 So I'd like to first start by thanking the organizers for inviting me to this important event and of course wish Boris many many years of Productivity and and happiness In this talk, I'd like to tell you about some of the recent work. This is unpublished work that we've been doing to explore the interplay between a superconductor and And and a semiconductor that has a very strong spin orbit interaction in it We're going to be exploring mostly the in-plane the in-plane Magnetic field effects on this induced superconductivity inside the semiconductor and what I'll show is that the induced pairing consists of finite momentum This momentum that the pair acquire is controllable in these systems furthermore the order parameter oscillates in space and the oscillations in the order parameter in space could be both Perpendicular to the contact as well as Parallel to them and and we understand quite well right now what the different regimes are and finally we can look at the data and understand under what conditions is the effect of spin orbit Interaction dominant in this system versus just plain Zaman The work that I'll be talking about was done by three graduate students in my group Sean Hart, Hetchum Ren and Michael Kosovsky The theory was derived in collaboration with Bert Halperin and this work is done in close collaboration with the group of Lawrence Mullen Kump who has provided us with materials as well as a lot of the A very good collaboration on how to fabricate these samples That is being developed in parallel in both groups So the materialist system that we're looking at is that of mercury telleride cadmium telleride quantum wells just very briefly Historically these materials were introduced In in order to demonstrate the first topological insulator demonstrating quantum spin Hall effect I'll mention that very very briefly, but this is certainly not the topic of this talk So mercury telleride itself here. You see the band structure is a semi metal It has the p orbitals Sitting above the s type orbitals in the in the band structure But nonetheless, it's not a semiconductor and so in order to generate a semiconductor out of mercury telleride You sandwich it between barriers of cadmium telleride and you see the effects here this is a sandwich structure and If the width of this mercury telleride sandwich between the conventional semiconductor cadmium telleride is narrow Then this inversion that appears in mercury telleride gets undone and the electron states Reside above the whole like states This occurs for very short Well widths less than six point three nanometers But as you increase the well width What happens is that you can reach a situation where band inversion occurs Nonetheless, but a true gap occurs because of science quantization and here you see that in this specific situation the Conduction band consists of whole like states very close to the bottom of the band and the valence band consists of electron like states so this is the inverted Type of structure and this is why this 2d material is a 2d topological insulators and it supports a quantum spin-haul Effect when the chemical potential is set into the gap the focus of this talk However, we'll be at chemical potentials that are in the conduction band We will tune density, but we're not really interested in the quantum spin-haul nature of this thing just to understand What the interplay of superconductivity and spin orbit in this material will be Here's a schematic of how the device looks we pattern out of this quantum well a Mesa structure And then we remove we remove the quantum well everywhere except in this green region We have these superconducting contacts that are these gray Gray contacts here. They make reasonably well Superconducting contact to the 2d electron system and then we deposit also a gate that allows us to look at this Induced superconductivity in this geometry as a function of carrier density and note that the gate covers the two electrodes So that we can induce rather uniform change in density as best as we can Now the knob that will be or the tool that we'll be exploring is a simple frown-hofer Interferometry, namely we're going to look at supercurrent flowing through this device between two superconducting contacts and a narrow Strip of the semiconductor here just the 2d electron gas And we're going to look at this at the critical current as a function of perpendicular flux And I want to emphasize the role of this flux is just to generate the frown-hofer Interferometry it is not playing any important role in terms of its alignment of spins and so on it's very weak Magnetic fields on the scale of a few millitesla that we're exploring with a perpendicular field parallel fields will be very large Very large in the Tesla range And so you see that this is just a canonical case in the absence of any parallel magnetic field you see that If you look at the critical current Or zero resistance in this system as a function of the DC current you see that it's very large When the flux is zero and then you get this traditional frown-hofer Interferometry that corresponds to uniform current flow through this channel and here is The corresponding current flow distribution through this structure as analyzed from this frown-hofer pattern So I want to emphasize that here what has been measured is basically