 You hear me all right? Yeah, so a really great pleasure to be here and Have the opportunity to talk about talk about recent work and Also to you know talk about Boris let me good that he's not here So Yeah, so let me start with telling you about what I discovered last week I Have a friend who asked me you know where where you're going and I said you're going to Boris birthday She you know said Boris birthday. She laughed at me and said that There's no way you you're kidding. They cannot be a person Who is named Boris Boris is a Everyone knows Boris the name of fictional character an animal from a book and I I mean I didn't believe her first because she's a little strange but But then I I looked it up and indeed and indeed Stranges as she is there is there is a book that everyone knows and except me Which you know, which is this one and in this book there is a character named Boris and This is an owl and the Owl that when I saw it, I thought you know it looks strikingly familiar and in some ways and when I Looked closer it turned out that you know even more familiar than I thought because because this is our Well, it's a nocturnal bird nocturnal bird which is a Antiquarian bookseller in a book and it gives instructions to you know to bugs and to To to butterflies where the butterflies Yeah, the butterflies and it got got assistance a lizard and professor leap frog and Yeah, so and the whole the whole setting looks very familiar so Looking close and close I I I got convinced that there is indeed, you know something here about Boris and And you know finally I looked at the publication date and I discovered that publication date Also agrees, you know published on the same here as Boris's first paper and so and and so so that that means that now we Know Boris is true nature and That's that's the That's my big discovery. I wanted to share with you Yeah, and so now You know, let me talk about less important things So, yeah, so so the story Today is is about how to use Gherkin to create Various interesting charge neutral modes and then you know excite them and it detects them Using non-local non-local measurements and there will be two parts to the story Part one is about valley currents. So valley currents is Is something that arises when you have a band structure with with several several valleys like here in graphene or here in silicon or in in Decalcide unites if you have valleys you can imagine the situation that carries on the valleys are in use to move in the opposite directions and then Then there is a current, but it doesn't transport charge it transmits transmits a valley index So it's like spin current in materials with spin orbit Charge neutral, but it you know, it's a real real current of something some quantum numbers And you if you if you have control over it, you can use it to you know to do interesting things And maybe potentially useful things And so that's that's the appealing appealing thing here another appealing thing here is that in In in graphene and materials of that family You can as we'll discuss shortly you you can induce Berry's curvature and if you have Berry's curvature you can control this current by Valley whole effect which arises without Magnetic field just because there is Berry's curvature you get the whole effect in in this case something that you don't get in silicon For example because in silicon there is no Berry's curvature in a band structure and and so so there is no Option to create valley currents like this So if you so this is an idea in one slide if you have two valleys that that Have Berry's curvature then You know red means positive curvature blue negative curvature if you are if you have your Fermi level here somewhere then then Positive curvature means that you will have if you apply electric field carriers will move in a skewed way to the right and Negative curvature means they will move in a skewed way to the left and if you have both valleys You apply electric field then as you can see there will be transverse to the field motion To one direction one valley the other actually that Charge charge currents will cancel and and valley currents will double So there will be a valley current Transverse to electric field and that's that's valley whole current and if you if you if you write it Just write what I said then it gives a relation between valley current and transverse electric field and Proportality constant called valley whole conductivity and now if you do this using equations of motion then what you you know What you get is is quite interesting you in addition to the to the ordinary quasi particle dynamics that we write deriving velocity as a group velocity from dispersion relation and writing Lawrence force equation for momentum you get an extra term which is called the normals velocity that that It's proportional to berries curvature and as you can see there is striking similarity Striking symmetry that it it creates to the equations of motion playing the same role in momentum space as magnetic field plays in position space And so with berries curvature things become Looking very nice and and symmetric and so that's that's that's that's another Reason this this is appealing So, okay, so now going on to to discuss how we create berries curvature and graphene Want to mention collaborators students from MIT and collaborators in UK and on the on the last part collaborate the Grisha Falcovich from Witesville Institute so so to create berries curvature we need to We need the logical bands and the logical bands are something which is familiar in topological materials materials that that have bands that have Bear berries connection and curl of that as berries curvature and the berries curvature flux Threading the surface is a non-zero integer Chair number than it's a topologically material band and these materials will support topologically protected states and Graphing pristine graphene is not like that But it turns out that you can combine graphene with other two-dimensional materials and build Topological materials and so that's what we'll be trying to do the reason this is interesting is that graphene has excellent transport properties and so you can You can benefit from that if you add topological properties on top then