 Well, maybe let's wait two minutes more, it's just 30 now, but I think that many times there are some people connected on streaming as well, so I would say that in like 2-3 minutes you can start. But the YouTube channel is already on. No reason. So Stefan, we take 5 minutes, so we start at 35, okay? Thanks. Okay, let's start. Welcome back to the Max School Quantum Espresso Mass School. Good afternoon. So it's a great pleasure for us to have this special lecture by Professor Stefan Brügel. Thank you, Professor, for accepting our invitation. So let me briefly introduce him. Professor Brügel obtained his PhD in physics in 1988 in Akan University, followed by a post-doc position at the University of Tokyo, Japan. Since 2002, he is director of the Institute of Solicited Research at the Department of Quantum Theory of Materials, and he is also professor for theoretical physics at Akan University. He received several prizes and honours such as the Friedrich Wallen Prize at Akan University, Einzmeier-Leibniz Prize for Excellent Affluent in Research Work. He is now leading an ERC theory grant about three-dimensional magnetization textures. So his research interests are mainly about computational physics in electronic properties of more real materials, in particular spin-orbit related phenomenon to the materials, topological insulators, rush by effects, all effects, skirmishments and many other topics. Professor Brügel is a worldwide recognized scientist in computational magnetism. As shown by his publication track record, just a few numbers here, he is also about 520 publications and many of them on Science and Nature Top Juniors, with H index of about 72 and a completely cetacean of about 22,000. So today, he will speak about ab initio-spin orbitronics, how the tini-spin orbit interaction saves energies in information processing. So I leave the world to Professor Brügel and I'm very sure that we will all enjoy his talk. So thank you very much. Yeah, thank you very much, Alessandro. I would like to start my talk thanking my chairperson, Alessandro, for the nice introduction of the Mac school for advanced materials in molecular modeling with quantum espresso for the invitation. It is my true pleasure to present this lecture, because I really think that you are the next generation in this world that carries on all the things into the future, at which many people of my generation and the previous generation spend their entire scientific life on, including Stefano Baroni, which you heard this morning, and myself, for example. It is a true pity that we cannot meet in person here at a wonderful place of Trieste. Yeah, I really enjoyed the Italian espresso, and I think it would be good to have a real Italian espresso after the quantum espresso. Wait a second, how to continue. I really think that you are the rather diverse community who is attending these lectures that are interested in many very cool problems. And maybe some of you have never heard about the word spin orbitronics before. But you maybe notice the word spin orbit coupling in this word spin orbitronics, and the spin orbitronics is a compound word actually developed by Ferd for spin orbit coupling and electronics. So the spin orbit couplings mean maybe you vaguely remember from your lectures at the university. The spin orbit interaction is the coupling of the spin degree of freedom to the orbital motion of the electron. And the orbital motion of the electron in a crystal means of course the crystal motion, the motion of the electrons in the crystal. So maybe you remember this form here of the spin orbit interaction, which is the coupling of the orbital moment to the spin moment. And I have written it in the radial symmetric form. You might have seen other forms, but this is the radial symmetric one and maybe from the atomic physics you remember this interaction best. You see here the velocity of light, the speed of light, and obviously it is a relativistic effect. And this tells you already that this is probably a very small effect. But maybe you don't consider this to be very important. Often this spin orbit coupling in the literature is simply abbreviated as SOC because spin orbit coupling to write it all the time and to speak it all the time is a little bit hard. So when you hear SOC, it is the spin orbit coupling. Maybe you have seen a key in the Espresso code or Espresso configuration file and you may be able to switch the spin orbit interaction, the key of spin orbit interaction on and off. And maybe you have switched it already on and you see hardly any difference between switching on and switching off. And this I have done for you. So I have calculated here a germanium without spin orbit coupling and germanium with spin orbit coupling. And if you look at it, it looks really like the same. But if you look a little bit careful with spin orbit interaction, you need good eyes. If you look a little bit careful, you see here, just at the valence band maximum or minimum, you see here that here you see basically suddenly a small spin splitting, sorry, a band splitting, which you have not seen here. So if you look very careful, you see here the empty band, you see here the valence band, and the valence band splits off into a split off band here and a heavy hole and light hole band. And now you may say, this might be 100 milliEV. This is nothing. No, that's not true, because 100 milliEV are already 1000 Kelvin. And if you remember that maybe many transport experiments are maybe done at 10 Kelvin or 70 Kelvin or at 200 Kelvin or at room temperature. On the level of these experiments 100 milliEV is a huge number. So we're dealing here with small numbers, but we're dealing here with numbers which are absolutely irrelevant. I would like to come to a second example. And if you do not understand everything what I'm saying today, it is no problem because next week, Alessandro, our chairman will give you an introductory lecture to the subject. And maybe I give you a little bit more an overview while he goes into the nitty gritty details and explains you all the details. So here I show you another example, which is the band structure of ferromagnetic iron. Iron is ferromagnetic. And maybe you remember, then you have a pair of bands which are basically the same, but shifted by a certain energy. So for example, you see here these black bands, when you see here the red bands, and you see here the band structure of these DX and DY states, DXY and DYZ states. When you see these states again here, these are the minority states and these are the majority states and between we have an energy difference, which is basically what we call the exchange splitting. Everything is super fine, everything is great. But now we switch on the spin orbit interaction. You know, if you look at it, first of all, it looks rather the same. You have here the bands here, you