bert7.txt

 So good morning, everybody. And I hope that with the microphone, it will be better. Someone asked me to use the microphone, because when I move, this one doesn't capture so well the audio. Anyway, what is the problem of today's lecture? Essentially, we will start so that we continue the discussion of the micromagnetic approach. And then we move to dynamics. we start seeing the Gilbert equation and Brown and Gilbert equation. Just to remind you which was our final point last time. So using this expression for the GL, I was able to find some parameters, especially is it, this k parameter, which is now parameters which is a dimensionless parameter, which is the range between the anisotopy field and MS. So let's start seeing the meaning of this parameter. So if you have a material with a large anisotopy, which means with a large anisotopy field, So it is typically a good material for memory. So it's like cobalt platinum, some iron cobalt, iron boron. These materials, they display very large anisotropy field. For some iron cobalt, you can have five Tesla, and iron boron, five, two. Huge coercive field, huge anisotropy field. And this parameter is parameter telling you how the anisotropy energy compares to the magnetostatic energy cell, which is connected to the saturation. So it's a ratio of these things. So that you can have now different kind of material. If you have the ABCs much larger than one, so you are in a situation of a hard material. ABCs are, say, not so hard, which means much lower than one, okay, your soft material. So this means that here you can place, I don't know, cobalt, cobalt platinum, iron platinum, neodymium, iron boron, some more cobalt, and so on. And here, if you have to typically make a list of salt material, so the most used one is permaloy, which is this alloy here, is nickel 80, iron 20, which is also written like this, PY, permaloy. It's an alloy, and there are different alloys made of nickel and iron. Depending now on the relative security, you dramatically change the properties. And this peculiar ratio, 80-20 in mass or in atomic percentage, it doesn't change too much because the mass of nickel and iron, that's quite similar. So in this peculiar proportion here, you have a material with no magnetostriction. We didn't see so far magnetostriction, but it's a property of the material which responds to the application of strain with some anisotropy. So it's very soft. and there is another one which is widely used which could be for instance technologically relevant is an alloy for cobalt zirconium tantalum these are the two most widely used soft materials they are used for magnetic flux concentrators integrated transformers conduits for propagation of domain walls to different applications, but especially when you need a soft material. Especially these two, they are used also in integrated inductors. And in the Apple computer right now, Intel is placing some inductors with some magnetic core, which is something which makes a difference in terms of the inductance, of course, because the inductance of a coil is largely increased when you put a magnetic core in. side and the typical material which is used is cobalt zircon tata the susceptibility here could be very high so typically what is susceptibility of course when you approximate m equal chi by h so when you have a linear behavior which is before you reach saturation which means before the anisotopy field so when you have when you have a linear behavior don't forget that they will always display a low flat V so in the linear range you have now the stability and I here is only in the order of I don't know a few thousand few thousand so if I look at our material in polypharm this one is 6000 of permeability. With material which can be used for magnetic fluxes and trade cores, for magnetic inductors, many other things. And not only this, it can be used also for propagation of spin waves, for magnetic logic, many applications. Every time you need something which is soft, soft means the anisotropy can be neglected in practice in simulation when you run a simulation even in oomph including permalloy you set the anisotropy constant equal to zero because it's so small it's smaller than 10 to the minus 3 that you can 10 to the 3 sorry general permitter Q you can safely neglect it and this is a parameters which is widely used for characterizing now the kind of material that you have, soft or hard material. Memories, flash concentrators, media for propagation of solid like domain walls, skirmish, something like that. And now what about the exchange length? The exchange length is another parameter coming out from the normalization of the GL to this energy, mu0 ms squared by V, which is connected to the magnetostatic energy, the polar energy. And so the Lx is the square root of A divided by mu0 ms squared. What's the meaning? The physical meaning is here. It's the shorter length scalar on which the magnetization can be treated which means now the rotation by 180 degrees, so it's a full magnetization reversal, so it's a twist. If you consider just exchanging dipolar interactions, so without considering walls and isotopes. So this is a problem which is different from what we have seen in the case of block domain walls. in block walls we consider the early contribution of two energies, not dipolar energies, but anisotropic energy and the exchange. If you remember, if you go back to the lectures devoted to block walls and to the calculation of the width of the domain wall and also the energy of the domain wall, we carried out calculation to contribution exchange and anisotropy. So the idea is that you are considering a material like permalloy, in which the anisotopy is so small that you can more or less neglect it. And you just consider something which is extremely soft. It's the limit of a very, very soft material in which you just have exchange and a polar interaction. And this is the definition of the exchange length. Now, it's a relevant length because, for instance, when you have to run a simulation, using ohm, for mu marks, any kind of magnetic simulator, you have to divide your body Magnetic body into cells. Now if you want to use the machine of Micromagnet, you want to have a small cell that inside that cell only spin are parallel. So essentially this is a kind of Constraint that you have to set. So the size of the cell must be smaller than the range length. Okay but now we have to prove somehow that this is a good definition that this expression for the exchange length corresponds to this kind of definition of physical meaning and to do that something which is okay a sort of phenomenological story but this is the idea of the story and in the end you will understand that it makes sense you can just refer to what we did in case of calculation okay which is strange because my computer is still working but okay for some reason there is no interaction they lose the anyway so let's go back to our calculation for the block domain wall width if you remember we obtained a formula which was like this for the block wall delta w was pi by the square root of a divided by K which was the anisotropic constant for our system and in that case we were considering the uniancial anisotropy the origin was not clearly stated but was essentially magnetocrystalline anisotopy in this case okay now because we consider that lattice and a chain of hand atoms and so on now one could say okay but here we don't have magnetocrystalline anisotopy we have to consider magnetostatic anisotopy which is really strange I'm using this one, but if I move there, I don't know. I have no idea. Anyway, even without magnetocrystalline anisotropy, just because you have a body with a definite shape, you can have shape anisotropy. So when you take, for instance, this situation, which is a cylinder, so it clearly has an easy axis, which is along this length on the cylinder. So if you have now to characterize the anisotopy energy of this cylinder, you can use the description of the anisotopy in terms of the magnetizing tensor. And this is something which is reported here. When we try to write an expression for the magnetostatic energy for a body for which the