 Okay, so thanks for being present at the picture, I've been told that everybody was smiling so it's good. And for the last two hours of the day, we have a lecture by Andy MacKenzie on hydrodynamic transport in electronic systems. Thank you very much. Thank you. Well first of all I'd like to, could we turn down the volume, I have a pretty loud voice. You okay? Is that? Sounds booming to me, okay good. So first of all I'd like to thank the organisers for asking me to come along and give you these talks. The well-handled is that we'll talk until about maybe 10 to 5, then take a break and then carry on for the rest of it to give you guys some chance to keep your concentration going late in the day. So I'm an experimental physicist and the bulk of what I'm going to be telling you today will be an experimental story. It's the story of whether or not hydrodynamic regimes of electron transport are observable and why these are hard experiments and the progress which is being made towards turning this into a more routine thing. My own work on this subject is done in collaboration with a number of people, I probably haven't listed everybody there, but they are experimentalists from extremely important people like Xiong-Hun Kim and Palavi Koushwaha who grow the crystals. Experimental measuring people like Maya Bachman and Nabanila Nandi who are two very good graduate students and also what's really important to this, well there's Phil King and Veronica Sunco who do photo emission which has been very helpful in setting up the materials that we work on, but particularly important to me has been an interaction with theorists, Joel Moore from Berkeley, Burkart Schmidt from our institute and really in particular Thomas Scaffidi who's currently at Berkeley but is moving to a faculty position at Toronto. So I've known Thomas since he was a graduate student at Oxford a few years ago and yeah I was going to say he's taught me a lot but in some senses we've also learned a lot together really focusing on this issue of how can you honestly do a theory experiment comparison to really make progress in this field as opposed to at times maybe exaggerating what you're doing. So what I'll do then, what I hope to do is to give you some background and then I'll tell you a little bit of very general stuff about hydrodynamic flow and what's called the minimum viscosity conjecture, that minimum viscosity conjecture was made outside electronic systems altogether but it's part of my own motivation for getting interested in electron hydrodynamics so I wanted to tell you something about that. Then I'll tell you a bit about the historical development of the field because the minimum viscosity conjecture stuff came about 12 years ago now and it stimulated a lot of research on electron hydrodynamics, a lot of that research seemed to have forgotten as often as the case an earlier wave of theoretical research and as far as I know the first people to look at these problems theoretically were in the Soviet Union in the 1960s. So then I'll go on to telling you about experiments, if I can work my pointer, I'll tell you about experiments in graphene, I'll also mention the old ones in semiconductors and then I'll tell you something about the material that we're working on which are very high conductivity oxides called the Delafocite and hopefully by the end I can draw some conclusions. Throughout all of the talk I'm going to have quite a few references on my slides, those are designed to help you get an end to the field if you get interested and you want to read some of the source literature but it isn't a proper review talk so I'm almost certainly missing out some papers and that would unintentionally upset people but this is really just meant to be a for example to give you guys a way into the field. So let me now start if you like right at the beginning and remind you about what hydrodynamic flow means. So let's imagine we just have a pipe and we're putting a hydrodynamic fluid through it, it's called hydrodynamics because it was developed for water but it could be for many fluids and I'm going to draw this pipe not as a pipe but as a two-dimensional channel because all the experiments are being done in two dimensions. So if you have an empty two-dimensional channel and you're trying to push this fluid through it by putting a pressure gradient across it what you would find if you had a microscope that would measure the velocity profile is that it would be less at the edges than it is at the middle and that's because the fluid in a hydrodynamic fluid the only way it develops flow resistance is through its interaction with the boundaries. Everything that happens internally is irrelevant to the flow resistance unless there's a boundary that acts as a momentum sink. However, not quite everything's irrelevant because you might then say well why isn't that element of fluid just flowing with no resistance at all and the point is that any fluid contains some transverse coupling from the part in the middle to the element next door to the element next door etc. out to the edges and the way I think of it is that that transverse coupling is the quantity which is being parameterized by the shear viscosity eta. So eta is often quoted as a kinematic viscosity where you take out the mass density so I'll kind of flip to and fro but usually I'll be using the kinematic viscosity when I'm discussing viscosities. The other thing I didn't say in the preamble at the beginning of course is that I completely welcome questions during the talk I don't really know what level you guys are at and so I'm very happy to answer whatever you need. So let's look at this a little bit more microscopically how was I able to make those statements. Well whatever the fluid is it's flowing through this channel and that does not mean that there are no interactions going on inside the fluid the fluid particles are of course having multiple collisions with each other all the time whether that's water or liquid helium 3 or any of these fluids. The point is that when the particles bounce off each other from the point of view of momentum they just exchange momentum and the resistance is determined by the net momentum of the whole assembly so if all they're doing is exchanging they're not relaxing that momentum of their whole assembly so you know shockingly or surprisingly if that wire that channel were infinitely wide or there were perfectly slippery boundaries there would be no flow resistance no matter what the viscosity was. So these electrons or these particles and later to be electrons are bouncing off each other like crazy and then when they get to the one element gets to the wall it will bounce off the wall and if you have rough boundary conditions there's a chance that it will bounce backwards in other words relax some of its momentum to the wall and it's those wall momentum relaxing events which are so important to us. So to parameterize this situation and understand all the physics you needed to you would need some boundary conditions you would need to know the mean free path between those momentum conserving collisions and the flow resistance would still depend in some way on a characteristic width dimension of the channel. Now most of you guys are solid state condensed matter physicists I guess like I am and if you're a condensed matter physicist there's something which is very non-intuitive about viscosity when you think about it microscopically and it's the following. Imagine you had an element of fluid here with a very short momentum conserving mean free path it's quite hindered that element or the molecule in that element from reaching the boundary if however the mean free path is much longer the element near the middle of the of the fluid can reach the wall much more efficiently and therefore relax its momentum more efficiently so that means that the shear viscosity which which is talking about this transverse coupling is proportional to the momentum conserving mean free path and that's very surprising because we're always really used to the idea that resistivity in a real metal is inversely proportional to a mean free path but what we're going to get to is that there are two kinds of mean free path you need to consider so and then in these circumstances if you knew in general if you know the number for this momentum conserving mean free path you can straightforwardly estimate the viscosity if you know the viscosity and the width and the boundary conditions you can calculate the flow resistance based on the Navier-Stokes equations so that would be the the hydrodynamic kind of pathway that you would be using now what about a quantum fluid then so let's think about helium 3 so this is helium 3 going from being a classical fluid of order 3 degrees that's roughly where it's degeneracy temperature is so it's firmly temperature is of order a few degrees and then as you go down in temperature its viscosity diverges as 1 over the temperature squared and again that's a slightly surprising thing to see when you first see it but it's just the same physics what's happening is because it's a Fermi liquid the scattering is getting cut off as 1 over T squared so the internal mean free path is going up as 1 over T squared so it's getting more viscous now I guess how many experimentalists are in the audience yeah we're in the minority here aren't we well the experimentalist should know this but the theorists should know it as well because you've all looked at results which have come from measurements made in a so-called