bert6.txt

 So good afternoon everybody and we start today again with the story of the basis and the model of the model. Last time you have seen this model, we solved the problem in case of applied field which is parallel to the easy axis of the uniaxial analysis of the preparatory accuracy. And you remember that the solution was in the end the explanation of history. It was the loom, the artistic loom. And I told you that the second option, responding to the same problem in which you keep just the perpendicular component of the applied field with respect to the easy axis. So this is actually the problem of the anisotopy field that we have already solved when I first tried to use the GL for the solution of the micromagnetic problem. And we expect something like that. I'd just like to remark one thing which is interesting. The coercive field that we have found last time, so the coercive field for the loop, when you have, when you solve the problem for H A parallel, and this is M parallel, you find a loop, which is a square loop, and this field corresponds to the coercive field, and it's exactly equal to the anisotropy. The coercive field hc is equal to the anisotropy field. It means 2 times k1 divided by the real field ms, where k1 is the anisotropy constant of the linear term. Now, when you move down to the other situation which is that stated over there, you discover that the situation is very similar is M, say, is H applied perpendicular, is the M perpendicular, and you find this. You have a linear behavior. And this one is the anisotropic. So the frequency field, when you have when you apply the field parallel to the easy axis is equal to the analytical field that you find as a saturation field. The neurons still need to saturate the magnetization in direction which is perpendicular to the easy axis. And that's all. Analytically you you cannot solve all the other problems that in practice you can find. The other problem corresponds to the application of the field, not just parallel perpendicular to the easy axis, but at any angle. So the problem is usually this one. You have your easy axis, and now you apply your field, h applied at an angle that in our model was called theta h. So this solution here is for theta h, which is equal to zero. And the second solution, this one, is for theta h equal to pi over two. Another question is what happens for all the other possible angles that you have between the two and the easy axis. And as a matter of fact, this cannot be solved easily in an analytical way. So the best thing is that we move now to MATLAB and we use now just this tool in order to solve all the other problems. So to do that, now let me move to MATLAB. So let me use this one and now I can show you Yalarki. So essentially this is just a very simple code which implements now the solution for the general problem. So it's a function which is a function of theta a. So without going into the details of the calculation, but okay, it's the minimization of the GL that you might have. The expression for the GL is exactly the Javis-Magnesi, 1-out sine squared of theta minus h-A micro sine of theta minus 1. Exactly the model that we have solved analytically. Okay? And then it's just a minimization. You can go to the code if you want, I can share this code with you. is a line of code, but okay, it calculates now the evolution. The idea is that for any angle, it minimizes the GL, and also look if the minimum or stationary point is really the minimum, if there is not another one. So otherwise, we can go to create, find the solution to the problem. But apart from this, which is just a matter of a few lines of code, now let's see what happens. So now in the code, at the end of the code, I ask to make a plot, okay, a plot of the applied field and the magnitude. So now if I just ask him to calculate what happens for theta h equal to zero, theta h equal to zero should be exactly this one, and the result should be visible here the one and you see the result is quite clear it's just a very small and this is the reduced field so the small age is H divided by the go back, go back, and you find the square. Now, just to check if things are reasonable, you can check what happens when you repeat for pi divided by 2, and you find something that is like this. okay definitely what you see is the anisotropy field minus one plus one and the linear v now what happens midway so what happens when you have now for instance a field which is slightly tilted with the easy axis so this is quite easy to see so let me check you can make them okay let's say that instead of five five over eight okay and we'll see what happens right now There is a sort of very very smooth transition corresponding to the flow rotation of the legalization. We are under the assumption of a single domain. There are no domains, there's just one. And all these things are repeating coherently. the coherent rotation. But what you see here is that, still to this critical field, when you see there is an abrupt transition, you have a gradual rotation of the magnitude. When you reach this field, there is a sudden rotation. And the mass that remains here is down to a field in a very, very small range of a volume. When you go back, of course, and you see that when you increase now the field let's see for instance I think you see that this is slightly please oh sorry yes okay you see it 5 by 4, and gradually you move towards what you expect for 93. Now, twice 5 is this. Gradually the load tends to close, and we'll reach this equation which is exactly this one. But last time someone asked me, I don't remember who among you asked me, but what is happening here, what is happening? I'm sure, very sudden rotation, which is quite a vertical part of the moon. And you can easily understand, you can figure out what is going on if you go back now to the energy profile. Okay, so let's see now, let's start now from something which is this first case, which is the easiest one for understanding what happens. Theta h is equal to zero, so we are applying the field exactly along the crazy axis. But let's have a look to the landscape, to the energy, and how the energy is modified when you change now the applied field. okay so the idea is this one so there's a function here that should be called energy profile a check the ers is energy so just to have a look energy profile is this it's very very simple it's just the plot of the gl you see the gl is one l sine of theta square minus h a cosine of theta which is exactly the expression for the gl when you have a theta h which is equal to zero and it's a plot for different values of the applied field so you want to see what happens for instance when you start for situation which is this one okay and you gradually think you feel towards positive field just to see what happens just to follow now what what is energetically the explanation for this is the original only you ask me this question yeah okay good so you are the response no no Okay, let's have a look. Energy profile for let's start from zero. What do you expect to see for zero? Which kind of energy profile do you expect to see? For H equal to zero. Which kind of energy landscape do you expect to see? At a constant. Don't forget that you are dealing with this equation. The GL now is 1L sine square of the angle theta minus HA cosine of theta. This is the expression that you have on