magnetized tensor can be defined, we found that expression here, in which you find a pre-factor mu zero half ms squared, and then minus nx, but in this case it's negative because z is along this direction, and as the body's elongated in this direction, nz is smaller than nx. You remember that there is this inverse relation. The longer the body in that direction, the smaller the component of the demagnetized tensor. So in this case, you clearly have that nz minus nx is negative. So this is a negative pre-factor here, and this is the reason why you have a uniaxial anisotopy with an easy axis along the center of h. If you put here cosine squared of the angle theta, which comes from the definition of the magnetostatic energy. But now what you can say is that in the end, in this expression this part here mu0 divided by 2 ms squared and z minus nx plays the role of the anisotropic constant it plays exactly the same role of the k which appears here in the same role that you have so that you can say ok in some sense you can replace now in this expression for the domain role a k which is given by this kind of expression okay this prefactor here which is coming from the anisotopy related to the polar energy which is exactly the assumption of the calculation for the exchange length and when you do this you find something that will be pi divided by a and that's about the k the k will be okay mu zero mu zero ms square and so clearly and then something that will be okay but in terms of if you want n z minus n x that could be possible but this means that clearly this Delta wall will be proportional to apart from some pre factors okay that can now change also with the way you make this calculation. This is the normal magnitude of course. It's not a precise calculation. But this would be proportional to the square root of a divided by mu0 ms squared. All the other they are numbers that can vary depending on the shape and so on. But if you have to find dependency on some characteristic parameter, it will go like this. okay, the square root of a divided by mu zero m square, which is exactly the definition of the exchange length here. So what you can say is that if you just consider exchange interaction of dipolar energy, the minimum distance over which you can observe a twist of the magnetization is in the order of this, okay? Just to give you an example, if you run this calculation for permalink, You will discover that you find something in the order of 5 nanometers 5 nanometers, which means that typically if you want to run a simulation Which is really safe in terms of the fidelity to the real system. You should use some cells both sides It's smaller than 5 nanometers. In practice, this is extremely demanding in terms of computation power that you need So very often also in our paper, we run simulation with 10 nanometer size of cell, even 20. And of course, what you do is that you run, just run simulation with five, you check if with 10 you obtain the same, and then you leave with 10 just because of the computational power that you can have at your. Francis and so on. Okay, so that's the explanation for the definition of the exchange length that you have to take into account. This is exactly reported in the tables for each material. And this is relevant because when you have to set your simulation, okay, for each material, you take the exchange length and you set the cell, if possible, smaller than the exchange length. Other parameters that are used sometimes are a combination of things, but more or less they are just some definitions. So you can have, for instance, this one, LW, which is essentially the square root of A divided by 2 times K1, which is also this one. It's the domain wall with the block wall. So when you have the square root of A divided by K1, apart from the pi, which appears here, apart from the constant, but it's something which is connected to the anisotropy coming from magnetocrystalline anisotropy origin. So this is the block wall, L wall. and another parameter sometimes is this one which is Ld so depending now on the paper so you will find some different usage but okay if you have to really stick to relevant constant I'll say k which defines now if the material is soft or it is hard the exchange length and typically the block wall okay which is always given by something proportional to the square root of divided by K. And this is definitely enough, I'll say, for treating most of the problem. Then if there are other definitions, you just consider what is written in the beginning of the paper. And now, so I would like to see in a very crude approximation how you can use also some analytical treatment in order to make some estimates for the behavior of some objects. This is just an application of this dimensionless energy that we have found by dividing the GL capital GL by mu zero ms squared multiplied by v and it's a very simple example this one It's that of a sphere. It's taken from different pages and also some textbooks. So it's a sort of exercise, this one. Very crude approximation. Fermi style. Fermi style means like Fermi was used to do. The first lecture in the United States, you remember. So let's estimate the number of what? Young tuners. So this is physics. Let's make an estimate of this number. This is really the capability of modeling with crude approximation giving you the order of magnitude. The problem is very simple. Let's take a sphere. Let's consider the possible micro-initiated configuration that you can observe when you have a sphere of a given material. Okay, and the material is characterized by a uniaxial anisotropy that corresponds to an easy axis, which is along the z-direction. And now you can figure out that there could be a different configuration. The A is that of the single domain. The B is that of two domains, which means you have two halves of the sphere. One with magnetization pointing up, the other with magnetization pointing down. and a domain wall in between where you observe a rotation. And here, in this representation, you see that from the left with a magnetization point upwards, you move to the right with a rotation which is like this. So these arrows are the red one in my sketch. They represent the projection of the magnetization on this axis. So you see that at the center is just a point because the magnetization is pointing towards you. Okay. And then you see the gradual rotation. So it's a black wall, this one, by definition. Magnetization rotates in the plane of the wall. And then there could be another possibility, which is that of the curling. And the curling of the vortex, which is not an exotic situation. if you take for instance a disc a disc of permalloy when you have a critical when you become a critical size you will find exactly this curling the vortex it's something that we have used for instance in some papers that we published around 10 years ago for some specific application and it's used also for the properties of this vortex because this vortex can really display an apex and you can see the procession of this as a max of speed, so that's an interesting oscillating system that could be also useful for radio frequency application, computing application and so on so it's a configuration that can be found not really in the spheres they are used in magnetism, for instance there are some producers of spheres and they are really not cheap because the sphere of Yig has some interesting properties, for instance, the insensibility to variation of temperature, so it's widely used, but it's not so easy to find a good producer of Yig sphere. But, okay, in most cases, we are dealing with two-dimensional objects, especially if you're working with plant technology, producing something which is based on the silicon wave. Anyway, in this case, it's a sphere, like a Yig sphere, if you want. Now, the point is always the same. So you try to identify the range of parameters, especially here would be the radius. So in which you can find either A or B or C, or not only the range of R, the range of the radius, also the range of the result. So the problem is to understand in which region of the space of parameters you can have A or B or C. Okay. What are