 After talking about magnetism in nanoparticles, let us take up the topic of magnetism in clusters. Early on we had said that we could actually divide into materials into three size regimes. One is what we call small clusters, one we call large clusters and one we call nano crystals. And this division of properties was based on a variation of a property and with the vertical addition of removal of more number of atoms. And in fact, magnetism in some sense is a ideal property to make this kind of a distinction between what you might call a small cluster, a large cluster and a nano cluster. We said that clusters or small clusters are that size regime, wherein the addition or removal of a mere atom can cause a large variation in the property. And here of course, we are talking about a property like a ferromagnetic property of a cluster of atoms. This is important note, because even you are in the nano crystal size regime, a mere addition of or removal of atom does not cause a serious change in the properties. In other words, there is a small change, but that is not significant with respect to the already existing value of the property. So, these clusters are size regimes which are usually small and less than about 5 nanometers. Typically, we want to consider clusters wherein we can actually count the number of atoms like a 13 atom cluster or a 25 atom cluster or a 50 atom cluster. So, these are the real size regimes of clusters, wherein we are really interested. And in the limit, what you might call in the large size limit, what we obtain is called the bulk limit, wherein you consider a large nano crystal. Now, when you are talking about magnetic behavior or ferromagnetic behavior of clusters, a few important points have to be kept in mind. This is of course, the atomic structure. And here we may note that this atomic structure need not be crystalline. In other words, this could just be a cluster which has its own structure. And this is not necessarily a crystalline kind of a structure. We have to worry about the nearest neighbor atom distance in the cluster. And of course, there could be more than one nearest neighbor distance depending of the size of the cluster. Or of course, there could be one nearest neighbor, there could be one other distance which is very close to the nearest neighbor distance. Then of course, the purity and the defect structure of the cluster is also very, very important. Because manufacturing some of these clusters is very difficult. And getting a right number of atoms and later on of course, separating those cluster sizes based on mass spectroscopy and studying their properties is indeed an important challenge. When you are talking about ferromagnetic clusters, we identify the regime of small clusters with about less than about 20 atoms. And here there is large oscillations in the magnetic moment of the cluster. And of course, we are talking about the magnetic moment per atom. We already noted before that when you are talking about a property like magnetic moment, when you decrease the size of the or the scale of the system, the magnetic moment increases above the bulk value. Now, we will be noting that how this behavior becomes really drastic when you are talking about small clusters. And when you are talking about a large group of atoms like more than 600 atoms, you typically notice that bulk like behavior emerges. That means, with increasing number of atoms, the oscillations in the property here of course, the magnetic moment dies down. And therefore, there is no sharp variations with change in number of atoms and actually the bulk behavior emerges. So, we have 2 or 3 size regimes that means, small clusters having typically less than about 50 or 20 atoms. The big clusters from 50 atoms to say 600 atoms and finally, of course, the bulk behavior wherein you can think of what you might call the magnetic moment approaching that of the bulk value of ion. Now, we already noted that ion can have a maximum possible magnetic moment of about 6 mu B per atom, 3 arising from orbital and 3 from spin. We had also noted that typically in bulk ion most of the orbital magnetic contribution is quenched because of the strong interaction with the lattice. Now, in the if you take up individual clusters to highlight how the property changes and this is schematically shown here in this graph here. You are studying the number of atoms in this cluster. This is no longer a dimension like nanometer, but it is purely a number because we are really in the small cluster regime. And you are studying the magnetic moment per atom on the y axis. Now, so therefore, we can see that clearly there are 3 regimes what you might call the small cluster regime which you can see where in there are the small cluster regime where you can see that there are sharp variations in the property. Then there is the medium cluster regime wherein you have oscillations, but they are smaller oscillations. They are oscillations spread over a large number of atoms that means over a large collection of atoms. You see these oscillations are taking place, but when you add many number of atoms not a small one or two atoms. So, you have this regime which is of interest to us now. You notice that nearly by adding one atom and going from this value you can see that this is the Fe 12 cluster and you go to the Fe 13 cluster. That means you are just adding one atom. You notice that the magnetic moment changes drastically. There is a serious severe precipitous fall in the magnetic moment and this is what is we call the small cluster regime, a signature of the small cluster regime. Further when you add one more atom, there is again a sharp increase in the magnetic moment of the cluster. These oscillations continue in what you might call in a sharp fashion till you reach about 25 atoms. After that of course, we said that the variation with cluster size gets a little more smoother. Finally, of course, in the bulk regime you will notice that the variation becomes even more smooth. You have the bulk behavior emerging. In other words, the iron in the large crystal regime actually approaches a magnetic moment of about 2.2 mu B per atom. In this graph of course, the lower limit is the bulk limit and the upper limit is what you might call the atomic limit which is possible, which we said is about 6 mu B because an iron atom cannot have more than or an iron cluster cannot have more than 6 mu B because each atom only contributes about 3 mu B from the orbital motion and 3 from the spin contribution. Therefore, the maximum limit is 6 and all, but nevertheless it is important to note that all these magnetic moments which are noting in the graph here are above that of the bulk limit. That means, there is an enhancement in the magnetic moment in the of course, nano crystalline regime, in the regime of the large cluster and also in the regime of the small cluster. So, the Fe 12 cluster has a magnetic moment of about 5.4 mu B per atom. That means, very little of the orbital motion is quenched in this cluster. So, this is would be like approaching the atomic limit when you have a I in 12 cluster. On the other hand, just one more atom making it Fe 13, we notice that it has a much lower magnetic moment of only about 2.44 mu B. Now, if you compare this Fe 13 cluster with for instance nickel 13 cluster, it also has an abnormally low moment and this can be attributed to the icosahedral structure of the cluster. In other words, the cluster is densely packed. That means, atomic separation is reduced in this cluster and we have noted earlier that 3 factors coming into play while deciding the magnetic moment of the cluster is the nearest neighbor distance in the cluster. And therefore, if the nearest neighbor distance is coming down, then you have a reduction in that means, there is an overlap of the atomic orbitals and therefore, there is a reduction in the magnetic moment. With larger and larger cluster size, the orbital contribution seems to be low, but there is still an enhancement of the magnetic moment over the bulk value. This we had noticed before, because this is purely coming from the reduction in the size. The structure and packing fraction seem to play or packing method seems to play an important role in the net magnetic moment obtained when you are talking about these clusters of magnetic atoms. So, to summarize this story of the ferromagnetic clusters, we can divide the ferromagnetic clusters into nano crystalline regime where the oscillations are small. The small cluster, big large cluster regime where there are oscillations, but this spread over large number of atoms and the small cluster regime typically less than about 20 atoms, wherein there are serious variations or sharp variations when you add or move an atom. And you can see that the ion 12 cluster actually goes very close to the what you might call the theoretical possible limit for an, which is close to the atomic magnetic moment of ion. On the other hand, there can be clusters like the Fe 13 cluster, which are close packed cluster having icosahedral configuration, which give you actually a low magnetic moment. After talking about the ferromagnetic material like ion, let us talk about what you might call the anti ferromagnetic clusters. Now, when you are talking about anti ferromagnetic clusters, these are actually not anti ferromagnetic in the cluster zone, they are actually what are anti ferromagnetic in the bulk. So, for instance we know that chromium is one of the anti ferromagnetic, one of the few elements which are anti ferromagnetic like manganese. And therefore, in the bulk form chromium or manganese is expected to be anti ferromagnetic, but here we are talking about anti ferromagnetic materials in the bulk, which have behaving like ferromagnetic ferromagnets in the cluster zone. That means, when you are making the small clusters out of these anti ferromagnetic materials in bulk, they are actually getting ferromagnetic property. Again a very interesting observation that because we have these materials do not behave like ferromagnets in the bulk. In anti ferromagnetic materials, we do not expect any net magnetic moment in the bulk, because the up spins and down spins cancel themselves out. However, there is a