the even normalized units of the dimension that units like. 1L sine square of theta minus HA by the cosine of theta. And so when you have that theta, sorry, HA is equal to zero, you just have the square of theta, which is the energy profile associated to the uniaxial and interoperable with what two equivalent minima corresponding to this state or the other state. they are totally degenerate in terms of energy. And in fact, this is exactly what you see. For zero, and for pi, and for 2 pi, you have a minimum. And your system can sit in one of these minimums. Either in zero, by using the equation of pi, or or in pi, or in pi. Okay? Do you see? This is the energy profile. The maximum, of course, is the maximum possible situation. Maximum of the reaction on isotopes when the magnetization is 90 degrees, especially in these gases. And then quickly, the two states, the Zn1 used for memory application with magnitude of the space and magnitude of the movement. There are two possible states. The degenerate dynamic does, as we have seen in part of the story. As soon as the height of this barrier is large enough, as compared with what? KBT, the system is stable. It doesn't jump in the other minimum. So the probability of jumping into the other state is proportional to the exponential function function of what? Minus is, it's so small that essentially it stays in one of the two limits. So no way. This is energy problem, we can't help, no way. And then you start doing what? So let's decide on which minimum we see as an initial point. For instance, if you measure the frequency, it is that you start from this point here. Here it is. Because this is essentially, if you want, this could be also the cosine of the angle theta. You take this one divided by the separation of the equation, so it is the equation of the cosine of the angle theta. So you start from here It correspond to the V part here. Okay, and now Let's increase number field remove this way and let's see what So what does it mean? So don't forget that one in my units correspond to The course is filled. So let's let's now go to One else. Okay I'll see how the energy profile is slightly changed. Okay, there's a big change. Big change is that now two minima are no more equivalent. They are no longer equivalent. But if I was sitting here and at this angle, I'm still in the minimum. There is still an activation energy to be overcome, to jump into the other absolute minimum. You see my point? As soon as you increase the field, the minimum in which you are is no longer the absolute minimum. But the relative is a local minimum. the system will stay there because there is an activation energy here which is much larger than kbt and so the system system stays there cannot escape or probably to escape is so low that in the end they stay there in a statistical sense but then what happens again so let's assume that you increase again let's move to 0.8 okay it's not more heavy there is no more the absolute minimum but still there is an inflection here it's still a minimum so if you imagine that you are working very low standard or imagine that again okay it is not so hard the system will It stays there for less time than in case of the great pronoun minimum, but it stays there. And then what happens at 1? When you move now at apply field, which is equal to 1, what is going to go? Okay, this is no more minimum. There is no minimum at all. So now the system goes naturally, falls down into the absolute minimum, because there is no more activation energy. But even at zero Kelvin, the system will jump into the absolute minimum. And this is exactly what happens here. It's a very abrupt transition, very sharp, sudden transition, because when you destroy So, why not the conditions with the system deterministically go into the absolute of the state. Okay? And then what happens if you increase again the field? Of course, if you increase and grade the field, so let's see what happens when you go to, I don't know, 1.5. So, we are just, okay, making more evident the fact that this is the wrong meaning. So it's a maximum, a maximum, and then you have the other state. But the state now is down here. It will stay here if you increase the field, and now we are moving on over there. OK? And now you go back. And now let's go to the system to go back. You decrease the field. with the crucifix you go back, which means go back to what? To 1, when you are seeing this transition. But okay, what happens? You are here. There's no reason why you could jump over there. It's a higher energy state, so the system will stay here. here, okay? And then what you do, you go to zero, you go to zero, and moving this way right now, you go to zero, but okay, in zero you see that there is another one, but hey, there's a minimum, which is not an equivalence, but you wish analysis, so I bet you stay here, and you safely stay here. And then you start decreasing the field. Minus 0.3. Here. No longer the absolute minimum. The absolute one now is this one. but there is still a huge activation energy to give a chunk. And this is the same mistake there, which means that you are here, you're staying here, okay? You're staying in the upstate of the system till you reach the other critical condition, which is that you find that minus one, At minus one you see that, okay, as before, this is no longer a minimum. And this difference, four, falls down to this minimum, which is what corresponds to this second transition system jumps here. Okay? Is it more clear now what is going on? This is the reason for hysteresis. So what is now the key ingredient for hysteresis, not the extract, is anisotopy. No anisotopy, no hysteresis at all. And indeed if you really work with very soft material like turmaloy, where the anisotopy constant is negligible, it's very, very hard to measure hysteresis. So, just to tell you, the machine for 80-inch weight is now very thick in terms of nickel-ion, and the compensators and converters are the recursive phase. The recursive phase is the use of less than one oyster. Essentially, in a reasonable range of applications, the device is based on the least self-material is no hysteresis at all, okay? Well, I understand, just to give you also something which makes sense in terms of experiment, is the resolution of our electromagnets. So you cannot go lower than when it's in terms of steps. Okay, another way for having a look to what is going on. This is the case of the hysteresis in this situation. So if you want to see what happens, there is another representation which is a 3D representation, which is this one. It's the landscape function of zero. Again, it's a landscape. Now you will see what is going on. So it's a plot in which you see that is a function, so it's the gl, it's a function of what? The angle between m and v is this, okay? Here the theta angle and the applied field, okay? So now we can have a look at the same story and And this is done, if I correctly remember, so let me have a look. So where is this gl function? Okay, this one. Okay. And what is the value? Okay. Okay. okay is exactly the GL for all possible things so now let's have a look to this one is very boring now for me so this is the angle and now let's see This is the function of the dimension. Okay, this was our representation. Okay. Now, exactly profile that we have seen This is for instance for What is happening? is function of