dilution fridge so if you wanted to do milli Kelvin measurements today typically you use a dilution fridge to get to those temperatures and here this divergence of the viscosity really comes in because there's no fundamental reason no fundamental limit as far as we know on the base temperature that a dilution fridge could have right the thermodynamic process is valid as far down as we know but a dilution fridge even if you're going to pay a lot of money for it will have a base temperature of 7 to 10 milli Kelvin and that's simply a pumping issue because by then helium 3 has about the viscosity of motor oil and you're trying to pump it through capillaries as part of what you're doing in making that dilution fridge so the actual temperature dependent viscosity of helium 3 is what means that you cut off your dilution fridge operation way before helium 3 goes superfluid so now that so that was the basics of hydrodynamics without thinking of any electrons now let's go a little bit further about motivation for why we should care and a lot of interest has been applied or has been generated recently by a very simple uncertainty based principle based argument that tells you that there's a time and exactly characterizing that time and knowing the meaning of this is more complicated but it's easy to see that in the system where temperature is the only energy scale there's a time which is going to be of order h bar over kbt and people now have put and so this was the significance of this was mentioned in such death's famous book on quantum phase transitions and by some other people earlier as well very recently people are putting more theoretical flesh on the bones in terms of defining it as the most the shortest time by which you can equilibrate it's kind of a shortest equilibrium time or it's the shortest time of it's like the that this paper is in terms of the Lyapunov time in quantum chaos right but in in many ways that you look at this you get lots of observations and lots of theoretical calculations that tell you that this if you like there's a minimum time of relevance in a thermal system and it's given by that so that same idea in a few lines can be turned over to turn over to give you something about a bound on so so in other words if this becomes a bound in other words tar must be greater than or equal to that number that is equivalent to saying that the viscosity divided by the entropy density must be 1 upon 4 pi must be greater than 1 upon 4 pi times that ratio okay and most of you are from condensed matter this was a paper that came out of string theory and was first talking about the quark glue on plasma because when they measured the viscosity of the quark glue on plasma at very very strong coupling when the scattering was as high as they could make it they realized that that was extremely low number they were getting for the viscosity and they were they were interested about why not only is it a low number it's a number pretty close to this proposed bound so this is not a particularly well known paper in condensed matter physics but it's extremely well known across physics one of these papers that probably has a couple of thousand citations when I so that's not exactly minority work now that then goes that's where the interest then goes over from that field to electron systems because we have lots of ways now of sex selecting electron systems and putting them in circumstances where something like that minimum time is applying and therefore could it be that there's a minimum viscosity conjecture to go with the viscosity of electrons as well as other forms of matter so thoughts like that stimulated some fantastic theoretical papers which are you know extreme they're interesting they're thought provoking it's as I'm going to tell you not yet clear that we can do experiments to really test what they're saying but so but maybe one of them the paper on graphene which has a schmalion muller and fruits on it they made some concrete suggestions some of which do appear to have been seen now okay so so that generated this idea of going into very strange metallic fluids and looking for very low viscosities is pretty topical alongside that came the rebirth of lots of other theories which are saying we're not going to go to a strongly coupled fluid we're going to go to what these guys would think of as a weakly coupled fluid in other words a Fermi liquid so we're going to go to a Fermi liquid with well-defined quasi particles are we going to think about electron Fermi liquids and we're going to think about what the consequences would be if we could drive those electron quasi particles into a hydrodynamic regime so with all of this going on and some experimentalist being friends of some of these theorists I guess a number of groups believed in about 2014 that they were asking themselves whether electronic hydrodynamics was observable actually now that I know literature better I realized that we were re-asking that question but it's certainly been worth re-asking and we've managed to extend now the systems where it's being seen yeah question we'll get there so why is it a challenge why don't we just turn it on and do it and this next five slides is going to be attempting to answer that question the thing is that when you're passing an electrical current through your electron channel made of something the pipe isn't empty right and so in in these empty pipe things for water and helium it's been very easy to say to yourself okay all of these internal collisions of momentum conserving actually in a solid arguably exactly the reverse is true because solids have impurities and the the if you bounce off an impurity in the bulk of the solid you're straight away dumping your momentum out in the bulk rather than having to get to the boundaries and any viscous effect is real with the absolute start point of any viscosity measurement is that the boundaries have to be extremely important relative to the bulk okay but now it gets worse so I mean the impurity question is a really nasty one most of our materials are far too dirty but then we have to think about other forms of scattering as well that take place in solids so recapping to get resistivity in the traditional way you must relax the total momentum of the conduction electrons I'm now meaning an infinite sample without boundaries so impurities will do it as I said but also solids vibrate creating phonons and electrons can scatter from those phonons so one process you have is what's called a normal electron phonon scattering process where some electron on the Fermi surface will either absorb or emit a phonon the energetics means that that's a quasi elastic process and so you could certainly create resistivity with these normal electron phonon processes and in almost every case that's what you're doing almost always normal electron phonon processes are momentum relaxing and anything in bulk which is meant momentum relaxing is the enemy of hydronomics we'll come to the exception later now here we get what people regard as the good guy particularly in semiconductors and graphene electron electron scattering in so-called normal processes is just the analog of helium three quasi particles scattering from each other in helium three electrons scattering from each other in an isotropic system do not relax the electron assembly's momentum right and so if you're a hydro fan those are good things except that we then get into the problems with unclap scattering so unclap scattering is best thought of in the following so you imagine you have an electron sitting on the Fermi surface and you have some scatterer whether it's another electron or a phonon which is able to scatter through a mod k or mod q which is that radius that gives you a circle of allowed scattering events and if you were just in free space if you were of this radius because you need to conserve energy you could only scatter onto two points on the Fermi surface right however that's not true because in a solid you have crystal momentum so you can think you have to think about the Fermi surface in the repeated zone scheme and that same radius of circle can be scattering you from the first zone to the second zone so it's scattering you from there to those two points as well those two points are equivalent by periodicity to those two points so that unclap process even for electron electron scattering involves the reciprocal lattice and so it involves the crystal momentum and it becomes momentum relaxing so both the electron phonon unclap and electron electron unclap are always momentum relaxing so as we first went through the scorecard we have impurities normal phonons electron unclap and electron phonon unclap which are all trying to kill the observability of hydranomics and at the first count only electron electron normal processes are helping it I'm going to show you in a minute that there's one exception to that so this is part of why looking for the right regime is so difficult so given all of these momentum relaxing collisions let's go back to velocity profiles and if you calculate the velocity profile for a typical material from its boundaries right across the sample to the boundaries again yeah the boundaries will come in and give you a slightly lower velocity at the boundary but basically as soon as you've got any small distance away from the boundary scaled by the momentum relaxing mean free path then the velocity is just going to go to constant and it's going to be constant all the way across the wire and that constant velocity