Good morning. We are into the last day and my last talk on this thing. So, this is the continuation of the magnetism lectures that we have been seeing. So, as I mentioned we in the last lecture we have seen what happens to the magnetism of insulating materials mostly insulators were discussed, which was some way it was a continuation of the atomic magnetism and I was highlighting the differences between atomic magnetism and solid state magnetism especially when it is insulating solid. In that context I mentioned this point that in the case of rare earth compounds insulating compounds the difference between the atomic magnetism and the solid state magnetism in difference is not very much whereas in the case of transition metal oxides the difference is huge. In atomistic picture atomic magnetism kind of a treatment is absolutely wrong in the case of transition metal oxides when we discuss the magnetism because of the fact that you are having the crystal field interaction as a very important contributor which is not in the case of atoms and the crystal fields the extreme case of crystal fields we have seen is that the orbital angular momentum is completely quenched. So, you have only the spin part of the magnetic moment. So, this is a big difference compared to atomic situation where you have a this J as a good quantum number J representing the total angular momentum and correspondingly you get the magnetic moment. Whereas, in the case of transition metal oxides we have seen that only spin contributes that is what the experiment data and the theory finally, tell you. Now, we are actually going to the other part of solids that is the solids which are actually metals or alloys that means they are good conductors. Insulators we talked about. So, there the features we have seen, but now we are going to see what happens to the ferromagnetism mostly in the case of metals and alloys. When we tell metals and alloys it can be again transition metal based or rare earth based or combination. So, it does not matter as long as they are metals the treatment is different as we are going to see. The understanding or the way one has to really understand the magnetism in these materials this class of materials is different from the picture that we have seen earlier. The main difference in this case is that magnetism is not localized. In atomic magnetism is localized, the rare earth magnetism is localized. In some sense the transition metal oxides also there is some kind of a localization. But in the case of metals what we have to use is what is known as a band magnetism like the band picture we have seen in the case of solid state yesterday. Those features have to be brought in to understand the magnetism of metals. Now, the problem is different. The band picture is very important in this case, but the band structure or the band theory that we did yesterday or the free electron model before that there was no magnetic field. So, it is purely the effect of band we have been seeing at the most we applied an electric field and saw what happened, but there was no magnetic field that was applied to the bands and see what happened. Today we have to do that. In this lecture I will also cover some aspect of magnetic materials which are actually used for applications, a flavor of different applications some of them I cannot cover everything. Some of the applications will be covered in this lecture. Towards the end as I promised I will also talk about some very very brief introduction to the so called high temperature superconducting materials. So, this much we have to finish in this lecture. So, this was actually kind of delayed the magnetism of metals and alloys this discussion was delayed because we wanted to see what happens to the band structure. Yesterday that was done now we can build it up further by seeing the effect or the response to an applied magnetic field. So, the first thing that one has to keep in mind before we really start is this one. In the case of atomic magnetic moments we have seen the magnetic moments are actually the atomic magnetic moments are integer numbers or most of the time it is integer numbers. In the case of oxides the insulating materials there again it is integer half integer kind of numbers. On the other hand in the case of transition metals and alloys as this table suggests what is shown here is a QD temperature that means most of them are ferromagnetic at room temperature. So, first three are ferromagnetic at room temperatures. This is the GD, GD is a gadolinium which is the rare earth which has got a highest QD temperature among the rare earths and it is close to room temperature. This is a compound an alloy and these two are similarly other materials. If you compare what you can see is that iron has a maximum QD temperature that is why most of the applications we use Fe. I am not talking about that right now we are looking at this. The magnetic moments if you see they are quite different compared to what we have been seeing they have non-indiger magnetic moments which is 2.2 to 1.7, 0.6 these kind of non-integers we never saw. So, a Hohm's rule picture which is valid for atoms and the picture where the crystal field interaction came I mean is taken into account in such case also you have a quanta number instead of J you have a quanta number S but G factor being a kind of an integer or a half integer you always got numbers which are not like this. In the case of transition metals and their alloys the numbers are not there they are all these kind of non-indiger values. This is a characteristic of these kind of materials or these alloys and metals. So, this has to be understood why this kind of a thing happens. This is a very important difference compared to the insulating solids. Now, what are the key features of these things in addition to this? As we mentioned magnetic moments are generally non-integers. So, which and here as we are going to show the band structure has to be taken into account to understand this thing. As I am going to show there is an actually there is a good overlap between the 3D or the 4F, 3D in the case of transition metal, 4F in the case of rareites the bands and the so called conduction band which is essentially the S band or the P band. In fact, this is more in the case of 3D 4F is still more kind of an inside or more localized thing. So, that means, the band overlap or the band formation is not affecting very much the 4F electrons even in the case of metals, but 3D is very much affected. The therefore, as I mentioned magnetism in these materials should be treated as something like a band magnetism. In contrast to what we saw in the other case which is more of a magnetism that is localized magnetism in the case of atoms and a crystal field dominated interaction magnetism in the case of transition metal oxides. Not only is a simple band structure will do in our case we have to do one step further and see what is known as a spin polarized band structure which is very important. From the, these are all from the point of view of fundamental issues, from the point of view of applications there is a big difference. There is an addition here yesterday we talked about a hysteresis loss which actually is not wanted in most of the applications other than in permanent magnets where you have no cycling of the magnetic field only hysteresis loss is there I mean it is not a loss that is what is needed. In the case of alloys or conducting materials if they are magnetic. In addition to the usual hysteresis losses that we talked about yesterday you have also the eddy current losses because you are changing the magnetic field there will be Faraday's law there will be induced currents and correspondingly there is eddy current and these eddy current losses are all very important in the case of this one. So, when you are talking about metallic magnets in addition to the usual hysteresis losses you should also worry about eddy current losses. This is one of the reasons why whenever you have high frequency applications you always look for insulate materials like ferrites because ferrites are insulating and this at least is eddy current losses will not be there. If you use metallic systems this