can readily satisfy yourself that 26 of these atoms in a cube in this cube would lie at the surface. Therefore, the dispersion here is 26 divided by 27. Those lying at the surface is 26. Those lying in the interior in the interior is only one and therefore, the dispersion is 26 by 27. So, if you actually go through this exercise and do this for cubes with different number of atoms n on the edge then you can see that it is possible to come up with this general formula f is equal to 6 n square minus 12 n plus 8 divided by n cube. Now, the reason for this form 6 n square it defines the number of atoms on all the 6 phases together with the 6 phases of the cube. So, it is 6 n square because a cube has 6 n 6 phases. But, when you consider this 6 phases what happens is that you have to subtract your double counted the edges they are 12 edges your double counted them and therefore, you have to subtract 12 n. But, then when you subtract these atoms on the edge to avoid double counting you will have removed all the 8 corner atoms and therefore, you have to add 8. Therefore, 6 n square minus 12 n plus 8 is the total number of atoms of on the surface of a cube with n atoms on the edge. Therefore, the dispersion is 6 n square minus 12 n plus 8 divided by n cube. This can be simplified as you can see and for large n you can readily see that this f would go as 6 divided by n. Now, n the small n is equal to n cube rather the capital N is equal to n cube the total number of atoms is equal to n cube. And therefore, you can see from this simple example that f scales as n to the power of minus one third which is really what we had earlier also. Therefore, the scaling law for the dispersion is n to the power of minus one third. Now, one can actually plot this dispersion function for this cubic arrangement that I have just mentioned as a function of n the number of atoms on the edge of this cube. So, you have the case where f is equal to 1 for n is equal to 2 that is capital small n is equal to 2 f is equal to 1 and so forth. And it declines rapidly. So, when you have a cube with 100 atoms on the edge then f in that case is 0.059. In other words 5.9 percent of the atoms in a cube of 100 atoms on the edge would light the surface is still a pretty large number. But for bulk materials where you really have Avogadro number of atoms of the order of 10 to the 24 let us say then n the small n the number of atoms on the edge of a cube would be about 10 to the 8 and therefore, f the dispersion is 10 to the power of minus 8. Therefore, you can see how much of a difference it makes when you go from the macroscopic world to the microscopic world namely that the dispersion increases greatly when you size it down. So, here is an example of the case of palladium clusters of palladium made of different sizes varying from 63 micrometers all the way down to 1.1 to 1.2 nanometers and in the clusters of palladium measuring 1.2 nanometers you can see that 76 percent of the atoms are at the surface of these clusters. So, this is a practical illustration of what happens when particles are size down one can look in a very elementary way at the scaling that happens consider a cube of edge a its surface area 6 a square if the cube is divided into smaller cubes of edge a divided by n then the total number of cubes will be n cubed each with a surface area of 6 into a by n whole square. The total surface area of n n cubed cubes is therefore, equal to 6 n a square that is we have gone simply by dividing our large cubes in large cube into a large number of small cubes. We have gone from as total surface area of 6 a square all the way to 6 n a square where n is the factor of division. Therefore, this can shows readily that by sizing it down the total surface area of an object can be greatly increased and coming back to our finely divided material what it shows is that such finely divided material would have a large surface area which people understood a long time ago was very important for catalytic action. So, it accounts for the effectiveness of catalytic action of finely divided metal particles. Surface energy another important aspect of scaling now when particles are divided as we just did for a cube it is important to recognize that energy is necessary to create a new surface because you have to break bonds. So, this energy that is required to create a new surface is called surface energy and you measure it in so many joules per unit area. Now, as you can see because of the very large surface area that becomes possible when objects are scaled down the surface energy of nanoparticles is very high and it goes inversely as or the radius of the particle and what I have here is a table that shows how the surface energy of a particle varies when the size is reduced from macroscopic sizes that is centimeter level sizes all the way down to nanometers. So, it since it goes inversely as a dimension you can see the surface energy varies from 10 to the power of minus 5 joules per gram in the case of macroscopic objects all the way down to all the way up to 570 joules per gram when the size is reduced to 1 nanometer. Coordination number what is coordination number? Coordination number