have here the bands here, you have here the bands here. But if you look a little bit more careful, you realize, yeah, okay, without spin orbit interaction, spin up and spin down states or eigenstates of the SZ operator. But with spin orbit interaction, that's not true anymore. Spin up states and spin down states can mix. And for example, you see here, you have here a state here here, an unoccupied state here, which has a crossing. This is just the state between the DX and DYZ. And here you see you have the same crossing. But the equivalent state here has a gap opening. And the same happens here, for example, here you have a crossing between the majority state and the minority state. And exactly at this crossing, you have here an opening, but you have no opening here. Please notice this band structure is plotted along the gamma H line and the gamma H3 line. This is this line from gamma H to gamma H3. These are busy. The crystals is identical. But what you do now is, you have a magnetic structure, and due to the spin orbit interaction, you translate the magnetic structure into the crystal structure due to the spin orbit interaction, and you break the symmetry. You break the symmetry of the crystal. Your crystal looks cubic, but in reality it becomes tetragonal. And what you see is then suddenly due to this magnetization direction, the electronic structure along this direction and the electronic structure along this direction is different. And you see this tiny gap opening. Now you say, what is this tiny gap opening? Who cares about this? The difference is due to the gap opening and to the fact that the easy exit that the magnetization direction has different band structures. It means also that the different magnetization directions have different energies. And there is suddenly one magnetization direction relative to the crystal lattice, which has a different energy than this than the other magnetization direction with respect to the magnetic to the crystal structure. This we call the magnetic anisotropy energy. The energy scale is about 0.5 to 1 milliEV, not EV, milliV. This is not a typo. So remember this morning, Stefano Baroni, Professor Baroni has talked about phonons. There's also a small energy scale, but this phonon energy scale might be in the order of 100 milliEV at a prion zone edge or 200 milliEV at a prion zone edge. We're talking about 0.05 or 50 microEV to 1 milliEV. And here you see a totally over exaggerated picture of the magnetic anisotropy. So you see that in this, the length here is the energy. You see in certain energy directions like the 1101 direction, this particular case, the energy is high. And in another direction, the energy is low. So this is the magnetic anisotropy energy. And now you say what is the importance of this very small number? This is a very important number because it is important for all electromotors. So a modern car like an e-car has about 200 to 300 engines, electrical engines in the car. So you want to move the window, you want to drive, you want to use the viper plates, you want to change the mirrors. Everything has motors. And these motors require a core, a magnetic core to save energy. And this magnetic core needs a certain material which has a certain magnetic anisotropy because it's magnetically hard material. The same material which you have in your headphones. And there is a huge interest in these materials and there is a huge activity worldwide to produce the right material for the right energy scale, the right temperature scale, environmentally appropriate. So this is a huge field, but I'm not talking about this for the day. I only want to make you aware that 50 milli-EV or 50 micro-EV or 1 milli-EV can change a total perspective of an entire industry. So this spin orbit coupling, maybe you never heard of, maybe you have never interested in, has fascinating realizations and ramifications in the solid. And has also attached to it are also fancy names of interesting people like the Rashbar effect, the Dresselhaus effect, the Edelstein effect, the Magneto-Electric effect. We talked about the magnetic anisotropy related to these are the orbital moments. We have a spin orbit torque. We have the Jalochinsky-Muria interaction. We have chiral magnets and skirmjons. We have all the Hall effects, the anomalous Hall effect, the spin Hall effect, the chiral Hall effect, the quantum Hall effect, the topological spin Hall effect. We have topological and churn insulators. We have spin relaxation mechanisms like Elliot-Yaffet and Dirk Neufberell. We have the Gilbert damping. So there is an entire zoo of phenomena which are in this way or that way of great relevance which hinge on the spin orbit interaction. And I cannot explain all this in my lecture, but I would like to make you aware that based on this spin orbit interaction, there is an entire field which we call the spin orbitronics. And for example, the spin orbit interaction is finally responsible for creating an interaction between the spins of atoms which is the Jalochinsky-Muria interaction. This Jalochinsky-Muria interaction can form chiral magnetic textures. One of them is the chiral magnetic skirmjons. It's a finite sized object of 10, 20, 30 nanometers and behaves like a particle. It's ultra robust and has a very efficient electrodynamics or very efficient dynamics. There is the hope that such an entity may be an entity for an information carrier with which you can store information. Then of course, this is very robust because it has a topological nature. It can be described by a topological, non-trivial topological invariant in the real space. But the spin orbit interactions are also responsible for materials which have topological invariance in the momentum space in the electronic structure. They produce, due to the surface-boundary correspondence, they produce edge states and surface states which are also topologically protected and therefore they are very robust and have a very small resistivity. They can be described by the topology in the momentum space. But there is also an effect which we call the spin orbit torque, which is described by the topology in a mixed space in the momentum space and in the spin space. And this spin orbit torque is a phenomenon which is, if you drive a current through your system, you produce a magnetic torque on these chiral spin textures, for example. It allows you, for example, to move this chiral spin texture with high speed. And you don't need for this an external magnetic field. So you can basically integrate that into a technology. And you need very little power. It is scalable to a device. And this allows you the manipulation of this squirmion. It allows you, this topological materials allows you to control the detection of the squirmion. You can count certain squirmions which go through a wire. And at the end of the day, you can build up