 To summarize the story on magnetism so far, we noted that magnetism is virtually universal. It is found at the cosmic scale, the scale of the earth, the scale of the plates and also in the scale of many birds and bacteria, wherein it is in the nano scale actually. We said that there are two origins of magnetism. One is macroscopic flowing currents, another is the atomic origin which is primarily because of the spin of electrons and the orbital motion of electrons. Additionally, we noted that because of the orbital motion lattice coupling, often which is a strong coupling, often the orbital motion is quenched and the scenario might changed when you go to nano particles and lower dimensional systems. We have to understand two important effects when you talk about magnetism. One is the effect of external fields, the other is the effect of temperature and we also noted that often when we talk about magnetism, we also have to worry about the band structure of materials without which the picture of magnetism would not be complete. We had noted that there is one kind of magnetism which is universal, which is diamagnetism in materials and we said that we could actually use a quantity like the susceptibility to differentiate what you might call paramagnetic and antiferromagnetic and also diamagnetic materials. The effect of temperature is typically given by what you might call the Curie Law and the Curie Ways Law, which tells you that there is something called a molecular field or an exchange which gives a positive value of theta. That means there are ferromagnetic materials which become paramagnetic and these kind of materials have to be differentiated from materials which are paramagnetic always like oxygen. We also noted by mere arrangement of these atomic magnetic moments, we can have collective definitions of spins which we call ferromagnetic, antiferromagnetic and free magnetic ordering. We also noted that in a simple metal like iron, if you want to understand magnetism and we note that it is actually the localized electrons which are dominating in the system which is a large number, suppose you take the number of 8 valence electrons out of which 7.05 of them occupy a so called localized D band and only 0.95 of them are free and this 0.95 cannot contribute to the magnetic moment of iron. Additionally, we noted that the magnetic moment of bulk iron is about 2.2 Bohr magnetons and to understand this, we actually have to talk about you might call spin dependent density of states. That means the up spin and down spin electron density of states are shifted with respect to each other and this is what the way they fill gives you what you might call the non integral magnetic moments in a material like iron. We also pointed out the task of actually going from atomic magnetic moments to a magnetic moment of entire solids and hybrids is actually an adverse task and therefore, we use certain techniques to actually make this shortcut possible. We also noted what are the thumb rules for instance for ferromagnetism based on the band picture, wherein we said vacant levels, high density of states close to a Fermi level and the appropriate inter atomic distance are responsible for giving rise to magnetism in 3 D metals like iron cobalt and nickel. Having talked about the effect of temperature in to some extent, we will now take up the effect of an external magnetic field. On the right hand side is the very familiar diagram and which is called the B H or M H loop for an ferromagnetic material. So, we are focusing our attention on a ferromagnetic material below the curie temperature and we also said that often either B or M is construed or B or H is construed to be of what you might call a fundamental quantity, but here we will focus on the M H loop. So, we have H is the external field and M is the magnetization produced on the material and the M H loop is the one which is below. So, the curve which is the M H loop or the M H curve goes like this in other words when you increase the field initially you have a material in which you have 0 magnetization and we said it is actually the breaking up of domains which is responsible for the 0 magnetization. As you increase the field you notice that the magnetization increase and later on reaches a saturation value which is called M S. If you are plotting a B H loop then you would notice that even on saturation the curve continues to increase because B is also a function of the applied field H. That means, since H is increasing the B will continue to increase when you plot a B H loop and therefore, there is saturation at the material level, but there is no saturation of the curve. Now, in this curve lot of important quantities can be marked the what you might call the coarsivity and this coarsivity is an important quantity because this coarsivity depends on what you might call the microstructure. In other words initially suppose I am plotting a B H loop then the curve initially increases then when you reduce the magnetic field it does not follow the same curve that is why it is called hysteresis. It goes down and even at 0 field you find there is a net magnetization of the material and you actually have to put an opposite field to actually bring down the magnetization to 0 which is called the coarsivity. The field strength at that level is called coarsivity and further of course, we can go ahead and complete the entire loop. Now, the other point to note from this curve is that the slope of this M H curve is called the is an important quantity and the slope varies from point to point and the similar quantity if you note from the B H curve is actually called the permeability and similarly, the permeability also varies from point to point in the curve and the permeability is not the instant slope of at any point, but is actually the slope of the tangent from the origin. So, this is now what you may call the initial permeability and this is what you might call the maximum permeability this curve. In other words permeability is not defined as an instant slope at any point in the B H curve, but is actually defined as a slope of the line drawn from the origin. Now, another important point to be noted in this curve is that the field required to bring a ferromagnetic saturation at room temperature is actually small. So, it is about the order of 80 kilo ampere per meter, but suppose that is to this point the point till B S, but suppose I want to increase the coercivity even the magnetization even further then the fields required are very very large and this kind of magnetization is called force magnetization and in this lectures we will worry we