zero resistance For example along this cut up until this point where finite resistance develops and that allows us to determine what the critical current is As a function of perpendicular magnetic field Yeah, yes Absolutely Yeah Okay, so as we are going to apply relatively large magnetic field in plane and we're gonna study two superconducting materials aluminum and niobium I want to convince you especially with aluminum that has a relatively low HC one that if you make very thin aluminum Contact and you apply a magnetic field in the plane of that film that the superconducting critical field Of the aluminum film could be very very high and here you see a very old result from 1971 as you reduce the film thickness down to 10 nanometers for example the critical field can exceed several Tesla and this Parallel to the film exactly parallel to the film And here are measurements of Just two contact resistance measurement of each one of the leads showing indeed that our superconducting contact sustain Superconductivity up to magnetic fields in the range of 1.8 Tesla And so whatever effect we see as a function of magnetic field We now know is not a result of basically just eliminating superconductivity in the contacts Now in order to simplify things and acquire data faster I'm not going to show you these Fraunhofer patterns as I showed in the beginning, which is the critical current Versus flux. I'm just going to take a line cut Let's say at zero parallel magnetic field here Of the differential resistance when we're sending a very small current through the junction And what you see is basically that when the system is superconducting the differential resistance is zero And then at the nodes of the Fraunhofer pattern it develops a finite resistance This is the normal resistance and then it becomes superconducting again And then it rises again. And so what we're going to look at we're just going to look at this differential resistance as an Indicator whether the system is superconducting or not. We're not going to know from this measurement what the critical current at any flux is But at least we'll know there is superconductivity taking place. So we're just trying to map out Where is superconductivity present as a function of the in-plane magnetic fields? And so here you see these kind of differential resistance showing zeros Where there is superconductivity and the nodes where the resistance is finite So this you can imagine in your head just as a Fraunhofer pattern And then we're varying the parallel magnetic field and up to something like half a Tesla in plane Nothing really is happening. And by the way the direction of magnetic field is outlined here currently it's applied Perpendicular to the current flow between the two superconducting contacts But as you increase magnetic field further at around one Tesla or one point two Tesla We see that there is no superconductivity left So nowhere as a function of flux can we see a diminished differential resistance meaning the system is no longer superconducting But then if you increase magnetic field further it recovers and it doesn't recover as strong as it used to be And I'll give some indications. Why is that but The the surprising thing for us at least was that superconductivity Restores itself as you increase magnetic field. So it looks like there might be something Oscillating as a function of parallel magnetic field in the superconducting behavior of this junction Now I mentioned that this was aluminum contacts and there there are very few flux jumps So the Fraunhofer interferometry looks quite beautiful Here are the results for niobium here. We can study a larger flux range because the Superconductivity is stronger so critical currents are larger, but basically you see the same effect You see that the Fraunhofer pattern that you observe at zero magnetic field slowly Vanishes and then disappears at about one Tesla But then recovers again and then disappears again and in fact If you just saturate the colors a little bit more you see that it actually Reappears again and disappears again So this supports this notion that there is something oscillating as a function of parallel magnetic field In this device and that's what we'd like to understand here You can see just a line cut trying to detect what is the lowest resistance for any one of these cuts here And indeed you see superconductivity. It's lost At this roughly one Tesla range then it recovers It's lost and then comes back again and lost again and now here the resistance is really not zero So I'm not arguing that we see supercurrent But we believe that if we would to eliminate all noise in our system and cool the system even further below the temperature that it's Being measured we believe that this will become down and will demonstrate superconductivity So it's just an indication that the system wants to behave superconducting in this regime So now we can explore this behavior not only as a function of parallel magnetic field But also as a function of density and so we're taking these kind of Fraunhofer's at different cuts Either as a cut at a finite density as a function of magnetic field This is what you've seen already But here are some cuts where we fix the parallel field and just scan density and you see for example in this particular case That the central lobe