will not be the case and you will have large eddy current losses which should not be there. So, these are the some of the features which one has to worry about but looking at the main issues like what is the reason for thermo magnetism the hysteresis properties those things are domains why domains are there what kind of a number of domains what determines the number of domains all those things essentially remain same in this class of materials also. So, the first thing that we have to this I think I mentioned earlier but again you can see that when you have actually a solid which is in conducting these are the core levels which are actually atomic like as you reduce the distance and get into the solid state slowly the bandwidth develops and the core levels as I told you earlier the bandwidth is still very small. But when you go to these bands with a 3D level or the 4S level you can see that the bandwidth is very much. In fact, the 3D essentially overlaps with this 4S conduction band 3D is a magnetic kind of magnetic band whereas, 4S will be the conduction band. So, you can see that there is a good overlap which means that there is electron transfer between the 3D band and the 4S band this is not the case when you talk about an insulator if you take oxides transition metal oxides this is not the picture. So, that is why you need to get it the band formalism to understand magnetism in these materials. So, this is a very important difference compared to between the transition metal oxide magnetism and transition metal magnetism. The same thing can be talked about for the rare earths but the difference will be smaller because since 4F is very much inside it is much more localized the differences between a band picture and a localized picture the difference will be small. So, this picture tells you what is what kind of band you have this is we will look only at the 3D band of course, we should know that there is an overlap and there will be some electron transfer between the two as I just mentioned. But this picture that we talk about in the context of band theory of solids is without the application of any magnetic field there is no magnetic field that is applied. So, in that into one study magnetism or study ferromagnetism what you need to do is that you have to apply a magnetic field if you are looking at paramagnetism or if you are looking at ferromagnetism you have an internal exchange field the molecular field which is presented. So, let us see what happens to the band when you are having no applied field no internal field then what happens is to understand it what we should do is we should actually split the band the whole 3D band let us take the 3D band which has some overlap with the 4S band this has to be split into two sub bands one band consisting of only up spin electrons the other band consisting of only down spin electrons. Because magnetic field is going to distinguish between the two the energies are going to be different as you know. So, in this picture we have to separate the electrons into two categories the ones which are having up spins the one which ones which are having down spins remember again spin is half. So, you have plus half and minus half for the MS values. So, all those electrons of MS plus half you put in one kitty and minus half in the other kitty that is what is shown here in the absence of any field the energy is will be the same and the common Fermi level you can see something like this. So, up to this it is filled otherwise it is not filled. This is a case when there is no applied magnetic field or there is no internal field. Suppose there is an internal field as expected in the case of a ferromagnet what is going to happen is there will be a displacement of these atoms with respect to energy as shown here. The up spins are actually little push downward and here it is moving as relatively essentially this support and if you see the Fermi level there is a population difference between the up spins and the down spins. Here if the population difference is 0 you will tell that there is no magnetic moment the net magnetization coming from here is 0. Here on the other hand you can see that there is a magnetic moment and this is a uncompensated spins together will give you a magnetization in this direction. So, what you are seeing is the applied magnetic field or the internal magnetic field is going to make a relative shift of this up spin sub band and the down spin sub band giving rise to a net magnetization. Because of this reason the usual picture of using this arrows up and down at every atomic site either in the case of an atomic magnetic picture or in the case of an insulating picture where we have been showing up down up down that picture loses its validity completely. Here there is nothing like atomic or side twice side twice putting up side up arrow down arrow is has no meaning here. You can only take all the up spins together and give an up spin direction there similarly for the down spin. So, this individual kind of localized kind of picture will not work in the case of band furrow magnetism. So, all the up spins are treated together all the down spins are treated separately as a another sub band as shown here. So, in that respect that is why this is called the band furrow magnetism not the usual furrow magnetism that we talked in the insulating solids. So, one can actually work out what conditions will determine this kind of a behavior under what condition this feature or this scheme changes to this scheme. I am not going to the details of this one, but only one point that is important here to mention is whenever the density of states which talked about in the earlier classes the density of states at the Fermi level when it is very large generally the tendency for getting this kind of a picture is more. So, you will get this picture this displacement of up spin and down spin this is called exchange splitting. So, this is not exchange split this is exchange splitting. Exchange splitting means that up spin sub band down spin sub band they have a relative displacement with respect to energy. So, exchange splitting can be expected or band furrow magnetism can be expected whenever you are generally having this density of states devoid stands for the density of states at the Fermi level is more. Now, if you see what is going to happen let us see what happened we are trying to explain why the non integer moments come in the case of transition metals. As I showed you earlier when we talked about insulating materials where if I take 3D I know how many electrons are there and where if it is quench I will not worry about orbital part otherwise I will worry about the whole thing, but the number is fixed number is 3, number is 4, number is 5, number can be maximum 10. Whereas now there is a good overlap between the 3D band and a 4S band which is a conduction band. So, you cannot meaningfully talk in terms of pure 3D electrons and pure 4S electrons there is always a mix up between the two that is also fine because since the number is concerned it is fine, but the problem comes from the point of view of magnetism. Because of this picture because of this statement that is mentioned here the furrow magnetic tendency actually is determined by what kind of density of states you have at the Fermi level. If you see this earlier picture which I showed here you can see as this band goes from 3D to 4S the bandwidth is increasing you see this is the bandwidth this the bandwidth is actually not there here the bandwidth is this much 3D bandwidth is more 4S is even more. So, as you go to higher and higher energies the bandwidth is increasing because the spread is increasing when the spread increases and the total number of that number of electrons in a given band is fixed the density of states will decrease. So, the density of states at the Fermi level for example 4S band is very small compared to the 3D band. So, what happens is even though in principle you are telling that I have taken care of the redistribution I know how many electrons have gone to the between the 3D and 4S. But as far as magnetism is concerned the crucial parameter of interest is nothing but the density of states at the Fermi