you planning to do? Essentially, you write down the GL and you try to see in which condition you have the minimum for GA or GB or GC. And this will identify now the range of parameters determined in that configuration. so you start from A, A is very simple because you say okay in principle the G is made of different contributions so the G the GL is given by the exchange contribution plus G due to anisotropy plus G due to magnetostatic energy which is GM probably A, why is it not J? Yeah. Okay, there is no Zeeman contribution here because there is no applied thing. This is a remnant configuration. So in this case, this is from case A, and you realize that in case A, there is no exchange contribution. The reason is very simple. All the spins are parallel. There is no extra exchange contribution to turn some tilt between the spin. And the anisotropy also energy will be 0 because you expect that in this case, the magnetization is aligned to the easy axis. So k1 sine squared of the angle theta is 0. There is no extra energy contribution due to some deviation from the easy axis. But you have some what? So GA, we stand for G in the case A, is equal to the magnetostatic energy. But you have a sphere. For this sphere, you know everything. You know that if the magnetization points up, the demagnetizing field points down, and the strength of the demagnetizing field will be minus 1 third m. So when you put this inside the expression for the magnetostatic energy, don't forget that. So gm will be essentially what? Minus mu 0 divided by 2. And then you will have Watt. So H demagnetizing by M. But this one is essentially Watt. So this unit vector is minus 1 third. So it's mu 0 divided by 2 by 1 divided by 3, which means 1 over 6. Don't forget that we are dealing with some dimensionless fields and magnetization. the reduced magnetization. So A is equal to minus one third. And so it's one over six, which is the contribution for a sphere. Are you okay, Sofia? Good. So this is a dimensionless energy, and it's a dimensionless energy density. It's an energy density, this one. so it's a sort of as in this configuration you have that everything is uniform so it's really coming from the definition of the gl that we did which is here okay is the capital gl divided by this so but okay you're assuming that you are in a uniform configuration and you divide an energy by an energy so that I have to correct my statement it's not a dimensionless energy density because if it is a dimensionless unit it is dimensionless so it's not a density it's not in units of volume okay but it's dimensionless and it refers to something which is exactly connected to something which is normalized to the volume so it's something which is quite uniform now you start from A where you have 1 over C and then you move to B in B you have a domain wall so which are now the energy contributions so clearly here you will add there the domain wall brings two contributions coming from exchange because in the domain wall there is some regular rotation So there is a tilt between nagel speed, and there will be also contribution due to isotopy. So clearly, they will give an extra energy contribution here. So you can start saying that in B, you can simply say that as you have a sort of closure domain configuration here, the magnetic energy will tend to be zero. so in a crude approximation I will put the gm equal to 0 but I will have to compute this two contribution arising especially from what? from the domain wall. Let's do this in a very very simple and crude approximation just to obtain an expression which gives you the dependency of the characteristic parameter so you start from this situation. Now you have to evaluate what? Don't forget what is the expression that is left here. So in principle, not in principle, say, this is the expression, okay, is Lx squared divided by 2 less than this, okay. Now, so if you want, let me write it here. Gx is Lx squared divided by 2 and then the gradient of mx squared plus the gradient of my squared plus the gradient of mz squared. Oh, sorry. It's the density that, in the end, must be integrated over the volume and then divided by the volume. okay that's the interesting story you integrate over the volume and then you divide by the volume okay now let's do it in a crude way in the case of our domain now what you can see is that here there is a rotation from in from the the left edge to the center which means over delta divided by 2 and its rotation by 90 degrees so if you look for instance at the variation of the mx delta mx here on the left is 0 and at the center it would be 1. so there is a variation of of mx so delta mx which is equal to 1 over delta divided by 2 okay so the gradient is 1 over delta divided by 2, but as you have the gradient of mx squared, you have this, 1 over delta divided by 2 squared, and this is the gradient of mx. For my, nothing happens because there is no component along the y, but you have a similar contribution for the gradient of mz, because you start from mz, which is equal to 1 on the left, and at the center, the magnetization points towards you, so mz would be 0, so the delta mz will be 1 and again this variation is over delta divided by 2, it gives you the same contribution so we have this factor 2, so in the end you have that this contribution within brackets is giving you essentially what? is giving that expression 8 divided by delta squared okay it's a very crude approximation but it's just to understand the third dependency and now you have to pay attention because this is just what is inside the integral so essentially what you are planning to do right now is to make the integral over the volume of the domain wall okay over the volume of the domain wall and then you will find what the total expression is. If you want to compare now what you find out now you have two possibilities either you multiply this one over six which is a density by the volume of this figure which is 4 divided by 3 pi r to the cube. Or you multiply now this one by the ratio between the volume of this and the volume of the sphere. It's the same story. If you work with this density like this, it's a sort of uniform story, so it's OK. And then you take this value, you multiply this by the volume of the wall, and you divide by this one. Or the other possibility, you multiply this by the volume of the sphere, and you multiply this by the volume of the domain wall. Do you see my point? It's not so easy, I understand. So let me end. OK. You have two options right now, OK? You can say that. You start from this expression here, and you say, this is just what? Now you have to integrate over d tau, and then divide it by the volume, okay? This is the definition that you have found before. Okay? So you integrate, where is it? Okay, here. You integrate the whole volume and you divide by the volume, okay? so what you can do now is exactly do this so you integrate over the volume but this term is 0 oh good morning you send me an email telling that you yeah but you recover in a fast way that's very good so if you follow this definition you make an integral of this and clearly this will give you a non 0 where where there is the domain wall and then divide by the wall volume of the system and so if you do this you will find that this will give rise to essentially 8 divided by delta delta square and then is 1 let me write some proportions 1 divided by 4 divided by 3 pi r to the cube and then you multiply by the volume of the wall that should be proportional to pi by r squared by delta. Okay? Do you see this? It's like a slice of your sphere around the equatorial plane. So it's pi r squared multiplied by the thickness of the slice. okay and when you do this you find that apart many factors and so on but you will find that the gb in your case so is exactly what's reported here the in the end apart from the pi and so on all these kind of things okay but in the end of the day it will be proportional to delta divided by R because this is R to the second and this is R to the third. Okay? So when you do this, you find that gx will be proportional to lx squared divided by 2 a divided by delta squared by delta divided by R. Okay? But