is too fine not which is what you expect for a conventional Fraunhofer Interference of uniform current flow, but as the sample depletes it actually recovers a simple interference a cosine Interference pattern, and this is actually the signature of the quantum spin-haul effect that I'm not going to talk about I'm just pointing it out. This is something that we've studied in the past What this diagram is showing us is the reconstruction of where the lowest resistance of the system is as a function of gate Voltage density and magnetic field and you see that This entire region here of magnetic field and density is strongly superconducting the resistance is zero Then it goes away along this kind of yellow. This is normal. This is normalized to the normal resistance So you see that you recover normal resistance and then superconductivity reappears in this place here in this place here It's there is a little bit of tendency towards superconductivity between these two pockets and this regime here I'm not going to talk about this is the regime of the quantum spin-haul effect when the bulk is actually depleted So the question is what is giving rise in particular to this node? And what is the general picture behind this this thing if we repeat the experiment with the niobium Contacts one could see a very similar behavior again This is just the lowest resistance Normalized to the normal resistance and you see as a function of density and magnetic field again the reappearance of Superconductivity here and it actually reappears a little bit here. That's this this pocket at high magnetic field And we just pointed out that the aluminum device maps onto this parameter space in this corner here You see that it's a very similar behavior in this corner What we don't see in the niobium device is this particular pocket But we definitely see this one very much extended up to very high densities I also want to point out that the boundary between these two pockets of superconductivity is not just dependent on magnetic field But it depends on both magnetic field and density and the question is what is the origin of that? So in order to get some insight to what is going on We need to address the band structure and I already alluded to The fact that the system that we are exploring here is an inverted system namely we have holes in the conduction band This will become very very important because the g-factor of the holes is very strongly anisotropic That of the electrons as well, but in particular the g-factor the in-plane g-factor of the holes Because they're composed of spin three-half The in-plane g-factor of a pure hole state is zero So it's hard to understand in that limit why the magnetic field is doing anything But that is only when the density is absolutely zero as you move away from zero density You see that the the states become mixtures of electrons and holes, and so they do acquire a g-factor All right, so this system has been studied quite extents extensively originally by Bernoulli-Cuse and Zhang And what I'm going to be discussing is an extended version of this model that takes into account the various effects of spin orbit interaction So the bulk just the band structure effect without spin orbit Interaction in the system has this kind of shape you see that it's mostly a Dirac like Hamiltonian Linearly dispersing dispersing with momentum where a is the Fermi velocity There are some mass terms that correspond to the opening of the gap That's the gap in this semiconductor and the mass can be negative that corresponds to the inversion of electrons and holes But the important terms that I'll be focusing on are Vulcan version asymmetry So that's a Dresselhaus type spin orbit interaction shown here where the angle theta I'll spend some time later talking about it, but it's basically the angle That is the angle between the momentum directions that we're considering and the crystallographic orientation of the sample So for Dresselhaus that angle matters This is the rush buspin orbit terms structural inversion asymmetry due to some electric field in the system And finally just the same on effect and here I've indicated what the g-factor matrices looks like For the same on effect and note that first of all When the magnetic field is applied perpendicular to the quantum well along the z-direction All four g-factors exist, but they're very different for the electrons and holes and The parallel component of the g-factor you see it's a very sparse matrix It indicates that there are only matrix elements that Take the electron state which is a spin half to a spin minus half and they do not couple at all the whole states Which are three halves to minus three halves and this again is important To factor in all right, so let's let's take a look at this band structure in the absence of spin orbit interaction It just looks like a parabola or something that actually is You know it's more straight. It's it's nearly a dirac like cone at high energies If you include the spin orbit interaction, of course the Fermi surface splits into two Fermi surfaces And here I've outlined what happens with rush buspin orbit interaction Note that the spins orient themselves in the plane in the absence of a magnetic field and you have two Fermi surfaces And with the bulk inversion the Dressel house the way the spins wind