an entire technology with low power consumption, topological robustness. You can even go into Majorana fermions and quantum computing. So this is the big scope of the spin orbitronics. And in this lecture here, I will pick out a few things to make you give you a feeling for the field without going into all the details. Spin orbitronics fits into a technology based on magnetism, which already exists. And maybe you know that I would say 90% of all information worldwide is stored on magnetic hard disks. It is cheap. It has a high capacity. It's extremely reliable. And it is capable of further developments. And without this hard disk, all this cloud computing would not be possible because the hard disk is where all these big data centers are. They are using this hard disk. Of course, not these small ones, but use hard disk farms. And if you look at the basically the ship capacity in exabytes, you see that the hard disk, which is this blue section here is basically, it is still expected to grow. And what you also see, it's a huge fraction that all the information is stored on this hard disk. And what we realize is there's also a Moore's law for digital data. The doubling of the data of the created data increases every two years. And therefore, even now, you have already 10% of the worldwide electricity use goes for information technology. And if you think about computing, you probably realize also that computing takes quite a amount of energy. And the size of the computers, which we can afford are actually limited by the electric power. So there is a strong motivation to develop a technology where we can take data from a memory into a compute unit or store data on a device with much less energy. And therefore, we need novel paradigms for energy efficient memory and computing. And these scumions, which I've talked about, they might be a path to this disruptive change in the information technology. So how could you think about a scumion? Think about a ferromagnetic state. So this red thing is a ferromagnetic state. And on this ferromagnetic state, think of this ferromagnetic state like an ice hockey field. Of course, not everybody in the world, you have seen an ice hockey field, but maybe on TV. So you have this ice hockey field. And in this ice hockey field, you have an ice hockey puck, which is a hard rubber disk. And this hard rubber disk resembles this scumion. And if you have to spin up a torque, they run around here on this ice hockey field. But these scumions are not arbitrary magnetic structures. No, they are topologically protected magnetic structures. And what happens is that each magnetization points in a unique magnetization direction. And you can put now these scumions on basically on a memory, which is not a hard disk memory, but a racetrack memory. And then you can save a normal, yeah, you can use that type of registered type memory to save enormous amount of energy. And the huge unique properties of the magnetic scumions is that you have nano-sized particles which guarantee high data density. They are topologically protected. They provide thermal stability. And they are very fast. This enables high data flows. And you can also use the two dimensionality because you can not only go in one direction. But you may go in various directions. And so these properties give you an idea of a new generation of magnetic memory, which is the solid state memory. So no mechanical parts are moving. And therefore it would be a low power memory. And you can also do programming logic with this memory. So you can reconfigure the memory to have different OR and XOR structures. You can do that massively parallel in memory computing. And what is a great advantage is you can also do it. You can also use this object for neuromorphic computing, which guarantees low power consumption. And it is a scalable architecture. So there's a lot of future into this. But now before I get carried away by all these possibilities, which we have in the future, I come back a little bit more to the basics. I remind you if we have a non-magnetic state without spin orbit interaction, you may have a band structure of a certain band on a certain K vector. And these are typically block states. And these block states are too for degenerate because we have a spin up and a spin down state with the same energy. But if you switch on now the spin orbit interaction in your code, then you will realize that this spin up and spin down are actually not good quantum numbers. Instead due to the spin orbit interaction, you can mix the spin up and spin down states. But of course the spin orbit interaction is small. So we call this one state spin up because most of this state is spin up, although there's a little bit spin down. And we call a spin down because most of it is spin down and there's only a little bit spin up. And so basically this factor A is large and this factor B is small. And this is called the Elliot-Jaffet parameter. And actually since the spin orbit interaction is small quantity, you feel you can do perturbation theory. And what enters in this perturbation theory is the spin orbit interaction. And this of course you remember that you have this sigma, these poly matrices. And you have off diagonal elements of the poly matrices which are responsible for the spin flip matrix elements. So you have a spin here, maybe a spin up can flip into a spin down due to these off diagonal elements of the spin orbit matrix. And in some instances you have crazy Fermi surfaces. And in some instances you have so-called hotspots where this denominator becomes very small and this pre-factor here B becomes suddenly very large, maybe 50%. And this determines the lifetime of spins. So if you imagine you have a solid here, for example, osmium and then you have a spin current. Somebody externally produces a spin current for you. For example, you run the current to a magnetic materials, then you have a spin current. And you put the spin current into the solid. Then the spin orientation matters. And you see that in some areas if you inject the spin like this, in some areas you have so-called spin hotspots where this B becomes very large. And if you integrate now all this B over the Fermi surface, you get the inverse of the spin lifetime. And I'm saying in this configuration the spin lifetime is roughly 1 over 20. But if you have now another configuration, for example, you shoot in your spin in a different configuration with a different magnetization axis to have a different spin current, then the same Fermi surface will produce at other places a spin flip hot area. And then you can do it the same. You integrate now this Fermi surface, which is the same Fermi surface, but the matrix elements are different on this Fermi surface. And then you see suddenly your lifetime has changed. So the inverse of the lifetime is not 20, but here the lifetime is 