it's really the expression giving you now the dependency of things. Okay. So in the end you find something which is proportional to that's better by 28 by Delta square by Delta divided by huh okay and this is the G the exchange now you have to consider also the contribution of the anisotropy energy and in this case you still have the rotation the sort of the block wall and again is the sort of K multiplied by the average sine square of theta divided by 2 and Again, you have this factor Delta divided by R that now should be clear It comes from this integral over the volume of the wall divided by the wall volume and then you are left with chi the average value of sinusoidal theta is 1 so you have chi divided by 4 2 by 2 here 4 and then delta divided by half so gm is equal to 0 as I discussed before because it's a flux closure configuration so it's almost negligible straight field and so in the end for the case B you have this expression so the the GB is 1 over 4 times LX squared divided by delta plus K divided by 4 but the question is what about delta and in a very similar way to what Bloch did you say okay let me find now the minimum for this expression by finding the minimum with respect to the delta and so it's easy so it's just a few forms of algebra so you find now the stationary point where you find that it's a minimum and you find that delta is equal to 4LW, which is, anyway, is 4 times LX divided by the square root of K. So different way for finding it, but okay, it's okay. In the end, when you put everything inside, then it's not just some algebra. I would not like to discuss too much. But you find this expression that the GA is 1 over 6. the GB is equal to this expression so two times LD divided by R which makes sense LD is one of the parameters but it's inversely proportional to the R, that's the important thing, this is the constant this is inversely proportional to the radius okay, and now you have the KC what is the KC is the curling and now here is a very oh my god it was blocked the curling so the idea is that the curling which is a vortex invading the wall sphere it's really a very nice situation in terms of what magnetostatic energy because this is the best situation the magnetization is always tangent to the surface and as it is tangent to the surface it doesn't produce any kind of stray field so the magnetostatic energy in this case will be minimum so it's a very nice configuration for minimizing magnetostatic energy and in terms of the other contribution of course here you will have some field on the spin so there will be some exchange contribution some It's not a big contribution and so on. So how can you evaluate it? It's a very crude way of doing it, but you can say okay in some sense This one can be viewed as a sort of expansion of the domain wall which invades the wall sphere So in this very crude but in the end also powerful approximation Just say okay let me take the expression that I found for B, and I will put inside that delta is equal to 2 times R. So I take the expression that I found before, which is this expression here, before minimization. And now instead of delta, I put delta equal to 2 times R. And when I do this, I find this expression here, which contains now an interesting story a term which goes like 1 over R squared plus a constant okay plus a constant and now let's see what happens this constant is a K which defines now the Arden's of material and so you have now two possibilities the first one is that of very soft material so that you can neglect this term here and now you're left with three different possibilities the case A single domain where the G is 1 over 6 which is this value here this line here is a constant doesn't depend in principle on the radius of the it's really the configuration the single domain and then you have the case B which is that of domain rule and the case B is 1 over half so is the green contribution here then you have the KC which is the case in which capital K here which is small but you have this dependency like one over R square so you get this line and if you go into the mathematics you will discover that these two curves the red and the green they have no point of contact which is a double point of confidence in the end, which is placed here, which is the practical implication, so which is now the physical method of this calculation. This is telling you that according now to this sort of plot of the energy as a function of the radius normalized to the exchange length, but okay it's a function of the radius in the end, you have different regimes. When you are below this first value here clearly the winner is the situation corresponding to this blue line okay so the system will stay in a single domain configuration the case a when you now increase the radius you overcome this point this crossing point clearly the more convenient state is that corresponding to the green one that correspond to the domain wall and this is not this is not unexpected because we have seen that due to the competition between volume and energy contribution and surface energy contribution, when you overcome a critical radius, you move from the single domain to the multi-domain configuration. But the interesting point is that in this case, there is also this critical radius here, R2, at which you see that the case C and D, they tend to be degenerate. What does it mean? It means that at this point, you can also find the curling, okay? You can also find the curling, the vortex configuration, which is also a possibility. And when you move towards, again, large value, again, it turns out that the domain wall seems to be the most stable. But this is the case for K, which is much smaller than 1, the capital K. so for a soft material, what about a hard material? The situation changes completely because now if the anisotropy becomes relevant, so this means that mathematically this curve here shifts up and up, so the C will be here, which means that it doesn't come into play any longer. It's a very high energy state, and you understand why, because the curling is extremely expensive in terms of anisotropy energy. If the anisotropy strength is very high, having this curling implies that in the wall sphere, you have the spin tilted with respect to the easy axis, and this has a very huge energy cost. So this is not affordable at all. So now the game is in between A and B, not between A, B, and C. And again, now is the usual one, which means below a critical radius, you have a single domain. When you overcome this critical radius, you jump into a multi-domain state, which is a two-domain state. It's a very crude approximation, this one. But if you go through some simulation, and you will see something during the informatics laboratory in the presentation, you will see when even in case of a dot you can find some trends that are really well described by this very good model and with this now this is the end for the first part of the lecture we take a break and then we start with the magnetization because this is now to make spins moving which means processing around the magnetic field let's take a break Okay, let's move to dynamics, which is good. Let's move to dynamics is a good expression. This lecture is devoted essentially to Brown and Gilbert equations, which are the solution of the micromagnetic problem in the static and dynamic condition. so let's start from the very beginning which is the brown equation so we go back now to this problem that you've already seen so the GL which is Landau free energy for the system that can be written as we did so far I'd just like to notice here that there is an additional contribution appearing here that was not present in previous slides which is something that we didn't see so far, but it's also very relevant. So here there is a contribution which is an integral over the surface of the body, and it contains now a surface energy density. When does it appear? It typically appears when you have, for instance, a multilayer, which is the basic of spectronics of magnetic devices. You have now a multilayer like that you have in the tunneling junction, cobalt iron boron, MgO, cobalt iron, or in the GMR or in any other kind