profile of course is implicit in ohms law so when you go and use ohms law to study conductivity as we all do that's what you're assuming that the viscous effects really aren't playing a big role and that the boundaries are far away and the other thing so then that brings out that ohms law is an empirically derived law it's not a fundamental law of conduction solids because if you change the parameters to ones where the electrons looked more hydrodynamic you could get a quadratic velocity profile of the same type as I showed you before so what do we do when and yeah I guess up here I would say 99.9999% of metals are in this regime and you know if you're a hydro guy that's a bit sad so how do we handle that well we do it in the usual way that you handle theory and condensed matter you say okay we just forget the outliers and we use theories to analyze our data which are based effectively on ohms law so in a Boltzmann approach it's saying we throw away all the interesting terms that you would have to think about in hydro and you only keep the ones that are in this normal standard case and that's a very dangerous thing when you're analyzing experiments because it's a very hard wired philosophy in the minds of experimentalists or condensed matter theorists thinking about experiments in condensed matter and basically you're in the usual case if you make an observation and you only you only analyze it with one kind of a theory you're not going to find out that another kind of a theory would would apply in until you generalize the way you're doing things but most of me you know there's this old joke I don't know if you know the guy that invented QCD multiple or got Nobel Prize for QCD Murray Gellman he refers to our whole field is squalid state physics that we are studying the physics of dirt no matter what we're trying to do right and you know these these are the very nasty joke for us but it's a bit close to the truth which is why it's a good one so in most of the squalid state here's where we sit so lucky for the development of the field there's always somewhere in the world somewhere in the literature the point 001% guy and this guy so it was a guy called Gurgi working in the Soviet literature working in Soviet Union and publishing in the early 60s and he began to think about he wrote these papers about because of these type of helium-3 effects the idea that in a Fermi liquid viscosity would go up again at low temperatures he wrote that you might find that in metals you got a resistivity minimum at very low temperatures because they were so pure because what you were seeing would be a mixture between ordinary omic flow and a hydrodynamic term right and he was probably actually thinking about the history he was probably not misled but he was probably motivated there because there was a huge amount of experimental study at the time of resistance minimum solids because of the condo effect so the condo paper is probably not long after that it's about the same time and you know it goes he was probably thinking well condo type physics isn't the only way that you might be able to get a minimum so those papers are certainly of historical interest and relevance and encourage you to read them if you if you like this field but what Gurgi really did was just to lay out in a very simple couple of pages the fact that instead of dealing only with with one mean free path and one device scale if you add the second mean free path one for momentum conserving collisions one for momentum relaxing ones and then your device scale you were likely to have enough parameters to construct a theory that would allow you to see the crossover as well between a hydrodynamic regime and an omic regime and obviously there's these two limits actually there are more than two which complicates things but we'll forget about that for today if you're in this usual regime where you're so squalid that LMR is very short then you're omic if you were able to engineer it this is great can't use my own technology right it's been one of those days this lecture nearly didn't happen because I got lost in that park sent Roderick an abusive email saying you need to send a helicopter to come and get me let's see right now can I get myself back to my yes I can right so if you can money if you can achieve this regime where the momentum conserving mean free path is much smaller than the other two in the problem then that would be electron hydrodynamics but for lots of very good materials reasons it's tough and if you're a sensible person we're going to see that my own research is not in this regime so that's why I can make that joke if you're a sensible person the place to try and do this is in semiconductors or semi metals for a range of reasons so the first thing is that heterodoping these techniques and high purity semiconductor electron gases they allow you to achieve very long mean free paths with very low impurity scattering the other thing is I didn't really labor it but if you go away and play with these diagrams for electron electron and clap you realize that if your Fermi surface is very small you can be below a critical size where no unclapped processes can take place no electron scattering can find the Fermi surface in the second zone because it's so far away on the scale of kf so in semiconductors you can suppress electron electron clap which is a very good thing and there are also tricks you can play to try and suppress the electron phonon scattering so you can play tricks to essentially get the electrons so much higher temperature than the background vibrational temperature so and then the fun went and this is really important because if you've got the electron gas up to a rather high temperature then you're getting up towards its Fermi temperature and you're hugely increasing the quasi particle quasi particle scattering rate by doing that so all of these things say there may be a chance particularly since in the semiconductor two decks you cannot they were the first system where experimentalists really developed these techniques of mesoscopic struck device fabrication so that led to a number of successes in the 1990s that were forgotten and have now been re-remembered so there are how there were hydrodynamic predictions of of Ted or hydrodynamic explanations or hydrodynamic theories were used to account for plasma oscillations in the terror hurts these were actually important papers for electrical engineering and they're very well cited in the electrical engineering community because at the time terror hurts radiation was a very difficult thing to get and work with but much more directly relevant to what I'm going to be telling you about the modern experiments were some fantastic experiments done by Lawrence Molencamp and his graduate student the young right back in the early 90s and they just did the semiconductor equivalent of the simple flow experiment that I've been telling you about so they made with gating and micro fabrication they made wires with W of order for microns with care impurity mean free paths of tens of microns can be achieved in semiconductors when I say with care I mean after 25 years development so and in only a few very specialized labs and then their key idea was that because they had one two-dimensional electron gas buried in a lattice of lots of you know the background gallium arsenide or the background silicon so they just have all the dopants and all this stuff and then this one rather pure layer they showed that you can put push a high current through that very pure layer and heat it up to considerably higher temperatures than the background so you try and keep your whole sample with a standard cooling thing at say 10 millik then you heat your electrons up to much higher than that and that's this trick of getting yourself quite close to the degeneracy temperature of a low-density system and therefore allowing lots of electron electron scattering so in the graphs I'll show you in a moment we won't talk much about this one I have nicer ones in a second what what you have then is this current axis because they're heating their electrons with the current they didn't know how to calibrate it in those days now they're doing noise spectroscopy where they can but from these 1990s paper that current axis the mod of the current is related to the electron temperature maybe not linearly but there's some relation and then the temperature that is that these these curves are being drawn on top of each other at different lattice temperatures so that's the overall thing how did you how what was the temperature your dilution fridge was at and then they studied the differential conduct and the differential resistance but don't worry that's just like the resistance so what they then did was to combine these and they got a lot of curves looking like so and basically what's happening here is that you're starting with some ordinary resistance and you notice that these effects are pretty small on the background right so because everywhere where people are studying hydrodynamics and electron systems they aren't studying real hydrodynamics they're studying the point at which viscosity begins to play a role in what you observe so you're always trying to observe a viscous effect beyond the background so what they're seeing is that they have a certain amount of resistance and then as they heat the wire up first of all the mean free path I yeah they heat the wire up and they create the usual thing that's happening of the resistance going up as you heat as you heat a wire I