30, so 13. So it is much, much smaller and therefore you have a giant anisotropy in the lifetime of a spin current. And this you can now think of a Gedankinen experiment or a real experiment. So imagine you have here a magnetic material and this magnetic material has the spin polarization pointing up. Now you run a current through that system. And the current through that system, electrical current goes through that system and becomes spin polarized. And now you have a spin polarized current entering a non-magnetic material. For example, this osmium. And then you have a spin current in this circle. But parallel to this you have a diffusive current which diffuses into your solid. And now you can detect this current by your detector. So what you do is you measure the voltage difference as function of the position. And this I basically have done for you here. And you get such a curve because you get a typical diffusion behavior and the diffusion length depends on your spin lifetime. And now you can do the same for a material with different direction. And what you will see is that you get a different lifetime. So that means basically you can polarize spins. You have to get a spin diffusion which is different for different directions of the spin current or polarization at the spin current, I should say. So this was one example. But let me come back to the importance of spin orbit coupling in relation to the inversion of the spin inversion symmetry. So imagine you have spin inversion symmetry. Space inversion symmetry. Then you can show that these two wave functions are corresponding to states. These are block wave functions corresponding to states which are degenerate. This is this particular case. How do you approve this? You take this wave function here. You apply time reversal symmetry. That means you have to take this poly-matrix sigma times minus i times the conjugate times the conjugate. And what it does for you is your time inversion means k goes to minus k. And you also have to flip the spin. Time inversion means spin flip. You go from here to here and you go from here to here. And now you remember that you have space inversion symmetry. You go from minus from k to minus k. And you're ending up at this block vector again. And you end up in the wave function with opposite spin. So these two are totally degenerate. But the energy, the state is totally degenerate. So what I have shown you is that for time reversal and space inversion symmetry, the energy of the k point spin up and the energy of the k point spin down is the same. But if you break the space inversion symmetry, then it is not anymore the case. Because then you have time reversal symmetry on it. And the energy of k spin up is the same as the energy of minus k spin down. But the energy of the eigen energy of k spin up is not the same as the eigen energy of k spin down. So that means if you are now looking at a particular k point, the degeneracy of spin up and spin down states is lifted. What does it mean? It means this electron, maybe this electron might have never heard about the spin orbit interaction. But what this electron understands is that its counterpart, the spin down electron or the spin up electron, has not the same energy for the same k point. In other words, the spin orbit interaction produces locally a k dependent magnetic field, a k dependent magnetic field omega of k. And that is very important to realize that if you have a broken space inversion symmetry, your spin orbit interaction has the feature of a local magnetic field in your solid. And this local magnetic field is very large. So a typical example of a broken space inversion symmetry symmetric situation is the surface. On the surface, you have electrons running around the surface, maybe they are delocalized. They are wonderfully characterized by the k parallel wave factors. Everything is fine. But I would like to remind you that we have in this surface, of course we have our Coulomb potential, one over our potential and exchange correlation potential and electron interaction. But we have these basically symmetric potentials. But if you come closer to the surface, you have this symmetric potential is broken. What you have is a finite gradient of this potential. And this finite gradient produces a gradient of a potential is something like an electric field. And this electric field in the rest frame of these electrons, we don't think about this electric field in the rest frame of these electrons. This electric field is Lorentz transform, it becomes a B field. And this B field interacts with these spins of the electrons. And this produces besides the original interaction of the electrons, an interaction which is has this notion here. It is a spin orbit interaction. The effect is called the rush by effect. And basically you what you see here, you couple the spin degree of freedom to this time, not to the orbital motion, but to the kinetic motion of the electrons. And this is also a spin orbit action. And this parameter here is called the rush by constant. And this rush by constant depends on the spin orbit strength, the asymmetry of the wave functions, the orbitals and the essence of S, whether it's S or P or D and so on. And of course our motivation to understand the strength of this rush by effect. And here I show you now an example. This is basically the rush by effect on the gold surface, gold 111. You see here the bulk states, which are totally unimportant. You see the famous L gap of this particular surface. And what you see is here the surface states and you see that the surface state, the degeneracy of the surface state is lifted, except for K equals zero. And here is the corresponding experiment to it. I think it's from Reinhardt. It's a photo emission experiment. And please notice the scale here and the scale here is slightly different. So basically I rescale here this experiment. And in photo emission, you can only see occupied states. So you see here it is a good comparison between theory experiment. And if you look a little bit more with a zoom into it, you see you have here a degeneracy at each K point. The degeneracy is lifted. You see here you have one, you have a relation between the spin and the momentum. Here you have the spin up states. Here you have to spin down states. And here it's exactly vice versa because this here is the spin down state and this here is the spin up state. So basically you have a spin momentum locking. Now this spin momentum locking has great consequences. I should say first, so where is actually this rush bar strength coming from? Now I think if you look a little bit more carefully on the surface, of course in reality you have atoms on the surface. Not only homogenous electron gas where everything moves. You have atoms on the surface. Or maybe we can describe these atoms by spheres. You call this a muffin tin