possibility that a small cluster in a small cluster, there are up and down spins do not cancel out leading to a net magnetic moment. Of course, this magnetic moment is not because this is now a small residual non cancellation and this is not expected to be as high as that you see in a ferromagnetic cluster, but nevertheless this is an interesting possibility. In these small magnetic clusters, ferro frustration is also an issue and this frustration leads to the fact that the spin on a given atom does not know which way to point. In other words, if I had a large system, then I know in a anti ferromagnetic material, if I had spin arrangements ups and downs they will go like this. But in a small crystal, there could so happen that there is one spin which is introduced here, there could be an atom here, which does not know that if this spin has to be up or down and this automatically implies that it is not following the anti ferromagnetic order and therefore, there could be a net residual spin up or spin down coming from the fact that that atomic position has been frustrated with respect to the remaining of the lattice. We will take up two examples to understand anti ferromagnetic clusters. One is the example of chromium and one is the example of manganese because these are elemental, but having said that if they are elemental, it does not mean that their overall magnetization states are easy to understand. Now, if you take up the example of manganese and study now like before, the number of magnetic atoms in the cluster along the x axis and the magnetic moment along the y axis, again you do see what you might call three regimes. One regime which we might call the, what you might call the small cluster regime, one you might call the large cluster regime and then of course, the nano crystal regime and finally, you have the bulk limit. The bulk limit in this case clearly is the case where there is zero net magnetic moment because you know this is an anti ferromagnetic material and therefore, you do not expect any magnetic moment in the bulk form. So, ultimately this curve has to go down to zero. In the small cluster regime, again you notice that there are sharp variations in the magnetic moment and you can observe that these sharp variations again can be understood in terms of certain geometry that for instance, the icosahedral cluster gives you a very small magnetic moment, net residual magnetic moment while then by adding just a lot of atoms, you get the n equal to 15 cluster which is having a very high magnetic moment. But having noted that this very high is only about 1.5 mu b per atom unlike the case of the very high in the case of the ferromagnetic cluster, where we went up to a value of 5.4 mu b per atom because these are not inherently ferromagnetic material. These are anti ferromagnetic materials which are behaving like ferromagnetic materials in the small scale regime. Therefore, again you see that in these cases, there are what you might call minima corresponding to the icosahedral cluster and the n equal to 19 cluster and we will talk about this in the in a few moments. But these oscillations tend to die down when you approach the large cluster regime and then of course, you obtain the bulk limit where in of course, you ultimately get a zero magnetic moment. Now therefore, in manganese clusters show some similarities with ferromagnetic ion cluster with regard to the cluster size dependence and this is in the regime of more than about 10 atoms. Compact this 13 atom clusters which is icosahedral configuration and the m n 19 which is actually a double icosahedral cluster. That means, there is one icosahedral shell and about that there is a second icosahedral shell which is called a double icosahedral construction gives very low magnetic moments compared to the neighboring clusters like m n 15 which has a high magnetic moment of about 1.5 mu b per atom. So, the low magnetic moments are about 0.4 mu b per atom if this is in this range 4 and 4 to 0.5 mu b per atom while the high one correspond about 1.5 mu b per atom. So, in the case of manganese you notice that there is a variation between about 0.4 and about 1.5 mu b per atom based on the cluster size and again like before in the case of ion we can understand this in terms of the geometry of the cluster that if there is a close packed cluster having in some kind of a single shell or a double shell icosahedral configuration then the magnetic moment is small. The case of chromium is slightly more complicated in fact it is significantly more complicated and the reason happens to be that the anti ferromagnetism in chromium is supposed to arise from what is known as a spin density wave. So, an often the phrase used is that it is an incommensurate spin density wave anti ferromagnet. That means that this is not a simple kind of an anti ferromagnet which you can be drawn like a picture using this one we saw before this is the underlying spins have a spin density wave and this spin density wave as I pointed out can actually be incommensurate with respect to the underlying lattice. And therefore, you can have an incommensurate spin density wave which can which is actually giving rise to this anti ferromagnetism in chromium. And therefore, since this is a very rich system this also means that there are lot of complications and understanding such a very rich system like chromium. Though we said that it is an elemental anti ferromagnet this does not make it a simple anti ferromagnet though it is elemental. This is akin to the example of manganese in crystal structures though manganese elemental crystal often you know that alpha manganese and beta manganese have actually a very complicated crystal structure having very number of atoms. The plot of magnetic movements oscillates with size like in the case of the manganese or the case of the ferromagnetic clusters. A cluster size of CR 9 is expected to exist in multiple magnetization states and this multiple magnetization states actually makes the analysis very difficult for us to understand that what is really giving rise to this effect. That means in addition to this multiple magnetization states the issue is of there are many structural isomers also possible that means that CR 9 cluster not only exist in a single structural form it exist in many other structural isomers. And therefore, these structural isomers each one of them will have a different magnetization and the net effect on magnetization you are seeing is actually when you do an experiment is actually an average over all these structural isomers and all these multiple magnetization states. The CR 9 magnetization can actually be small it is only about 0.65 mu B per atom or it can be 1.8 mu B per atom that means a single cluster I do not mean a single structure structurally a single number of atoms 9 atoms in the CR 9 cluster can exist can give you a minute magnetization of something like 0.6 mu B or 1.8 mu B which means this is a large variation we already saw that the maximum minima it is between the almost the maxima and minima in such a plot. So such large variations can it is actually difficult to understand it and the underlying mechanisms are supposed to be as I pointed out the existence of multiple magnetization states and also existence of structural isomers. So the essential message of studying anti ferromagnetic clusters is that there can be materials which are anti ferromagnetic with 0 magnetization in bulk, but which behave like ferromagnets in the small cluster regime. When and in the small cluster regime like in the case of ferromagnets ferromagnetic clusters there are large oscillations and these oscillations finally die down and you obtain what is called the large cluster regime and finally the nano crystal regime. Additionally we noted that to understand the variation in magnetization in these small clusters we can invoke the principles we used in the case of the ion clusters. In other words the icosahedral or close packing in these crystals is expected to give us these large variations in the magnetization. So this is a very interesting possibility that actually you are seeing ferromagnetism in small clusters of an anti ferromagnetic material. We had seen that the magnetic properties of clusters is a very sensitive function of the number of atoms in the clusters. Before we go on to other interesting magnetic properties of other nano structures and hybrids, let us consider briefly here the experimental production of clusters and how we can measure the magnetic movement of these clusters. Typically a gas phase super saturated in metal vapor is ejected into a flowing inert gas which is cooled and at the heart of or the schematic of this kind of a production technique is shown in the figure below wherein you see that there is a liquid metal bath. There is a gas which is flowing into the chamber. This chamber is at a high suction rate, but in the low vacuum. Further this liquid matter which is in the vapor state is atomized through a nozzle. It passes through a skimmer and goes on to a chamber where there is an high vacuum pump which of course has a low suction rate, but it is a high vacuum pump. So essentially we have a super saturated metal vapor which is ejected into a flowing inert gas. The metal vapor itself could be produced by thermal evaporation, laser ablation, sputtering or many of the other techniques which can produce this metal vapor. Further we know that now this metal vapor consists of clusters of diverse sizes that means they may have a right from a few atoms to a few thousand atoms and we require mass separators which can actually separate out clusters of various sizes. And for this the clusters typically need to be charged and if the clusters need to be ionized if they are not charged priorly. So this is a very important point because the mass separators will not work most of the mass separators will not work on non charged particles. Here we list the name of some of these mass filters though we will not go into the details of their operation or their physics behind them. These mass filters include radio frequency quadrupole filters, vane filters, time of flight mass spectrometers, pulse field mass electro etcetera. But as we pointed out all