of device, even in couple weight guide for Magnolias. So you have now an interface. And now when you create an interface, there are some very interesting phenomena arising that we will see like linear coupling, exchange bias, that introduce some additional energy terms, but what is relevant is that there are no more volume terms. They arise from the interaction at the interface, so they can be modeled in terms of a surface energy density, not as a volume energy density. And this means, for instance, that when you have a system made of, for instance, I don't know, a layer, and then interface with something else, the effect of the interface will go down and down as soon as you increase the volume of the body. The relative weight will change. You are not changing the surface, but you're changing the volume. So the volume energy contribution will overcome at some point the surface energy contribution. But the point is you have this surface energy contribution which can arise from different phenomena that we didn't see so far, but we will see them in a while. Because after seeing the dynamic of the magnetization, the bridge towards spintronics will be exactly this. So the investigation of interfacial energy contribution arising from the interface with other . Now, this is the functional to be minimized. And Brown, at some point, tried to solve the problem using a variational approach. So the message is you have this functional. Now we chart the condition for minimizing this functional. It's a functional in the sense that here is not a single variable problem. So now here the variable is the field, is the M field. M is a function of R. This is exactly what you are expected to find in a given condition of applied field for some peculiarity of your material shape and so on. Now the question is, which is the spatial configuration and the magnetization leading to the minimum RnGL, which is what is expected to be found after some irreversible process leading to the equilibrium configuration. Don't forget the thermodynamic approach that you are following right now. And so the approach by Brown, this one, is a variational approach that you have seen in your courses on mathematics. Okay, let's start from a ham. Let's suppose that you have equilibrium configuration. Okay, which is the equilibrium one and now You want so the zero one now you apply now a perturbation So you change slightly the MX by having this U which is a function of R Any function of R with a to the epsilon that will be used for the variational approach The same for the my it will be my 0 plus epsilon by V Which is another arbitrary function and for MZ of course, it's not a free component because in the end of the day MX square plus my square plus MZ square must be equal to 1 So in the end this relation holds true. So in the end you just have these two function and one variational parameter epsilon must be small of course and using now this variational approach, if you want to see exactly how it works, my suggestion is that you go to the Avroni textbook, pages 175, 177, and after three pages of calculation, partial derivatives, and so on, you find this result. So let's keep to the final result. So you have now that M is giving a minimum of the Landau free energy, provided that this equation holds true that m cross the effective field is equal to 0 at any point in each point, okay? So m is a function of r cross h is a function of r. Let me write it in a very explicit form. m cross h effective must be equal to 0. This is the Brown equation. And now, the point is that what does it mean? So let me first comment on the structure of this equation. This equation is something that you can understand. So from your knowledge about magnitudes, what you know is that when you have a field H and then you have a magnetic dipole like this which is now the mechanical action on this dipole it's a torque okay now the the torque, tau, will be equal to mu 0 mu cross H. So there is no torque acting on your dipole if m cross H is equal to 0. This is the condition for no mechanical action on your dipole. And this macroscopic physics is really the dipole corresponding to a loop of a wire, okay? A magnetic dipole, whatever the origin. And there could be also some force, but you have a force only if you have a gradient of the magnetic field. In absence of a gradient of the magnetic field, which means in a uniform external field, the unique action is given by that torque. And that torque is equal to 0, which is the equilibrium condition. The torque will produce what? A motion, typically a process. So the torque is equal to 0 only if this cross product is equal to 0. So what you find at the end of this variational principle is an equation which makes a lot of sense in terms of this classical analogy of the story, But with a big and huge difference that here the field is the applied field introduced by an electromagnet and so on. Here the effective field has different components, different origins that must be clearly outlined right now. the HF as you see is the applied field H is the plus the demagnetizing field that produced by magnetostatic charges but you have two additional contributions the first one what is this? the exchange Tifner divided by mu0 and then nabla square of M where the definition of this nabla square of M must be clarified because The nabla square applied to a vector. So it's a vector whose components are what? The nabla square of mx, okay? Nabla square of my, nabla square of mz. But when you see this, you can easily understand that the origin for this contribution to the effective field is? What? Exchange, okay? of course because in the G-algebra is the exchange so if you now find a very elegant condition for the equilibrium the magnetization is in a condition of equilibrium if also the exchange energy is minimized and in terms of M cross H the H effective must contain something arising from the exchange and this is something coming out from the variational calculation by Brown and you have another term which is this one where you have minus 1 divided by 0 ms the derivative of the anisotropy energy density with respect to a given 2m where you see this definition so the derivative with respect to m is made of is a vector where all three components are the derivative of f with respect to mx, my and mz this is anisotropy field so that the variational principle states that the condition is given by this locally M and the effective field must be parallel that's a condition giving you the torque equal to zero but the effective field contains both exchange, anisotropy and in addition of the applied field and the demagnetizing field, full contribution to be taken into account. But you understand now the story. Locally, everything is dictated by the value of this. But the value of this field is dictated also by the distribution of the magnetization. So it is a self-consistent problem. You have to find a condition, an arrangement of dm, so that locally, the effective field, which contains also the sum terms, depending now on the M, locally the effective field is parallel to M. So it's a self-consistency equation, this one. It's not that given something external, you find the M parallel to it. No, you have to find a self-consistent situation in which the M is always parallel to the effective field, which is also the sun of the M. Because the M appears here and also in the second one. And there are also some boundary conditions arising from the minimization and so on. But the main message is this one. So the Brown equation states that what is relevant is not just the external field. It's not just the external field plus the demagnetizing field coming from static charges. It's also something which takes into account the distribution of M, which creates some exchange interaction and also some anisotopy. Very famous result, but now what's next? In statics is like that. You just find the solution for the variational problem of Brown, which leads to this equation and you're okay. But you immediately realize that, OK, this is the static situation, the asymptotic situation, the equilibrium. But what happens if at some moment, in a dynamic vision of the problem, the