will only talk about a material magnetize still only the saturation and the coercivity in the B H loop is called the normal coercivity. On the other hand the coercivity plotted in a M H loop is called the intrinsic coercivity and usually is added an additional I in front or in back which therefore, it is called M I C or M C I which implies that I am not referring to an M H plot in which I am describing the coercivity and we will be repeatedly referring to this coercivity because as I pointed out this coercivity is a structure or what you might call a microstructure dependent property. Therefore, to summarize this curve the magnetization in a ferromagnetic material shows hysteresis that means the magnetization curve and demagnetization curves are not what you may call it if you started that completely demagnetized material and you for instance if you had heated the material above the curie temperature and cooled it down and then you start magnetizing the material then you would notice that such a material when during magnetization follows a path which goes from say for instance the origin to a point x to a to b s then while demagnetizing it follows the curve from b s to a point y to a point z to finally, to the h c or h c i. So, it is in other words it shows a distinct hysteresis which is what you might call usually use as whenever a material shows a hysteresis like that you add a prefix to the term called ferro. So, let me go down to the board and write down a few of these ferro quantities like you can have substances like which show hysteresis like ferro and you can have ferro electric you can have ferro magnetic you can have ferro elastic and so forth. And here the word ferro originally of course, originating from the term implying that there is iron in the case of magnetization, but here the word ferro in common sense actually implies that there is actually hysteresis behind the behavior. Now, how does this magnetization come about in other words I have a material which is as initially a zero magnetization and later on as you magnetize a material you see that actually you see that there is an increase in magnetization how does this come about this comes about at the microscopic level by preferential alignment of domains and this is brought about by the external magnetic field. During magnetization the domains oriented favorably along the field direction grow at the expense of unfavorably oriented domains. And to give us schematic picture of how these domains might look for instance suppose I have a picture like this here where in suppose I am applying a magnetic field in the z direction then you would note that these domains are all oriented favorably these domains are not oriented favorably and therefore, these the domains pointing with the north south up will actually grow at the expense of the ones which are pointing north south down. So, this is a schematic diagram and real domain structure is actually much more complicated, but the essential point to note is that the domains oriented favorably grow at the expense of the domains oriented unfavorably. And this can occur by two important mechanisms what you might call domain wall motion and the second is by rotation of magnetization of the domains. Now, which of these two mechanisms is operative is actually a detailed you need a detailed picture of the material you need a detailed picture of the microstructure is a polycrystalline etcetera, but crudely speaking if you look at the m h curve then you can sort of differentiate that initially the magnetization takes place the initial portion the magnetization actually takes place by domain wall motion this is of course, as I pointed out the some kind of oversimplification. Later on close to saturation it actually takes place by domain wall rotation. So, both these mechanisms are possible and for the domain wall rotation itself there are many sub mechanisms like coherent rotation, incoherent rotation there is that you will come across terms like fanning curling etcetera. So, there are dot of detail things which do take place, but in the end what happens is that the favorably oriented domains grow at the expense of the unfavorably oriented domains in other words you are magnetization. Now, even this domain wall motion has lot of details in it like it can be smooth or jerky and this domain wall motion is happening, because now what is happening is that external magnetic field tends to align misoriented spins on the domain wall. We will have a detailed look at the domain wall picture very soon, but domain wall is a region where the spins are not aligned along any one of the domains it is misoriented with respect to both the domains. And therefore, it is a region between the two domains it is like a some kind of an analogous term you can use the grain boundary. And this means that if spins oriented favorably are going to grow at the expense of spins which are in unfavorably in other words the domain wall is going to move. And this is going to lead to the displacement of the domains and as I pointed out during quite a bit of the magnetization process some regions of the sample may actually be experiencing domain wall motion while other regions in the sample may actually be undergoing domain wall rotation also this is also possible. And therefore, the actual picture little more complicated. Now, the rotation of spin is opposed by the increase in anisotropy energy. And we went back to an early slide where we defined anisotropy is by the fact that now my magnetization is going to be direction dependent we even pointed out like for instance a material like iron the 0 0 1 direction is the favorable or the easy direction of magnetization. And the 1 1 1 direction for instance is the unfavorable direction that means that there is an inherent magneto crystalline anisotropy in a material which means that when spin tries to rotate there will be an opposition to it. And we additionally pointed out that other anisotropy factors like shape and stress also play an important role. And an additional point to be noted is that during rotation all spins need not be parallel to one another and this makes the picture actually somewhat complicated. So, the broad take home message from the slide is that magnetization occurs by the growth of the favorable domains at the expense of those which are unfavorably oriented to underline mechanisms can be thought of for this 1 is called the domain wall motion which essentially comes about because your magnetic field is trying to align those spins which are misoriented along the domain wall to a direction which is aligned to the magnetic field. And