these techniques typically depend on the clusters to be charged and if they are not charged as we shall see in a coming slide that we will charge the clusters before they are sent into a spectrometer like for instance time of flight mass spectrometer. At the end of the separation our goal is that we have a narrow distribution of masses of these particles or these clusters and if you are talking about small clusters we would even like to get a precise number of atoms in these clusters. Because as we have noted before that in small clusters addition or removal of a single atom can lead to very steep changes in terms of their property magnetic properties like we saw that for the case of the ferromagnetic clusters where we noted that if you have an icosedral cluster it typically has a low magnetic moment. But if you add an atom to it then the magnetic moment increases steeply. Similarly, so for anti ferromagnetic clusters that we saw that in the small cluster regime actually we have severe oscillations in terms of the magnetic moment of the particles. Therefore, in small clusters we would like to isolate a precise number of atoms and as I pointed out some of these techniques like radio frequency quadrupole filters, wind filters etcetera help you in separation of these different mass particles which also imply different number of atoms in these clusters. We shall also briefly take up here the measurement of the magnetic moments of clusters that is we had already noted that these plots which we have seen are actually experimental plots. That means people have actually produced clusters of atoms consisting of magnetic material and further have gone on to measure the magnetic moments. So, these are experimental results and therefore we need to know how we can actually measure the magnetic moment of these clusters. Though our lecture here would be limited to a brief outline and students interested in these techniques may consult extensive text on these aspects further. Now, first we shall talk about experimental results presented for free clusters. Clusters can be of three types they can be free clusters, they can be clusters which are deposited on a substrate that means they are on the surface of a substrate or they can actually be embedded within a material. But here we first take up or we will present in a little detail how can we measure the magnetism or a magnetic moment of a free cluster. For instance this could be iron ferromagnetic clusters or it could be chromium and manganese antiferromagnetic clusters and the typical experimental setup is based on the classic Stern-Gerlach experiment which was first used to detect the electron spin. Now, typically this Stern-Gerlach kind of an experiment is coupled with a pulse laser vapor deposition techniques and we shall consider some of these details in the next slide for instance. So, typically such an experimental setup is what is used for the detection of the magnetic moment at the heart of such an instrument is what you may call the time of flight mass spectrometer and the whole setup starts with a cluster source. So, before we take up this setup in detail let us go with go ahead with the further points that typically a collimated cluster beam is guided into a magnetic field gradient that means we do not have a constant magnetic field, but a field gradient which is quantified as d v by d z. The magnetic field gradient will deflect a cluster with a magnetic moment mu by a distance d and this is given by the equation as here. So, if you have the distance of deflection d under the influence of a magnetic field gradient d v by d z then if the particle has a mass m then this is the equation governing the deflection of the particle. Now, in this equation l is the length of the magnet d is the distance from the end of the magnet to the detector m as I pointed out is a cluster marks and v x is the entrance velocity which is assumed constant. Of course, there may be many situations in which the entrance velocity is not constant and therefore, this would be an simplifying assumption which is used. So, that we can approximately find out the distance of deflection as I pointed out. There are other kinds of magnetic particles those which are embedded in a second phase in a matrix or those which are deposited on surfaces and for clusters deposited on surface other techniques of measurement exist. That means the technique mentioned above is not suitable for that purpose as you will see that in such a technique you need a stream of these magnetic particles and these magnetic particles will then be deflected by the magnetic field and will be put through a mass spectrometer. For particles or clusters deposited on surfaces we can use techniques like extra magnetic circular dichorism, dichorism in photoelectron spectroscopy, surface magneto optical care effect, UHV vibrating sample magnetometry etcetera. And here we are just mentioning the names an interested reader obviously needs to go into text which will go into detail regarding the operation of these techniques and the physics behind them. For embedded clusters techniques like micro squid measurements, micro hall probes etcetera can be used for the measurement of the magnetic movements. So, a single technique is not sufficient for us to understand all kind of magnetic clusters, but here we take briefly the one which is used for the measurement of pre particle, free particle magnetic movements. Just to summarize this slide at the heart of the experiment is the setup which is based on a technique like this term Gerlach experiment, the classic experiment which measured the electron spin. The physics behind is that you have a magnetic field gradient which deflects the particle based on the mass of the particle and its magnetic movement. And based on the deflection we can then back calculate the magnetic movement on the particle assuming that the velocity of entrance of these cluster particles is independent of their size and it is a constant quantity. So, this is the brief outline of the experimental setup for the measurement of magnetic movements. First metal clusters are produced by pulse vapor deposition of course, other techniques may also be used for the production of these metal clusters of a target material into a jet of helium gas. Of course, helium you know is inert and therefore, will not react with the metal cluster vapor. The cluster for glass mixture is undergo supersonic expansion on entering vacuum. So, this is the next stage after the production of these metal clusters that they undergo supersonic expansion when they enter the vacuum. Then the beam produced so that means here you have the nozzle which produces the beam. The beam produced is collimated and the apertures play an important role in this collimation process. Then further the magnetic field gradient which is produced by these field gradient magnets is used for the deflection of these collimated beam. But, as I pointed out even though now you have a deflection produced unless these clusters are ionized we will not be able to do mass spectroscopy on them. Therefore, these cluster need to be ionized and the next chamber in down the line is the ionization by laser. So, you have a laser is specifically used to ionize these clusters and this laser is different from the laser which is used initially for the production of the vaporized metal. So, this laser has a different purpose this laser is used for the ionization of the cluster. Finally, after the cluster is been ionized now their time of flight will depend on the mass and the ionization and therefore, the mass dependent deflection is measured and is perpendicular to the beam in a time of flight mass spectrometer. That means initially the beam was arriving in this direction, but now the beam is deflected into the mass spectrometer at right angles to the original direction in which the cluster source was producing that. So, to summarize this experiment we are trying to measure the magnetic moments of free magnetic particles and for that we are using a equation like the one listed here below. We are also using a modified version of the Stern-Gerlach experiment and the flow chart for this experiment is as follows that we have a cluster source produced by a laser. This cluster source is collimated and this cluster source is in mixture with a gas and therefore, this gas and the cluster source goes through series of apertures and then a field gradient as shown here. So, it goes through a field gradient and this field gradient deflects the particles depending on their magnetic moment and their mass and finally, the to detect the particles we ionize these clusters and finally, use a time of flight mass spectrometer to deflect it and finally, back calculate the magnetic moment on these clusters. Now, obviously as we can note that there are issues regarding the measurement of magnetic properties of nanomaterials which are quite unique to nanomaterials and they are quite different from their bulk counterparts. So, needless to say that the measurement of magnetic part properties of clusters and nanostructures is challenging as compared to their bulk counterparts. In clusters magnetic moment is a sensitive function of the number of atoms in the cluster that means we need to know the number of atoms precisely. In the case of a bulk it does not matter to us that if the bulk actually is a 1 centimeter bulk or 1.1 centimeter bulk, but in the case of nanoparticles and nanoclusters we need to know that what are the precise number of atoms in the cluster. Coagulation and contamination of these clusters and nanocrystals is a serious issue and this contamination may take place either during production or during measurements and has to be avoided. Because small amounts of contaminants can severely alter the properties of these magnetic nanoparticles a good example of this would be the surface oxidation of the cobalt magnetic particle. On surface oxidation you actually form a cobalt oxide on the surface and the cobalt cobalt oxide system has a very different magnetic behavior which we will consider shortly and which is at the heart of a phenomena known as the exchange anisotropy. Therefore, we should avoid any kind of contamination or any kind of