you can get the material which is you know excellent and Many different ways and that that will be That will be quite interesting So Yeah, so let's go on so sorry Great, so so just a quick reminder what you know berries curvature if I have if I have block states in you know in some band I can consider berries connection that Standard vehicle when introduced when deriving deriving berries phase and Derive take a curl of that that will be berries berries curvature And I want I want this quantity to be non-zero in in graphene We we do have berries phase. We know that if we take berries connection It defines the vector potential in graphene brilliance on which has non-zero Circulation about every direct point and so there is there is berries phase. However, it's curl less it has zero curl and so there is no berries curvature and That's that's a problem, but it can be If we fix it then we'll be able to benefit in Several ways so Right, so so in order to fix it What we want to do is to open to open a gap and graphene spectrum and if we do that Then then near each Dirac point with gap opened up. There'll be a pocket. There'll be a pocket of berries curvature and and Then we'll have this is you the magnetic field in you know in the absence of real magnetic field in the system With in which there is time reversal symmetry. So so this is I think this is very familiar in From other systems for example from the From the anomalous whole effect that people Studied for a long time on magnetic materials the difference here However, is that is that here berries curvature arises without breaking time reversal symmetry and magnetic materials due to spin orbit With magnetic field with time reversal symmetry broken by by magnetic polarization You you don't have time reversal symmetry constraint here here you do because of that berries curvature is non-zero But it's such that it cancels Contributions and valley k and valley k prime Opposite opposite and so they cancel each other okay, and so if you take a full graphene band and compute Sharon and Varian because of this Because of this time reversal constraint, it will be zero. However, if we as we'll see we can define valley mini bands and and in this case for each valley, there will be a you know Non-zero churn and variant for a valley mini band and so that's what we want to do Practically this is done as follows we can take graphene and pair it up with another atomically thin material boron nitride and if you put one on top of the other they they They are nearly let us match so that they form a structure which is which is nearly in register There is a slight Incommensurability, so there is a long period modulation and this long period modulation Take slightly different forms depending on the twist angle if if there is a non-zero twist angle Then it's a roughly sinusoidal modulation corresponding to some kind of more rare Structure that I will see the next slide if it's a if it's an axis aligned Arrangement then then you get commensurate commensurate domains and and the main walls in in in between and so in these two yeah, so they have two different states controlled by the twist angle and Change my position. Yeah, so so you get You get more a type structure at non-zero twist angle and commensurate structure zero twist angle and if we if we now compute mini bands and focus on you know focus on one valley We will discover the following you can construct Can construct the mini band theory very easily under being you know different people People working on it including our chairman and and and many others so you can you can Using the fact that a long period modulation is much longer way length than lettuce constant you can You can benefit by going to continuum theory and and working near the vicinity of which drug point Considering continuum drug model in in the presence of a long period modulation and then Taking taking account of what kind of modulation you create in each case more rare or commensurate you can determine the side the value and the relative sign of these coefficients and If you do all that and calculate berries curvature in the incommensurate case you get You get berries curvature Which is sign changing in the mini brilliant zone So this is a pocket of berries curvature direct point and these are opposite sign pockets and in at the corners of mini brilliant zone And in the commensurate case berries curvature is not in the incommensurate case It's not changing signs So if you integrate berries curvature in the first case you get zero in this case you get one so this is topologically on trivial band and so So not very important for what I'm going to say next But it's interesting to mention that that in this case we have we have two types of structures with with Non-topological and topological bands and they there is a transition which is tunable tunable by by twist angle And so now we you know now we have that and now near each near each direct point We have a pocket We have a pocket of berries curvature and we can now play our our game of inducing valley currents And we have valley whole coefficient We can now we can relate that to berries curvature and so it's all under control So how we excite currents and detect them The so here is the idea what what you want what you want to do is to you know make sure that your Charge neutral currents don't interfere with charge currents and the easiest way to do it is to use non-local geometry imagine a long bar much longer than much longer than this one and and Then we have a pair of contacts here where we apply charge current and then another pair of contacts at the remote location where we measure voltage if my system has valley whole conductivity charge Charge current will induce transverse valley current which will be charge neutral because charge neutral it can propagate Without any charge build up or electric field build up As long as inter valley relaxation allows it to go in the valley relaxation is quite slow and graphene so you can you can you can put this many many microns away and Then then here there will be no absolutely no charge currents but there will be valley currents and then these valley currents by By the reverse valley whole effect at this pair of contacts will