this automatically will lead to a domain wall displacement. The second process is what you might call rotation of domains and a rotation of magnetization within the domains and which leads to a rotation of the domains. And we also point out that both these process might be simultaneously occurring during the magnetization of a sample. Let us return to this topic of effect of temperature once more. We broadly pointed out that at the curie temperature a ferromagnetic material will become paramagnetic. But actually the real complicated picture is that this demagnetization process actually occurs by a process which involves spin waves and the quantized version of these spin waves is called magnons. So, the reason behind is that suppose I have all spins oriented in one direction and I suddenly say that there is one of these spins is going to be pointed in the opposite direction. In other words, this spin is going to flip. In other words, it is not it is randomly going to flip because of the thermal energy. Then such a isolated spin flip actually costs more energy than a distributed spin flip which is what is the origin of spin waves. In other words, now my spins have a certain orientation dependence. They are not just one spin flipping, but there is a disturbance which is carried across the crystal. And such a spin wave is actually carrying out the demagnetization process on heating. Though we are not going to details of magnons for now, but this is equivalent to phonons for thermal disordering which is positional for the atoms. Here it is a orientation dependent disordering which we call and the quantized version is magnons. Now, the important thing to note that at T c the susceptibility becomes infinite. And the reason for the susceptibility and of course, we are already pointed out though we talk about a sharp curie temperature. In reality, the curie temperature is not that sharp. In fact, even after the curie temperature, there are localized region within the sample where there are local magnetic clusters which are still oriented. That means, the domain structure is not completely been broken down. And this means that there is some magnetization which is still remaining after the curie temperature. And since there is an inherent propensity for the material to get magnetized, at curie temperature you might say it is a critical temperature at which that thermal forces are just balanced my magnetization forces or the internal exchange forces. Now, that means that if I now apply a little extra magnetic field, the material will tend to a light. That means the susceptibility is going to be very high. And in fact, you might say that at the curie temperature my susceptibility is going to be infinite. And I just pointed out there are beyond T c there are local clusters or splitting clusters of magnetically aligned movements. Obviously, maximum magnetization is obtained when all magnetic movements have a parallel orientation. And this is corresponding to the highest magnetization state m 0. And sometime you can also use the per unit mass sigma 0. And therefore, if I plot my sigma s by sigma 0 which is s stands for saturation. And T by T c where c is the curie temperature, then we will expect that there will be a master curve which you can plot for demagnetization of all materials with temperature. So, in other words you can have a master curve for various materials using two normalization quantities. One is the saturation magnetization m 0 and the other about the maximum magnetization possible m 0. And the other is your T c which is the curie temperature. So, we see that effect of temperature is very very important on a material. And we will return to this when you talk about magnetism in nano scale materials. Now, it is important note a few more things regarding the domain structure and the magnetization process. And here in we will note that even in normal bulk materials there are certain entities which you cannot avoid which are in the nano scale. Like we noted in a bulk material also we had a grain boundary. And the grain boundary was typically what I call ascribed width of the order of about 0.5 nano meter to 1 nano meter which is the nano scale. Now, the reason that the material as a ferromagnet actually splits into domains is because of a combination of various energies which we will see now. And this in effect overall reduces the magnetostatic energy or the overall energy stored in the external magnetic field. Now, what I am saying is that suppose I had a material and the whole material. Now, this macroscopic material I am talking about and not an nano scale material. Then if this material were a single domain then the energy stored in the external magnetic field will be large. And the material this energy of course is called the magnetostatic energy. The material will try to reduce the magnetostatic energy by splitting into domains. As you can see if you split into two domains pointing in the opposite directions then the overall external field reduces. If you split into four domains it further reduces. And you can reduce this quantity even further by actually having what you might call domains which are the normal domains plus what are called closure domains. And therefore, now I can form continuous loops of B. So, using these magnetic domains and closure domains I can actually reduce my magnetostatic energy. Now, the region separating two of these the region here is the domain wall. And you can see in the domain wall you have of course all spins oriented in of the up direction on the left hand side all spins oriented in the down direction on the right hand side. That means that the spins are switching orientation at the domain boundary or what you might call the domain wall. So, this is my domain wall. Broadly two type of domain walls can be differentiated. And the one which is usually found in bulk materials is called the block wall. Additionally there are other kind of walls which are called a needle walls. And needle walls are typically found in what you might call thin films. And they are not found typically in bulk materials. Now, other walls apart from block walls and needle walls are also possible. And they are given names like cross tie walls and etcetera. And there are even more complicated configurations which can be present in a domain wall. The important characteristic of a block wall is that the spin vectors rotate out of plane in the wall itself. So, there is a schematic diagram here. There is a domain one which is on the left hand side. There is a domain two on the right hand side. The spins are as shown here. Now, in the domain wall the