reaction with the atmosphere during production or during the measurement process. As you know temperature plays a very key role in the magnetic behavior of nanoscale systems because reduction in size is equivalent to increase in temperature. And therefore, when we assume that the temperature of the system is a particular number we have to assume it is precisely measured and is constant at the level of the cluster and not merely at the is the average temperature of the chamber. So, not only we should precisely measure the temperature, but control it at the level of the clusters. The spin alignment in nanoscale systems which we will consider in detail in the coming slides could be very different from their bulk counterparts. And hence when you try to back calculate using a model or use a model to understand experiments then the precise geometry of the system and the spins with orientation therein has to be taken into account. That means, we should know the geometry of the nanostructure we should know the alignment of spins therein in the nanostructure. And we should also take into account the surface effects else the model will not be able to satisfactorily either understand the experiment or predict properties based on the experiment. In the case of particles in a substrate or embedded in embedded in magnetic nanoparticles we have to remember that this is no longer a free system. That means the properties of an particle which is in intimate contact with for instance with a substrate or a matrix is very very could be very very different from the free standing nanoparticle. And the role of these interface of the substrate we very very pronounced. And often this interface could play could alter the properties of the nanoparticle all together. And also we should remember that if you talk about a single isolated free standing nanoparticle it is magnetic properties could be somewhat different from that of a many clusters placed in proximity. In other words now the system is the nanoparticles are not free and they talk to each other. And this collective behavior could be very different from that of the single isolated nanoparticle either free or deposited on a substrate or embedded in a matrix. And therefore, but also we have noted that deducing the properties of free standing nanoparticles from those measured could be very difficult. In other words suppose I make measurements on a nanoparticles which have been deposited on a substrate. And from there I try to back calculate that what would be the properties of a free standing nanoparticle this could be an adverse task and sometimes will not give the correct results. Therefore, to summarize this slide issues regarding the measurement of magnetic particles are many a few of them we have taken up in the slide. But as you can see that unless some of the issues regarding for instance the purity of the nanoparticle the temperature of the system the actual knowledge regarding the number of atoms in the system the kind of models which have been taken into account in the underlying model like in terms of the spin alignment etcetera have to be specially constructed for these nanoclusters and nanoparticles. And we cannot merely extrapolate from the their bulk counterparts. And also we noted last, but not the least that isolated nanoparticles could be very different from those which are in proximity to similar kind of nanoparticles. So, nanoparticles are different types and therefore, the surrounding medium is or the surrounding environment to the nanoparticle or the nanocluster is also a lot of importance when you try to deduce the properties. Next we take up magnetism in thin films and hybrids and we had pointed out while we talked about thin films that there are couple of important things that in thin films of magnetic materials ferromagnetic materials. You actually observe what you might call nail walls which are not found in typically in bulk materials where you actually observe what you call domain walls which are called block walls. And interestingly we will talk about we talk about magnetism in hybrids we will come across very interesting phenomena later on. First let us talk about this dimensionality of the system. When we talked about previously about the thin films we noted that the magnetization is in in plane and there are no components of magnetization out of plane. Of course, this is only true for very thin films and when the system size grows you would note that there is an out of plane component of magnetization or the orientation of the spins which means the system starts to behave like a 3 D system. So, when is this transition from a 2 D behavior to a 3 D behavior is what we are noting. So, people have studied of course, these thin films cannot be free standing thin films because they would tend to warp and they are difficult to study. Typically these thin films are grown on a substrate and in the case of nickel films which have been studied on a copper 1 0 0 substrate. Of course, it is copper is chosen because copper is not a magnetic material. Often you may also notice that in some of these systems studied the film may actually be epitaxial with respect to the substrate. Now, in the case of the nickel films on copper substrates and the copper 1 0 0 single crystal substrate has been used in the experiment. When the thickness of the nickel film is greater than about 7 monolayers that means this is very very thin film regime. The systems behaves like a 3 D Heisenberg ferromagnet. In other words it becomes a 3 D ferromagnet that means with spins out of plane when already the system size is only about 7 monolayers. On the other hand below 7 monolayers it behaves like a 2 D system. In other words all the spins are in the plane while in the 3 D system spins can be out of plane. So, this implies that you can have a transition from 2 D to 3 D behavior in these nickel films which are grown on copper substrates for a very thin size of film like about 7 monolayers. So, below 7 monolayers you actually have 2 D system when all the spins are in plane and above 7 monolayers the system starts to behave like a 3 D ferromagnet wherein the spins have an out of plane component also. Which means that now correspondingly then the neal wall which we talked about wherein all the spins are rotating in plane would also have a sudden out of plane component in such kind of systems. If in of course we are now that means that we are talking about a multi domain system in the film if you have to consider that. Next thing we talk about in these thin films is the case of the curie temperature in thin films. We had noted earlier itself that the curie temperature actually varies with size in the case of iron 1 0 0 films grown epitaxially on the silver 1 1 1 films that means my substrate is now silver 1 1 1 substrate and you are actually growing a iron 1 0 0 film on it. And again we are in the what you might call the very thin regime wherein we are tracking the layers layer by layer that means it is 1 monolayer or 2 monolayer or 3 monolayer. Now when you are talking about a these thin films we notice that the curie temperature can actually be reduced with respect to the bulk value. Because we already told that decreasing the size in some sense is like increasing the temperature and when you increase the temperature what happens is that there is a ferromagnetic to paramagnetic transition which means that actually you are crossing the curie temperature. So, this is clear. So, when I plot the thickness of these monolayers with respect to T c and of course now I am normalizing this T c with respect to the T c of the bulk material which is what we normally called the curie temperature. You would notice that of course for large sizes of monolayers more than about 50 monolayers you approach the bulk limit that mean T c by T c bulk becomes 1. But as you go down to the very thin monolayer regime the curie temperature actually falls. In other words in this very thin monolayers less than about 20 monolayers you can see that the exchange coupling is not able to fight against the thermal disordering that effectively because they are not just not enough number of atoms in this layer which implies that the curie temperature is going to be reduced. In the case we are talking about here the ion 110 film on silver 111 substrate the curie temperature can actually be depressed by about 100 Kelvin with respect to the bulk that means you can approach about 10 percent of the bulk value when you are in the regime of about 1.5 monolayer thick films. That means that in these thin films again like because it is a reduced dimensionality system the thermal disordering effects become very important on magnetization. So, therefore though there is this possibility we introduce that in a reduced dimensionality system there is this enhanced magnetism per atom we can obtain that means the number of bore magnetons I get for an atom increases that possibility is there. But then this is also fighting against thermal disordering effects and therefore we will have to cool the system to actually obtain that value of the what do you call the or the potential magnetization per atom. Before we take up what you might call magnetism in hybrids let us summarize this slide by saying that in very thin films you have a 2 D kind of a behavior and this very thin we mean is in the order of a few monolayers and here as I pointed out we are not really studying free standing thin films because free standing thin films are difficult to study. So, these are typically films which are grown either a bit actually or otherwise on some kind of a substrate typically the substrate used in of choice is a non magnetic material like copper or silver which is a single crystal and these single crystalline over layers when they are in the small size regime typically behave like 2 D system and later on they start to behave like a 3 D system for thicker sizes. Additionally these very thin films can actually have a depression in what you might call the curie temperature with respect to the bulk of a large value that means it can actually reduce to about 10 percent or more for very thin films for when you are trying to find the curie temperature of these thin films. Next we take