produce a voltage by which you read out read out Presence of the valley current. So that's that's the idea and Can make a loose analogy with with With the radio we have charge circuit by which you excite charge neutral Photons and this charge neutral photons on the other end are detected by charge circuit And so that's that's the idea of the experiment and experiment has been done two years ago Manchester Geometry Very similar to what I showed so it's a long bar with which is about 15 microns long in this case with with many pairs of contacts and what you can use one pair of contact to to apply current and another pair of contacts to detect voltage and What you observe is a signal that as a function of carrier density which you consume by Changing voltage in a bag gate Shows Shows these peaks the red red curve is the non-local voltage Shows peaks which are aligned with peaks in in in resistivity which appear Which appear at the densities where you know very very phase Berry curvature pockets are Situated and so from that we see that non-local response only arises where at the at the points where our Fermi level is near lines up with with berries curvature and The geometry is such that if you estimate the you know effect of stray on the currents That is very small. There are always some currents that that are transmitted by Current lines that move very very far away From the nominal pathway, but they're their contribution to voltages smalls this exponential well-known to people in transport and So the the distance it exponential of pi times distance over width Racial and in this case, you know distance over width ratio is about five at least and so times pi makes Exponential negative power makes it a really small number. So so it can be excluded And so we because of this because of you know modeling that we did and you know various other other reasons maybe Yeah Not much time to go into it. We're pretty pretty certain that we we're seeing here non-local response a long-range response due to due to the fact that That I described One can also by measuring at a different pairs of contacts one can also determine length over which Decays on the length length in this case characteristic length in this case is about a micron Which in this device coincides with it with its width So that that would be consistent with valley valley scattering occurring when whenever carriers heat heat the boundary and Yeah, so that's and I mean there is a checklist of what what you expect For valley currents and what you don't expect and we went all over it and I'll be happy to discuss And and it all all agrees you you expect from for valley current You expect to see a scaling of valley whole response, which is which goes as cube of resistivity and so that's that's That's a check on that and and there is a reasonably good agreement And you know a few other things so so the conclusion here is that we we have a way to induce charge neutral Currents and detect them by by the local response. And so this is a bonus bonus picture Top gate which which is added on top of this device can be used to shut shut this non-local signal on and off and And the sensitivity to the top gate voltage is pretty pretty strong and you can shut it down fully by applying a very strong very weak potential on that top gate Right and so this is a little bit like for you know for people who who are who know spin tronics. It's a little bit like the Proposal that the dust proposal of a spin transistor where you put put top gate put a gate to control spin orbit interaction and you know spin rotation and So so this is the same thing essentially from symmetry point of view It's the same thing about for valley currents except that it This thing works much better in in in the case of that of dust Proposal, I think the the swing was never in the experiments the best experiments that That that that that are published which which are these the swing was never more than you know 50% and here It's at least a factor of a hundred so Everything works very well on perfume. Well, I mean the Valley quantum number is more decoupled from Everything there is no There's no analog of Zeeman interaction. No analog of you know spin orbit interaction. No So so there I mean there are weak effects that Disorder would produce into valleys get rain. That's the only thing So if you if your system was very clean, then you are you're you're safe And it appears that that it is very clean So this I mean just maybe for curiosity. It's been repeated in other systems For example in graphene bilayer, you can open a totally different system No, no more nitride substrate and graphene bilayer can open a gap by applying transverse electric field and then at that bilayer direct point you you also get berries curvature and then you Similar effect and in graphene bilayer you see an even better non-local response going over You know twice as long as three times as long as here, right? So so yeah, so Summary for this part is that we have we have a system where we can produce valley currents charge neutral currents and they Because they can be quite being charged neutral. They can mediate non-local electrical response We can use valley whole effect to excite them and detect them and And in this case we use graphene superlattice as as a platform by Creasing block curvature in medium bands and it's you know, it's an interesting system because That particular system is interesting because we have topological and non-topological bends So if you manage if you manage to find to create a system where because of an homogeneity or something else that we Stangle will be changing Then you you you will have a system with with domains where you know Where bends are topological and neighboring domains non-topological and then at the boundaries of between these domains, they'll be topologically induced age states and so that's that's maybe the way the way to go from here Okay, so let me it's roughly yeah, right, so let me now switch to a different kind of neutral mode so so Electron viscosity and maybe start with this discussion of hydrodynamics and in in electron systems and when when is it relevant and What do we how do we test it? you know subject to a long story even if I if I first