spins rotate, but they do not rotate in plane. You can see the spins actually rotate out of plane. So, this spin is slightly misoriented. This plane is misoriented little more. This spin is misoriented little more and so forth. Finally, you obtain the new orientation of the other domain. Now, of course, you may say that why not put up an abrupt wall, because now these misoriented spins are actually going to cost you in terms of the anisotropy energy. But an abrupt spin change would mean that I am going to pay a higher price in terms of the exchange energy, because exchange coupling is actually trying to orient two of these spins, neighboring spins parallel to each other. Therefore, the system actually comes to an equilibrium in a competition of these kind of various kind of energies and we will list them in the next slide soon. So, overall configuration actually consists of normally block walls, where the orientation in a block wall is or the misorientation takes place gradually. Of course, here this is a schematic and you should actually note that the domain wall is much larger, much longer and actually the misorientation between the neighboring spins is much smaller than that shown in this crude schematic. So, domain wall is actually much larger and typically you would note that these domain walls have a width in the nano scale and we will take up some numbers for this in the coming slides. In the passing, it is important to note here the Neal walls, which are seen in thin films and these thin films I am talking about nano scale materials of the order of about 40 nanometers and in these, the spin actually rotates in plane. In other words and if I have now a material, which is a thin film, such a system can actually if you look down from the top, if you are looking down from the top can be thought of as a 2 D system and here suppose spins are oriented here in this left hand side, the spins are oriented downward on the right hand side, the spin misorientation the rotation actually takes place in plane. So, now this rotation you can see is actually in plane and this kind of an in plane wall is called a Neal wall and this is found in basically in thin films on nano structured 2 D nano structured materials. And we have to note that this domain wall is also a nano structure in its own right, because it has got a thickness of about few hundred atomic diameters. So, when you are talking about domain walls two things are important. Number one the equilibrium bit of the domain wall is determined by a competition of various energies. The reason that the material splits into domain is to overall reduce the magneto static energy. Number two is that the domain wall actual misorientation between the spins is small. In normal materials that kind of wall you find is called a block wall. In thin films which are of about 40 nanometers thick you find this Neal walls where in the misorientation or the spin rotation actually occurs in plane in over the whole system remains 2 D because of the configuration of the thin film. And the domain wall itself is a nano structure in its own right. Now, few more words about this domain wall and the energetic reasons behind it. The domain wall represents a region of high energy at the spin vectors are not aligned in directions of easy magnetization. Hence, thicker walls represent higher energy and materials with high magneto crystalline anisotropy like for example, their error materials the domain walls are typically thin they are about 10 atomic diameters. If the magneto crystalline anisotropy is small then you will find thicker walls on the other hand. There are other source of anisotropy as we pointed out due to shape and residual stresses. A competition between the magneto static energy and the magneto crystalline energy essentially decides the thickness of the wall. Now, the word essentially has been used because the other factors like magneto elastic energy which is coming from the process of magneto restriction. Now, in other words if somebody were exposed to an Hartman diagram this is what you might call a cross coupling term wherein because of a application of a magnetic field you are actually having a change in dimension. This is called magneto restriction and because of this change in dimension there is actually an elastic energy associated with the process of magnetization. Now, this implies and the familiar example given for magneto restriction giving rise to sound is in the case of magnetic cores and transformers there is a hum and this hum is coming from this phenomena of magneto restriction. And this magneto restriction energy is also going to contribute to the overall energy and now when the optimization takes place the system tries to optimize putting together all these energies. In other words the total energy can be given by the fundamental exchange energy between the spins the magneto crystalline anisotropy energy or coupled with of course, the other anisotropies the magneto elastic energy coming from these strains associated with a magnetization process and of course, the external magnetic field which is the magneto static energy. So, the overall equilibrium width of a domain wall is determined by the competition between these energies and therefore, the system tries to minimize this overall energy. So, at the heart of the magnetization process we note that is the fact that there are domains these domains want to orient themselves in the process that there is there are domain walls these domain walls are nanostructures in their own right and the motion and rotation of these domain walls are important and finally, lead to the magnetization of the sample. Now, let us take up another important topic which is the concept of magneto resistance again this is like a cross coupling term in other words there is a change in resistance the electrical resistance as you impose a magnetic field like in the previous slide we have seen there is a change in size or there is a strain associated with the magnetization. So, this is a cross coupling term normally when you apply a magnetic field and the natural response is magnetization or not the strain right. So, this is a cross coupling term between what you normally expect stress to be causing strains and not magnetization to be causing strains. So, this is a cross coupling term between stress and magnetization similarly, there can be a cross coupling term between magnetic magnetism and resistance which is called magneto resistance the resistance of a conductor can change when placed in an external magnetic field this effect is called magneto resistance. In other words in the absence of