the angle or look at this curve of the case of a negative applied field See there is just more just a minimum And you have two maxima, which means if you have a negative of point H, the unique possibility is to stay in the corresponding to the magnetic ratio point in darkness. And now, of course, when you move in this direction, you gradually change the story. Now let's have a look to what is happening from the top. So you realize that as soon as you increase the field, you are moving towards the situation in which your field is no longer there. And at some point you are in a situation in which there is no need to move, this will be pulled up either on one side or the other one, depending on the way you are sweeping your field. But what you see is that the magnetization doesn't move. This means that when I make a scan of the field, from very negative to very positive values here, the minima always stays at pi. It's always here, the minima. You are now decreasing the energy points to jump into the other one, but the minima are stable and are at pi. And at some point you start to see the appearance of another minimum at zero or two pi. But the position of this minimum is there. So you are not changing the magnetization. This means that you move here, but you do not change the cosine of the angle. So you see my point. So I move here, what I say, what I said so far. If you look now at the difference between this and the landscape, and that was only to the other situation, this one. When I now try to apply the field at 90 degrees, let's have a look how this energy landscape will change. So let me see this for pi divided by two. Okay, now again, Let's see if we can find something which makes sense. Yeah, sorry. Okay, now it's... This is quite an efficient way. Okay, as a function of the angle from different applied fields. And what you see is that in this equation here, there is just one possible minimum, Which is that for, I don't know, this is, it could be this one, this one. So pay attention now. What is this value? So don't forget that now we are applying the field at 90 degrees, okay? So when you have a very large negative field, very large negative field, this means, so let me sketch it. So we are in this situation here. This is the easy axis. We apply a field like this. For a large negative field like this, the angle between the magnetization and the easy axis would be minus pi divided by 2. Or this one, which is component to pi. And the other example is here. Okay? This is one beam. And then you have the other maximum, which corresponds to the other situation. And plus pi divided by 3 is exactly this one. It's 90 degrees. This is 3.14 degrees. You've got here. Okay? You see what is going on. And now you increase the field. the field. So you start from that situation with the magnetization point like this, very large negative field, and now you're decreasing magnitude of the field, which means we are increasing that component of the F-1 field. And what happens now? Let's have a look to this 2D representation. So you can probably, yeah, this is quite good representation of the story. Do you see that now the minimum is moving in this direction? You see, it's no more stable. The minimum of your function is moving towards what? Towards pi. Yeah, because you are here and gradually the magnetization moves like that. So it's no more in abrupt change. It's a gradual change which corresponds to this gradual rotation of the magnet. Okay, this is what is going on. And the different landscapes justify the idea of the historical radius loop. Did I answer your question right now? I hope so. Okay, and this is the historical radius in the Stone of Orphan model. Do not forget that when we talk about historical radius, we always refer to single, in this case, in the model, we always refer to single domain configuration. No domain mode, nothing, okay? Single domain configuration. Just to give you a number, which also is interesting in terms of the impact of my statement. If you take now iron, you take iron, FCC, no, BTC, standard configuration, and iron And you take the anisotropic constant and the exchange, and you find out in the saturation magnet system, you estimate the coerced field. You find something which is in the order of 250 years, but in practice, when you have a bin field or island, it actually grows on some region, for instance, MGO, and you measure the coercive field of this wind is low, do you know what you find? Depends on your capability. How good you are. If you are a very, very good blower, which means that you are able to blow a single crystal, the coercive field can be on the order of the third field 4.5 percent. Just to tell you, when I was a PhD student, my first supervisor was Franco. I was working on the polarimeter and I was growing my very beautiful fields of iron on top of MTO 10 by 10 mm. At that moment, I was not aware of micromag, there's nothing. It was just a PhD student, second year PhD It was very big, 10 mm by 10 mm and in my mind that system should stay in a single domain configuration Which is impossible and in fact So the accuracy field was so small So small that depending now on the sample I had now that the system was broken into two free domains and so on and each kind of information was In the end of the day that that this is This is right now the best system for building up a polarity. Then some companies, they could have a comp and they did it and so on. But at that time, I was, my idea was, okay, because it will be something in the order of, I don't know, 100 or, no, absolutely no, because you immediately have the main propagation of the main growth. This is no more valid and the growth is beautiful in the order of 4.5% growth. If you grow something to very, very high quality and our teams at that time, they were very, very high quality. Because it was nothing. Overcome the result. But not in this. Just just to make a domain more moving. There are no pinning sites. 4.5% just to stress the fact that the term Warfarin is very nice, very interesting also because it helps you to make some also some quantitative estimate with a model but this is not the end of the story. Okay. Now we can go back Well, we're like two and now I like to discuss with you something which is related to the super power magnet is problem Okay, sorry. Let's move now to the presentation mode So far I It with you, okay I I stressed many times this fact that a minimum is a stable minimum as soon as activation energy is small as compared with the thermal energy of the system. So every process that jumps into the other minimum is a statistical process, it's a room-wide, quantum statistic. So what I mean is that when you say, okay, I have a minimum, this is the minimum, but what is now the activation energy to jump into the absolute minimum? This activation energy is very, very high. The kinetics of this process will impede you to see the jump into the other minimum. you could take a very very good, a very big magnetic object. So the duration of the stability of one state would be so high that the system could stay in that state for thousands of years. I will not see the end of this state, but it is possible. So the statement is this one. So you have always to check the stability of the minimal as compared to the thermal energy. And this is very interesting, because now the question is what happens when