is in the opposite direction to the Structural inversion asymmetry so you get patterns that look like that again the Fermi surface is split And depending on the angle that theta the crystallographic orientation you get different types of Orientation of the spins in the plane I don't want you to focus too much on the bulk inversion asymmetry because Experimentally it turns out that it's a weak effect, and we don't really see it So the main effect that I'd like you to consider is the rush buspin orbit and the Zeeman case. I Rather mentioned that the g-factor in this system is zero at the band bottom because it's a purely whole like state And then this g-factor in the plane increases up to the bulk value as the momentum increases in the plane Okay, so so far this was just the single particle properties of the system now what we want to consider is what happens when we're trying to pair Electrons on these Fermi surfaces as induced by the superconductor, so we're trying to do some singlet states and The problem is that once there is the spin orbit interaction. You see that what pairs are Basically, and so this is pure Zeeman. Okay, so let's imagine. We don't have any spin orbit interaction We're just considering Zeeman the Fermi surfaces Of course split one corresponds to spin along the field and one corresponds to spin opposite to the field So that's the spin arrangements, and if we'd like to pair electrons in this case Note that we would pair one electron from the outer Fermi surface It has a spin up and the other one from an inner Fermi surface It has a spin down, but note that that pair now has a finite momentum. It's not centered around zero Similarly its counterpart the down-up version has the opposite momentum so the wave function begins to acquire an interesting phase and And we're gonna explore the implications of that phase I Want to note that this kind of physics is very similar to what happens in superconducting ferromagnetic superconducting junctions Where they're a Zeeman effect effectively exists due to exchange interactions But I want to point out that in this system the nice thing of course is that you can tune the magnitude of that effect It's not just fixed by the exchange interaction Also want to mention that this kind of finite momentum pairing is something that is being discussed in the context of F F Flow states Here it's not an intrinsic effect that we have a pairing amplitude with finite momentum But it's the induced superconductivity that has the finite momentum, but very much an analogy with the F F Flow state so that was the case of and also I want to point out that in the case of Zeeman Note that the momentum of the pair is always in the direction of the propagation of the electrons so every and Every pair here has momentum And it's always aligned with the direction of propagation of the electrons Now let's consider the case of spin orbit interactions strong rush by spin orbit So in the case of strong rush by it's very easy to see that the application of an in-plane magnetic field Shifts one Fermi surface relative to the other in a way that's perpendicular to the direction of the magnetic field So again what happens now? Spin orbit interaction wants to pair electrons within each Fermi surface because the spins are winding So we have opposite spins within the same Fermi surface But the application of magnetic field now shifts one Fermi surface relative to the other again generating a phase But note this phase exists only for electrons moving perpendicular to the magnetic field Whereas for electrons moving along the magnetic field There is no net momentum change because the Fermi surfaces are just moving transversely one with respect to the other and Finally, I'm going to skip the effect of the bulk inversion asymmetry It basically has similar features the pairs acquire a finite momentum as a result of the shifts in the Fermi surface All right, so in order to understand what is the implication What how does that affect superconductivity the critical currents in this device? one can try to consider the situation where One has a tiny speck of superconductor at point x1 and wants to know what is the induced pairing amplitude at a point x2? so the pairing amplitude at point x1 would have some Some magnitude delta corresponding to the superconductivity of aluminum Let's say it has some phase that's given by the phase of the superconductor itself as well as the phase that has to do with the Applied magnetic field in the z direction the weak flux that we're using in order to modulate the interference So that's what appears here and then We need to consider how that order parameter decays in space as we move from point x1 to x2 and What that entails is this kind of Green's function That has the phase of the Cooper pairs as they propagate along the trajectory from x1 to x2 so Phi is just the Overlap the dot product of the shift in momentum the momentum that the pairs have acquired Along the trajectory from x1 to x2 and if we want to know what the Overall energy functional of the system so at point x2 It's going to be just the order parameter lambda 2 Times the induced order parameter at point x2, which is lambda 1 times this Green's function f and So the nice thing is that this phase of the Cooper pair enters this Green function in a very simple way