a magnetic field there is one value of resistance and in the presence of a magnetic field there is a different value of resistance and here we are talking about electrical resistance. The resistance is higher if the field is parallel to the current and lower if the field is perpendicular to the current and in general the resistance depends on the angle between the current and the magnetic field and this effect has a general name called the anisotropic magneto resistance otherwise called AMR. An important point to note that this magnetic resistance arising because of the magnetic field is typically small in most materials it is does not usually exceed the value of about 5 percent. And this magneto resistance arises from a larger probability of S D scattering of electrons and we will take a more pictorial picture when we actually go down to nano materials and talk about this enhanced version of this magneto resistance which is called the giant magneto resistance. But essentially we note it comes from a larger probability of scattering and this is now spin dependent scattering which is important and this arises in a direction parallel to the magnetic field. This AMR effect itself has been put to good use in magnetic field sensing devices and when whenever you have an enhanced value of AMR which is anisotropic magneto resistance it is beneficial in using it in the form of a sensing device. To summarize this slide in normal materials in the presence of a magnetic field there is usually about a 5 percent change and in some very special compounds like uranium compound this value can reach as large as 50 percent. But nevertheless this is usually a small value this change in resistance caused by an external magnetic field and this is coming because of a larger probability of S D scattering electrons parallel to the magnetic field coming from what you might call the spin dependent scattering. And in general the value of this resistance depends on the angle between the current and the magnetic field and that is why this is called the anisotropic magnetic resistance. So this is an important phenomena as this finds important applications like in magnetic field sensing devices and once you have a magnetic field sensing device you can put it to various kinds of good uses like you use it like a counter you can use it like a sensor and so many possibilities. In other words suppose I have a sensor here and in the presence of a there is a magnet here and if this magnet comes close to this then this is going to click one because a magnetic field resistance is resistance in this conductor is going to change and therefore I can keep on counting the number of times this sensor is actually passing through this region. So we will find that there are even more important applications of this effect when it comes down to nano materials. Now let us switch from magnetism of bulk materials wherein it was we studied some of the important effects the origin of magnetism the effect of temperature and the effect of magnetic fields to now magnetism in nano materials. When you talk about magnetic nanostructures they exist even in bulk materials as we had pointed out like the example was a domain walls for instance 60 nanometer in iron. We even said that even though we are talking about domains in bulk materials some domains like small green boundaries could exist even in a bulk large green size micron green size sample. So some domains could be very small especially so close to the surface or in a close to the green boundary in a poly crystalline material. That implies that even in a bulk material some domains could actually be nano sized because of the distribution of domain sizes. We also noted that spin clusters about the paramagnetic temperature close to the paramagnetic transition temperature could also be nano sized. So there are three entities which are clearly nano sized even in bulk magnetic materials which are the domain walls some domains in an overall distribution of domain sizes and finally spin clusters just about the paramagnetic transition temperatures. We have to additionally note that when you go from bulk to nano only the microstructure sensitive magnetic properties like coercivity is expected to change the overall the for instance the microstructure insensitive properties like saturation magnetization is not expected to change much. And we will note now that there are extremely what you might call interesting possibilities in magnetism when we go from the bulk to the nano and these include that the ferromagnetic particles could become single domain. That means the entire particle is a single domain. We will encounter a phenomenon like super paramagnetism in small ferromagnetic particles and there is no bulk analog of super paramagnetism. We will also encounter giant magneto resistance and we will note that this is a phenomenon exclusively found in or mostly found in hybrids. In other words we can construct hybrids synthetic hybrids where you will find phenomena like giant magneto resistance. Then additionally and a very equally interesting phenomena is that you will find that materials which are anti ferromagnetic in bulk. That means they are they show properties which are purely anti ferromagnetic. There is an equal and opposite magnetic moments aligned in the material therefore all the magnetization actually cancels out. But they start to behave like ferromagnets though with us reduced amount of they do not have very high amount of bore magnetons per atomic magnetic moment but still nevertheless they will start to behave like ferromagnets. So, we see that when you go from bulk to nano in magnetism there are very very interesting possibilities and many of these do not have any analog in the bulk world. So, next we will take up the topic that how the magnetic moment and here we are talking about magnetic which is intrinsic to the material depends on the dimensionality of the system. It is to be noted that there is an increase in the magnetic moment per atom as we decrease the dimensionality of the system. This implies that there is a fundamental differences in the magnetic behavior between nano structures and bulk materials. This effect is all the more noteworthy because if you take a nano scale material we would notice that the spins on the interior could all be aligned parallel causing for instance in a ferromagnetic material. But then the surface spins will actually be somewhat misaligned with respect to the bulk. In other words the surface spins are actually contributing less to the magnetization of the material and the overall saturation magnetization or the overall