the power here becomes comparable to the thermal energy. And there is a very, very simple thing to do in order to achieve this situation. Just shrink the dimension of your body Why very simple let's take a very simple system, which is a single domain So you are in the limit of the storm So exactly the theory that we have seen so far But now you have to ask yourself. What is now the landscape the Landscape of the energy of the chair is exactly the shape that we have seen Okay, but in terms of the absolute value for a body with the volume, don't forget that the reason for this gradient is always a result of it. So you should have what? K times square of the angle of theta. But it must be multiplied by the volume in order to get not the energy density, because K is an energy density. K is measured on the alpha meter cube. So you have to move now to the real energy. And to get the real energy of your body, you have to multiply now the energy density by the volume. What I mean is that in the air, the barrier part is K multiplied by V. This is just an energy density, K by sine square of the angle theta. But the real energy would be K by V by the sine square of the angle theta. So for a small particle, let's assume that you are talking about a spherical particle, I don't recall what meter you want, for a spherical particle, now the energy landscape due to an isotopy will be a sine squared with an amplitude given by volume, I'll say with an activation and it will be given by k by v, okay? 0, and here the value would be k by b. And this is the energy to be compared to kb, the force line caused by the temperature. And now you can have different scenarios to see. Here you can see that. One scenario is that depicted schematically here, which is corresponding to the possibility that now kbp much smaller than k by v, so that the term of fluctuation and of the real the producer jump with high probability, high probability system is moving a little bit what we in the surrounding. They, they are not enough to make system jumping into the other side. And we said we gave up the memory, the good memory. Indeed, the producer of memory, LU for each cell, which is the ratio between this minor height and KBD, is a factor influencing the stability of the memory. Typically, even the producer of mRNAs, they won't get the ratio he said at least larger than 5. When you have something which is larger than 5, taking into account that helping to do this conventionally, okay, so the stability in terms of time is good enough in order to use it, use this cell as a method. But otherwise, now if this ratio goes down, if you have now the k by b approaches now the k by d, what is going wrong? So you are moving towards a situation like this, this is a particle presentation, so now the thermal activation of the The jump makes the jump not impossible at all. And the system, what are you expecting that the system does? Jumps. From up to down, from down to up. In a way which is a statistical way, okay? It's not really, it's not a sinusoidal, okay, is not something which has characteristic frequency, okay, but is something which happens randomly, that randomly you will observe water jump from up to the outer line, which is thermoactive. And statistically, now you just refer to this kind of equation describing water. So this is the sort of residence time in one of the two minima. So it's a phenomenological expression which takes into account the Boltzmann, the Boltzmann probability for the transition. So the probability of transition, P of JAN, will be proportional to water. exponential function of what minus KB divided by KB by T is like that because now it's not the other I always this problem minus the bigger this one for me yes it is like this okay this is a transition probability so this is the activation value, the larger this ratio, the smaller the transition probability that you have. Now you can move from the transition probability to the residence time, the average residence time in one of the two minima before you see the system jumping into the other one. And so it is given by something like exponential function of plus KB divided by multiplied by the characteristic time of the input, tau zero, which is experimentally integral to the nanosecond, the value of one nanosecond. And now you understand this. The stability of the memory depends now on the comparison between this value and the requested duration for the storage of your information. So if you want to have a memory which keeps information for one year, so this means that this tau must be larger than one year. Okay. And this is exactly how you describe it. Now, historically, at some point, there has been a definition of the so called blocking temperature. What is now the definition of blocking temperature? It is the temperature below which you see in an experiment that the system, so the single particle, behaves like a single domain with the nice hysteresis loop without fluctuation. But now the question is, so this is not enough for the definition. We want to say, tell me please, which is your measurement time. If you have now a characteristic partition time or residence time, so clearly any kind of definition of working temperature is intrinsically related to the way you see, you measure that you're talking. What I say is there. Let's imagine that you have a combination. And now you have a characteristic residence time of the order of one millisecond. Now, if you measure an extended loop with a technique, the most typical required time for the measurement of a loop is one minute. In one minute, there are so many milliseconds that you do not see an average magnetization. You will always state the average between up, down, up and down. The system fluctuates, and the characteristic time of fluctuation is one millisecond. You are running a measurement where the integration time is one minute, so essentially you are just taking the average between up, down, up, down, up, down, with a frequency which one over one nanosecond, which means that we will never measure a magnetization, non-magnetization, nor the evolution of the magnetization. You see my point? Now, if you have an high-tech plane with the resolution of one nanosecond, you will have the opportunity to measure an hysteresis loop of that particle. You see my point? So everything is connected now to the measurement time. At the time of the definition of this dropping temperature, the characteristic time for the measurement of the increase of the group was in the order of 100 seconds, something like that. And this gave rise to the standard and widely accepted definition for the dropping temperature, which is based on this total. So you could say, let's assume now that your measurement time is on the order of 100 seconds. So what you can define the blocking temperature according to this equation. You say that the measurement time is equal to tau zero, the exponential function of k by v divided by kv, Tb, which is the blocking temperature. So the blocking temperature is the temperature at which the residence time is equal to 100 second. And if you take now this expression and you put here one and second, It turns out that in the end, this ratio corresponds to kV divided by kV by Tb is in the order of 25. So this is the conventional definition for the walking temperature. Here's the temperature for which the alpha, which is