think about Systems other than the electron systems and everyday life. We know very well that hydrodynamics is Is is really very very relevant and everything around us a few especially if you go to the to the beach Everything everything there the boats the waves the winds the sand everything is controlled controlled by by by hydrodynamics so this question doesn't do and doesn't even arise at this point but if you really want to think about it then then the hydrodynamics emerges when when you have a system like a fluid or a gas where where collisions between molecules or atoms happen very quickly and and They are momentum conserving and energy conserving so you get you get conservation laws of energy momentum which are not You know particle specific momentum is very quickly energy are very quickly detached From from from individual particles and become collective variables And then they have their own life as collective variables and that life is described by by hydrodynamics and If you put it in a mathematical form you you have to introduce Quantities that control it Discosities and thermal conductivity and they basically everything everything you see in this picture Can be can be described in principle can be described if you know these three three three quantities, right? so So what about what about electron systems? So in electrons in electron systems You know, we do have electron electron collisions, which are I make many cases Have a very fast rate. However electrons also exchange energy and momentum would let us so they they they sketch on disorder and And also they couple to phonons and that provides mechanisms for energy and momentum relaxation And so one might one might think that hydrodynamics maybe is not so so relevant in this case however, there are there are cases where hydrodynamical treatment Is is is justified and one so one such case I think was described, I mean historically not the first one, but the one I like most was described by by Anton Andrejavan collaborators in in in high mobility to dimensional systems where where the disorder is long wavelength Has correlation length, which is much longer than than elastic means Sorry, then two particle Then the means repass due to two particle collisions Right. So this this condition then then then you have Hyrodynamical picture established on length scales much much smaller than the disorder correlation length scale and then in this case in this case they they they predict that Transport will be controlled by hydrodynamics that should be applied at the intermediate timescales between between the collision length scale, sorry immediate length scales between the collision length scale and the correlation disorder correlation length scale and and Then the transport coefficients like resistivity and various other coefficients will be directly expressed through through thermal conductivity and Discosities and so so this is a situation when when the viscous gas flows over over some metric over some curved curved space with Some angle of curvature provided by slowly varying potential and then in in this case the usual transport picture through the like transport picture should be replaced by a high dynamical by a dynamical picture and And of course then there is there is another case when You simply don't have any disorder and that that's that's has a that has a much longer history It's been discussed by by by people a long time ago starting from Guruji 1968 so so in this case a few if you have a system with essentially no disorder and that's that's what graphene will be like we Have two regimes. We have Knudsen regime when you're I'll say width of your channel is much smaller than mean free path and we have We have something like Poiseuille regime or Poiseuille-Gourje regime and it's called when when it's wider than mean free path and And and then then in this case viscosity is viscosity controls transport and and and one can show that momentum relaxation Momentum relaxation because it only occurs at the boundaries momentum diffuses Momentum imparted on the system by electric field has to diffuse out to the boundary to to actually dissipate the Momentum relaxation time essentially becomes momentum diffusion time with viscosity playing the role of momentum diffusion coefficient and so in this case as as as Guruji predicted the momentum relaxation will will slow down as a function of system width and As a function of temperature because temperature will increase the collision rate collision rate will make Viscosity coefficient go down and that will make diffusion time go up And so momentum relaxation will decrease and so the striking manifestation of this regime was that when we increase temperature and It can be done for example by by applying higher current resistivity resistivity should go down because momentum relaxation time will go down and That that's the grocery effect that has been tested in ballistic wires by Mullen company collaborators So so it's a non linear non linear IV and there is this drop in in resistivity at when current is When current is high enough and that's that's that's the signature of Discosity so in in in graphene we have also have a strongly interacting system and Interactions even become you know very very strong near direct point because they become on screened and and Because there is low density of states and and because of various other reasons So so in graphene near direct point is in many ways an ideal place to look for electron viscosity Collision collisions are very fast interaction of interaction very strong plus graphene is a very clean system So So disorder will not play will not play that much role And so this has been realized by you know by by many people and you know There is a detailed theory that has been worked out of what you know what what viscosity and graphene is and And even you know comparison of viscosity and entropy Racer was shown by Marcus Miller Was sure and collaborators shown to be very close to the limit to the limit that was conjectured conjectured by high-energy community as a Fundamental limit for you know for that ratio And so so because of that you know and other other reasons Graphene and viscosity and graphene is is is is interesting Question is