eat the electrons but as they're doing that the electron electron length is coming down so the things getting more viscous and eventually the resistance turns over so that as the wire gets hotter the resistivity goes down again and that's that like that Gurgi minimum so basically what's happening there is the wire is getting hotter so the viscosity is getting smaller as the viscosity gets smaller the resistance goes down the flow resistance goes down for the same given boundary conditions and what they did which was very appealing to us was they did a development of Boltzmann theory to go beyond and include beyond the standard which is called the relaxation time approximation and they then included higher order terms in the analysis which gave them sensitivity to be able to analyze hydrodynamic effects if they had them and within that theory they were able to publish pretty satisfactory qualitative agreement between the theory they were doing and the data they were taking I guess everybody's work this is a very young field and everybody's work certainly including our own work has it's hidden secrets and the hidden not so nice bits and the hidden not so nice bit of the young and Molochamps work I believe it must be correct I mean it all agreed too well but they don't actually physically understand their boundaries very well and anything that you're going to do with these hydrodynamic calculations whether it's based on Boltzmann equation or whether it's based on Navier Stokes you've still got a boundary condition in there you still have an assumption of how you relax the momentum of the boundaries and basically the one they've used there is the one that matches the data the best I think right and and you know one does always have to just think about that I think in any of these flow experiments you always have to be aware that there are hidden things going on about boundary conditions that aren't being discussed as openly as they might be. Now Jörg Schmalion is a deliciously open guy so he and his student Egor Kizilev have just published a really nice paper about all the hidden secrets of what boundary conditions can do to you in electron experiments. So I'm you know feeling like I've talked for a very long time even though it's only 35 minutes does anybody have any questions that are going to liven this up with is your question answered yet you know you say no okay good so you define you define hydrodynamic transport as you you do an experiment you get a data set you find features in that data set that you can model by taking into account the viscosity of the electron fluid instead of the resistivity see the resistivity that we're all used to is the property of the medium that the fluid is flowing through the viscosity is a property of the fluid and then they have fundamentally different signatures in principle in experiments so what you're always looking for is an experimental signal ideally that you can model with a viscous theory that you cannot model with a conventional one and the claim in this paper is that that turnover of resistivity where that resistivity is going down only by you know what 20 and 400 only by 5% that 5% correction as a function of this current is their evidence that they are getting into the hydrodynamic regime of transport okay good yes oh I can ask him then I'll ask you yeah yeah so okay so graphing there's two kinds of hydrodynamic claim because you have tuning we're just about to see that when you dope graphene to semi-metallic doping similar to the densities used in the semiconductor experiments you can see very similar things in the differential resistance of graphene you can also though try to tune graphene right towards the charge neutrality point you run into some disorder effects but there is a second regime of graphene where people have looked at thermal transport where they claim that they can see huge differences from Fermi liquid thermal transport that they associate with being right in at the charge neutrality point I'm going to mention those experiments but I'm not going to describe them to you in much detail what we are basically always consider considering today because it allows us to compare different systems our systems with at least semi-metallic densities why don't you do a very very good question a very good question for with the answer being just the thing I didn't include in the talk so I was trying to give you guys a chance at this time in the afternoon by saying everything's about omic and hydrodynamic but actually there's a third regime which is relevant to your question which is called the ballistic regime so if you make your wire say what you're wanting is a wire which has lots and lots of internal momentum relaxing collisions as soon as I draw on the board you see why I PowerPoint my slides right I and so that you've got a lot of scattering going on inside your wire on the scale of your width if you get into the regime where you make your wire so narrow that none of the internal length scales is shorter than the wire size then you're in a regime called a ballistic regime where you get these effects where an electron between scattering events is always going to find a boundary basically for almost every direction it's on okay and that's an entirely not an entirely but it's a an interesting but different non hydrodynamic regime of transport if you want it but a lot of these things are about words because in the old theory of gases so you know the original kinetic theories of viscosity were done by guys like Maxwell and people working for gases going through pipes and what we call the ballistic regime in electrons is called the Knudsen regime from everybody who studied hydrodynamics in a hydrodynamic system it's where one mean free path takes you to the edge when you get with this negative differential resistivity that's going to what's called the Poiseuil regime and the Poiseuil is the hydrodynamics I'm telling you about acceptable for now good other questions yep in in high-frequency experiments or what do you think what okay so my I'm a DC guy microwaves counts as high frequencies there are certainly hydrodynamic effects in fact one of our organizers I'll reference one of his papers on that later so yes you can think about what would happen at those frequencies in a in a hydrodynamic fluid but there's very little experiment so far because making a hydrodynamic fluid of electrons is so tough so hard yeah well you could but but what you could do is even worse you could go with the same geometry of the channel and then you can think of what is what happens when an electron meets the boundary right you've got two limits one well three two limits and in typical hydrodynamics textbooks they use sticky boundary conditions so they just basically say the velocity all of the x velocity is stopped when you get to the boundary right so the velocity is zero at the boundary but if you had totally specular scattering then you would have essentially no momentum dissipation at the boundary of the x momentum so the reality is always going to be somewhere in between this there is there are no quantities are all semi-classical calculations in the regimes where the experiments are being done they seem to be perfectly capable of matching what people see but that's the main motivation for saying that the quantum the quantum corrections aren't important I mean this is a very young field obviously that is an important next phase it's also probably an important next phase to start doing the experiments at lower temperatures than they've been done at already to try and see whether quantum corrections are observable experimentally to be honest experimentally we're still at the stage of seeing fairly small corrections to omic theory so you know where we're not there yet we're also not there in as I'll say at the end of the talk in going to most of the really strongly coupled fluids yes indeed you hope that you could yes and and particularly our theory colleagues are putting a lot of thought into doing that indeed probably yeah the issue is you then have to find ways of really being confident that when you fabricated your sample into those unusual shapes you haven't done anything to it as you've been as you've been doing that but certainly in our group working with Roderick mercenaries department next door you know ideas are coming up for ways that you might do precisely that enhancement and in some senses I'm going to show you an experiment done on graphene later which involved specially tailoring the ways that the blockages you made the fluid flow through that was more than I expected that's excellent so now I had better say a few more things before the break so yeah so this is the graphene equivalent of the 1994 the young and mowing camp experiment it's pretty cool though the geim group who published this only put it in their supplementary information they were saying okay you know just repeating what's gone on before in graphene isn't so interesting to us we want to discuss something really new and what they did was the following they studied a situation where they put current in the end of a bar and they sunk it over here and they studied the voltage as a function of distance for various devices as they moved away from that current sink but staying very close to it so this is a two micron scale bar by working with and that means that those contacts are probably a couple hundred nanometers wide and they're spaced by about a micron from each other they're managing to keep those separations of order