sphere. And in this sphere, you have this one over R potential here and other contributions to exchange and correlation and so on. But you remember that this spin orbit interaction is somehow related by the strength proportional to the derivative of the potential with respect to the radius. Now you should think the following. Your wave function has a tail and has a head. The tail is always out there in the homogenous electron gas as doing it as a block wave vector is propagating. But it's also a head. And his head feels this one over R potential and he has strong oscillation here. It means it feels the depth of the potential. It's running around like crazy and producing a heavy mass and producing this spin orbit action. And sometimes this electrons is far away with a certain probability in this homogenous electron gas picture. And some probability it is in this tail where it sees the strong spin orbit interaction. And if you put this all together, you can ask yourself where in the hell is the spin orbit interaction created? And then you see 99% of this spin orbit interaction comes from a very small region around the atom. And I have plotted this here. As you see here, a typical sphere radius might be 2.5 atomic units. And you see 99% or close to 90% or 99% you have already at 0.3 atomic units of the sphere. So the radius might be 10%, the volume might be, I don't know, 1,000. This you should keep in mind. And then the next question is, I told you it depends somehow on the gradient of the potential. That is true, but it depends also on the asymmetry of the wave function because at the end you have to calculate the matrix elements. And therefore you see the contribution of this rush bar parameter here or of this spin orbit strength or maybe better the rush bar parameter. You see 60% comes from the surface atoms, but the other 40% come basically from deeper lying atoms. So you have some depth profile which contributes to the rush bar effect. So why I telling you all this on the rush bar effect? Because it has a strong implication to magnetic interactions. And this I would like to discuss with you now. This is the Gialoginsky-Moria interaction. Gialoginsky-Moria interaction, I would like to discuss now what happens is, imagine you have two atoms on the surface. And if you have two atoms on the surface and you have your conduction electrons, the atoms are magnetic, they have a local magnetic moment. The electrons come, the electrons scatter on these electrons, sorry on these atoms with this spin. The electrons propagate to the other side, to this second atom and maybe they propagate back. And this propagating to one side and propagating back this causes an interaction and this we call the Heisenberg interaction. And this you can formally describe something like the interaction is the conduction electron scatters and it's propagated, it scatters, it's propagating back and this leads to the interaction. But if I have now rush bar electrons, then I get an additional term here and this rush bar electrons due to the fact that this rush bar electrons have a certain behavior that leads to the spin orbit interaction, produce an additional term here, which is this Jalogen-Skimorie interaction. And let me remind you, the symmetry breaking of this at the surface produces this rush bar electrons and therefore these propagating electrons have not only a term which comes from the ordinary electrons, it has also a contribution which comes actually from the spin orbit interaction of the rush bar electrons. And so you have here these greens and we call this the greens function. You have here the greens function twice and the spin orbit interaction twice, spin orbit interaction is small, the square of the spin orbit interaction is even smaller. So that means, forget it, you only state the linear terms in the spin orbit interaction. I can write out the linear terms in the spin orbit interaction. I remind you at the fermionic rules of Pauli matrices. So sigma times A and sigma times B gives you AB and sigma cross B. And if you put that in, you see you get this term here, but this looks like this term, but you also see you have a propagating term from one to two and from two to one and they don't cancel because propagation of a K vector means propagation of a minus K vector and the spin is down and therefore it adds up. And this gives you exactly the remaining contribution here. And we cannot only also get this algebraic form of the gelatin scheme orbit interaction. You can also get a description of the gelatin scheme over here in terms of the electronic structure. So what I have shown you is there's been orbit interaction in combination with the structure inversion symmetry produces a new interaction and this new interaction we call a chiral interaction. Why? Because if you look at the magnetic of this pair of atoms with this particular magnetic structure, you see that the cross product of this points in the y direction. So maybe this is the direction towards you whereas if you rotate, so here the spin structure is rotating clockwise, but here the spin structure is rotating counterclockwise and counterclockwise and this cross product maybe points into my direction. And what you see is if you rotate left and if you rotate right has a different energy and the difference in the energy depends on the sign of this gelatin scheme of your strength and the sign of the gelatin scheme of your strength depends on the details of the electronic structure. So I have showed you that the break of the inversion symmetry is a path to chiral magnetism and of course we had to learn a lot about this strength of the gelatin scheme of your interaction as function of the electronic structure and so on. And this gelatin scheme of your interaction is really the path to chiral magnetism which you can see for example that ground states the magnetic ground states can become a spiral because if the chiral interaction is strong enough the ferromagnetic state is not any more stable and you have a spiral which spirals only in one handedness in this case for example counterclockwise whereas the mirror image where the magnetization spirals clockwise does not exist in the experimental proof of this situation. It is also responsible for these are one dimensional magnetic structures it is also responsible for the two dimensional magnetic structures and here you see an example of a skirmjorn which is calculated and measured experimentally. It is also responsible for chiral domain walls these chiral domain walls are extremely stable because they don't flip to the energetically equivalent one and therefore you can move it with high speed it is also responsible for the magnetism it also changes the magnet dispersion basically you have a non-reciprocity in the magnet dispersion a right rotating magnet and a left rotating magnet has a different