up magnetism of hybrids of hybrids and in this context we will take up the important phenomena known as giant magnetor resistance or called as GMR for shot. It is important to know we had earlier noted a concept known as the anisotropic magnetor resistance and we had noted that that magnetor resistance value is typically about 5 percent. That means in the presence of the magnetic field and in the absence of the magnetic field the resistance value changes only about by 5 percent. We are also talked about an important concept in that context that was the concept of the spin dependent scattering that means the resistance is coming from a new kind of a phenomena which is not the usual charge scattering arising due to phonons or impurities but what we might call the spin dependent scattering. The important thing in giant magnetor resistance the word giant implies that the change in resistance could be of the order of 80 percent or more. So, this is a very important point known that is why it is called giant magnetor resistance and there is also a related phenomena or a different kind of phenomenon known as colossal magnetor resistance which we will not talk about in this course but this giant magnetor resistance is not the same as colossal magnetor resistance. Another important point about giant magnetor resistance is that is actually found in hybrids. That means it is found only in composite structures it can there are cases where giant magnetor resistance has been reported in other kind of systems also but the classical system is that of an hybrid. So, what is the configuration which is giving me this giant magnetor resistance it is typically a non magnetic layer in lying in between two magnetic layers. This is of course the simplest configuration you can think of let me talk about for instance in this case example given here you have two layers of iron which is ferromagnetic and there is a non magnetic layer like chromium. The thickness of this chromium layer is very very important as we shall see soon and typically it is of the order of about 9 angstroms. The thickness of the iron layer is about 30 angstroms in instead of making one single sandwich structure like this you could actually repeat this structure to form a larger layer structure consisting of these ions and these chromium. And this kind of a multi layer super lattice structure can actually give you an even higher enhanced value of giant magnetor resistance. But the overall physics can easily be understood by considering only a three layer system consisting of two ferromagnetic materials and a non ferromagnetic interlayer between the two. Of course it is as I told you the non ferromagnetic non magnetic layers thickness is very very important and it has to your of a very specific size. The reason is that depending on this size of the non magnetic layer there is an anti ferromagnetic coupling between the two layers of iron. That means the two layers of iron one layer has magnetization pointing in one direction and the other layer has magnetization pointing in the other direction. I have chosen a configuration known as the magnetization in plane and there are other configuration where magnetization perpendicular to plane also is possible. But the essential physics again does not change. So, in this case your one magnetic layer which is pointing in this direction that means all the spins in this top layer is pointing to the right. While all the spins of course again we are ignoring the interface spins which could be little distorted and the surface spins which could be distorted we leave that out. And in this layer all are pointing towards the left. This anti ferromagnetic coupling is a weak coupling coming from a phenomenon known as the R K K Y type of coupling. And in R K K Y coupling depending on the distance between the two ferromagnetic layers the coupling can either be ferromagnetic or can be anti ferromagnetic. That means if I have my two layers at a certain distance if it is of course joining together then it will be ferromagnetic coupling at a certain distance it will become anti ferromagnetic coupling at a certain larger distance it will become ferromagnetic again. That means there is an oscillatory coupling between these two layers and this is called R K K Y coupling coming from others whose name start as Ruderman and Kittel. Therefore, so you see that this is at the heart of the understanding of the what you might call giant magnetor assistance. In other words there are two layers and in the absence of any external magnetic field the two layers of the ferromagnetic material are anti ferromagnetically coupled. I am not saying they have become anti ferromagnetic these individual layers continue to be ferromagnetic, but the coupling between these two layers which I call layer 1 and layer 2 are anti ferromagnetic that means they point in opposite directions. Now what happens when you are passing a current and when you pass a current through a ferromagnetic material then the current becomes spin polarized. And when the current that means when you are passing a current in a normal material the spins are pointing randomly while if you pass the current through a ferromagnetic material the spins get polarized because now all the spins in the ferromagnetic material are pointing in the same direction. That means now you have a spin polarized current that means not only is the charge being transported to the ferromagnetic material spins are also being transported to the ferromagnetic material. Now let me draw this schematically like this. So now you have a ferromagnetic material in which all the spins are pointing upwards say for instance and now you have a material which is non-magnetic for now I will assume that this is again a long piece of material which is non-magnetic. So when I pass current through this material this spin gets spin polarized the current which is flowing this of course the fixed charges on lattices this fixed magnetic movements in the material. But this is the one which is carrying the current and that thus I am drawing suppleately in lead and therefore the current the charges become spin polarized and this is travelling. Therefore when it crosses the interface this material is non-magnetic and this spin polarization would actually die down from after a certain length and this length is called the spin diffusion length. In other words in the magnetic material when the current is flowing it becomes spin polarized but since this material is non-magnetic there is no internal field or no internal spin polarization to keep the current as spin polarized and after a certain distance when it travels in the non-magnetic material it loses its spin memory. In other words it becomes after the spin diffusion length it rather its spin memory is lost and it becomes the spins become randomly oriented. Now the important point to note is that this intermediate non-magnetic layer as a very small thickness that means typically its thickness is less than the spin diffusion length which implies that the the spin current flows across the non-magnetic layer without the spin memory being lost. So that is a very important point. So I have a spin polarized current coming from the top layer which goes down the bottom layer. Now what is happening? Initially since the magnetization of the first material is in this direction if there are any spins in the current originally pointing in the wrong direction like for instance you might have a normal conductor actually injecting electrons into into this ferromagnetic layer those spins will get scattered preferentially which are pointing in the wrong direction. That means this kind of a spin which is pointing to the left will get scattered more preferentially as compared to a spin in another electron which is preferentially oriented with respect to the magnetic field. So spins which are wrongly pointed get scattered and now this resistance which that means if they are scattered that means it is contributing to resistance and this is purely coming from spin polarization related scattering that means it is spin related scattering and not the usual phononic or impurity kind of scattering which we have talked about before. So this is a new kind of phenomena and therefore what is happening here is that you have resistance arising from that. And therefore when the when the charges cross this interface and come to the chromium layer you have a spin polarized current in which all the spins are pointing towards the right direction which is the direction of magnetization in the layer one. And this when polarized as a point of the chromium layer is thin enough such that it is smaller than this spin diffusion line that means that when it goes to layer number two you have another magnetic layer but unfortunately this magnetic layer the magnetization is in the opposite direction with respect to the first one that means there will be a strong spin dependent scattering in the second layer. The layer two will scatter all these electrons very strongly in other words you will have an high resistance state. So this is very very interesting now that I am producing a high resistance state just by anti ferromagnetic coupling two layers by using a spacer layer and using spin dependence scattering as a mechanism for increase in resistance. So you have a high resistance state in the absence of any external magnetic field. Now suppose what happens when I put an external magnetic field H what happens now is that in the presence of an external magnetic field the magnetization vectors in the bottom layer are also pointing in the same directions. And we already noted the spin polarized current in the middle layer when it crosses is pointing towards the right. So when it comes down to the bottom layer there is nothing which is impeding it in fact it is going to reinforce the spin polarized current and the current can flow unimpeded that means you have obtained a low resistance state in the presence of a magnetic field. Of course why is this external magnetic field easily able to switch the magnetization direction because we said that this R K K Y coupling is actually weak coupling it is not a very strong coupling. So when you apply an external magnetic field this