level. As far as this quantity is concerned the major contribution or significant contribution comes only from the 3D band and not essentially from the 4S band. So, whatever has gone to the 4S band that will not be able to contribute significantly to the magnetic moment and only things which are actually physically present in the 3D band will be able to contribute. This number is not exactly well defined and hence you can see these numbers are like this you can expect depending on how much how many electrons have been transferred from 3D to 4S or other way that this number can be non integer and that is what experimentally is seen 2.2 more magnet on for Fe metal. So, that is because of this the two problems one is the overlap between the 3D and 4S and this charge transfer electron transfer between the two and the fact that the 4S band essentially is not in a position to contribute significantly or as strongly as in the case of a 3D band is giving is responsible for getting different non integer values as far as magnetic moment is concerned. So, you can see them giving an example here actual contribution is only instead of 10 it is only 9.4 if it is a number a total number is 10 and hence correspondingly the magnetic moment will be smaller compared to what is expected from a insulating picture or from an atomic whoms rule picture. So, this is going to happen in all metals and alloys because of this overlap of conduction band with the magnetic band when I when mean magnetic band I tell I mean the 3D band or the 4F band, but the 4F band being much lower in energy this chance of this overlap is still smaller. So, you do not have to really worry too much about it, but when it comes to transition metal the way transition metal oxides are subjected to strong crystal fields because the fact that it is outside the 3D shell is outside in a different because of a different contribution because of this conductivity of the system because of the conducting nature of the system and the fact that the 3D is outside 3D metals like the 3D oxides the 3D metals are also subjected to any kind of external influence very easily in this case the influence is nothing but the band effects and that is what is making this particular thing whereas a crystal field is still giving you the not this kind of funny number band picture gives you this funny numbers of 9 I mean various non integer magnetic moments like 2.2 or on 0.6. So, this is purely the result of band magnetism. Now, the question is what about orbital part here the orbital part in this case is almost zero because when we tell this is essentially treated as a free electron essentially the band of course is not really free, but the orbital contribution is very very small. So, generally it is not is not taken into account. So, essentially it is a spin case and the spin is purely distributed we do it spin up and spin down 2 sub bands we worry about it. So, as far as metallic magnets are concerned you can just represent all the up spins as just one sub band all the down spins in the down sub band and when they are not magnetic that means they do not have the ferromagnetism then you will tell that the number is the same and you do not see any magnetization in the bulk scale. Otherwise you see the difference between the two giving rise to some finite non integer magnetic moment or the corresponding magnetization. So, this is the situation for the 3D metals I mean as the metals change from Mn to a feel like this how the Fermi level position happens and you can see the density of states plotted here this is for the 4s you can see the density of states very very small here this is the density of states is very large. So, the difference is very important this is not able to contribute a strong ferromagnetism whereas this one is able to contribute a strong ferromagnetism the band ferromagnetism. This is the reason why you are getting into this non integer moments because whatever has gone to this part this will not be able to significantly contribute to the magnetic moment whereas only these fellows will be able to contribute to the magnetism. So, that is what is written with an example here you can go through this this is an actual calculation we are taking the some number of electrons and how the non integer values actually come about. So, please go through this if you have any problem all these details are written you can send an email. Almost all types of magnetic materials have been discussed by this time atomic where the application is actually very small that is only a building block to get into the solids or molecules or these days than nano systems, thin films, multilayers all these things you need to get some idea about atomic magnetism. Then we saw what happens in the case of insulating solids what happens in the case of metallic solids in one case you have localized magnetism one case you have kind of a band magnetism very essential. Now, what we are going to do is that what are the main applications of these materials as I mentioned in the beginning we will not be able to go and see all kinds applications because magnetism has n number of applications. So, it is very difficult to talk about all of them what I am going to do is I am going to take some of the important properties and how these properties are actually getting translated into applications very important applications. So, basically the applications would be are selected in such a manner that there is a direct correlation with the main magnetic properties those applications I will highlight. So, this actually will give you some correlation between what the properties that we have been seeing and the applications which are actually seen probably by us almost on an everyday basis. So, first and foremost I will talk about the permanent magnets which all of us know and all of us use everyday. They are as I mentioned earlier that the classification hard magnets versus soft magnets these magnets are actually extremely hard magnets where hysteresis loop must be as large as possible. So, that your remnants and cohesivity are maximum. So, there are a large number of materials and most of these materials are actually metallic based the one which I talked just now. So, they are all mostly magnetic materials which are based on the alloys most of the alloys and most of the time the alloys are formed between rare earths and transition metals not oxides. Transition metals like for example, a very standard material magnet that all of us use is what is known as samarium cobalt 5. Samarium is a rare earth cobalt is a transition metal. Samarium cobalt 5 is purely a metal it is a very good conductor and this is a very standard permanent magnet made of it is what generally used in many cases. There are other cheaper this is slightly expensive there are cheaper magnets like alnico magnet. One should not forget that we have also have ferrite magnets but ferrites are not I mean conductors they are insulators. So, ferrites as I mentioned can be used in high frequency applications also but these magnets these materials they like there are some CO5 materials cannot be used in high frequencies because of their eddy currents problem that will be there because of the conductivity being very large. So, there are one or two important points I will mention there are started with a steel steel was the starting point of this whole journey it has gone it has many many materials have been developed and I mean the kind of the field is saturated as far as sub bulk magnets are concerned but recently people actually have been working since many of you are interested in nano systems and other things lot of nano research is going on to get and actually people are getting better magnets from the nano composites and things like that I will not be able to go into the details of this. So, there are lot of new magnets coming out because of this nano technology or the nano materials being developed nano magnetic materials being developed many times the nano composites many of the nano composites carefully chosen are giving very good permanent magnetic properties. So, and the problem is many of these materials are very costly these days. So, there is a need to