dispersion. So now I'm looking a little bit at the time but maybe we have still 10 minutes we started a little bit later so let me say a word about this skirmjorn simulation in the description. So I have introduced to you without much undo this Heisenberg interaction this Geologinsky-Maurier interaction I mentioned the magnetic anisotopy there are also some other parts which are currently not of my interest it is a dipole-dipole interaction and you know basically if you have calculated all these parameters G, I, J and D, I, J and K then you can put it into an atomistic spin dynamics program and you can calculate all these magnetic structures and you can simulate the lifetime and the dynamics and all this and you can do it even on your mobile phone you can download this code it's free for you and you can play with it. Originally the code was made for a high school teacher but experimentally it's like to code so much so that we develop this code further for our play with this spin dynamics but you know sometimes all these spin lattice models is a little bit complicated you want to do also some analytic work and therefore you often go from this atomistic spin model to the micro-magnetic spin model and this you can do if you for example take a long wavelength approximation you take the magnetization as point J look at the magnetization at point J and expand the magnetization at point I and you do a theta expansion and then you come basically to this micro-magnetic model that is well known it developed in the 60s and of course we all want to relate these parameters A and these parameters G they are related due to these approximations but at the end of course we want to calculate these quantities from first principles and so there are methods and ways to do to relate the total energy which you can as function of the rotation of the magnetic structure for example you introduce a spiral and the spiral changes the energy like in this behavior with a spin-spire with a certain Q vector and then you can basically calculate the spin stiffness which is this A and you calculate the derivative of the energy with respect to Q if you we have strength which we call here the spiralization and you can calculate the magnetic anisotropy and you will learn this how to do that next week and then you can for example calculate these your parameters or you can do a principle called infinitesimal rotations you take the magnetic state you basically change a pair of atoms slightly and then calculate the change of the total energy with respect to these small changes and this gives you basically these Heisenberg and we have parameters and you can do that with different codes for example the one special code we are doing it with the flow code and the KKR code because we are more used to that code now let me say one word about this energy functional this energy this micro-magnetic energy functional I have shown here here is the exchange the magnetic anisotropy and for example external magnetic field and the question which I would like to address is I am putting now in this micro-magnetic energy functional a squirmion which I showed you before and what I am doing now is I take this squirmion into this equation and scale a little bit the radius of this squirmion and look at the stability of this squirmion so I take my squirmion I basically squeeze the squirmion a little bit by basically scaling up the radius of this magnetization or make it a little bit larger this one this way or that way if I do that the energy depends somehow on this stretching parameter you see here here at the first derivative square due to the scaling this becomes one over lambda square then I have here due to this dr square a lambda square and if I plot now the energy as function of this lambda I see I see I see that the energy has a parabola behavior as function of this lambda and what you see is the minimum of the energy is that lambda equals 0 in other in other way it means if you put your squirmion in this micro-magnetic energy functional your squirmion is squeezed to death when it doesn't exist because it has the radius 0 this changes in one dimension because in one dimension if you have structures in one dimension this still scales like one over lambda square but here you have something which depends only on lambda scales only with lambda and therefore you see your energy functional looks like something like it has this form it has a minimum and this minimum is actually your domain wall so a domain wall is a stable object in one dimension but if you add now the Jalodzinski-Muriya in action which has this form of the chiral symmetry breaking you get here an additional term which goes proportional to the derivative and therefore you have here in your energy function in a different term which is one over lambda and this guarantees you a minimum and that means the Jalodzinski-Muriya in action stabilizes these two dimensional objects maybe I skip this part and come to the squirmion radius so we have studied this micro-magnetic energy functional what I can do now I am saying my squirmion is actually symmetric it doesn't have to be but I assume it is actually symmetric and if I put now these axial symmetric conditions into my equations in the energy functional I can ask myself what is the optimal profile such that I can minimize which optimal profile minimizes the the energy of the squirmion if I do that I am coming to a squirmion profile equation which is terrible complicated very non-linear but I can solve it for example numerically by certain boundary conditions saying that the spin inside the squirmion points down and outside of the squirmions it matches to the ferromagnetic state and what enters in this equation is a dimensionless parameter which is basically this kappa which depends on all these parameters which I have first calculated by up-initial and my equation is written in the units of this domain which I showed you before and what you see is this is the solution this is the radius as function of this parameter kappa first of all you see not much it is a very non-linear functional but if you zoom in into this kappa you see small changes of this kappa small changes of this material parameters give you large changes in this radius and now my time is a little bit short I am running out of time I have done the squirmion applications a very important material stack to put this squirmion into a reality is the magnetic multilayer because the magnetic multilayer offers you a great versality what you can do to massage the magnetic structure such that your squirmion and all these properties which you need are at the right curie temperature, at the right size at the right stability all the conditions which you want to have for a squirmion in a technology room temperature, fast speed stability, all this can be massaged by such multilayers so you need