magnetization I get from such a particle is going to be small because of the misalignment of the surface spins. But in spite of this effect we actually find that in nano scale materials or in reduced dimensionality systems we actually find an increase in the magnetic moment per atom. So, if you look at for instance iron and you look at the bulk iron it has got a magnetic moment of about 2.2 Bohr magnetons per atom. But if you took at a 2 D iron you see that the magnetization has already increased to 2.96. You go to a 1 D system made of iron then it goes down to about 3.3 and in a 0 D system it increases even further to about 4.0. This trend line can also be noted for nickel where in again you can see the bulk magnetization is about 0.56 Bohr magnetons per atom and this increases to 0.68 for 2 D, 1.1 for 1 D and 2 for 0 D. In other words there is an increase in magnetic moment per atom as you reduce the dimensionality of the system from bulk to 2 D to 1 D to 0 D. So, this could be for instance an example of a quantum dot this would be for instance a quantum wire this could be a quantum well like this and this of course, a bulk material. Now, the important comparison number is of course, the maximum possible magnetic moment possible for an ion atom because you have 3 mu B arising from the orbital contribution 3 mu B from the spin contribution and in the bulk material we had noted that most of this orbital motion is quenched and therefore, you land up with a small number for the overall magnetic moment which is about 2.21. So, bulk materials do not have very high magnetization, but if you note now for a 0 D system you can see that the value is increased. This implies that that in 0 D nano crystals very little of the orbital magnetic moment is quenched. That means that now because of the reduced dimensionality the lattice orbit coupling which we have talked about which is a strong coupling was actually reading to this reduced magnetic moment you are finding that that is weakening up and therefore, you have an increased magnetic moment. So, this is very interesting that purely by reducing dimension now I am changing one of the fundamental properties of magnetization which is the number of bore magnetons I can achieve from an atom in a material. So, this is changing and this is changing progressively with a reduction in the dimension and this trend line can be seen and these are I think calculated values this can progressively be seen both for a material like nickel and iron. This implies that there is a fundamental change when you go down to the nano scale and to reiterate the important point here there is always this influence of the surface spins which is now what you might call disordered or not that well oriented with respect to spins in the bulk which implies now that this effect is all the more startling when you go down to small sizes. So, there is there are fundamental changes fundamental differences in the magnetic behavior between nano scale structures and bulk materials and we will take up a few more examples as you go down to more and more systems. One of the important effects when you reduce the size of a system is the effect of super paramagnetism to understand super paramagnetism. Let us talk about what you might call a change in coercivity and here we are plotting log of intrinsic coercivity and we said that intrinsic coercivity is a quantity when you are plotting an m h loop and not the b h loop and you typically include a subscript i for indicating that it is the intrinsic coercivity are plotting and we are plotting log because now we are talking about a variation in coercivity of 2 to 3 orders of magnitude a large variation in coercivity. It is not a small variation, but an extremely large variation in coercivity and this variation in coercivity is coming again of course, from a large variation in particle diameters. So, we are talking about 5 orders of magnitude in particle size here going from about a nanometer to about a micron about 5 orders of magnitude in sizes and these are talking about freestanding particles and about 3 orders of magnitude changes which is occurring in coercivity. In other words the coercivity of bulk materials is very very different from coercivity of nanoparticles this is the important thing to note. Now, what is happening to the behavior of a material when you are reducing the size starting from a micron sized particle of course, here this micron sized or bigger particles can be classified as bulk. In other words we are tending to bulk behavior close to the right of the diagram and as you reduce the particle size you would notice that the coercivity is increasing it reaches a peak value and at this peak value you draw it you give it a value d s and further to that this coercivity decreases on further reduction of size and finally, the coercivity falls to 0 and this is another critical number which is called d p. Now, this behavior has to be understood and this is understood in terms of the domain structure which is changing as you are reducing the size. In very large sizes of course, you know there is a multi domain structure and domain wall motion is one of the important mechanisms by which magnetization is taking place. In very small materials very small particles the mechanism actually is a domain wall rotation which is a mechanism and the structure actually changes from what you might call when in the bulk scale it is multi domain and in this says the coercivity increases with decreasing size that is what you are seeing in the trend line here. Finally, of course you obtain a peak in coercivity which corresponds to a single domain and so therefore, there is a peak coercivity corresponding to a single domain. Now, further when you reduce the size of the single domain particle the coercivity begins to decrease and finally, the coercivity vanishes when you obtain particles which are of the order of for instance tens of nanometers. And throughout this process as you pointed the whole curve can be ascribed to an important property which is now my change in the domain structure. So, here it is a multi domain structure wherein I have domain wall motion predominating and here it is a single domain structure in the low sizes wherein I have what you might call domain wall rotation which is predominating. And this overall effect is that I have a material below d p which behaves in a way which is called super paramagnetic. Now, we will talk a little more about super paramagnetic material soon, but we will take up what are the things which we need to note before we take up this concept. As the particle size is reduced the whole