replace these materials or substitute for these materials that is actually is achieved from the many of this nano systems that are being worked. So, very important physical property that is needed here a permanent magnet actually is as I mentioned it has to have a very high large hysteresis area and it should have a very high query temperature. So, for example, if you have something as a query temperature of 400 Kelvin that is not a very good magnet because you are operating temperature is room temperature which is 300 or 310 Kelvin. So, for that you the properties will start degrading the ferromagnetism will start degrading if the query temperature is 300. So, one should look for materials where the query temperature is may be 600 or 700 Kelvin. So, that room temperature is fairly stable ferromagnet that is a very important point. So, materials of this kind only be used and as I mentioned earlier this must these are all hard magnets hard magnet means the magnetic anisotropy the so called magneto crystalline anisotropy must be very large. The magneto crystalline anisotropy is very strongly related to the crystal anisotropy the crystal structure anisotropy as again I mentioned earlier a cubic system even though you tell that there are different access of magnetization possible the difference in the energies is very small and hence you will see that cubic systems are not good permanent magnets all the permanent magnets which are actually the potential magnets you see here none of them actually belong to the cubic family they all belong to some uniaxial crystal system where your C axis is quite different from your A B plane A axis or the B axis. So, the crystal built in anisotropy is there and that actually gets translated into magnetic anisotropy and this is very important. So, you cannot choose any material and try to make it permanent magnet that will be a wasting your time because you will not be able to get good hard magnetic properties if the symmetry is very good. So, the crystal symmetry the crystal the cubic symmetry will never allow you to get a very good hard magnet out of it. Another very important property that determines the quality of a magnet the so called a quality factor is nothing but what is known as an energy product this is very important point because energy product is very quality factor because if you remember a magnet how is a magnet made a magnet is made from a material you take the material you identify the material and then it is magnetized and that becomes a magnet. But you cannot take a bulk piece of a material and try to magnetize it what is done is you take the material identify the material you get it in a fine powder form then these powders are magnetized and then in the same magnetic field you press it and so that you get a compact. So, any magnet that you use a bar magnet it is never a single bulk piece it is a collection of powders sintered pressed together in presence of a magnetic field it was actually field pressed. So, it was pressed in a magnetic field this is a process by which it has to be made. So, and the thing is when you are actually applying the magnetic field it gets magnetized and after that because of the very large hysteresis properties the field is removed and it has to retain that is why it should have large cohesivity and large remanence. So, that the magnetization whatever it got does not go away and it is retained. This is why we tell that hysteresis must be very big in this material not only that after that what happens is the field is removed this is subjected to you cannot tell that this is not subjected to any magnetic field now because I removed a magnetic field. But it this is subjected to its own demagnetizing field as you mentioned because of one as I told you once it takes birth it is actually coming with its own enemy of the negative field the reverse field or the demagnetizing field that is why we always talk in terms of the second quadrant as far as hard magnets are concerned. See second quadrant is one where the field is negative this is positive field positive magnetization this is negative field this is magnetization because you think that I have made my magnetic field to 0 here. But actually there is a reverse field the negative field that will be coming and that is why the discussion must be in this one. So, one can actually show the energy product the so called energy product which actually is product of the B field and the H field B is because the material actually has a B field and H field is a field that is using used to magnetize the material this product is called the energy product this BH is called a energy product basically tells you how much energy this magnet can provide you. Unless you actually make a correct design your demagnetizing field will make it smaller and smaller and as is plotted here if you take any magnet the BH this product actually has this variation it increases for a particular value of this applied field or a particular value of the shape you see that corresponding to a particular value of the shape because the shape is one which actually is going to determine what kind of demagnetizing field it has got. As I mentioned earlier it is essentially minus Nd the demagnetizing factor times the magnetization. So, this Nd the demagnetizing factor is critically dependent on the shape of the magnet and hence you can see corresponding to a particular shape or corresponding to a particular demagnetizing factor only you have the maximum value for this BH the so called the BH max or the maximum energy product. So, you have to choose the shape in such a manner that this energy product is maximum so that you get maximum efficiency out of it. So, this is a very important point as far as the design of a magnet is concerned just because you have a good material just because you magnetize it properly does it mean that you have a good magnet because you have to optimize the shape so that you get the maximum energy product out of it. Why we talk about permanent magnets every time because there are N number of applications where permanent magnets are used. I am just giving you one example which is a very standard example given just a car all these arrows which are shown here all these things are magnets the permanent magnets are sitting all at the point of these arrows. So, many magnets are there in a given car itself. So, there are many such applications where permanent magnets are there and there is a huge demand. So, one has to really have to have very cheap magnets because otherwise all these costs will go up and these days actually there is a tendency for this price rise because of the availability of the material unavailability of many of these rare earths especially is creating a problem and that is actually increasing the cost of magnets which eventually will get translated into the device cost. The other extreme now I will go to the other extreme where the hysteresis is not at all expected because I need a very very thin hysteresis loop that is nothing but the so called a soft magnet and all of you have know that a transformer has what is known as a soft iron core this is a soft iron core which is this rectangular frame or a square frame. So, this is actually called a soft iron core because it is a soft magnet. What you need here is you have a primary circuit and a secondary circuit you want to have a good connection linkage between the primary and secondary. In the absence of a any core just air also it will work there will be a coupling between the two the magnetic coupling will be there but if you want to have a good coupling enhance the coupling between the two what you do is you use a conductance of a magnetic circuit that we is done with the help of a ferromagnet. So, the ferromagnet when you put you can see instead of air core if you have a soft iron core the flux linkage between the two will be more and that actually gives you a better transformer. So, here for these kind of applications you are looking X for extremely soft magnets. So, the soft iron the pure iron that is something which is a very soft magnetic material that is used but there again I will not be able to go into the details iron has this problem because