AMP is luckily not aligned to the effective field? Recession. That's a unique possibility. So when you have a system like this, mu will process around the H. So physically, if you solve the problem, which is a mechanical problem, so it's a problem of the first year of your course here at Politecnico, the mu will process around the H. It's a torque. That's very relevant. So the torque doesn't push the mu to align to the H. Just causes the mu to process around the edge Is it clear? Because people say the alignment no, it's not the origin for the alarm Yeah, the origin for the alignment to age is the dissipation Not dissipation. There is no alignment Okay, now let's move to the dynamics and let's start from a single spin just to understand now the level of single spin. Why? Because in the end the magnetization of 3D for magnets like iron, nickel, and copper and so on is coming from what? Essentially it's the spin angular momentum which is giving you now the magnetic moment. It's not the orbital angular momentum because it's squenched for 3D materials. So it's the spin angular momentum which makes the difference. In an isolated atom, so for iron, you will have 6d electrons and in the end you will be expecting the isolated atom 4-bore magnetism okay arising from the spins where for the first spin which are uncoupled so they are aligned with the easy axis of your system in reality when you are in a solid so due to the interaction with a The lattice is 2.2-bore magnetism. But OK, the origin of the magnetic moment in these three different magnetisms is really the spin angular moment. So let's start with the spin. Now, if you have a spin, what happens? What is the equation first? What is the equation of motion that you are expected to use? This very simple expression here, which comes from the definition of the general magnetic ration. So essentially you say that mu, oh sorry, mu is equal to gamma multiplied by hair. Every time you have a angular momentum connected to a particle like spin or a charged particle, a charged particle you also have a magnetic dipole associated and there is a proportional factor which is the general magnetic ratio okay so clearly the equation of motion will be that this torque is equal to uh what is written there okay in terms of b okay within terms of b this could be mu cross B. So what you should say is that tau from this, the tau is equal to the derivative of L with respect to the time. But as L is mu divided by gamma, you find this equation here. The time derivative of mu is equal to gamma and general magnetic ratio mu cross B. This is a fundamental equation for the motion of the magnetic dipole in a field which is a field B. Okay, now the question is what is the value for the general magnetic ratio for the electron? And so this is a spin moment. So it's a spin moment. So you write the spin moment like this. One half G by mu B or mu B is the ball magnet which is this expression here. E h bar divided by 2 times the mass of the electrons. And the g is almost equal to 2. So that's essentially mu is roughly equal to the Bohr magneto, to the mu B. And now what is going on? Don't forget that the electron is negatively charged, which means that if you have the spin up, what does it mean, spin up? This means that when you take now the spin power operator, we will see in a while the spin power operator in more details. But essentially, if you take now the Sz eigenvalues and eigenvectors, so for the eigenvalues, you can find just plus h bar divided by 2 and minus h bar divided by 2. And then you have these two states, up and down, 1, 0, 0, 1, so in the representation of the spinors. Spin up means that the expectation value for the Sz will be plus h bar divided by 2. This is the angular momentum, in terms of the angular momentum, in terms of h bar. The mz, if the spin is up, so it will be negative because, okay, the spin angular momentum points up, m points down. We will see it in spintronics part of the course, so it's one of the basic ingredients. The spin is up, but the m is pointing down, okay? And so mz, you can write it as minus 1 half g mu b, minus 1 half g mu b, where mu B is E H bar times M E. Now, if you factorize things in order to see something which is M like this, or is it, oh my God, M proportional to the angular momentum, so this means that M must be proportional to H bar divided by 2, the angular momentum expectation value. And so you find that the pre-factor is minus G E divided by 2 times the mass of the electron, which is the gyromagnetic ratio for the electrons, minus G, the charge of the electron divided by 2 times the mass. Now, depending on the textbooks that you plan to use, now there are different definitions of the values. So sometimes you find that the gamma is written like a value without the minus sign in front. So the gamma can be either positive, negative. So there are many different options. So you have to, or the physics in order to recognize what is correct. But for this part, I'm using this text, which is the Getsla, fundamental of magnetism, which is described quite well, this part of dynamics of magnetization. In this case, for instance, it's written like gamma is written as minus gamma, where gamma is the magnitude, G by E, modulus of the charge of the electron divided by two times ME. OK? So but different possibility, different options. But what is clear is that this one is negative. And now you see which are the implications of the fact that this is negative. So moving now to this equation, you have this expression for single spin. And now you move to the magnetization. The way you move to the magnetization is very simple. If you start from iron, for instance, 2.2-bore magnet, then you have a density of atoms per unit volume. So if you multiply the single magnetic moment by density, you get what? The magnetization. The magnetization is, by definition, the magnetic moment through unit volume. So you multiply the magnetic moment of each spin by the density of the spin, and you find the magnetization. So you jump from the equation of motion for a single spin to the equation of motion for the magnetization locally, where you have that inside that cell, the spin are parallel. And so you jump from this equation to the equation for the magnetization. you keep exactly the same structure nothing changes apart from the multiplication by the density the end which is the density of magnetic dipoles per unit volume and you get this equation for the dynamics of the magnetization the M over d tau is gamma mu zero M cross HF yes why HF yes because if Brown equation tells out that the equilibrium conditions that M is parallel to H effective. So it's really, so the most easy thing to start with is that if the effective field, the field around which you will observe the dynamics, which means the perception, which means that if effective field is that, it's appearing in this equation, in which you see also the formalism of the torque and so on, it's a natural idea that the dynamics should be a dynamic governed by what? Not the external applied field, but the effective field, the same field appearing in the Brown equation, which is, again, in practice, you find that it's exactly like this. But it makes definitely sense. It's a very natural extension of the Brown. So you expect that the dynamics will be described by this equation, describing a precession, in which the field appearing here is the effective field, which is the sum of the external field, the demagnetizing field, the exchange field, and the anisotropy field. Okay? And now you introduce this expression for the gamma, so minus GE2 times the mass of the electron here inside. and you manipulate a little bit the thing and you see the appearance of this constant gamma zero so If you want now, let me write something that could be useful afterwards, which is G I went to see the doctor of Polytechno saying I'm not very well. No, you are okay. You're okay So I had the impression that the idea was now you are ready to go to war Don't say that you can't go to work. Anyway, I trust him. If he says that I'm OK, I'm OK. Even though I'm not very, very OK. So it's g, German iteration, charge of the electron, mu 