that we can then analyze Semi-classically to get some assess get a feel of how critical currents should look like in the presence of the magnetic field So let's consider the case of only structural inversion asymmetry remember when we apply the magnetic field in this case Along the particular direction the Fermi surfaces move Perpendicular to the direction of the field so momentum now is only acquired along the y direction And what this means is that regardless of what trajectory we're taking here the phase is always the same It's just Delta K y Times W doesn't matter whether the trajectories are diagonal or not. It's always Delta K w so this turns out to produce a term that factors out of the Energy functional and therefore the current as well as the energy is Modulated by this cosine term Delta K times W and this is very much Analogous to what we're observing so here is a calculation based on that simple semi-classical theory of How we expect the critical currents to depend on the parallel magnetic field for the given width in our device and indeed We see that there is an overall cosine dependence and this is the node of that cosine Superconductivity goes away. There is a destruction effect here and then recovers again, and that's exactly what one is seeing here So this happens at high densities With the niobium device we can look to higher magnetic fields, and I want to point again. It doesn't look very much yet Similar to one another but nonetheless this reappearance There is one node here another node here and the simulation for the ideal case shows this kind of periodic behavior just this cosine factor that factors out and this Oscillation is a direct manifestation of the momentum acquired by the pairs as the in-plane magnetic field is applied. I Also want to point out that one can look at the order parameter as is being induced by one contact and One can easily see that it is oscillating in space and one can understand this kind of nodes Simply as having the width of the device corresponding to the place where the order parameter vanishes So not only in principle one can explore this as a function of magnetic field which changes the period of this real space Modulation of the order parameter. In fact one can study devices with different dimensions for a same field and one would then see again The same kind of oscillatory behavior Because this order parameter is now oscillating in space for this particular Case the application of magnetic field in this direction and spin orbit interaction You see that the spatial dependence of the order parameter is only Oscillating a long one axis However, if we look at the Fraunhofer patterns Somewhere at lower density. We see that they they have a slightly qualitative Difference and that is that the central lobe disappears, but the side lobes don't remember that here It actually looked pretty nice in the sense that all lobes more or less disappeared both center and side lobes At low density. We see something a little bit different Central lobe disappears side lobes do not and it turns out that that can be generated if one forgets about spin orbit interaction and includes only Zaman in The case of Zaman remember the momentum acquired is always along the trajectory. So every path here has a different momentum And has a different path length and if one factors this kind of phase factor in the green's function in this Energy functional one can compute what the interference pattern should look like and indeed you see that the center lobe This appears but the side lobes do not and they basically fold into the center one very much like we see here So what we believe is going on here is that at low density where Zaman dominated spin orbit somehow is not so important at large Density spin orbit is dominating Is dominating the physics? Just to contrast the two spatial dependence when we only had the spin orbit interaction I mentioned that the order parameter is oscillating in space only in one direction for the Zaman case It looks like the order parameters oscillating in two directions both along perpendicular to the contact but also Parallel to the contact and these oscillations parallel to the contact are induced by the edges of the semiconductor without edges Those would not exist. It's really an edge effect Okay, what happens now if we rotate the direction of magnetic field? So so far we've applied the magnetic field in this direction Let's apply it now in this direction and again consider only Rajba spin orbit So when we apply magnetic field in this direction Fermi surfaces move perpendicular to the magnetic field So the momentum is in this direction So note that a trajectory that now goes straight the shortest trajectory the one that contributes most Does not acquire a momentum at all or a phase difference but all side trajectories do and again It's very easy to run the calculation and see what one expects in this particular case the Fraunhofer patterns should split And the order parameter in space is oscillating only as a function Parallel to the contact due to the due to the boundaries of the sample One can understand this splitting very simply as if you recall that the Fraunhofer pattern is a Fourier transform of the current distribution Note that when the order parameter has this kind of oscillation the current distribution