iron is a conductor it will have the eddy current losses and all as you would have seen I mean even in schools people discuss this to reduce the eddy because there one has to use iron is a very soft. So, it is a very ideal choice but somehow you have to reduce eddy current losses hysteresis losses almost zero the eddy current losses are there that has to be reduced because it is a conductor what is done is you take this iron core in the form of laminations and this laminations will make sure that the circuit is not complete circuit is not very continuous which we actually will offer some resistance. So, this resistance will make sure that the eddy current losses are small similarly we can also put some kind of elements substitute into this one like silicon and things like that one can add into this one. So, that the conductivity reduces or the resistivity increases without affecting the magnetic property the magnetic softness very much but the electrical conductivity is reduced so that the eddy currents are reduced. So, without really manipulating too much of magnetic properties you are able to manipulate the electrical conductivity property and you can actually reduce both the hysteresis loss and the eddy current loss. So, this is an application which has it is a huge market demand because most of the transomers we have they are all silicon substituted iron which acting as a core very very important application. So, the two extremes you have seen one is extremely soft other thing is extremely hard hard in the sense from the magnetism point. A very important billion dollar industry application of magnetic materials is magnetic recording I think I do not have to tell all of us know about this I will not worry too much about this but I will come to a very important technique in that. So, magnetic recording involves writing a information in the form of a magnetic storage and then it has to be read. So, you need to have a writing mechanism and a reading mechanism. So, this many material media have been used for this purpose mainly you should have a system where you should be able to differentiate between 1 and 0. So, that the binary information can be stored without any noise. So, for that purpose you need to have magnetic states which can easily switch between large magnetization small magnetization or between 1 and 0. So, 1 representing 1 state that is highly high magnetic state may be 0 representing the other one. So, the very important characteristic or property required for that particular application is that you should have a hysteria loop which actually is not very curved which actually has a rectangular shape. So, it actually goes very abruptly between. So, if I take H which actually I do not distinguish between H and B as I mentioned. So, this is B or if I take M. So, what should you should have is something like this. So, this is an abrupt jump between the two. So, this is my plus M this is my minus M. So, this change should not be very curved like the usual hysteresis it should have a sudden jump let us say from 1 to 0. So, these kind of materials must be there for this application. So, this is again a very careful choice only very few materials you can get into the situation or you can do lot of things externally and you can get this kind of hysteresis loop. And in this context many of them will be in the thin film form magnetic thin films as you know will be used for this purpose. So, this involves a writing mechanism and a reading mechanism both are needed. So, here what I am showing two things one is what is known as a longitudinal recording where you can see the information is stored in the plane of this medium you can see this one whereas, here it is a perpendicular recording you can see the bits are like this one and zeros are shown like this. So, this is something has been written this here also something has been written there are some advantages and disadvantages for both. So, here the recording density you can get is much more, but there are practical issues I will not go into those details. But my interest is to understand something regarding the reading of this one. So, this is a writing is you magnetize it locally whatever means and do a writing the thing is how do you read it out. So, there are various mechanisms to read it out earlier many mechanisms were there, but these days what people use is actually what is known as a magnetor resistive reading. So, the reading is done with the help of a magnetor resistive sensor kind of thing what does it mean. So, what does it mean is that you have a small circuit like the one shown here as this moves from ones and zeros depending on the ones presence of one or zero that is depending on the magnetization state of a particular region the so called bit region. The resistivity of this circuit changes by noting the resistivity of this circuit the change in the resistivity of the circuit depending on the magnetization one can actually understand what was written here whether it was one or zero and this information is actually read using the so called giant magnetor resistive heads they are called GMR heads this GMR heads are actually used for reading. So, writing is one process reading is done with the help of this GMR sensors this is a very recent thing and this actually is I mean of course, because given dumbbell price not belong back. So, this is a very new idea that has come into the whole topic of magnetism. So, I will spend some time after this to show you what is actually meant by magnetor resistance or specifically giant magnetor resistance. So, this is the way you actually write and read the information magnetically. So, there are various materials available for this medium as well as for this part I will not go into all the details of this in this limited time. Another very important topic in magnetism is magnetostriction. What is magnetostriction? The dimensions of a ferromagnet actually change as you apply the magnetically the way magnetization changes something like that you can see this is called the magnetostriction magnetostriction initially is low actually increases like that. So, this is actually defined as the fractional change in length that is what is written here actually is very small in these materials most of the materials what shown here they are mostly this is a rarer this is a transition metal they are rarer the transition metal alloys they were this material sometime back. So, they were actually having these kind of fractional changes recently maybe the last 15 years or so people have found out a new set of materials which are actually much showing magnetostrictive properties much larger than this what is shown here. They are actually called ferromagnetic shape memory materials there this strain the so called the magnetic magnetostriction or the magnetic field induced strain the length change for example is huge compared to this one and they are becoming much better and they are being used in various applications. One of the important applications see something like what is shown here this is a transducer what is a transducer as a field the length changes when you apply a magnetic field if you apply an alternating magnetic field the length keeps changing increasing and decreasing this actually sets this let us say rod in the form of a magnet in the form of a rod actually subjected to vibration. So, it vibrates that means you are supplying magnetic energy and you are getting mechanical energy. So, this can be used for energy conversion that is you are converting one form of energy into other any device that does this job is called a transducer. So, what is shown here is a magnetostrictive transducer which has got lot of applications I will not go into those details, but a lot of applications are there. So, the main thing here is a magnetostrictive material magnetostriction is very strongly connected with what is the as we talked about the magnetocrystalline anisotropy. So, one has to really play with the magnetocrystalline anisotropy it should not be very small at the same time it should not be too much. So, that you are able to get decent magnetostriction in reasonable fields low fields. So, then it becomes a good