is the number of atoms number of nearest neighbors that an atom an atom is bonded to. So, it is the number of bonds it has got and that is the coordination number. Now, atoms at the surface are in a way partially orphaned because they do not have atoms to bond to above the surface. Therefore, as you can see that the bonding of a an atom at the surface of an object is weaker in other words its coordination number is smaller. One refers to therefore, the average coordination number of a material because the surface atoms have a lower coordination than those in the interior of a solid. What is shown here is the mean coordination number of magnesium clusters that is calculated as a function of the cluster size. So, the mean coordination number is on the y axis and the number of atoms in the cluster is on the x axis and the linear scaling here shows that the coordination number goes as n to the power of minus one third that is it scales the same way as the dispersion function. Now this can be expected because as the scale of the as the size of the object is scaled down then the surface area increases and therefore, the number of surface atoms increases therefore, the average coordination is diminished. In the limit of bulk objects then you can see that the coordination approaches the value of 12 which is what it is for a close packed magnesium crystal. Cohesive energy this another one that scales the way that the dispersion does. Cohesive energy is the energy that holds the solid together you could say that it represents this strength of bonding in a solid. What is shown here is a graph of the average cohesive energy of magnesium clusters as a function of the size of the cluster. What is shown here is a linear plot again that is the scaling is now the same as the scaling for the dispersion and for the average coordination number. Therefore, all the quantities dispersion average coordination number and cohesive energy they all have the same scaling law they go as n to the power of minus one third. In this graph you can see if you extrapolate for large values of n which represents which is on the left hand side of the x axis for large values of n this approaches the cohesive energy approaches the value 1.4 electron volts which is close to the experimental value for magnesium crystals. Now, for just a moment let us look at liquids as condensed matter. What is shown here is the vapor pressure of different liquids as a function of the radius of drops of these liquids different liquids like chloroform toluene water are shown here. What you can see is that as the radius of the drops or droplets of these different liquids comes down to very small values the vapor pressure increases exponentially. What is vapor pressure? How is this generated? It comes out of the boiling off of atoms at the surface into the vapor phase that is from liquid to the vapor phase. Now, as you size it down as you size a liquid drop down what is happening is that a large fraction of the atoms of the droplet are now at the surface and these are less well bound than the atoms in the interior and therefore they can escape or they can boil off which is really what vapor pressure is due to. So, this illustrates again that in the condensed liquid phase as well the effect of the coordination being low at the surface leading to higher vapor pressures. Now, to repeat the average coordination number as well as the cohesive energy scale similarly go as n to the power of minus one third. This makes you intuitive sense because for a given atom each neighbor forms one bond. So, it has if it has fewer neighbors then it has fewer bonds and therefore a lower cohesive energy atoms in the interior are more fully coordinated and they are more strongly bound. Those on the surface are less well bound even less well bound are atoms on the edges of a cube for example that we just went through and those in the corners are least well bound because they have the fierce coordination. All of this is actually illustrated by experimental evidence for example if one examines in an electron microscope microscopic samples of let us say metals one finds often that the atoms at the corner are missing even under thermodynamic equilibrium conditions these are defects and these arise because of the low coordination at the corners. For same reason corner atoms are favored sites for adsorption that is if you are to develop a catalyst then one of the things that is important is that you really make sure that you have particles that have a lot of corners in them because these corner atoms are hungry for adsorption which is what is required for better catalytic action. One of the consequences of particle size is related to the cohesive energy which we already mentioned but a different physical property is affected by that. Actually this is a well known experimental result as well as a theoretical result dating back to 1871 the Gibbs Thompson relationship where the variation of the melting point of a solid is related to the particle sizes. So, it is well known even such a long time ago that the melting point would be lower if the particle size is smaller. So, I want you to note the one over all dependence