the activation energy k by V, divided by kV by Tv is equal to the value of T. That's the definition for the... Okay, according to your request, there will be no break today because I know that you have to move to the other class. Okay, now, this is just to understand what happens with volume, but K should be already well visible. This is just a representation, graphical representation. If you have a look to this expression, what is clear is that now you can play now with two things, either the K, you change now the anisotropy of your system, but if you are using the same material you can play with the volume. And it's immediately clear that as soon as you decrease the volume, you are decreasing also the working temperature. What I mean is that now if it's evident from here and this is just a different graphical representation of the story of the third particle. Take now the logarithm of tau, so it will be the logarithm of tau zero plus kv divided So it's a linear slope, and if you increase now... Sorry. If you increase the volume, you increase the slope, and so the unit has a higher intensity. But let me now build up a sort of phase diagram for a magnetic particle, just to clarify what do you expect to see when you start with a particle. So let's assume that we have now a particle radius, with a radius which is r, made of the material. It could be iron. So you fix the material. And now you ask yourself what happens when you increase the radius. kind of magnetic regime that you find, that you encounter when you increase down the radius. So this is zero. At some point, so for very, very small values of the radius, what do you expect to see? Oh, sorry, I didn't mention something which is very relevant. Here's the title. You didn't pay attention to the title, superphotonagly. Why superphotonagly? This regime in which you have the fluctuation of the magnetization around the earth, this goes superparamagnetically. In which sense, as we have seen a while, the macroscopic behavior of an ensemble of particles behaving like this is similar to that of a paramagnetic. And it's called superparamagnetic because the response in terms of susceptibility is very, very high. And we will come back to this just in a second. But just to tell you that this regime of fluctuation is called superparamagnetic regime. Now moving, let's move here just to appreciate this story of the change of the behavior with the volume. Now if you start with the very, very small particle, what you are seeing is that you are in a super-paramagnetic range. And then there is a characteristic transition radius, m1, at which you jump into mode. the radius is large enough so that the volume is large enough so that and this is a this is done at fixed temperature okay t equal to a constant at some point you are constant temperature you are increasing the volume you are increasing the height of the energy target and you jump from the super-pharmacognitive activity, the magnetic thing of the magnetic field. That corresponds to the story of water. You may use this note. You are familiar with the scene, but you are excited for the system. But this doesn't subsist up to infinite radius. At some point, something else happens. The characteristic radius of creation. You see another change in the degree of the system. Which one? The more simple domain. You see multi-domain separation, domain walls and so on. So for the example that I gave you related to my experience with PSE students, here you measure a quiescent field of 250 oesterns. Immediately after you measure something, you do a term. That depends now on the quality of your PSE. 10 oesterns, half oestern, and so on. So the radius really makes a difference. And this is one of the problems of any manufacturer of magnetic mass. On one side, you are asked by your customer to shrink now the cell size, because you want to have a very high density for back-end information, but as you are a magnetism guy, you know that you shrink your cell, you are moving towards the super-paramagnetic regime, so that you are reducing the instability of your memory. And so what is the other possibility? Because your customer is asking you for reducing the cell size. I saw the reply to you. But in general, it happens. I have to like a resolution. Yes, you can reduce the temperature. Yeah, okay. You can say, okay, don't don't don't want just to be inside the class. inside the cryostat. I knew you would do that. But the customer could be not very satisfied because the customer wants to put it inside the mobile phone. That would be a problem for the human cryostat. Ah, everything is written in blackboard. Yes. I failed the software that you cannot do because it's a memory. Otherwise you cancel the information. Turing K. Remember your method? Turing K. Turing K. If you can now change the material or change the properties of your material by increasing K, don't forget that the body height is a product of the volume by the anisotropic cost. So if you increase K, you would increase now the stability of your memory. That's why, in open memory, you always try to have material with very large k, if possible, compatible with the fact that you have also to write the information. So the custom web is demanding many, many things. We want also to write information with no power words and many other things. To increase now the stability, the key point is to increase the k. And now, yeah, this is exactly, no, this is not, it's another story, another diagram. What do you see over there? It's another kind of diagram. Here I was working at fixed temperature. It's the case of the memory that we have seen together. But there is another point of view, just better for one thing, okay? Now I have my cell, let's go from the, the component by the temperature, and let me play now with the temperature. And what you see is quite clear. You start now from very low temperature, system is born with the activation energy. So essentially you are playing with the, here I was playing with the numerator, now I'm playing with the denominator here. So if you start with very, very small temperature, you are in a block configuration, single domain. You increase the temperature, you overcome the blocking temperature, and you enter the super-thoramide ability. And at some point, you cross also the three temperatures. What happens there? Now, the thermal ventilation is able to recover the interaction, to destroy now the long range with the root-weight set into action, and you have to define the function. So don't forget that there is also this equation responding to what happens when you play with the denominator of that expression there. And now, the justification for the fact that it's called superparamagnetism. Now, let's imagine what happens when you have a system, an ensemble of particles, and the superpharmacognitive regime, which means that they are fluctuating from alpha to delta. And we apply a field. We apply a field and the physics is exactly the same. So the equations are exactly the same at the end of the paramagnet. When you have a system with two possible states and you apply the Serrano field. The Zeeman interaction promotes one of the two states, but okay, the system still tends to fluctuate, you have different population of different states. This is described in the end by an s-shaped function is that corresponding to, speaking to the large-band pool. So