magnetostrictive material and then you can actually get into various devices the most important one is as shown here a magnetostrictive transducer. Another again a recent thing is somebody was asking about entropy some questions I thought I will talk about it also. Magnetic materials also have a role to play in refrigeration then it becomes magnetic refrigeration. What really happens is you are actually playing with the entropy of the system any refrigeration system what you are doing is that you are playing with the entropy of the system. So, here you are doing is that you are playing with the magnetic part of the entropy of the system. So, as I showed you here. So, this is a low field situation where this moments aligned randomly remember I can take it as a ferromagnet these moments are like this. When I apply a magnetic field let us say as I am applying a magnetic field and if I do under adiabatic conditions if I do it and adiabatic conditions what happens is the magnetic field will force them to be in the same direction or the entropy will the magnetic part of the entropy will try to reduce because this is a high entropy system case this will be entropy minimum less. But you see I have taken the process to be done in an adiabatic manner an adiabatic manner you know the total entropy has to remain constant. But now my magnetic part of the entropy has decreased this cannot be allowed as far as the total entropy is concerned it has to remain the same. So, if there is a reduction of two units of magnetic part of the entropy somewhere else these two units must increase. So, there the total entropy does not change this is a very important requirement as far as the process is done in an adiabatic fashion. So, how can you increase I mean make an equal increase of two units of entropy this can be done from the lattice part of the entropy because entropy has lattice contribution that means a positive ions related to the vibration of the positive ions. There is a small contribution from the electrons and a contribution from the spin the magnetic moments if it is a magnetic material. So, you can depend on the lattice contribution to contribute or to get this plus two units in this case in my example. So, that means the lattice has to vibrate little more vigorously. So, the lattice vibrates little more vigorously. So, that the entropy remains constant. So, the lattice entropy increase will give rise to more violent vibration is corresponding to an increase in temperature. So, t original temperature now it becomes t plus delta t. In the same argument I can tell if I do an isothermal magnetization and then an adiabatic way of demagnetizing it. So, isothermally if I magnetize it. So, here it will be t here it will be t. Now, if I do an adiabatic demagnetization reversing my argument I can show that the temperature t becomes t minus delta t. So, at the end of the whole process of isothermal magnetization and an adiabatic demagnetization you will see that your original temperature is reduced by a some amount. So, you are actually able to produce cooling. You can do this in the form of a cycle there are lot of rather restrictions I will not go into all those detail. But in general this principle can be used to reduce the temperature in any system and this is this technique is not new. This was actually known for many years many decades this principle was known this principle was actually known as adiabatic demagnetization. But at those days people were using paramagnetic materials these all these moments were paramagnetic materials and it was used to get or produce temperatures below 1 Kelvin. Now, this technique is extended to materials ferromagnetic materials not just paramagnets and ferromagnetic materials and you are able to get these cooling done even at high temperatures. You can you are you are looking for refrigeration means you are looking for or air conditioners means you are actually trying to use these materials to produce a cooling in this room for example or newer household refrigerator. So, people are working on it lot of things are being I mean happening and hopefully a magnetic refrigerator will come to the market not in a distant future. Adding to that since you have been talking about electronic magnetic moments and sometimes we talked about the nuclear magnetic moments also whatever we are talking about here electronic magnetic moments for a paramagnet the so called standard adiabatic demagnetization. One can do the same thing for the nuclear moments then it becomes nuclear adiabatic demagnetization a technique which was again known for many decades. This nuclear moments as I mentioned being smaller than the electronic moments nuclear adiabatic demagnetization techniques produces temperatures even lower. So, one this was also used along with the adiabatic demagnetization and the nuclear adiabatic demagnetization people were actually going were using this technology for refrigeration at very low temperatures. Nuclear will be lower than the usual adiabatic demagnetization. Now, you are actually getting this effect happening even at elevated temperatures as high as room temperature itself. So, this is a very applied thing and it has got very strong physics connection here. Now, I will come to this thing that I talked about earlier with little more details regarding the magnetor resistance method of reading the information that is written in a magnetic medium. Magnetor resistance straight away is defined is nothing but a change in the electrical resistance of a material under the influence of a magnetic field. So, that is why it is called magnetor resistance. So, magnetor resistance can be negative or positive depending on the system. This actually is shown by most of the materials the bulk metals it will it shows in metallic multi layers the effect is little more and it is called the giant magnetor resistance the one which I mentioned. You can also get a totally different thing but where the change in resistance is huge orders of magnitude change happens when the magnetic field is applied or the when the magnetic state changes. So, this is called a colossal magnetor resistance CMR. CMR is not very much applied at this point of time whereas, GMR is very much applied today as I showed you in the other picture. In all these cases one of the main important concepts behind this is what is known as a spin dependent transport somebody ask this question sometime back also. So, that means so far we have been talking about the in solid state when we talked about conduction electron scattering and other things we never worried about what is a spin of the electron. We only worried about the charge of the electron and how it gets scattered and so on. So, that will determine what kind of a resistivity it has. As far as this particular phenomenon is concerned we never worried about the spin of the electron and we know that electron can have either up spin or down spin MS is plus half or minus half. So, what is found is that the scattering rate actually is a function of the spin direction and that is giving rise to what is known as a spin dependent transport or a spin dependent scattering or this is a very important issue as far as this particular thing is concerned. This actually is very important this idea is very important in the so called spin tronics which actually is the recent thing and this actually one step above our electronics. The electronic devices are actually getting modified or replaced by these spin tronic devices. This is actually dependent very much on a class of ferromagnets known as half metallic ferromagnets I will show you very soon. So, the main thing is like this when you have a system generally what happens is so what is expected is if you take a non-magnetic or the diamagnetic material that is what I mean if I apply a magnetic field because of the Lorentz force as you have seen there one can expect an increase in the resistance that is for initial magnetic fields and so on. But when a ferromagnetic material what is found is that as the temperature is reduced and it enters into the ferromagnetic region the resistance actually decreases as shown here in this for example, this nickel. So, it was coming