of some of the microscopic line scales in graphene of the electrons and what they say and was also backed up by other independent theory by Levitov's group what they point out is that if you only have omic conduction and you just start solving Laplace's equation for what you done when you were injecting a current through a point contact the only way that the currents can flow would be to fan out like so however the neat argument is if you're in a hydrodynamic fluid the viscosity will allow some kind of backflow and certainly allows a non-local relationship between current and voltage and the prediction of theory is that as you can go away from this point and you study the voltage as a function of distance you should get a point where you get a negative voltage here essentially because of that backflow and the argument they make is that you can never get that negative voltage in an omic situation and therefore that this negative resistance measurement is an absolutely killer couldn't be wrong identification of electron hydrodynamics so they do extremely nice experiments this is one of those experiments where any of the old hand guys like Molenkamp who've worked on ballistics they know fine that there are ballistic effects that can also give you negative resistances so this paper is in you know exactly that thing of ignoring the third regime and it's not totally clear and I'll have a couple of things to say about that there would be some checks that maybe one would like to see done there but it's an extremely interesting experiment okay so my voice is beginning to die we're just before 10 to 5 why don't we come back at 5 past 5 and anybody who wants to ask me questions can do so in the meantime okay yes we better get moving again so we've reached this point and up to this point this is graphene but doped to be a semi-metal so what you're looking at is not it's deliberately joked tuned away from attempts to reach the charge neutrality point as I was advertising here's an experiment where what they tried to do was to get exactly to the charge neutrality point and then cool the system down and what they see here is I hope some of you remember there's a relationship in a Fermi liquid called the vitamin France law and that relates the electrical conductivity to the thermal conductivity it turns out and this is beyond what I'm going to talk about today that one of the predictions of hydrodynamic transport is that the vitamin France law is expected to be violated because charge and heat are transported differently in a system conforming to the hydrodynamic paradigms so what they did a very clever experiment in Philip Kim's group at Harvard and they found a very clever piece of graphics which always helps you sell your results where they show that blue here is the vitamin France law being obeyed and red is and yellow and light blue of violations with this nice deep red being the biggest violation so they're seeing about a factor of 20 violation of the vitamin France law in this region of their phase diagram very near the charge new round clustered around the charge neutrality point and at about 60 Kelvin and this then helps to illustrate some of the more general problems with the experimental field if they go down to lower temperatures impurity scattering well when I say impurity it's this charge puddle scattering that they get they get static disorder which kicks in and is and kills their observability of the of the vitamin France law violation they go to higher temperatures electron phonon scattering comes in and dominates the momentum and that momentum relaxing electron phonon scattering dominates the momentum conserving scattering so they actually have this hotspot for the observation of what they attribute to a hydrodynamic violation of standard transport and this is again general to hydrodynamic experiments it can be that you need to find just the right temperature and density window to see the effects that you're looking for now though I'd like to talk more for a while about our work and I'd like to tell you why it was kind of insane to think about trying an experiment here and then tell you about the get at the loophole that we were trying to exploit so it's insane to try it on a full methyl because the Fermi temperature will be very high and that means that at your experimental temperatures the electron electron mean free path will be extremely long say copper at 10 Kelvin the electron electron mean free path is centimeters so you know whereas what you're always wanting is a mean free path of hundreds of nanometers even if you could somehow find a way of getting that electron electron mean free path to be shorter by cranking up the electron electron scattering if your Fermi surface is big then the umklap processes for electro electron that I told you about will come in and spoil that as well so there's lots of reasons to not want to trust electron electron scattering to do what you want here and the expectation in a standard metal is that you'll always be stuck with the momentum conserving mean free path being much longer than the momentum relaxing one in other words standard metals you would think would always be stuck in the omic regime however we decided to try some experiments because we've been working in our group for a number of years now on a really surprising set of metals called the Delafosites that name is because there was a French crystallographer called Delafos and the series was named after him but nobody can say it that's you can tell whether you've entered this field or not whether you can say the name in the materials so the way the materials are is that they're very simple they have triangular lattice layers either in the metallic ones of palladium or platinum and in the simplest ones of all you have cobalt oxide octahedra which are completely non-magnetic just acting as separators to make the the metallic layers sit a long way apart and give you a two-dimensional material because they're two-dimensional and very pure they give fantastic photo emission and we've been able to categorize the Fermi surfaces really in a lot of detail with photo emission experiments they're always hexagonal cylinders very two-dimensional materials and the degree of faceting of the cylinders changes slightly from material to material but the whenever I'm looking for materials to work on in our research group the two criteria is whatever physics we're trying to study we want them to be very pure I want them to be very simple I'm going to show you how pure these are in a moment but this single hexagonal Fermi surface is more or less as simple as you can get in anything non-trivial so the reason we became interested in them comes from just looking at a chart of the room temperature resistivities of metals so here we have back to your textbook days the alkali series of metals with the nearly spherical Fermi surfaces these are all their resistivities turns out that the resistivity at room temperature of elemental platinum and elemental palladium are rather similar there are a couple of the most famous highly conducting oxides put in here but down at the bottom of the curve of the chart there are all the world's highest conducting metals at room temperature so they at the moment they include copper and silver coppers a bit cheaper one of the reasons we use copper in all our wires is that fantastic conductivity what's extremely surprising then and particularly to anybody who you know had a long history of working on oxides like me is to see two oxides in this cluster of extremely conducting metals what's even more surprising is that the way you relate resistivity to the microscopic mean free path always depends on the carrier density the carrier density is much lower per volume in these materials because they're layered so actually the carry volume carrier density is about a third lower than it is in copper or silver so in fact the proper metals you know one electron per can one conduction electron per atom the proper metals that we know of everything the Delafasites are the highest conductivity at room temperature per carrier which is not fully understood we're working on it but it's just straight away with to me it was a remarkable fact and I was aware that other people were work working on these materials only a few I thought okay we should be interested in these the next good point is that so the room temperature resistivity is extremely low because the conductivity is extremely high but as you go further down in temperature the room the resistivity falls even further such that you can do a low temperature resistivity measurement and you end up deducing that the mean free path for motion in the planes is about 20 to 50 microns absolutely huge and in the semiconductor business it took 30 years of refinement and special growth techniques to get to mean free path that long in these materials they're jumping out at you when you do a very simple crystal growth sometimes using a vapor sometimes using a liquid and we you know we shim shim Kim in our group grows these things nowadays almost on a daily basis because we're using a lot of them because we're doing many experiments on them so they're very true dimensional now there is some evidence which I'll discuss later in more depth for a phenomenon called phonon drag and phonon drag is the one exception that I was telling you to phonons being bad for hydrodynamics turns out that if you're dragging your phone on they can become good for hydrodynamics and that is what we were wondering about