in the end of the day, if you define this is a mensualness parameter, which is given by the sigma energy divided by the thermal energy, x, which is mu zero m. m is the magnetic moment of a single dipole, of a single magnetic particle. So in this case, the individual particle is lifting like a single dipole, can have just two states due to anisotropy, up and down. This is at variance with the case of paramagnetism or spin-1-out, where it's really the quantum mechanics say that you can just move up and minus 1 out. In this case, is now the anisotropy telling me that you can have just up and down. But in the end of the day, you write down the expression for the statistics of this system, and exactly the same thing. And the dimension is going to be the ratio between the energy and the thermal energy. Now, the average component of the element along the z, which is the direction of application of the field, will be given by the n multiplied by the long range of management factor, which has this expression, the exponent of x minus 1 over x. But But when you are in a region of field which is close to zero, you can approximate now the function as a linear function of x so that you get a linear behavior, a linear response. So the magnetization of the ensemble of particles will be proportional to the applied field. And if you look now at the story, what is the cut? Don't forget the definition. So M, the magnetization in M-sperm is given by chi, the volume susceptibility, which is a dimensionless number, multiplied by H, again in M-sperm. Now this is the volume susceptibility, the unit volume. So you obtain now the volume susceptibility taking now the average value for the MZ of each dipole multiplied by N, which is the density per unit volume, number of dipoles per unit volume, which is the volume susceptibility to be placed in this expression, where M is volume density and H again as a field per unit volume. And then when you do this and you evaluate the chi as m divided by H as it starts from this equation here, you discover that it's now proportional to the square of M, where M is the magnetic moment of each individual molecule. And N is the volumetricity of particles that you have. Now, what is the point? The point is that M, in this case, is no longer the magnetic moment associated to each atom, like in case of paramagnetism, of each spin, is the magnetic moment of what? Of your spherical particle, iron, containing thousands of atoms. So thousands of 2.3 volt magnets. And what is the impact? susceptibility really increases in a way that really makes the difference. That's the reason for this phenomenon called superparamagnet. So what I mean is that if you now put a paramagnetic, conventional paramagnetic material inside your magnetometer and you measure now the m versus h, you find something like this. Usually you don't see the inflection point, you don't see the change of curvature and so on. You just see a linear view. So we are always close to the origin. But if you have a superparamagnetic material, if you really see the S-shape characteristic of the large fan, And the reason is that the slope here is much bigger. Why? Because the M is no longer the magnetic moment as a single atom or a single speed. It's not a single bone mass. It is, in case of iron, 2.2-bone magnitude by the number of atoms composing the small sphere that you have in front of you. So it's a big difference, a huge difference. So that for paramagnet, the susceptibility is in the order of 10 to the minus 4, 10 to the minus 3. For superparamagnet, we can have something in the order of 1. So 3, 4 order of magnet. That makes a big difference. It makes a difference. Okay, and that's also an interesting point of view for the story. I will come back to this in a while concerning the possible application of this. Yeah, this is something that we already discussed clearly. But the interesting part is that it shows the impact of different materials, which are small, different and is often with the detail. It's actually the kind of diagram corresponding to what I sketched here. In this case, it's something that's coordinated. So the idea is that, let's take now some practical experience, and let's see now which is the impact of the diameter on the area that you have. So let's start, for instance, with robots. In the case of covas, the superparamagnetic regime corresponding to this dashed area extends from 0 to 10 nanometers. And immediately after, you jump into this black area corresponding to the single domain. ESD stands for single domain. And you see, when you move to nickel, the size, say the extension, or the size of the particle below which he observed a superparamagnetic beam is much bigger. And what's the reason? The reason is very simple. The anisotropy of nickel is much smaller than of cobalt. For cobalt is 5 times 5. For nickel, he is in the order of 10 to 3. Now it depends on how you use your nickel fill and so on. So you see here you are decreasing the K, taking the volume. The product K by V, which makes a difference. And this is for the different particles of the situation that you have. For instance, copper platinum, which is a very hard material, so it's something that use for memory is something which is different. We have this for global platinum, for iron platinum you see that you can really push now the size down to just a few millimeters, because still in the same domain which is good for memory. Okay but so far we have seen the point of view of memory producer. okay, your customer is someone who is interested in building memory. Now the question is, is there any other possible application that could take advantage of superparamagnetic, not being afraid of the so-called superparamagnetic limit, which is that encountered for memory when we go to the more science. But is there any application in which this superparamagnetic it is some advantage and the question is okay there is there is and there's something where is an application which is typically connected to biological application probably you don't know that but every time you go to the hospital i'm sure that i mean that does and then there's no longer okay I choose the bad presentation. Every time you go to the hospital for a blood analysis, you are really taking advantage of the super-fine. Are you aware of that? No? Of course you cannot. You must go for a CT. You must simply attend my course in the UK. Next time you go to an orthopedist, you say, hey, now I know why I put this on. Anyway, it's the problem of sample preparation. When you go to the hospital, they take out of your body a 2-milliliter of blood, and they want to isolate now some different target molecule. I don't know, could be, I don't know. It's a target model. Like creating NINA, all these kind of models, and they want to really see what is going on. So the idea is that you have a vial like this with the blood, and inside you have your target. Let's assume that your target is given by this target here. How can you capture it? right now. And typically what you want to do is first concentrate this target molecule. Concentrate means that instead of having just one molecule per milliliter, you want to have thousands of molecules per milliliter depending on the specific product. The way you do that is that you use some probes. So this is the target. And now you design a probe, which is a complementary