like this it decreases and I am comparing nickel here versus palladium palladium means more or less paramagnetic they will not much of a change whereas, nickel enters into a ferromagnetic region. So, that you can see a resistance decrease. So, this gives rise to a negative magneto resistance because here any positive effect that I mentioned due to the Lorentz force is small and the other effect is actually contributing to this reduction. This is a very simple manifestation of negative magneto resistance seen in the case of a ferromagnetic element like nickel. So, this idea what is needed is what is known as a spin dependent scattering. So, when electron is moving the scattering it is subjected to is determined by what kind of spin direction it has got. The scattering is different for spin up electrons and spin down electrons which means that when I am talking about a conduction of this thing and if I want to do an equivalent circuit what I should have is I should have two parallel branches one corresponding to an up spin the other corresponding to a down spin like the way it is shown here. So, these the resistance of these two equivalent branches will be different because the resistance encountered or the scattering rate encountered by the up spin and down spin will be different. So, this is idea behind this equivalent circuit needed when you actually have a magnetic material when the conduction is talked about. So, let us take straight away to what happens in the case of a this one I will come to the result later. So, what happened is people prepared various kinds of multi layers and people got reasonably good magneto resistance values this in fact this is a noble price work not long back it is 2007. So, you can see this is a multi layer you can see the multi layers are produced with iron chromium ions. So, iron. So, what happens is that you have a tri layer where a chromium which essentially is a non magnetic layer on the sense that it is not ferromagnetic layer and that is between the two ferromagnetic layers Fe and Fe. One very important point in this context is one can actually make sure that the direction of magnetization in the first ferromagnetic layer and the third layer can be parallel or anti parallel you can make parallel or anti parallel depending on or by changing the width of this non magnetic the so called non magnetic here I mean non ferromagnetic chromium layer for example. So, you can start with a situation as I am showing here where you can have. So, you can actually make this magnetization and this magnetization to be anti parallel by carefully choosing the thickness of this then what you can do is you can apply a magnetic field so that these two become parallel. So, and what is found is when they are anti parallel there is a certain resistance across the whole multi a trial air in this case and when you apply a magnetic field they switch over to ferromagnetism these two becomes ferromagnetically this two become ferromagnetically coupled and in that configuration the resistance is found to be smaller. So, this is a very important idea this device is something like this this is made of Fe, Cr, Fe, 3 layers here. So, one can actually explain with little more band picture one can actually explain how it happens that is what is shown here. So, initially what is done is they are actually anti parallel coupled this Fe and this Fe this is a spin polarized band structure I talked about this is up spin this is down spin here since it is anti ferromagnetically coupled this Fe and this Fe you can see here it is like this and here it is like this. So, this is changed and now what happens is this is anti ferromagnetically coupled you can see this is up spin down spin this is up spin this is anti ferromagnetically coupled in the beginning and this is a non ferromagnetically as I mentioned this will have equal population up spin and down spin. Now, assume that an electron an up down spin electron is trying to move across this. So, it can come here and it can come to this one. So, this is actually a down spin a down spin can up come here and then if you want to go from here to here it has to go to this one only because a down spin has to be accommodated in a down spin sub band only a down spin sub band here is this one and if you see the density of states the space available for the down spin actually is very small which means that a path from here to here is further going from here to here is more resistive it will not be in a easy way to go up all the way here. This is a situation when these two are anti ferromagnetically coupled. Now, if I apply a magnet this is done by choosing the thickness of this layer the non ferromagnetic layer in this case chromium. Now, if I apply a magnetic field what happens this applied magnetic field is equivalent to seeing a stored information where the locally stored information acts as a magnetic field. When that is the case when this layer and this layer will become ferromagnetic this is ferromagnetic will get coupled and now you see what happens this electron which was coming here no problem it can go to the next one also because there are enough place here enough vacancy here in the same sub band the down spin sub band and hence the conductivity path is complete that means it is encountering less resistance. This is a more resistive path this is a less resistive path which is equivalent to telling that I have this small law representing small resistance or capital R representing larger resistance. So, basically if you have a geometry of this kind this can actually be put into a region this can be a very small device it can be kept into a region depending on the magnetic field there depending on the information written there this can actually choose and tell what kind of information is stored there because it can sense what is the magnetic field that it is producing and correspondingly this signal that is coming out will be purely in the form of an voltage because the resistance change can be converted into voltage. So, the magnetic information change can be read electrically and that information will give you what is written there this is a way of reading the information that is written in the magnetic storage medium very simple idea. But very difficult to this same picture this was a very important discovery in not so distant future past this is a 3D actually use the bands which were using actually the transition metal the 3D band I talked about earlier. So, the main idea is that you have to depend on spin polarized transport because you see I am talking about a not an electron going there but a up spin or a down spin going. So, that is what is meant by spin polarized transport. So, spin polarized transport has given rise to various important things mainly the spin tronics which actually is nothing but spin electronics where in addition to the charge the spin of the electronics also becoming an issue this is because of the spin dependent transport and this gives rise to let me straight go into a very important picture here. So, we have been talking about metals insulated semiconductors in the last class. So, this is basically as I mentioned yesterday this is actually a metal this is the first band this is filled up to this point only there are empty states here. So, these excitations or the conduction can happen without encountering this energy gap. So, this is a metal this is insulator because this gap is more and this is completely filled the first band is completely filled here the same situation but the gap is small. So, this is a semiconductor. Now, this is all done with respect to a charge property of charge. Suppose now I bring in the band treatment whatever I did in the last class if I bring in the spins into account I can I should distinguish between spin up electrons and spin down electrons and there is a possibility of getting something like this what about the situation like for one type of spin I am getting a metallic like behavior something like this that is something like this the left hand side of this picture where I am representing the up spins the picture is that of a metal right side when I represent the down spins it is something like an