0 divided by 2 times the mass of the electron. In this way, you are compensating everything inside this pre-factor that will appear in the equation of motion. so you can say that gamma zero is now this minus g e d okay mu zero is appearing bigger and then two times the mass of the electron and now the equation of motion becomes this one derivative of m with respect to time is equal to this gamma zero but now the point it was with the plus here minus gamma 0 m cross h effective okay first of all this is really a suggestion for you when you have to understand if it is plus minus and so on don't forget this story which is quite simple If you have like this, the effective field, and the magnetization is pointing like that, the procession takes place according to the right hand rule. The magnetization rotates like this. you put now this finger, the thumb, along the H, and the sense of rotation of the fingers of the right hand gives you the sense of precession of the spin. So the precession will take place like this. And is it correct? Yes, because when you say N cross HF will give you both. To get this sense of rotation, the torque must be like this. You need the minus sign. Okay? Are you okay with this? So this is a very simple sketch, but every time you are in doubt about the minus, the plus, and so on, Don't forget the right hand rule and it is okay. Good. So this is really the physics of what you expect to see is a precession. Now we can also probably have also an exercise that we can solve. And the solution is an undaunted precession for this equation. Yeah, the solution would be an undaunted precession. I have an exercise here probably. Yeah. Okay. we can solve this exercise okay without damping so forget about this story here so let's find a solution for this equation which we have over there minus gamma 0 and so on let's put a reference frame which is this one x, y, z you put the h effective along the z so h effective will be equal to h let me say h0 otherwise it's too long by the unit vector of the z axis okay and now you have to write down the equation for the procession and so essentially you're left with this dot product, so What does it mean? That the derivative of MX with respect to the time will be equal to what? So you have a determinant here Let me use this part of the blackboard, which is IJK then you have, okay, Mx, My, Mz, and then you have H0, 0, and H0, okay, because the effective field is just along the z, okay. so when you solve for this you are left with okay they I I which multiplies now My by h0 Okay minus 0 and then you have minus J or k minus J and then you have MX by h0 so here you have left the derivative of mx with respect to the time will be equal to minus gamma 0 and then you have my h0 and for the derivative of m y with respect to the time you have minus gamma 0 and then by the minus becomes plus m x by h 0 ok and then for the z the derivative of m z with respect to the time is equal to 0 This is the system of three equations that you have to solve. And this means essentially that Mz is constant, which is really in agreement with the precession. And then for this, you see that here there is something which is interesting because the realty of Mx is proportional to My. The realty of My is proportional to Mx. So something that you can do is that now you take the second, you take the derivative of this equation, which means that the second derivative of mx with respect to time is equal to minus gamma zero, the derivative of my with respect to the time, by h0. But now, okay, you have this one which is here. Okay? Now you replace this expression into that expression you find that the second derivative of mx with respect to the time sorry is here is equal to minus gamma zero square by mx by h0 not sure did I forgot something h0 squared h0 squared yeah I was looking for a square the dimensioning was wrong thank you okay so you are left with this equation here so when we have this equation so you recognize okay I know the solution to this one the solution to this sine or cosine of something okay so you just define omega like omega 0 the gamma 0 by h0 and you find the solution will be what? mx will be what? now it depends on the initial condition so it could be it depends now on the original angle that you have here let's assume that this is theta you will have what? that mz here will be equal to ms by the cosine of the angle theta. And mx will be what? Saturation magnetization by the sine of the angle theta. And then you will have a cosine of something. Cosine of omega t plus phi. Of course, if you look now for the solution of my, you will find something very similar with a different phi. and the unique possibility is that the phi will be, I know, 0 for mx and pi over 2 for my so that you describe now the precession. But the interesting point is that the precession frequency is this omega 0. Now you can evaluate what is the value for this omega 0 which is very interesting because it gives you a constant which is widely used in physics. So you can say, okay, good now what is the value the gamma 0 is 2.25 meter divided by amps divided by seconds and the gamma is this value here so now it depends on how you want to play with with the constant that sorry oh my god so what you can find is that the gamma zero which is written in this way it gives you this expression two point two ten to five now the problem is is that this is gamma zero I want to have something which is in a good way so this is radiant per second per Tesla okay let's start from this which is much more meaningful which means okay gamma zero Yeah, probably. But it's always the problem of the museum. Now pay attention a little bit to this story. Okay. The gamma is 1.7 10 to 11 radian per second. Okay? And the gamma zero is mu zero by gamma. It was written somewhere here. Okay? When you write that the omega zero is gamma zero H zero, let me write like this. So gamma mu zero H zero. Okay? mu 0 H 0 is expressed in Tesla in the end of the day is in units of Tesla because it's a B mu 0 by H so it turns out that if I write gamma 0 as F by 2 pi the frequency by 2 pi so the F 0 by 2 pi is equal to gamma by B0. Now, this means that F0 is equal to gamma divided by 2 pi by B0. So F0 by B0 is equal to gamma divided by 2 pi, which is 28 gigahertz per Tesla. if you take this number you divide it by 2 pi you get this expression here which is extremely relevant because it tells you that naturally you are in the giga regime that's also the reason why I'm working on radio frequency application naturally because when you are dealing with the dynamics of magnetization 1 T means 28 giga ok in this situation for a system of free electrons. 28 gigahertz per Tesla. OK? And this is just the solution of the problem of the precession. But this solution doesn't explain why physically, when you have an applied field like this and the magnetization like that, sooner or later, you see that the magnetization aligns to the applied field, which is not inside this equation. This equation just can't describe the procession, but physically you have something else. And what's the problem? The problem is that we are not considering something inside, which is now the damping of the magnetization. There are some relaxation mechanisms in the dynamics of magnetization In fact, for instance, you have seen in the block of equation for the magnetic resonance, which gave rise to the loss of some components of the angular momentum, of the magnetization, that in the end leads to the alignment of the magnetization to the externally applied field. But this is not included in the physics of this equation, which is just describing a uniform precession, a persistent precession without any kind of energy loss, without any kind of dissipation. But this is not physically correct. And that's the reason why at some point, Gilbert, after saying, OK, there is something wrong, introduced this phenomenological term in the equation of the precession, which really accounts for the alignment of the magnetization. It's a phenomenological term appearing here, you see instead that just minus gamma 0 m crossed the effective field Gilbert added this to minus eta the derivative