heavy materials like platinum for example or iridium for a large spin orbit interaction, you need ferromagnetic material cobalt for example you have to organize this material such that your magnetization is perpendicular then what you want is the ferromagnetic layer should couple and the ferromagnetically to the next layer for this we have an archaic y-material it should be enter ferromagnetic to minimize the dipole energy and so on and you have choices of the thickness the layer composition the growth condition and the coupling strength and so you have a huge perspective available and what we have done recently we have created 14 different multilayers so we have taken cobalt platinum which is the technologically most important one and varied 14 different elements here iridium, zirconium, niobium, molybdenum, tisnetium, ruthenium up to cadmium, copper, zinc and looked which one produce room temperature which one couple enter ferromagnetically for example here those in grey couple enter ferromagnetically those in pink or pinkish orange they couple ferromagnetically and if you put all this together you we could calculate the skirmjorn radius as function of the cobalt layers you see we can reproduce this very funky behavior where small changes of the cobalt numbers give you a large difference in the skirmjorn radius and so if you want to say that for example I want to have a skirmjorn radius of 10 nanometers which is technologically good maybe these compositions depending whether you have copper for example 10 years of cobalt would be for example important choice for skirmjorn radius at zero field with this maybe I should stop at this time maybe I move on to here I should thank all my collaborators the bright young graduate students PhD students postdocs and of course also senior people and of course I benefited from all of them in particular Gustav Bielmayer Nikolai Kislev, Sami Aloumi Yuri Makruzov also Hongying Ja, Markus Hoffmann all the skirmjorn work Yidion Dufu to the spirit code Fabian Lux who did a lot of work on the chiral hall effect and the chirality of the systems and also on the skirmjorn stability I thank many organizations but in particular Max without Max we could not have our code on this level and without Max we would not be able to enjoy our collaboration with the Espresso group one Espresso group and with this I thank you very much for your attention and I'm happy to ask questions Thank you very much for this very nice talk very interesting so I would ask if there is some question from the audience here in Zoom I think there was some question before maybe one can raise their hand and ask directly so please Nikita Yes hello and thank you for your fascinating presentation I just have a simple question concerning the beginning you raised that the energy scale is just about one of me how about does this effect survive at room temperature and how about the magnet for an interaction does it affect I don't know the application is it destroy or is it sensitive to apply the magnetic properties for them the the effect itself is indeed very small but the query temperature itself the effect does not determine the query temperatures in systems of three dimensions so if you have a three dimensional magnet for example the query temperature is basically determined by the Heisenberg interaction and not by the spin orbit interaction and therefore as long as your magnetic system has a query temperature is below the query temperature has a magnetization the the magnetic crystalline anisotropy is there but I should like to mention that the magnetic crystalline anisotropy has a different temperature behavior than the magnetization it becomes smaller faster so the hardness of the magnetic hardness of the material is reduced faster and that is of course a problem if you have for example if you have an electric engine in a car if the car becomes warmer the hardness of the material goes down and maybe it does not perform so well anymore and therefore therefore there is a research development increasing the query temperature making the material harder at higher temperature now let me come down to two dimensions if you have a magnetic interaction that if you have a two-dimensional system a strictly two-dimensional system which is totally magnetically isotropic then you don't have a query temperature query temperature is zero then the spin orbit interaction which produces this magnetic anisotropy stabilizes actually the magnetic phase against fluctuations so it is then it is a very important ingredient for a finite query temperature and then you had a question about magnon phonon so the magnon phonon yeah there is so typically the magnon phonon interaction um uh so the magnon phonon interaction is the interaction where the magnon and the phonon hybridizes this typically is an interaction between the controlled by the Heisenberg interaction which controls the magnons sometimes also the dipolar interaction which control the magnon and the phonon the spin orbit interaction does not play an important role here but the spin orbit interaction has a slight impact on the magnon dispersion and maybe can shift the resonance of the magnon phonon interaction okay there is another question from Amelia please hi um so are these scourmions do they tend to be stable in temperature under normal conditions or do these have to exist at extremely low temperatures or something because I'm a first year I'm probably not going to explain this intelligently but the electrochemistry group I'm in has been seeing some very strange effects with our thin magnetic films that we don't know how to explain so I find this very exciting but I'm not sure if it's possible to explain to the audience that they're currently creating scourmions that are influencing our results um you have scourmions at low temperatures and you have scourmions at above room temperature okay typically if you work in the world of ultra thin films one layer iron or cobalt on your for nanometers and typically these films have a very low current temperature. So you also, your skirmish has a very low current temperature. Maybe I say a number, 20 Kelvin. But if you go to this multi-layers, which I have discussed, you can easily have skirmish at room temperature. It's not a problem. And actually that is the motivation to do skirmish at room temperature. But skirmish are larger, typically say 30 nanometers. 