insulator or a semiconductor where you see that this band is completely filled the lowest band for the spin down is completely filled whereas the lowest band for the up spin is only half filled. That means this is like a metal this is like an insulator and this is actually a ferromagnet because you see that there is a difference in the number of up spins and down spins. This is what is known as a half metallic ferromagnet because only one part is metallic other part is not the whole thing is a ferromagnet this is called a half metallic ferromagnet very important the advantage is at the Fermi level if you see you are actually seeing you can see only one type of spin you can see that is in this case namely the up spin. So, when you have a current it is essentially made up of spin up current alone so that is something which is very good for any of these sensors spin tronic devices reading all those issues where you have a clean one type of spin situation make a conduction mechanism then it is very very useful and that is what is happening one of the in this context one of the first materials where you could get this kind of a band structure is actually chromium dioxide which actually is a very very important material. So, chromium dioxide it is not a material but actually it has this kind of a completely spin polarized this is called a spin polarized situation many elements including Fe is somewhat spin polarized what you are looking for is a 100 percent spin polarized system this is the example of a 100 percent spin polarized situation of band structure. So, chromium dioxide is something which is very important in this context. So, I just wanted to show you another type of magneto resistance which is more important from the point of your physics there but not from the point of your applications the main point in all these things is that the connection between electrical resistivity and magnetism. So, I am giving straight away an example here I have take example like Fe 3 O 4 it is a ferrite. So, you have Fe 2 O 3 and Fe O together it is called Fe 3 O 4 Fe 2 plus coming from Fe O and Fe 3 plus coming from Fe 2 O 3. So, Fe 3 O 4 is essentially Fe O plus join with Fe 2 O 3. So, there are two kinds of Fe ions there Fe 2 plus and Fe 3 plus it is actually an insulator as you know.  So, you can see the this is a because it is an insulating solid it has a crystal field thing. So, it actually has this more or less degenerate here. So, these three are here and these two are here. So, this separation is large that is due to the crystal field. So, they are all like this and here there is there are two plus means there are six spins to be a spin six electrons to be worried about. So, they are like this the last one without violating the Pauli's principle it has to be like this. So, it start with like this feeling like this then it is like this and because of this thing this has to be antifurromagnetic the condition is for antifurromagnetism it is like this. This is an insulator as I mentioned this is an insulator. Now, the question is this last spin or the last electron let me call last spin if it wants to actually come closer to this one and it is essentially it wants to hope between or travel between the Fe 2 plus and Fe 3 plus. So, that this electron actually does not belong to this completely or this completely. So, if it actually hopes between the two which means that then one of them actually this we cannot tell that I have my valency of 2 plus it will become 2 plus half 2.5 this is essentially 3.5. So, basically this electron or this spin is shared between the two sometime it is here sometime here's situation, but this if it has to happen the valency of course will become 2.5 and 3.5 that is okay, but if to that to happen if it has this electron has to come here at any point of time what is needed this is a down spin. The last electron here is a down spin if this has to come here in this present situation the left hand side situation this cannot come here because they are all down spin for this to come here this will violate the Pauli's principle. And remember I mean I am not going to the details when this comes here this becomes a metal because now this is able to move between the two. So, you are telling that there is a conduction between this Fe 2 plus and Fe 3 plus this becomes a metal. So, because of metallic city there is an energy gain. So, it will be generally preferred that means somehow this electron spin will try to or this electron will try to go to the Fe 3 plus side for that to happen these spins cannot be in this direction then it will violate the Pauli's principle the system what does it do it actually flips this spin if this becomes up so that this down spin can come here without violating the Pauli's principle. Now, because of this Hund's coupling that is available within the atom within the ion in this case once this flips all these are the remaining four also flip and so this becomes all parallel all up now you see what happens these are all up these are all up what is happening this is these two are coupled to ferromagnetically unlike the anti ferromagnetic coupling that is seen between the two here. So, anti ferromagnetism has changed to ferromagnetism because of what because I wanted to change the insulating state of this to a conducting state. So, electrical resistivity change has changed the magnetic state or vice-versa I can tell if I apply an external field from outside the same thing is going to happen. So, this is basically a field driven magnetic field driven driven change in the conductivity. So, this is nothing again but magnetor resistance. So, it is a different kind of magnetor resistance than here what is happening is an insulator to begin with has become a metal. So, the resistivity change will be huge orders of magnitude. So, that is why this is called colossal magnetor colossal stands for very huge change the colossal magnetor resistance purely a I mean a fundamental effect at this point of time which is not obtained absurd at very high temperatures room temperatures and they are mostly seen in bulk material. So, the application wise this is not really to that level, but fundamentally this has contributed very much and lot of understanding has been obtained by studying this I give you a very very simplified picture of this, but the main idea is this one the magnetic state change can actually change the material completely from an insulator to metal or vice-versa very very important field topic when we discuss magnetism we should not forget this. This I think the details how it can be worked out you can go through it I think I will save some time for my super. So, this is a huge change that I will be talking about to start with an anti ferromagnetic or a paramagnetic insulator when it becomes a ferromagnetic if it becomes a metal if it is a ferromagnetic it has to be metal or if it is a metal it has to be ferromagnetic. So, these two are related you cannot in this case you cannot have ferromagnetism and insulating nature going together your choice is gone if you want to tell that I have to have a ferromagnetic it must be metallic. So, this is what is shown by this actual data on this particular very important class of compounds. Another example is there I showed you between Fe 2 plus and Fe 3 plus one can get the same thing between Mn 3 plus and Mn 4 plus these are the two allowed valence states of Mn. So, there again the same situation you can see. So, with this actually I touched upon the most important features of ferromagnetism these things can be extended to nano thin films in fact some of the thin film already the multi layers are actually thin films all those issues are there. So, it is very very difficult to talk about in detail all these things in a limited time there are many developments happening lot of nano work actually is coming and lot of nano materials are found to be much better than the materials that I mentioned in this talk. There are some amorphous materials which are coming very promising for many of the applications I mentioned the nano composites as I mentioned becoming a very big issue today and very very important from the point of view of applications also.