weak anti ferromagnetic coupling is overridden and you have both the magnetization both the layers layer 1 and layer 2 pointing in the same direction. So now we have a very interesting effect which is coming from which is definitely in the domain of nanomagnetism coming from composites or hybrids made of magnetic layers and of course you have to as I pointed out you have to include a non magnetic layer which could be a conductor as in this case or a could be a resistant or a dielectric as we shall see in the next slide. So to summarize this slide number one you could have very high values of magneto resistance which are called giant magneto resistance which is coming in hybrids which exceeds the value of about 80 percent unlike the case of normal anisotropic magneto resistance where you expect a value of about 5 percent. That means the state in the presence of the magnetic field is different in resistance from the state in the absence of a magnetic field and there is at this difference is of the order of 80 percent in these structures. A typical structure consists of two ferromagnetic materials which with a spacer in between with a very small thickness of about 9 angstroms and this spacer it plays an important role of anti ferromagnetic coupling the two layers which they themselves are very thin this of the order of 3 nanometers you can see that the layers and actually you can make a multi layer super lattice which will give you an enhanced value of this effect. The essential at the heart of it is two phenomena one is the weak r k k y coupling reading to this anti ferromagnetic coupling number two is the spin polarized current which is and spin dependent scattering giving rise to resistance and finally noting that in the presence of a magnetic field I have a low resistance that means now I have a control to switch this material from a ferromagnetic or a low resistance state to a high resistance state by using an external magnetic field. And therefore, I have control over the resistance of the material by using an external magnetic field this gives us lot of interesting rich possibilities and important applications. Now, some of these applications we will site in this slide I had pointed out that when you are dealing with the system I can actually replace this non magnetic layer in between with what you might call as a dielectric material. Now, before we come to that we will talk about another interesting structure which is based on this concept which is the concept of g m r in the concept of g m r in sandwich structures a spin valve g m r also has been devised in a spin valve the two ferromagnetic layers have different coercivities and can be switched at different field strength. So, we are getting additional control here in earlier we applied one magnetic field and we got the anti ferromagnetic or ferromagnetic coupling. But, here what we are doing is that we have different coercivities for the two layers and we can actually switch one of those preferentially. So, this is what you call the spin valve structure now and this spin valve implies that now I am able to control the spin dynamics which means that I can I enter a new domain known as the spin tronics where in I would like I can manipulate charge into electronics I am able to control the spin of materials and do spin tronics. So, this is a very interesting domain which of magnetism which is gaining prominence of recent times and as I pointed out that an extension of spin valves can be obtained by replacing the non ferromagnetic layer by a thin insulating layer a thin insulating layer is otherwise called a tunnel barrier. And in this corresponding case the effect is known as tunnel magneto resistance the critical difference between the two is that in the previous case of giant magneto resistance the interlayer was a conductor that means the electrons could easily flow. In this case it is an insulator that means that now the electrons can have to tunnel through this barrier and we have already discussed about the concept of tunneling that means that with the thickness of the tunnel region the resistance increases exponentially. But the important thing is that the overall impedance of the system is large and therefore, it can be matched to the circuit impedance to obtain you know a matched system. In TMR the tunnel magneto resistance phenomena spins travelling perpendicular to the layers tunnels through the insulating layer and hence is the name of the effect. So, this is very very similar to the giant magneto resistance phenomena we saw before the important difference being is now the current through the insulating layer is a tunneling current. And therefore, is much reduced in value as compared to the GMR case wherein you had a conducting material like chromium as the interlayer. The TMR effect and the other GMR effect etcetera upon important applications which include hard drives with increased aerial densities magneto resistive random access memory which is called the M-RAM another important applications. These also form a fundamental unit in spin electronics as I pointed out with applications such as repro programmable magnetic logic devices. So, these are important areas of research where in lot of intensive research is going on to build better and better higher density storage devices and devices which can be integrated with the current technology silicon technology. So, therefore, we have seen a two important effects the GMR effect and the TMR effect wherein magneto resistance is the heart of you know designing important devices. Next we take up we continue with the topic of magnetism in hybrids and we talk about a phenomena known as exchange anisotropy. Now in exchange anisotropy you again have a composite in this composite there is a layer which is anti ferromagnetic and there is a layer which is ferromagnetic. Though we are showing here it has a linear configuration in reality this could be actually a particle like what you might call a core shell kind of a structure which we have dealt with before and this could be the CO O particle and this could be the CO and this external layer is the ferromagnetic layer. So, this is the ferromagnetic layer and the inside layer the core of this structure is the anti ferromagnetic layer. So, we are actually now focusing on this interface region between the two. So, this is the region we are zooming in on this picture. So, this kind of a configuration is what we are seeing. So, we have an anti ferromagnetic layer and a ferromagnetic layer in adjacent to each other. Due to exchange of spins coupling of spins across the interface between the ferromagnetic phase and the anti ferromagnetic phase there is a preferred direction anisotropy that means there is anisotropy for field which lead to a shift in the hysteresis loop as we shall see why this happening very soon. In other words example you have cited the cobalt oxide particles are the ferromagnetic particles and they are covered within cobalt oxide which is a anti ferromagnetic system with a large crystal anisotropy. So, what we see in this phenomena of exchange anisotropy is that there is a large coarsivity increase and this coarsivity increase is coming because now you have made the system highly anisotropic that means the system the ferromagnetic system highly anisotropic by coupling it with an anti ferromagnetic layer which is shown schematically on the right hand side. So, now I obtain high coarsivities for this ferromagnetic layer on the left hand side and this is coming because of the anisotropy introduced by the geometric. Now, this is an externally enforced anisotropy by coupling it with an anti ferromagnetic material. The steps involved in creating this exchange anisotropy are as follows first of course you have a single domain ferromagnetic particle like cobalt in this example in contact with an anti ferromagnetic material like cobalt oxide and you would of course we have already noted that this anti ferromagnetic coupling is actually mediated by a phenomenon known super exchange by these oxygen molecule oxygen atoms. So, that means that this coupling between the two anti ferromagnetic coupling is mediated by a super exchange where in oxygen plays the important role. Now, what we do is that we take the system above the nil temperature of the anti ferromagnetic phase and saturate the ferromagnetic phase. That means now above the nil temperature the anti ferromagnetic phase has become an paramagnetic phase and above that we saturate the ferromagnetic phase. Now, when we cool the system below the nil temperature of the anti ferromagnetic phase we introduced a preferential alignment of spins across the interface in the anti ferromagnetic phase. That means that Originally, of course, if I did not have the left hand side of the picture, that means if I had only the right hand side and we did not have the interface, what could happen is that the anti ferromagnetic phase could have across the interface spins pointing either upwards or it could have been spins pointing downwards when you cooled below the needle temperature. So, this could be all spins pointing downwards. So, these two possibilities this or this could exist, but now since there is a ferromagnetic phase which has a preferred orientation you see that the spins are matched across the interface when you cool down below the needle temperature of the anti ferromagnetic phase. Now, the spins in the ferromagnetic phase will maintain the orientation even after the field is removed because how did you get that provincial alignment towards the top originally in this ferromagnetic phase above the needle temperature we used the external magnetic field. Now, because now when you remove the external magnetic field the spins in the ferromagnetic field do not sort of lose their orientation and that is because there is this anti ferromagnetic layer which is imposing the anti ferromagnetic across imposing itself on the ferromagnetic phase on the right. So, I mean the ferromagnetic phase now retains its orientation which is now coupled to the anti ferromagnetic layer. Now, we have made the desired composite or the what you might call a structure and we can now construct the usual MH loop. If the field is removed the spins