along this device is oscillatory And this gives rise to this finite momentum Effectively in the Fraunhofer interference patterns, and I want to point out that we very clearly see this kind of V-shape Fraunhofer's but there is a big problem and the problem is quantitative whereas before there was actually a quantitative agreement between the BHZ model and And our data here note that the scale here is 6 Tesla on the simulation and is 600 mili Tesla on the measurement or even less point 15 Tesla to get the same kind of slopes so there's a big discrepancy here and Unfortunately, it turns out that there is another mechanism that can give this kind of V-shape behavior that it's not very interesting So I'm going to spend just a minute on it When you apply the magnetic field in the direction along the current flow the superconductor screens the flux a little bit And so you see you get some per Penetrating flux in the z-direction that's pointing upwards on one side and downwards on the other side And it's very easy to show then that under these conditions The kind of phase factors that are generated by these fluxes give rise to exactly the same kind of V-shape and In this case you can easily get The scale to produce this V-shape on a lower field scale and the true effect of spin orbit interaction in this particular case Is to split the V And that is something that we're not seeing at the moment. It's a very subtle effect and it's very hard to see I want to also point out the asymmetry between positive and negative flux Note that these things are not symmetric and I don't have the time to talk about it, but it has to do With lack of rotational symmetry Of a hundred and eighty degree symmetry of the device itself if the device had a hundred and eighty degree symmetry These V-shape should have been perfectly symmetric Once we understood this kind of screening effect It has also bearing on the magnetic on the field effect When we apply it in this direction in this direction you see that we don't have this effect because the screening takes place here and here So we don't have this kind of flux focusing But if the edge is a little bit jagged it will produce some random fluxes random phases along the edge And again, it's very easy to include the random phase So if you just add some random phase and recalculate what the interference pattern should look like You see that the lowest field lobes remain more or less untouched The next node is strongly centered around zero flux and the third one begins to be very disperse just like we're seeing here So what we believe is that we're very Clearly seeing the signature of this finite momentum real space dependence of the order parameter plus some Unfortunate phase randomization that exists because of the jaggedness of the contacts and that jaggedness is only on a scale of nanometers Is enough to produce this kind of effects of these large magnetic fields Next I'd like to address the boundary this node So in the context of this picture, this is in the picture of the strong spin orbit interaction This node corresponds to the simple condition where the induced momentum of the pairs delta k times the width of the device is just pi over 2 And it's not hard to see that the induced momentum is just given by the ratio of the Zeeman energy to the Fermi velocity It's just the sift of the of the parabola with respect due to the Zeeman effect The fact that the node here this condition is moving to higher field means that at low densities Delta k becomes smaller at the same field and we need to increase the field in order to recover the Interference this node condition so again, we can go back to the theory and Calculate the g-factor the Fermi velocity and the induced pair Momentum at different magnetic field and indeed it has this dependence that I mentioned There is a large induced pair momentum at large density, but that pair momentum is dying off a note It's a complicated thing because the g-factor is zero, but also the Fermi velocity approach is zero How exactly the ratio depends is a detail and this is the outcome You see that there is less momentum induced at low densities Which means we need a larger magnetic field in order to generate the node and that's what these dashed lines are these dashed lines are the theory That tell you where the node should be as a function of density and you see that at large densities They fit very nicely. We don't understand the behavior in this low density regime so with this I'd like to summarize as my time is up and let and just Conclude that in these two-dimensional systems with strong spin orbit the in-plane magnetic field Controllably induces phase shifts Momentum Momentum in the pair in the pairing In the in the Cooper pairs and the order parameter that is induced in this system Oscillates in space in some very interesting direction and in fact one could use this kind of understanding to generate a particular Behavior zero pi junctions, etc. It's simply by engineering the position of the second superconducting electrode and I mentioned that at large density. We understand that this Behavior this momentum induced pairing is primarily due to spin orbit interaction But at low density it looks like it's just Zeeman dominated and the effect of spin orbit interaction is weak That would be thank you Thank you very much