which is similar to the dependence of these other quantities that we just discussed. So, the melting point of a solid as I said is reduced when the size of the solid is reduced side of the particle is reduced. One particular experimental manifestation of this or experimental verification of this is the measurement of the melting point of indium in glass pores of well defined size. So, what is plotted here is the melting point on the y axis as a function of the inverse diameter of glass pores of different sizes and what I wanted to see is the linear relationship that is exactly the same that is the functional relationship that is exactly the same as the scaling of the dispersion of the cohesive energy and now the melting point. So, all of these scales smoothly with the size of the particles and this is really the same result where the melting point is shown as a function of the diameter earlier it was the inverse diameter now it is the diameter and what we see here is the approach of the melting point to the bulk value as the pore size of the glass pores in which indium is confined that pore size increases and then the melting point of indium approaches the bulk melting point of 430 Kelvin's. Now gold just as a rather well known example gold melts at about 1336 in the bulk 1336 Kelvin's in the bulk what is shown here is experimental results of the melting point as a function of gold particle size it turns out that it is possible to make gold particles of extremely well defined sizes so called mono disperse gold particles where an entire sample has a very narrow size distribution. For example, you can make gold nanoparticles measuring 5 nanometers almost exactly and so forth. So, the melting point of gold of these different measurements has been experimentally determined and the melting point of gold falls dramatically when it is size down to about 5 nanometers or below. Now it is also a context in which we can recall that there is a different experience that we all probably are familiar with that is the lower melting point of surface layers of objects or materials that we are familiar with. For example, ice is covered by a liquid like layer even down to minus 10 degree celsius whereas it is supposed to solidify at 0 degree celsius under atmospheric pressure. This is because of the relatively poor bonding at the surface which makes it possible for a liquid like layer to be seen on ice all the way down to minus 10 degree celsius. Another aspect of another demonstration I should say of the melting behavior on the nano scale is this set of data that shows the melting of indium in pores of very well defined diameters. I mentioned this to you earlier what is shown here is calorimetric data on the y axis is the amount of heat that is absorbed for the melting process to take place on the x axis is the temperature and what I want you to see is that as the pore diameter is reduced from 100 nanometers all the way to 5.6 nanometers the endotherm that belongs to the bulk indium melting which is at 435 gradually moves to lower temperature that is when the pore size is 101 nanometers the melting point is below 430 degree celsius and then as the pore size is reduced this temperature of melting of indium within these narrow pores becomes lower and lower as well as these peaks are no longer sharp. What this is showing is that because of the low coordination that is available to indium atoms within these confined pores smaller number of atoms to bond to therefore the melting point of gold of indium diminishes regularly as you size down the pores and also it is no longer as sharp. Now this can be understood in the following way the melting of a solid is a phase transition is a first order phase transition and these phase transitions are collective phenomena that is it occurs through the participation of millions of atoms together when this collection is smaller when there is a lower number of atoms in the cluster for example a phase transition now is well defined is not so well defined is no longer sharp. Small clusters behave more like molecules than as bulk material one may think of them less as phases than as different structural isomers that coexist over a range of temperatures when these phases are less well defined as in this case and in the small clusters the Gibbs phase rule loses its meaning. So you could think of nano materials as a point where thermodynamics that we normally know can break down and of course that the reason for that is you no longer have a large collection of atoms and in effect thermodynamics deals with bulk material. So what we have done in this session so far is to have an introduction to nano materials in a rather general way but later on to go through the scaling that applies to certain properties of materials when you bring them to nano scale that is the so called smooth scaling. In the coming classes we will deal with the effects of quantum phenomena where there is abrupt scaling as opposed to smooth scaling and in the coming classes because it is necessary to understand the basics of quantum mechanics we will go through the elements of quantum mechanics in order to be able to understand nano materials.