here the other thing experimental development to tell you about which is relatively recent is that the other thing you need to to do hydrodynamic or ballistic experiments is the ability to make mesoscopic devices from the materials of interest for a long time you could only do that reliably with semiconductor to decks then graphene came along and great techniques were developed for doing that now people are being able to show that focused on beam sculpting is a way of taking as grown single crystals of almost anything and making exactly the device geometry that you want from them so there's an as grown crystal of palladium cobaltate it's hexagonal it's a triangular lattice so it has these hexagonal type facets there's another one you can see the outline of it but everything else here what's in dark is the crystal after you've cut away the crystal what's in light sorry is the crystal after you've cut away the crystal to give you all these almost black trenches so what you what we've created there what Nabanila Nandi created is current injections through a special meander which is for technical reasons into a bar a channel of a defined width we have eight voltage contacts on that channel so we can do multi contact measurement both of the magneto resistance if we apply a field and of the Hall effect but you know the points are general one it's really now technically becoming feasible to do experiments on interesting materials in ways that were impossible even 10 years ago and that's a great development so if you measure if you look very carefully at the resistivity of palladium cobaltate at low temperatures this is what you see now so you see that the resistivity if anything it may even drift up a little bit at low temperatures which isn't implausible in a hydrodynamic situation but by and large what you would say is it's more or less completely temperature independent below 10 Kelvin and then it turns on and although the difference is small if you're doing low noise measurement you can very clearly distinguish what the resistivity is really doing there from the red dotted line the red dotted line is the T to the fifth that you would have in what's called the black Runeisen law that many of you will have come across as undergraduates actually the function that you're able to fit through the resistivity is exponential and if you just do that blindly you fit out a characteristic temperature for your exponential meaning that you have some sort of a gapped process of scattering you get out a temperature T naught which would be about 165 Kelvin and the reason that this is happening is a very interesting one it actually goes back to an argument that Piles had with block in the 1930s so a blocking grenades or a block postulated the standard laws for the electron phonon resistivity of metals and that's involving a really big assumption because it's saying that you have your electrons scattering from your phonons just an equilibrium the materials sitting there then you apply a voltage and you give the electron assembly some momentum the assumption of the block Runeisen law is that the electrons are scattering off the phonons while this is happening but without giving any of that net momentum to the phonon assembly so the phonons are staying at zero net momentum decoupled enough from the electrons that they can equilibrate but coupled enough that they can scatter the electrons and when block wrote that down in the 30s Piles straight away said that isn't what's going to happen if the electrons are scattering strongly from the phonons the phonons will themselves gather some momentum from the electrons the electrons going out of equilibrium will drag the phonons with them and that effect of phonon drag was looked at looked for in many compounds it's been quite easy to see it in thermal transport but it's been very difficult to see it in electrical transport partly because the characteristic temperature associated with the drag is much lower in most materials than it is here but let me explain that for you a moment so let's imagine that we are dragging our phonons perfectly so that all of the electron phonon scattering that's just normal electron phonon scattering around our Fermi surface just results in dragged phonons and no observable contribution to bulk resistivity because the phonons just get the momentum with the electrons then the first electron phonon process that you're going to see will be an unclapped one when the phonon has enough wave vector to scatter you across the zone into the second zone to give you an unclapped process you can calculate the characteristic energy for that if you know the gap in case space and you know the sound velocity you can then calculate an unclapped temperature and when we did that and we compared it to our measured t0 it came out numerically very similar so that combined with the exponential onset makes us wonder whether we're dragging our phonons that with the normal electron phonon processes involve phonon drag and that's important because those dragged phonons are suddenly the friends of hydrodynamics normally phonons are just the enemy of hydrodynamics but dragged ones are the friends of hydrodynamics because that coupled electron phonon soup conserves its own internal momentum and that's exactly what you want for the for getting a combined viscosity from that soup that's also something that have been considered theoretically in the 60s by Gurgi it's one of the regimes he wrote papers about of course being honest it's always the best thing we were not looking for this I had never heard of piles as argument I'd never thought about phonon drag in my life we just made the empirical observation that we couldn't fit the traditional function through our data and we could fit a different one and then we started thinking about it yes yes okay so you yes you can do it or what you can do is you can check rather than that sort of fit you can always plot in log log to see whether what you've got is fundamentally a power law of any power or not and that's essentially what you're asking so even if we varied the power away from five and left that as a fitting parameter as well we still can't fit the data properly with any power yeah so with that in mind that was why I thought okay we have a very strange material here electron hydrodynamics is looking interesting at that point there were no modern experiments the graphene stuff hadn't been published thought why don't we try an experiment and the type of experiment that we were thinking of is motivated by the following here you have kinetic calculations giving you three different predictions the color scheme is pretty horrible but just look at the lines if you have lmc a thousand times lmr then you're just then you're in the ballistic regime right because for a given wire size whereas if you have lmc five thousandths of lmr you'll be in the hydrodynamic regime and the difference comes that the hydrodynamic regime shows this curved onset and quite a low resistivity at low values of this ordinate which is a dimensionless one that I think I shouldn't spend ages explaining and the ballistic one does two things it's higher than the hydrodynamic prediction for relatively wide wires when w is quite small and then it crosses to be lower than the hydrodynamic prediction at very narrow wires and before anybody asks that that's because in a narrow wire you will always have some electrons which are just going straight along the wire and if they're going straight along with a very long mean free path they short out the others so we thought we'll try that type of experiment the first thing we did was to take our crystals this is about six years ago now and Philip Marl sculpted them into meanders so that we could check whether we could see the Schumann Coffta has effect in them Schumann Coffta has effect is extremely purity sensitive the question there was could we do all this sculpting to those samples and leave them with a very long mean free path we now have 10 or 15 different ways of saying that we can do that so then we just set out to do the simplest possible experiment we created a channel we measured it it's flow resistance in other words it's electrical resistance for a certain width then we have that measured it again then we have that measured again etc and the expert the results that we came up with with the following so the red line is the prediction of hydrodynamic theory and the blue line is the prediction of omic and embolistic theory and the data clearly looked to be agreeing better with the red line than the blue line that in itself isn't so impressive but it gets a little bit better when you consider the blow up because the blow up predicts the hydrodynamic prediction as I'd said before is of some upward curvature at low width followed by the hydrodynamic prediction crossing the ballistic prediction at high for high width sorry and at very low width it crosses we see all of that we see the curvature we see the crossing point around the expected result and we see that the different values for the very low width wires all of that together was you know whether people wish to believe that line of argument or not that comparison between kinetic theory and experiment led us to say we're seeing a signal here which can be better described by including a viscous term than not and it's just what you were asking about early again it's a it's a relatively small extra effect which is being seen that we believe there's a hydrodynamic explanation for so we published as