molecule, which is like that. This could be a body and this one could be an antibody in the language of biochemistry. Or you can talk about a single strand of DNA and the complementary strand of DNA. This is the probe. Okay, let's imagine that you put inside and what is exactly what is done on your board. They put in the vial some probes. They will stick on the target, so they feel it. It's something which is very, very, very good in terms of the chemical reaction. But now you want to concentrate. And the way it's done is that these probes are attached to some beads. Superparavagnetic beads. They are spherical particles containing many, many superparamagnetic particles. So what is a bead? The radius could be on the order of, I don't know, 200 nanometers, 1 micron. You can buy them. They are produced by some companies, so there are Dynabeats, so different kinds of beads. But the characteristic feature is that they behave as we have seen before, like this. They display a very high susceptibility in yoga one which means that when you put them in an external applied field They become magnetic and highly material. They develop a very large magnetic moment. Okay, but when you now reduce the field When you put the field is not equal to equal the magnetization goes down to zero. So you can activate the magnetic moment they have, or completely suppress it, just by playing with the instrument. Now usually you want to have a very large magnetic moment, and also a quite good size. The reason is that you have also to counteract the Bronyan motion. If you have a very small particle, this is not very effective, so the force that you can actually as to counteract a very intense agitation due to problem motion so with us you you like to use cobalt particles or something like that but if you play down with them and it will be completely useless they just they will just fluctuate due to program motion inside the total bed. So you have to increase now the volume, to increase now the force and decrease the fluctuation. And that's why you're using typically some small particles which are embedded. So these are the cobalt nanoparticles, for instance, embedded in a polymeric matrix. It could be dextran or other kinds of polymers and we have all these molecules embedded in this matrix and we have a spiritual being in the end. But the interesting story is that in this way, you keep the separation between the particles, which is fundamental, because if they touch, they will start interacting by their own energy or even by some kind of exchange depending on the touch. And so you are losing completely the superpharmacognitive regime. So you want to have all these moon nanoparticles in the superpharmacognitive regime and a bead with them embedded in a polymer in the midst. So that the bead behaves as you want. And now what is the interesting story is where If you have these probes attached to the beads, now you take the vial, you put it inside the magnet, and they will stick now to the magnet. But now if you reduce, so if you push away the magnets, they become essentially inert in terms of forces they exchange among them, them and they will disperse again into the soil. So it can concentrate, but when you deactivate it sooner or later, it will have some issues. No more clogging, no more clustering. And that's another point because this kind of beads, they are used also on the human body. But one of the fundamental requirements is that you cannot inject some magnetic particles in the human body if there is the risk of clusterization. The reason is very simple, because otherwise, the risk for clogging, creating now some obstruction could be negligent. So for this kind of problem. So clearly, this is something to be avoided. So the unique possibilities that have been injected to the body. And what happens is that, for instance, in the cancer treatment, we already discussed this, it's called hypertonia. You inject this particle. The particles, for instance, are attracted in the part of your body where imagine that I have a cancer here. So they inject in my body this particle and they place magnet in this area. The particle will be attracted by the stick here. And then instead of just a DC film, they apply now a AC film. The AC film creates water, creates water, the eating of the particle, because again, at some point, there will be some energy dissipated due to the fact that there is a tiny series of nodes and also there's a problem connected also to the fact that they have to move in. There is not just a reversal, there is also a sort of movement which creates a heating, and so cancer cells in the proximity of the region will be healed by this thermal process. This is called hyperplasia. But for this, you need superfana. The reason is just to tell you that superfana magnetism can be a problem for the user of memory, but can be a real advantage, It's a potentially useful thing for people working in biology. And with this, we have seen superparamagnetism, which is, I'll say, a well-known phenomenon, problem, but potentially pertinent. Now, I'd like to jump, if my computer wants to collaborate, to the next lecture, possibly, yeah. Which is word number five. And before we jump to this lecture, I'd just like to remind you that next week, next Tuesday, one week you we will start with the lecture dedicated to the uh micromagnetic simulation platform which is also with uber mag as a sort of environment in which you can now use a so this is something which is uh part of the course so for everybody is a normal lecture but it will be fundamental for people going for the choice of the project instead of the written test. Okay, and now I would like to introduce the general framework of micromagnetism, which is red-hued by the simulation platform like ODA. So just to tell you what we are planning to do in the next lecture. I'm just going to use now the form of this micro magnetism and then I will move to the dynamics of magnetism because so far we have seen such a static situation. Today we have been looking for the minima reached by the system under applications of the thermal field. But the interesting part of this theory is the response of the magnetic body to an R wave field. What is happening when you are creating some perturbations of spin waves and so on. And we have to see all these dynamic things before you start also seeing the own flat. Anyway, let's start from the general formalism of micromagnetism. When you have an extended body, It cannot be treated in a single domain approach because the size is larger than the critical size R2, for instance, in this picture here. It turns out that in the end you can have not single domain but multi-domain and you are really interested in understanding which is the geometrical arrangement of the local reduced magnetization. You want to know locally if the spins are pointing like this, like that, where you have domain rule, how that domain rule position depends on the shape of the body, on the field that you apply, on the indentation that you create in the structure, all these kind of things. So you have seen that example of the corner, like we have seen probably last time, with the pinning of the domain rule. So you want to be able to predict which are the stable position for the domain role and how they can be influenced by Some external parameter you can play with. So this is really the school. So