particle become single domain as we noted below the critical size. Now, why does it become single domain there are two ways of understanding it number one is that smaller particle then the domain wall thickness itself cannot support a domain wall that is obvious. Suppose, I am talking about a domain wall width of about 40 nanometers and suppose the particle is 20 nanometers it is obvious this particle will not support domain walls. In other words the magneto the exchange coupling will win over this particle and therefore, you will have a single domain particle. Now, and typically of course, that we will note that the domain wall thickness may not be constant width size. So, that is also expected to change it is not that 40 nanometer I said is going to remain constant width size, but nevertheless we have to note that there is expected to be a critical size below which a domain wall cannot be supported. Secondly, we have to note that the magneto static energy is a volume term which goes as r cube scales is r cube the domain wall energy is an interfacial area like time which goes scales is r square. And therefore, we expect such a system like you know in the case of nucleation in the case of phase transformation which wherein you see a critical size you expect that there will be a critical size below walls domain walls is not going to be stable. Look at it either from a picture like 1 or 2 we note that a domain wall is not going to be stable at small sizes. In other words a whole particle is expected to become single domain when you reduce the particle size. And the general trend line seen is that this critical size below which it becomes of course, we will just come to it in a moment. So, we note that below a certain size the material is becomes totally single domain structure. Now, the reason that this coarsivity is increasing in this region is that because some of the along the domain wall motion there are other phenomena like curling which is a way of domain orientation taking place is also occurring. And this curling term is now dependent on size and that is why you are actually having increase in coarsivity. Now, why does the coarsivity decrease after it becomes a single particle size single particle this is where the thermal disordering effects start to set in. Now, implies that you have in a nano particle unlike a bulk material very few limited number of magnetic moments aligned that means the exchange overall is limited to a limited number of what you might call magnetic spins. And therefore, if I employed temperature on the system the thermal disordering effects will set in sooner. And this system can actually be disordered more easily because there are limited now it is like a smaller system effect wherein now this thermal disordering is going to lead to a slow progressive reduction in the coarsivity. And finally, of course as we have seen at a certain size the thermal disordering effects win over and finally, you have zero coarsivity. We had earlier pointed out whenever you are talking about reduction in size in some sense we said this can be thought of as an increase in temperature we pointed this out. So, now instead of increasing the temperature to cause a material to become super paramagnetic to become paramagnetic we are actually reducing the size to make it paramagnetic because we have in a different context we had noted that reduction in size can equivalently mentally thought of as increasing in temperature. Now, though this material has become paramagnetic we use the important suffix for this material super. In the words is normal it is not this material is not a normal paramagnetic like a paramagnet like oxygen, but this is a super paramagnet because if you look at any temperature slightly below this transition d p then you would note that this material is now become paramagnetic either one way of thinking by increasing the temperature we said even beyond the curie temperature there are local spin clusters and therefore, there is some partial alignment and when you apply a magnetic field these tend to become proper aligned system and therefore, we said the susceptibility is going to very high. Similar to this in this case the material has become paramagnetic by reduction of size that means there is this is not a normal paramagnet this is a paramagnet wherein there is a latent tendency because of exchange coupling to become ferromagnetic. So, there is a latent tendency and this implies when I apply a magnetic field quickly all the spins would align and therefore, the susceptibility would be very high. Therefore, now when I plot an m h loop for such a material and I plot instead of plotting an m h I plot m h by t because now I can scale this whole system with temperature then I note that such a material will show what am I call a magnetization curve which has no hysteresis. In other words the magnetization follows the magnetic field. So, suppose I am having a field in this direction the magnetization will point in that direction if I rotate my field then the magnetization will rotate along with the external field in other words there is no hysteresis another spin field is opposite direction it will align along the opposite direction. Such a material in other words shows zero coarsivity and this is reminiscent of a paramagnetic material wherein there is the magnetization will follow the if suppose you try to magnetize ion then it will follow the direction of the field. But noting very clearly this magnetization effect is much larger in magnitude than the normal paramagnetic material because these are inherently as I pointed out ferromagnetic materials which have become paramagnetic purely by the change in size. Therefore, this phenomenon because of the intensity of the effect is called super paramagnetic and if you look at this hysteresis curve breaks down into a line in other words this material this is a signature that the material has become super paramagnetic. This value of d p is I pointed out in the nano scale in if you look at it for ion it is of the order of about 16 nanometers for cobalt it is about 8 nanometers for nickel about 35 nanometer. Therefore, you can see that d p which is now my transition temperature transition size below which the material become super paramagnetic is of the order of few to tens of nanometers. To summarize the story so far for reduction in size initially and now we are we have to note that the reduction in size has to be a few orders of magnitude only then you obtain significant effects. Number two we observe a large variation in coarsivity and below a certain size the material cannot support below