well and actually the two graphene experiments and ours were all done totally independently I don't think we knew about them when they were all published in the same issue yes yeah so the width we did we went 60 30 15 8 4 2 1 and then we actually had a point at 700 nanometers width as well yeah this is only one of this is the first cut then he kept on cutting so by the end you had something where everything had been blasted away and you just had a very narrow wire left for each data the thing came out of the Christ that went back into the focused on beam instrument and got cut again yeah same crystal so that was our experiment and if you notice in the analysis of it one of the things is that the conductivity of our wire in the hydrodynamic regime is predicted or the resistivity is predicted to be lower than the ballistic resistivity for a given width over some range of widths so a really nice experiment done by Gimes group in the same spirit a couple of years later was they said okay we won't go for a whole wire what we're going to do is we're going to and this is back to one of the questions I was asked from somebody at the back earlier on different geometries they decided to just make constrictions and they made these constrictions of different widths and they studied at different temperatures and the idea there is that the correlations in a hydrodynamic fluid can actually in some circumstances make that hydrodynamic fluid flow through that constriction more effectively than a ballistic one would so the viscosity gives some channeling is the idea and what they publish is a paper saying and this was based on theory from levittorff who tends to be right on these things that you can do a ballistic calculation you would think that a purely ballistic super pure material with no viscous effects would be the most efficient way that you could get current to flow through those constrictions but he pointed out that because of this viscous channeling it isn't the viscous fluid beats the ballistic limit so you end up actually getting higher conduct conductance through those those restrictions because of the hydrodynamic effects than you would have had in the best sample you could have thought of and that potentially has technological ramifications as well as pure physics ones and so that was now there's always assumption everybody's experiment is measuring a signal analyzing the theory and then you know coming up with some claim one of the really impressive things that they can do in geime's experiments though is that because they're working at density levels where they think that electron electron scattering is absolutely the dominant source of their hydrodynamic scattering they can then back out the electron electron scattering length from the analysis they do and plot it as a function of temperature and they then did not just a total Fermi liquid theory they're a Fermi liquid theory with some corrections and they claim but anyway even so what they are seeing is that electron electron scattering length diverging not so far from one over t squared and that is exactly the divergence that was seen in the viscosity of helium three right back at the beginning of the first part of the talk so this is a very important self-consistency check you're doing these comparisons with theory but you then end up fitting out a parameter and that parameter should self-consistently check whether the assumptions that you put into the theory look reasonable and getting that t squared i think is a really important point and you know the specialists may want to quibble with some other parts of the analysis of this experiment but i think that point alone tells you that this result must be correct now we can't do that because we're saying that phonon drag is the source of our collisions the the phonon drag length is extremely difficult to calculate as a function of temperature because you're making a crossover from normal to unclap events you don't have a circular Fermi surface so we can get something which looks temperature dependent not so far off these kind of temperature dependencies but we don't have an easy theory to compare it to so the next then the next obvious line of investigation for experimentalists is to say what happens when you turn on a magnetic field that's because we all have magnetic fields in our labs it's not anything and because we can control them very precisely i wouldn't say it's anything more cerebral than that right what you could say though is that in classical hydrodynamics creating a hydrodynamic fluid that breaks time reversal symmetry is extremely difficult right you have to work very hard to do that all you have to do with an electron system is turn on a field and then you're studying hydrodynamics in the presence of time reversal breaking aha this is one of these great things where it's only visible on the screen so at the time that people were thinking about turning on fields there were no calculations of what you what the predictions of a hydrodynamic either Navier-Stokes theory or kinetics theory were going to be so those were done first of all by Alex Seyf in 2016 and then Thomas graffiti based on discussions but not much more with the rest of us did a very nice Navier-Stokes and kinetic solution of this problem so he is the guy in the world to now go to for what the hydrodynamic and ballistic predictions for an electron fluid are so we then went away and did some experiments to test what what he'd been coming up with and if we go these are deliberately chosen to be quite high temperature experiments to try to get our supposed electron phonon length down nice and short that means though that all the internal length scales are very small compared to the width of our devices so it means that the experimental signal is small but everything looked very exciting this is what we got experimentally and this is what he could do with you know minimal parameter fudging in a Navier-Stokes calculation so qualitatively the data and the theory looked to be an extremely good agreement and we came within moments of sending a triumphant paper out about this about four months ago he'd worked on the Hall effect which is another thing there's a viscous Hall effect which is now well understood theoretically Nabanila Nandi who did the magneto resistance experiments was also doing Hall effect ones the signal to noise is poorer for various reasons there but again the modified Navier-Stokes approach field modified Navier-Stokes approach can get a pretty good agreement with the type of things you see so we thought there's only one more thing to test and this is this is I think really worth if you if you're going to get into electron hydronomic calculations you have to be very careful if you use a Navier-Stokes based approach you can still put some momentum relaxing scattering in as a phenomenological parameter so you're very well capable of studying the crossover from hydrodynamic to ohmic behavior however that whole framework is not capable of including this ballistic regime that I had to admit to when you asked me and so what we thought was we really need to check whether is this this is clearly a signal consistent with hydrodynamics but is it a unique signature of hydrodynamics or could it also be explained by other effects to my knowledge the only way you can do that is with kinetic calculations they're very hard to do numerically in these geometries and in these range of parameters finally Thomas got it going and it had a very annoying result because it showed that you can completely turn off the viscous effect you'll change the numbers a bit but you don't change the fundamental shapes of either this or your predicted magneto resistance so that's actually quite depressing because that says that you can do experiments that if you want to be optimistic might be hydrodynamics these field experiments but you can't say they definitely are and that's of course what everybody would like to do so we've actually not published any of this yet we're just going to you know I don't know lick our wounds and then write a very honest paper trying to explain this point so yeah there's the conclusion of that side we may be seeing a hydrodynamic signal but we're not sure it's very sad now the graphene guys Gaim's energetic group have been doing these kind of things as well and they've tried to measure the the viscous Hall effect in a slightly different geometry to the one we were using they have not been subject to the same kind of naval gazing cycle that we've been subjecting ourselves to so they've gone to press saying we've measured the viscous Hall effect and again what they do have is this time over a much narrower range of temperatures they can fit out the green points and they say okay over this narrow range of temperatures they're going up it's not so different from theory you know there's a self-consistency check being applied but because the range of temperatures is much lower I think the confidence that you can give that self-consistency check is a little bit lower and here's what's come to concern us about these type of geometries that they're using see they're using these geometries and they're using analysis techniques which are only capable of looking at the hydrodynamic to omit crossover we for very naturally once they'd publish this beautiful negative resistance experiment we wanted to check the negative resistance experiments on our samples so this is palladium cobaltate made to very similar overall shapes and dimensions to theirs and we were