this is the final target How can you do this? So the first point is that you have an extended body which is in a fairly small space here And you divide it into small volumes and the volume must be so small so that inside that volume you can treat the magnetization as uniform. And this is something that we have seen since the very beginning, something arising from the exchange interaction. If the size of the mole volume, the d tau, or delta v, which is appearing there, is smaller than the characteristic length that would correspond to the exchange length, then inside that volume we can consider that all the scales of power, the magnetization, is huge. And now, once you do that, luckily the magnetization will be always equal to what? Inside the volume in which all the skin your part the magnification is equal to the saturation magnetization Okay Operation magnetization is by definition the maximum volume density of language titles in iron 2.2 both for The pattern okay no more than this 1.7 megaamps per meter in the units of the international system of units. But okay, this is the maximum connection. If you shrink the cycle, if you have a power limit, it will be worth 2.2 bm divided by the cell of your cubic lattice. That's it. Now, let's assume that we divide our body with more cells. cells, this cell, so we have the saturation magnetization. So what makes sense now is just to consider the reduced magnetization. There's more M-R appearing there, which is the magnetization divided by the saturation magnetization. It's a unit vector. You are interested just in the orientation, the space of this unit vector. Which is the end of the story. Once you know the field f, it's done. The field, the vector field, once you know that distribution, that field, in any point of the space, you're good. Now let's write down the Landau-Friedrich. And then the other day our guideline is clear. We have to write down the expression go for the minimization The old software doesn't do exactly this Because it follows also the dynamic evolution of the game. So far we are not talking about dynamics Let's stay with what we know And let's write down the geometry. You see the different contributions. The first one is this exchange a divided by two gradient of mx squared plus gradient of my squared gradient of mz squared exchange anisotopy this f anisotopy the small f stands for density of course all these contributions are density but then we will integrate over the water And then, what is this? Magnetostatic energy. Minus mu zero divided by two, m dot hm. m dot hm. Magnetostatic energy. And the last term is the so-called Zeeman energy, or the term arising from the magnetic field. in the expression for the T3 energy for magnetic system. Minus with your ms m dot h by P. Okay, so the magnetizing field is this. And now what I'd like to see today, before we stop just in a few minutes, is the definition of some characteristic parameters that are defined once you start from this expression and you try to work out a dimensional message, in which you essentially take this one, and you divide this total energy by something which is associated to magnetostatic energy. Okay? So, what is the expression for magnetostatic energy in general? In general, you will write down something like this, E m is equal to minus mu zero divided by two h dot m in detail. But this h m is proportional to what? You can use now the demagnetized tensor, for instance. And so you're left with minus mu zero divided by two m plus n x x m x square plus n y y m y square. plus, okay, n d tau. So in the end of the day, this is a sample proportional to mu zero divided by two m squared. Okay. Multiply by the volume. Okay, just as a rough approximation, when you study can and just proportion to that, and this is the reason why now what you're trying to do is to take the. And you can identify museum. ms squared multiplied by b. Okay? So you are normalizing the Landau's gradient to something which is proportional to to the magnetostatic energy for your system. 0 ms squared multiplied by b. And when you do this, it turns out that you jump from the Capital GL, so not only to be nice to everyone. There is nothing exciting in this definition, but as in literature you find these parameters, They must be aware of the reason why they are used. And the reason is exactly this expression here. Now you have the small gl which is the capital gl divided by k plus i t. Your zero ms squared multiplied by b. Now starting from the expression, the original expression, you find find that moving now to something which is essentially due to exchange and relativity by the static interaction and the Riemann field, you divide down the different contributions and you write down the different contributions. You are left with this. Now, it appears this definition of the exchange language is coming from this. We take an exchange, so the g exchange would be the integral, it was what? a divided by 2, and then the gradient of mx squared plus, and so on and so forth. Okay? Now, we divide by what? 0, m squared by b. okay let's assume now that doesn't need to assume almost anything so anyway let's start here you have a and now mu zero ms squared divided by two So essentially you have A divided by mu zero ms squared, and you are left with this one half, which is right appearing here, by the gradient and so on. This one is called exchange length, where the vector u y is a direction like this. One over v, which is vector p and vector v, and I'll put it there. LX2 is square divided by 3, with the square, with bracket, with the size of the square, the gradient of MX and so on. And you continue like this, and you see the appearance of another constant in front here, on the anisotopy. Here you see the appearance of a dimensionless field, HM, which is water. see over there is the magnetostatic field, the magnetizing field normalized to the saturation magnetization and the applied field HAE which is exactly what we have seen, which is the applied field divided by the saturation magnetization. So you normalize all the fields to the saturation magnetization, you find an expression here which is called exchange length and you find a parameter k here whose definition is interesting. So the k, which is this k here, is the ratio between the anisotropy field because apart from the expression is h anisotropy divided by the saturation magnitude. And anisotopy is two times K divided by mu zero ms, which is divided by ms. So in the end, it's two times K divided by mu zero ms. So these are the parameters that are typically found in the table describing the properties of the magnetic material that we find in the textbook. This k, which is an anisotopy, anisotopy, it's a parameter giving the anisotopy strength, the exchange length L, and okay the field normalized to the separation magnetization, so it's not something that you would find, but especially this one, which is the anisotopy strength and the exchange length which is given by the square root of this a divided by mu zero ms square so these are found in literature as really the fingerprint of each one that we are also to use in some applications now so for today it's enough but tomorrow we will see what is now the physical meaning of these quantities and how they are going to describe the model. And that's all for today. Enjoy your evening and see you tomorrow. Thank you.