of m with respect to time and now one could say why this phenomenologically it makes sense so if you wanted something which is connected to the idea of a dissipation coming from the viscous motion like the case of viscous force when there is a particle moving into a medium which is a viscous medium you have now a force which is proportional to the speed and there is something similar so the damping torque that you have because in the end this is the torque component the damping torque is something proportional to the rate of precession phenomenologically it makes sense and in the end it describes what is happening also physically and so the way you can understand what happens is this sketch here let's try to reconstruct it a little bit so that you can appreciate how it works now you go back to the physics of the procession and you say okay let's assume that we have a procession and let's see visually how the additional term of Gilbert adds up and gives rise to this tendency to align to the external field. Okay. This is H0, the external field or the affected field. Now, let me call H0, which is the magnet of the affected field. And now you have your magnetization, which is pointing like this. Okay. As we have seen, the torque coming from minus gamma 0, minus gamma mu 0, m cross h, the torque will be that promoting a procession like this. OK? A processional motion. So the torque will be acting like this. and minus M cross H. And it's gamma zero M cross H. Oh, sorry. Yeah, it's not minus gamma, it was gamma. Sorry. Yeah, so it's just not the best. minus gamma mu zero m cross h. Okay? This is the tool. But now you are adding up something which is on top of this minus so in the equation this means that the derivative of m with respect to the time is equal to minus m 0 m cross h effective minus theta, the derivative of m with respect to the time. So you have to consider now this, m cross minus theta derivative of m with respect to time. So the derivative of m with respect to time, what is this? So it's a vector which is tangent. to the trajectory of the tip. This is the derivative. And now you take m cross minus eta by this. Minus eta derivative of m with respect to time. m cross this one gives you a vector which is like this. And this is a vector, the torque, which is in the end a component of the time derivative of m moving with respect to time. So this brings the tip of the magnetization vector towards a0. What is the result? It's a precession that start to reduce the radius of precession till you will find something like that. Ending with the tip of the vector aligned exactly to H0. Phenomenologically, it makes sense. It's exactly what you expect to see. And also, the experiment tells you that things are going like this. because if you measure now with the pickup coil, what is going on? You can really record the signal which attenuates over time. You record something which is really coherent. So this means that your signal will be an oscillation that attenuates over time. And these oscillations are really connected to what? This could be the electromotive force inducing the coil, which is filling now the precession of the magnetization. if the amplitude of the procession decreases, also the amplitude of the signal will decrease. So it makes a lot of sense. It's not just something which comes from... So it's really responding to describing what is happening in practice in a real laboratory experiment. And now the problem... Sofia, you have a question. I made a mistake, yeah. You are right. What is the mistake that I made? Yeah, it's the other way around. probably let me check not sure so the air over DT is in this direction okay make sense okay okay well there is the minus here so that's a problem okay minus minus which means plus okay Probably the minus. What is wrong with minus? When we divide by zero, it's zero. No, no, no. I think this is correct, I think. What's your name? Sorry. It should be correct. Let me check it, but it should be correct. No, no, it's correct. So now I have to convince you. Sophia was correct if you don't consider this minus sign. because when you take the dot product M cross minus heater you will find something which is pointing like this but with the minus in front you you find something which is pointing toward the z-axis okay I'm pretty convinced that it is like that but if you are not convinced please go back and in case ask me I understand that with this minus sign minus gamma 0 but guys which is life I have no idea so the problem the gamma here is positive minus gamma 0 gamma 0 is minus gamma music so gamma 0 is gamma by mu 0 is a cost is a positive in this case okay i think it's because uh what's the problem is a mathematician said you love mathematics it actually contains the value of gamma zero so the eta doesn't contain it multiplied the gamma with you i know but it's expression in your lecture in your lecture. I made a mistake somewhere. You say, what is it? E times equal to alpha over gamma minus 0 times the n-magnon. No, no, after this. Yeah, it's not even possible. But I didn't comment that slide. So you are anticipating my lecture. So please. I didn't make a comment to that slide. Wait a little bit. If it's because something that you didn't comment, I say, well, please, just wait a little bit. So stick to what I told you so far. Everything should be consistent. Davide, minus by minus means plus, and cross eta by the time derivative is giving you something which is producing this vector here. So this is correct. If I'm not completely wrong, and it shouldn't be the case because I didn't drink any high cold before, it should be like that. Now, so let's move to this slide, which is the other slide, if you want. So just to comment something. This is the expression for the undone precession. You can write it like this, where this is the eta. But the eta is the value which is positive. It is alpha divided by gamma with 0m. This alpha is a very relevant parameter, which is called the damping parameter. It's a dimensionless parameter. It's easy to see that is a dimensionless parameter because gamma mu zero M is like gamma mu zero H So is a radian frequency is like the gamma mu zero H that we found here So it's a radian frequency H has the same dimension of M So at the denominator here you have a radian frequency, which means one divided by second This is the derivative of M divided by the time. So second divided by seconds OK, so you find exactly something which is coherent. So you have to find everything here should have the same dimension of h, same dimension of m. It's 1 divided by second. This is m divided by, sorry, this is 1 divided by 1 divided by second. This is m divided by second, second and second. They cancel out. And you find that everything here has the dimension of m, which has the same dimension of h. OK, so alpha is a number. is a dimensionless parameter. And we see something giving you the strength of the damping. The larger the alpha, the more relevant the damping. And in real materials, the damping coefficient is very, very relevant. The record that we have found so far is in Yig, in single crystal Yig, you can find in the bulk something in the order of six. 5, 10 to the minus 5. In the best film produced by liquid phase epitaxy, 2, 10 to the minus 4 for EG. OK? In good PLD film, 2, 10 to the minus 4, something like that. In cobalt hyaluron boron that we are currently using, unfortunately, 3, 10 to the minus 3. and what is the difference? the difference is that the persistence of the procession is strongly influenced by alpha the larger the alpha the faster the decay the faster the decay of the procession which is good if you want to reach this state in a short time but it's not good if you want to use now the dynamics of the magnetization for bringing a signal for making some analysis in the frequency domain and so on. So the damping is something which you want to minimize in case of a system which is supposed to carry some information connected to the gigahertz frequency that we naturally have as a result of the Gilbert equation. And for today, that's all. And see you on Thursday. Thank you. you are strange then there remains this the cut