delighted to see we're doing some field experiments here but the zero field result is what counts at zero field we are seeing negative resistance and we're seeing the size of the negative resistance go down as we make the contact separation from the contact point bigger and that's exactly what a viscous theory predicts so again we could easily have gone to press and say okay fantastic we can confirm this result from graphene however what happened to us was that just accidentally we started doing the same experiments on bigger samples or different samples and we didn't always see the negative resistance so we started wondering whether the negative resistance that we're seeing was due to details of the size of this block of sample and we can do a very direct check of that we could make a big sample Maya Bachman did this and then she made the same big sample small with the same contact and again to our sadness she's able to make the negative resistance seen at zero field go away and the sign of that size change is all wrong if you had seen the negative resistance in the big sample and then seen it gone away in the small sample you'd be inclined to say oh there's some ballistic effect in this small sample which is destroying my lovely hydrodynamic signal but when you see it this way around I think you really have to worry we certainly are worried we don't really know why it's happening but we think it might be a ballistic effect and what we don't know but it would be interesting to know is how much sample size checking has gone on the graphene work all right however the graphene work does have these back calculations to the temperature dependent mean free path and they look quite convincing so I don't want to be giving you the impression actually I don't think experimental groups in our field tend to criticize each other right and that's not always a good thing because we're all exploring something new we're trying to advance stuff I'm just trying to give a rational analysis of what could be wrong with our experiments and what could be worried about with other people's ones it's not that I'm trying to say that the graphene guys aren't doing a really nice job I think they are and if I were a betting man I would say that the evidence for hydrodynamic flow of electrons is far higher in graphene than it is in our experiments you know some of our stuff might be due to other things so where does this now these were the details let's broaden out a little bit to the bigger picture again this is one of these amazing things you can find on the internet now right you want to know about where do electrons sit in viscosity compared to other things I even refereed a paper the other day with a graph similar to this the x-axis here is chemical it's a completely meaningless x-axis my students won't even show this plot they have re-plotted it into something they could defend right and I received a paper with an axis as meaningless on it it's really terrible so here are all these classical fluids you will know that olive oil is more viscous than water you've got a good feeling for that you'll probably know if you did chemistry at school that ethyl ether is a lot less viscous right and now if you're a cryogenic person you'll know from looking at it that liquid nitrogen is extremely non-viscous and it actually sits down here liquid helium which you can't see even at 3 kelvin as much is about as viscous as water if you go down to 2 mK liquid helium as I told you before is like glycerol or motor oil it's much more viscous if you put the electron experiments on the two graphene ones I mean the flow experiments and the charge neutrality point experiments sit about there are the two deg stuff sits there are sits in the middle they're extremely viscous their kinetic viscosities are like those of honey right now there is an issue here though because engineers and people and in our everyday life we kind of go smoothly to and fro between shear viscosity that includes mass density and kinetic viscosity that doesn't in all of the fluids we're looking at that doesn't really matter because the density of everything is order of magnitude the same right of course the density the mass density of an electron fluid is 2,000 times smaller than that of a similar fluid of something else so getting a feeling in your mind for what that statement that electrons are as viscous as honey really means is maybe you can't just use what you perceive with the fluids you see to scare yourself very well I think but anyway there's the numbers something that I will cut off in the bud before anybody asks me could you have turbulent electrons in principle maybe you could but our Reynolds numbers at the moment are extremely low so the the viscosities mean that it's going to be very different difficult unless you can pass really a huge current density through something you have a chance of doing that in graphene if you work harder your cryostat and I'm hearing stories that pre turbulent non non linearities are being seen in graphene at very high currents so the kind of experiments that we're doing there's no chance of doing it as I mentioned there's work now going on in general on high frequency effects understand the question I was being asked earlier was slightly different but Rodrich one of your organizers has been working on these kind of things and has a paper just published this year about if you could image the high frequency ac conductivity as a function of position there would be things you could see and other stuff in that paper too that he could tell you about so then we I started the motivation by saying we were interested in seeing whether electrons could be starting to probe this proposed low viscosity bound are we doing that yet and the answer is no the most the least viscous electrons that we have are very near the charge neutrality point in graphene and they are still about a factor of 100 when you take into account that 4 pi away from this bound the other ones are more viscous still so at the moment and the trouble is you can't really get into the charge neutrality point at low temperatures where you would like to do it because of this impurity stuff so there's you know it may be possible to improve there it may also be possible to improve by going to other Dirac systems but Dirac systems where the fundamental bandwidth or the band gradient at the crossings is much smaller graphene is almost the free electron number and that very high velocity is not good for you here in getting the Reynolds numbers to where you want so it might be that you can not the Reynolds number sorry the viscosity low so it could be that that's going to be accessible in future who knows and then i'll just leave you with this final point when i talk to you about resistivity of correlator or interesting materials saturating giving you a time that saturated the bounded time what you find is that whenever you have a t linear resistivity which we're now able to produce in many unconventional superconductors and quantum critical systems in many circumstances very surprisingly you can do an analysis in fact we did that analysis first that showed that across a whole range of materials the scattering rate associated with that t linear resistivity seems to be bounded all right so that means that you might have in these type of materials the fastest scattering going on that you could ever get with electrons so the obvious thing to say is let's go to those and see if that's associated with low viscosity hydrodynamics unfortunately though all of the internal length scales the microscopic length scales are extremely small there and that means that you would have to go to fabrication of extremely small devices which are smaller than the current capability of experiment so that may become possible in the future i guess we have some ideas of how to do the experiments if we improve the technology we'll need to see how long it takes and maybe somebody will find some other smart way of using a different trick than just size restricted samples the other route of growth is to maybe go in future into systems which have three-dimensional electronic structure instead of two-dimensional ones now if you could do that and you could set up vorticity as we know from water and a swimming pool there's a tremendous extra richness vortex loops and all of these structures of vorticity that exist in 3d that you can't get in 2d so you wonder could you set up hydrodynamics in very pure metals with three-dimensional dispersions and the type of materials which are really being thought of for that of the so-called vial semi-metals so these vial semi-metals which are being studied a lot for other reasons at the moment could also be excellent three-dimensional hydrodynamic candidates because they have three-dimensional electronic structure so all of that is again well it's not for all for the future some of my colleagues are establishing that this phonon drag idea may well be occurring in in vial semi-metals so it could be possible to go to strongly dragging systems at very low temperatures and see hydrodynamic effects and there even some some claims that's been done already so it's hard to judge how quickly you're going to speak you probably won't hate me for finishing early here are the conclusions of these two talks i can't get bored reading conclusions out if the talk's been clear you'll you can read them yourselves and they'll be clear so i'll stop there and thank you for your attention at this late time and take any questions you have