in the ferromagnetic phase will flip to the field cooled orientation due to the influence of the anti ferromagnetic phase as the field direction is reversed the spins across the interface in anti ferromagnetic phase with large crystal anisotropy oppose the reversal of spins in the ferromagnetic phase and this exchange coupling leads to the large coarsivity value. So, now having made a structure like this I can construct what you may call an MH loop which we normally do and while this field is pointing in the direction of the ferromagnetic phase there is no problem the magnetization is very easy and you can easily obtain the saturation. But when you are trying to put the field in the opposite direction as in the graph below the fact that the ferromagnetic layer is now constrained by the anti ferromagnetic layer and the spins in the anti ferromagnetic layer cannot switch to the opposite direction of the field or of the spins across the interface because now this is a material with high exchange anisotropy or high anisotropy. That implies that these spins sitting across the interface will somehow try to what you call tell its neighbors in the ferromagnetic phase not to switch its spins directing towards the magnetization direction which is externally applied. That means that there is going to be a large coarsivity a large amount of field you need to employ so that all the spins start to point in the direction of the field and this implies that you are going to get a high coarsivity in an MH loop. In other words what you are seeing is that and of course when in an MH loop we already pointed out the coarsivity is called the intrinsic coarsivity. In other words this is as if the loop originally which was shaded in gray here has shifted to a left and therefore you are getting a high value of anisotropy. So, to summarize these slides to see how we are actually obtaining a new phenomena known as exchange anisotropy. So, we are already noted that anisotropy in materials in ferromagnetic materials can come from shape can come from stress or the inherent anisotropy which is called the magnetochrystalline anisotropy. Here we are creating an anisotropy by making an hybrid and in this hybrid there is an anti ferromagnetic layer in touch with an ferromagnetic layer and an example we saw was a cobalt particles in touch with the cobalt oxide particles. In other words you leave cobalt outside it gets oxidized and you get this combination. Now to get the desired high coarsivity effect or the exchange anisotropy effect what we do is that we make a single domain cobalt on anti ferromagnetic cobalt oxide layer. We heat the system above the nail temperature and apply an external magnetic field to saturate the ferromagnetic phase. Then we cool the system below the nail temperature and that implies that the anti ferromagnetic order is now dictated by the ferromagnetic phase and the spins across the interface pointing in the same directions. Out of the two options we said one of the options is chosen as the spin up option because the spins are up in the ferromagnetic phase. Then of course what we do is that we can go ahead and construct what we know as the usual m h loop. Well constructing the m h loop in the forward upward direction of the spins there is no problem magnetic field pointing upward. When the magnetic field points downward the ferromagnetic layer wants to switch to the down spins. But because there is a coupling across the interface and the cobalt oxide as high anisotropy this implies that the spins in the ferromagnetic layer do not want to change their orientation across the interface. They are coupled to the spins across the interface and therefore this is as if those spins are holding them in an upward direction that means that I have to apply even larger field to actually causes reversal of spins in the ferromagnetic layer which implies that my coarsivity value is going to be high. So this is the case of the what you might call the exchange anisotropy. So therefore we have seen two important effects arising in hybrids one is the giant magneto resistance and the other effect of the exchange anisotropy. Now let us take up some nano structures or materials with actually a certain specific dimension and let us see the spin states of some of these. It is important to consider at least a few examples here because the spin states of these nano dimension objects can be very different from the bulk counterparts. We already seen how the spins are oriented in a bulk that they are divided into domains and within domains spins are oriented parallel. And of course only the domain wall that you obtain a slightly more what you mean complicated configuration of spins which means that spins would actually start to rotate. So we talk about first we take up the example of nano disks wherein there are special spin arrangements with no bulk counterparts. In 15 nanometer thick permaloid disk with the diameter of the disk greater than about 100 nanometers. So these are thin disks having a large diameter. So these are like thin disks like these look from the side they are thin disks like these and the thickness of these disks is of the order of about 15 nanometers. And the diameter of these disks is about greater than about 100 nanometers. So they are thin disks. The spin arrangement in these kind of nano disk consists of concentric arrangement of spins on the outside and with out of plane component towards the center of the disks. Of course this is schematically shown here like this in the case of the spins on the outer side of the disk they are concentric, but in the center they are pointing upward. This z displacement does not mean the spins have come out of the disk it just implies that the magnitude of the magnetization is large. It is what we are implying by pointing the z displacement. The core radius where in the spins are out of plane is of the order of the exchange length. And exchange length is defined is a characteristic length of a magnetic material below which exchange forces are dominant over magnetostatic effects. We know that when magnetostatic effects are dominant then spins in a domain will be aligned and when and neighboring domains could be misaligned. When exchange with below the exchange length it is the exchange coupling between the spins which is dominant which means the spins will tend to align parallel to each other. So you can see that in the central region of the disk which is below the of the order of the exchange length then you see that the spins are all parallel line and the magnetization is very large on the outside you see that the spins are concentric. So there is no what you might call analog of this in bulk material. So this is a very special kind of spin arrangement and we also see that clearly in this example that in on the outside these concentric spins are like a 2 D system in the center it is like a transition it is like a 3 D system that means there is a transition between a 2 D to a 3 D ferromagnet right in a single disk. So this is a very interesting kind of an arrangement and we are talking about the large disk because the exchange length is of the order of 5 nanometers therefore we need a disk which is large enough in diameter to actually see this transitions. So this is a very interesting kind of an arrangement in nano disks. Similarly when you take up a disk which does not have a core obviously you would expect or you would have guessed that there is going to be no out of plane component in this because the core has been removed and in these nano rings then these kind of structures are called nano rings there are more than one possible way of spin arranging itself. This left structure is called the vortex structure where you see that the spins are going concentric in plane then there is something known as an onion structure wherein spins on one half of the disk is pointing one direction and spins in the other half. So you see other half are pointing in the other direction so that these kind of arrangements are possible. So one small correction is required here so this spin has to point in this direction and of course this misorientation of spins need not be across the entire half ring but it could be only in a small region defined by a sector like this. So you see that and this state is called the twisted state of this nano ring. Therefore you see that more and more interesting and complicated arrangements of spins are possible in these nano structures like the nano rings and nano disks and it is interesting to note that suppose I were constructing actually a m h loop or a hysteresis loop in a starting with a vortex structure then I would observe that I would obtain these other states like the onion and twisted structures during some part of the magnetization hysteresis loop that means this state would change from the one from a vortex to an onion or to a twisted state during the magnetic magnetization process itself. So we have taken up two interesting examples of nano disks wherein there are special arrangement of spins which have no what you might call bulk analogs and you can see that these are the concentric spin states and in the case of the nano disks we saw that there is a small region in the core of the disk wherein actually there is spins going out of plane. In other words the system behaves like a 3D system and like a 2D system which we expect a thin disk to behave like. So with this we come to the topic end of the magnetization in nano structures and nano materials and we have already seen that magnetization is a very rich field wherein you have very interesting effects like we have seen the effects of super paramagnetism, giant magneto resistance etcetera and we all it is important to note that many of these actually have what you might call very interesting applications in and the very important technological applications. Next we take up the topic of optical properties of nano materials we had noted that earlier that optical properties essentially come from the electromagnetic structure and we had noted some of the important concept like band structure which play an important role in the optical properties. We will see further aspects of optical properties of normal materials and then further