20 nanometers, 13 nanometers. So it depends a little bit. What is the size of the skirmish your colleagues in the lab are seeing? Thank you for the answer. So let's take a question from the streaming. Should we consider a sock for all 2D materials? I think it was related to surface effects of a sock. I would say yes. Sock can be very important for 2D materials. Because some of these materials, or many of these, not all, some of these materials, for example, contain selenite, telluride. These are of course heavy materials. And they are P materials, materials with P electrons. And P electrons are typically have very strong spinorbid reactions. So I can easily imagine that such interactions are important. And in particularly, they change the Heisenberg model to very anisotropic models. For example, like the Kittaev model. And therefore I think the spinorbid interactions are important that they are always the dominating one. I cannot say it depends a little bit on the details. It depends also on the crystal symmetry. It depends also on your substrate. But without considering it, it would be dangerous. Another question from the audience. Ignacio Martin, please. Thank you. Thank you very much for this very nice talk. I'd like to know, could you comment on the stability of the kermians against external magnetic fields? How the radius depends on the external magnetic field? I'm thinking on real devices where you want to stack the components as closely as you can and you may have stray fields. Ah, yeah. Okay, stray fields. You are right. You have stray fields. That's not good. The stray fields have the disadvantage of making the kermian larger. And you want to have small and compact kermians. So therefore, we try to avoid this stray field. One way to avoid this stray field is to work with this layered anti-ferromagnetic materials, which was probably not very explicit in saying that I was a little bit rushed at the end. So what you saw is actually some multi-layer stack. And in this multi-layer stack, you have ferromagnetic films, cobalt films. But the way you stack these cobalt films is that we stack them such that we have what we call synthetic anti-ferromagnets. So the one layer points up, the next layer points down and we choose the intermediate layer such that this coupling is anti-ferromagnetic. So that we have certain stray field compensation. And this stray field compensation helps us to keep this kermian small. Another object, another possibility would be to create a kermian in anti-ferromagnet. That's also possible. But then the detection might be a little bit more difficult. If typically, if you apply an external magnetic field, your kermian is typically becomes, yeah. So the kermians, which I mentioned to you had been kermians without external field. But sometimes you can add a small external field and you move a non-existing kermian into, or a non-existing magnetic pattern into a stability regime of the kermian. Of course, if it's five Tesla, it doesn't help you very much. It is an academic. But sometimes you have small bias fields which you can implement in your multi-layer stack, might move you into the right corner of the phase diagram where you can basically then stabilize your kermian under a bias field. Okay, thank you. So Christian is asking something, please. Yes, thank you. I want to ask Professor if we do a fact that actually I solved the problem of spin electronic on magnetonene. That's the origin of the basher effect, basher effect. Is it a perturbation on the magnetonene or the fact that actually I solved the spin electronic on the magnetonene or the core electrons? Sorry, I wasn't quite sure. I didn't catch your question exactly. So do you want to know about the origin of the basher effect or? Yes, sir, yes, sir. The origin of the basher effect is in some sense spin orbit in the action of plasmity breaking. So if you consider these two facts, rash by symmetry breaking plus spin orbit in the action, then you can basically derive perturbation theory and contribution to your Hamiltonian, which is the rash by effect. But the symmetry breaking cannot be arbitrary because there is a, depending on your symmetry breaking, you have either the rash by effect or the Dresler's effect. For example, if you have a symmetry breaking like the symmetry breaking of a surface, which have, yeah, say called C2V, C4V, this type of a point groups, then you get a rash by effect. But if you have a different type of symmetry breaking like D2D, then, which is typically for gallium arsenide, for example, so that you have a broken bulk inversion, then you get a different type of spin orbit Hamiltonian, which is the Dresler's Hamiltonian. So if you have structure inversion symmetry breaking, you get a rash by, if you have a bulk inversion, you get a Dresler's house, both of them are linear in spin orbit interaction, yeah. And you can get more complicated thing depending on your symmetry in orbit. So Stefan, there is a question from the streaming. Can you please comment on the role of Salk in designing the photomaniotic systems? So back to the previous comment. In which systems in? Photomaniotic systems, that means manipulating spin through light pulses. Yes, here's of course, the light enters basically as a vector potential, and the vector potential covers to the spin. And through the spin, vice to the spin orbit interaction to the ladders. That is my picture. Okay. I think we can get this last question. So when we put non-magnetic material on top of material with large spin orbit coupling, does the strength of the Cheroshisky-Moria interaction depends on the thickness of the each material? I showed in the talk the dependence of the Rashbar strength as function of the depth. And you get exactly the same behavior for the, not the identical behavior, but an analogous behavior to the Cheroshisky-Moria interaction. So if you have, for example, a magnetic material in the vicinity of a material and its strong spin orbit interaction. So the wave function of the, so the electron of cobalt of the magnetic materials moves the electron hops to the platinum and maybe hops back and produce the Cheroshisky-Moria interaction. But the deeper it hops, the less is the contribution. So that means that the first, maybe the first platinum layer contributes most and then you have additional contribution from the second, from the third, from the fourth, from the fifth. But the contribution, the deeper you are, the thicker you go, the less is the additional contribution. And sometimes you have also a slightly oscillatory behavior. So maybe the first and the second platinum layer that add to positive Cheroshisky-Moria interaction. The third platinum layer has a negative sign, but of smaller amplitude. It's like an archic-y interaction in the DMI contribution. Okay, I think we can stop here and we really thank Stefan for this wonderful talk and for all these explanations. So thank you very much once more. It was a real pleasure, I really enjoyed it. Sorry, I mismanaged the time a little bit. Okay, sorry about this. It was okay, okay. So thanks, see you soon. Yeah, have a great time, a good next week. Yeah, thank you. And sorry that I was a little bit too long. I misplanned a little bit. I was explaining too many things. I thought it was very clear, especially at the beginning.