certain size like d s the material cannot suppose support a multi domain structure. In other words the material becomes a single domain structure and after it becomes a single domain structure thermal disordering effects play start playing a very important role and therefore, you reduce the size even further then you note that the coarsivity falls down to 0 and below a critical size d p which is a the material becomes super paramagnetic and this d p has an value in the scale of few to tens of nanometers. So, in other words there is this beautiful effect called super paramagnetism which comes from reduction in size a quick comparison between paramagnetism super paramagnetism is necessary and to do this we will assume that we are applying a very strong magnetic field of the order of about 20 into 10 power 6 ampere per meter and we are talking about magnetization of oxygen which is about 2.85 bore magnetons per molecule. And we know that in a mole of oxygen there are about 6 into 10 power 23 molecules and if you divide by the atomic mass you get the number of or molecular mass you get the number of molecules oxygen molecules in this scale of matter. Now, we are asking the question what is the magnetizing effect of this strong field we know that this susceptibility of oxygen is actually very very small and this susceptibility is of the order of 10 power minus 6 it is 1.36 into 10 power minus 6. That means that if I apply a magnetic field and see how much magnetization takes place I would notice that I can multiply chi m by h and I would get h of course magnetic field which I have given before I would get a value of 27 ampere meter square per kg. But if you compare this value with the overall potential magnetization that means I assume that all the molecules are parallely oriented then I have this mole of molecules of course per kg divided by this 0.032 then I would notice that actually I will get a magnetization which is of the order of 497 ampere meter square per kg. Therefore, only a small fraction of the molecules are actually effectively contributing to the magnetization of the material that is which what is basically reflected as a susceptibility we know that only about 5 percent of the molecules are effectively contributing to magnetization. That means in other words even strong fields are poor in aligning magnetic movements in a paramagnetic material and now we are talking about a pure paramagnetic material which is not a ferromagnetic which has been heated above the curie temperature. So, it is important to note that very strong fields cannot also align a pure paramagnetic material very effectively and thermal disordering is actually what is winning in this case. But suppose I talk about an ion nanoparticle about 15 nanometer in size and we assume that this particle is now in the regime of super paramagnetic sizes it is below dp I will assume and given that of course I am here assuming a bulk value of ion I can correct it for by using a different value of ion for magnetization for a reduced system which I should do actually. But I just take some number which is the magnetization value for a ion for an ion value I would notice that if I calculate the volume of the particle which is about 1767 angstrom cube and volume per atom is this number this lattice parameter cube divided by 2 because there are 2 atoms in a b c c unit cell. Then I would note that this 11.82 angstrom per atom in a b c c unit cell the number of atoms ion atoms in the particle is about 149 atoms and I would notice that the magnetic movement under particle saturation is 329 mu b of course I have pointed out this may have to be corrected for the fact that in a nanoparticle the effective magnetization could be higher. In other words in a super paramagnet you can get high magnetization values because all the particles except for the ones on the surface actually are aligned along the direction of the magnetic field. While in the case when you apply a magnetic field and because now this is a single domain structure there is no question of domain wall motion it does all and for now I will assume that the magneto crystalline anisotropy is small. So, that the magnetic magnetization vectors or the spin vectors can actually rotate rather freely. In that case I would notice that I obtain a magnetization which is commensurate with all these spins aligning parallel to the field. While on the other hand you would notice that the effective magnetization for a normal paramagnet is going to be a small number because thermal disordering effects are stronger. Now this is an interesting example we have taken up earlier in the course we had talked about how we can understand various properties and performances which can arise when you reduce a size. We said that there are four classes we classified them and we said that change in size can lead to a change in structure which can lead to a change in mechanism which can lead to a change in property and finally any change in property is going to perhaps lead to a change in the desired performance of the material. So, now if you look at this case of magnetization in the case of reducing sizes then reduction in size as we saw can lead to a change in domain structure. Here of course, we are talking about not change in crystal structure, but the domain structure a microstructural kind of a parameter. This we saw can lead to a change in magnetization process we said that even the what we call the mode of magnetization changes to domain wall rotation from domain wall migration predominantly and we see phenomena like super paramagnetism. That means that properties are going to change we are going to have a net coercivity and a density equal to 0 and this means a high sensitivity of this paramagnet to an externally applied paramagnetic field and needless to say this kind of a material can be put to good performance because now this material shows a extreme sensitivity to an external magnetic field as compared to a normal paramagnet. So, this would be a nice example where in you are noting that the whole gamut of possibilities which we pointed out that change in size can lead to a change in structure either crystal structure or in or the case may be some kind of a microstructural entity. Then change in magnetization which is leading to change in coercivity and finally, of course, the performance of any kind of a device which will be based on super paramagnetism therefore, magnetism is important to understand from a what you call a mechanism perspective apart from the what you might call the femmological perspective as well. The next topic we take up is magnetism of clusters.