proceed to optical properties of nano materials in this topic or in this chapter. People who are interested in advance readings regarding the optical properties of nano materials may concept the book by Zhang wherein he talks in details about not only about optical properties, but also spectroscopy of nano materials. First we start as I pointed out with the broad overview of what we mean by optical properties what are the possibilities then we take up specific examples in nano structured materials many of the optical properties are closely related to the electrical and electronic properties of a material. So, this is the optical properties and natural consequence of the electrical and electronic structure of a material this is as expected. But then other factors also come into play and when we are dealing with optical properties which includes for instance the arrangement of atoms and other what you might call the spatial distribution of the scatterers etcetera and this plays an important role in determining the optical properties. When one is talking about optical properties one is usually referring to the interaction of electromagnetic radiation with matter. And though the word optical would imply that we are talking about visible radiation that means which is visible to the human eye. But in general we have to talk about electromagnetic radiation covering the entire spectrum starting from gamma rays to radio waves and this all this interactions with matter can be thought of as optical properties though often we are restricting ourselves to only the visible part of the electromagnetic spectrum. The simple picture one can start with is by considering a ray of an electromagnetic wave and you know that the ray is actually a geometrical construct and has no physical reality and we consider a single frequency entering a medium from vacuum. So, this is starting picture of what you might think as an interaction of electromagnetic radiation with wave. Now, this ray implies that all the wave fronts in the wave are actually perpendicular to this ray. This ray could be reflected this ray could be transmitted into the medium and refracted and when I am talking about a medium this is my medium which I am shading in a low. And I imply that this medium has a higher optical density or an higher refractive index with respect to the medium surrounding it which we consider for now to be vacuum. So, this ray could be reflected it could be refracted and it could also be absorbed. In other words this ray entering the medium could lose all its energy within the medium and therefore, this does not propagate beyond the what you call even this a finite medium then it does not propagate beyond it. And therefore, you will notice that the wave dies down ultimately of course, all the energy in the electromagnetic radiation will have to be converted into heat. And therefore, this absorption essentially implies that the energy in the electromagnetic radiation is converted into heat. When you are talking about reflection it essentially means that of course, there is no system which is 100 percent reflecting or 100 percent reflecting or 100 percent absorbing. So, we typically have a combination of all these three effects and one of these may be dominated like you take a highly polished silver mirror surface then you would expect that such a system is strongly reflecting. And most of the energy is actually sent back into the medium from which the original ray originated. And here we have seen in the case of refraction that we have drawn a certain configuration which we consider to be what we call the material having a positive refractive index. And we will have a few more thing to say that how we can consider construct structures with negative refractive index as well later. Now this reflection itself could be specular or diffuse. So, if you take a case of a polished mirror then you see that the reflection would you follow something like a Snell's law and you have a reflection. But a rough surface like any wooden surface would have typically a diffuse kind of a scattering which means that you cannot see images in a case where there is diffuse kind of a reflection and not specular reflection. But suppose I consider from this whole process of interaction of the electromagnetic radiation with matter from a more fundamental perspective there are only two possibilities of interaction of a medium with electromagnetic radiation. One is scattering another is absorption and transmission can be thought of as a forward scattering phenomena. So, you send electromagnetic radiation into wave or array into wave and there could be scattering or they could be absorption. So, these are more fundamental perspective. And if you are talking of not about a single frequency entering a material but there is a wider spectrum of frequencies. In other words you have what you might call a pan chromatic radiation then some part of this electromagnetic spectrum could be absorbed while other frequencies could be scattered. In other words you could have a medium which absorbs some frequencies which transmits other frequencies and which could actually reflect certain other frequencies. So, this is a possibility when you have a radiation consisting of a large variation in the frequencies. Now, when you are talking about absorption it essentially means that certain resonance in the material has been excited which means that the material now can be taken to an excited state by this electromagnetic radiation. And that means that it needs to have a certain bare minimum frequency because frequency is the handle on energy we know that E equal to h nu for a radiation. And this frequency is able to excite some kind of a resonance in the material and therefore, this absorption is taking place. What is the kind of excitation which this material can undergo? These processes include the electronic excitation, the vibrational excitation or it could be the rotational excitation. A nice example of for instance the rotational excitation would be the case of microwave heating in our oven wherein we note that suppose I send microwave radiation into the microwave oven then the water molecules which have a natural polar character that means water molecules are polar and therefore, they have a net polarization in some direction. So, now what happens is that suppose now I send an alternating electromagnetic wave which consists of alternating electric and magnetic fields then this water molecule will tend to rotate its electric field will tend to follow the electric field of the incoming radiation. If the frequency is too high then it will not couple with it if the frequency is too low then the heating is going to be too small. But if at the right frequency which is of the order over 3 centimeters or so the wavelength of the electromagnetic radiation the 3 centimeter wavelength then what happens is that the water molecules are able to follow the externally applied field and which is coming from of course, on the radiation itself. And since the water molecules are rubbing against each other this whole process is going to be dissipative and therefore, overall the water in the food or the water in the tumbler actually gets to heater. So, this rotational excitations can be possible and this is a nice example of a rotational excitations. If you are talking about for instance phenomena typically like an electronic excitation which can also take place if there is a critical amount of frequency available then this absorbed energy could actually be reemitted when the system relaxes to the ground state again. And finally of course, we have to note that any energy we have noted before that any absorbed energy is finally, dissipated as heat and this is called dissipative absorption. Now the important thing to note between these 3 kinds of mechanism the electronic vibrational or rotational excitations is the fact that the electronic energy levels are highly separated. So, you can see that the e 1 and e 0 are separated and this corresponds to wave number of about 10000 per centimeter. So, you need lot of energy to actually excite and you can find an equivalence between the wave number and the e v's. So, there is a larger wave number associated with the electronic excitations that means, you need lot of energy to excite an electron from the lower energy level in an atom or a molecule to a high energy level. And of course, this could also be a solid you could be describing. On the other hand you would notice that the vibrational levels which is the next level is actually closely spaced. And that means that I require a lower amount of energy and corresponding wave number is of the order of about 1000 per centimeter that I and of a multiply this 1000 of course, by this 1.970 power minus 5 I can obtain the value in electron volts. But I require a lower amount of energy to excite the vibrational level that means that if I am having an electromagnetic spectrum the electronic transitions are typically excited by the high energy radiations while the vibrational transitions are excited by a slightly lower energy radiation. Finally, the rotational excitations have a even smaller spacing which is shown here or for clarity I will show it here. And that means that I have to impose a very small energy to excite a rotational excitation. So, even though there are three possible mechanisms by which I have I can excite a material which of course, again this excitation my relax to the ground state again by reemission of radiation. But these energies are all not the same and therefore, depending on the frequency of the electromagnetic radiation one of these might be excited. And therefore, the absorption would be in a certain frequency regime based on the what you may call the energy which is required to excite this. So, and again to go into the detail little more about these excitation the electronic transitions themselves could be from atomic orbitals could be from molecular orbitals or could be to a band that means an electronic energy level lying in a band. And we have already noted whenever there is a relaxation of a level from a band then the range of frequencies usually brought because brand is a semi continuous set of energy levels.