how do we measure it or how we even prove that it's it's a viscous regime, so there have been several proposals and So one one is in that same paper by Marcus Basically say I mean this picture, but if I reinterpreted what you should do is you should look at the scaling Because resistivity will scale inversely with widths of your channel and and in the in the Poiseu-Gerger Regime resistivity will scale inversely with a square of the width So if you measure scaling if you measure scaling then you will be able to see that it's viscous versus omic But for that you need to compare, you know several different samples of different widths and that that may not be very convenient There was another proposal Use carbina geometry with time time varying flux applied through it and so that not so Also also possible measurement, but you know hard to do because time varying measurements high frequency Difficult so so so you know the challenge from my point of view is to try to do it on a single device and try to do it without time varying signals and in a linear response regime so that we were not subject to heating and so that's that's the That's the challenge and that's the this is the solution so the idea The idea is that we will consider exactly the same geometry in the non-local geometry current current applied from source to drain like like here and And then in the case when the flow is viscous There will be a drag of the current flowing along the straight path on on carriers on the sides and and it will Launch vorticity on the sides of that flow and and vorticity means that there will be a backflow over here and backflow means means that there will be an Opposite sign charge build up Across across the boundary of the strip on the sides on the sides of you know line from source to drain and So from this picture you predict that the non-local Non-local voltage measured somewhere here will be negative rather than positive as you expect in the omic case And if you do if you do calculation you find that this is indeed true. So this is So what you see here are streamlines. These are these white white curves and The color is potential blue means negative and red means positive or Maybe the other way around but I mean what you see here is that is that There is an opposite sign potential on the sides as compared to potential on the drain and that that's that's a signature the mechanism here Mechanism here is very similar to What you what you get in in in the situation which is more familiar to you know some people Coulomb drag if you have the Coulomb drag geometry when you have two layers and active layer and the passive layer you apply You apply current through active layer Then there is interlayer momentum scattering and interlayer momentum scattering in parts momentum on carriers in the second layer But carriers in second layer electrically decouples. So they will they will You know with the same momentum as in the first layer They will induce voltage which is of opposite sign to the voltage by which you induce current in the active layer so Yeah, so this is essentially a Distributed version of that Coulomb drag geometry and If you compare it to the Omicrysium and Omicrysium Current is always along along the gradient with potential. So you get You know potential is of the sign that you expect everywhere near the source and drain and away from it and Now if you if you do modeling Introduce viscosity and write down write down hydrodynamic equations with with with with Viscosities and boundary conditions no time to talk about that Ask me if you like and then and then solve it convenient way to solve is to introduce flow function of which velocity is is is A curl then you solve in incompress in compressibility condition And you are left with a bicarbonic equation that you have to solve with you know boundary conditions And after you do that you get you get an expression for voltages function of distance That looks like a Fourier transform of some function now notice that this function is positive So there is no minus sign no obvious minus sign in front of you However, if you if you study what this function looks like it looks like two peaks in k-space And so if you have two peaks in k-space with zero in the middle It's very easy to do Fourier transform on your head And then you see that there must be a must be a wide range where Fourier transform is negative And and that's so that's indeed what What it is? You get so this is what what voltages function of distance looks like looks like In the model I described also in a model where you add omic resistance on top So so the red curve is a pure viscous region the blue and then going up means adding Higher and higher resistivity and so you see that non-local responses fully negative in the viscous regime and then And then changes sign in the case of when there is omic response Also it changes sign and becomes positive far away And then this point where it changes sign depends on the ratio of the resistivity and viscosity and if if if If it's detected you can get Numerical value for viscosity if you know resistivity which you usually do From from that type of type of measurement Lastly there is an experiment that you know appeared Simultaneously with that work and was just communicated to us a few weeks ago and in Manchester where people people do see What measurements do show a negative? Non-local response in you know in in the geometry similar to that to that I described This is not the only lab where Negative non-local response is being seen. I think there are similar data in Colombia and also at Harvard And perhaps other places Yeah, so summary is that you can I mean you can see You can see viscosity a signature viscosity Arising because because of vorticity generated on the viscous flow and vorticity Creates back flow and you know that leads to negative negative non-local voltage And so that that gives a direct a direct way to detect viscous regime and to measure viscosity and And then you know thinking about experiments experiments of course we we should be aware of other neutral mouths because The viscosity is about momentum transfer and momentum momentum is a neutral mode but it's not the only neutral mode and there are others the one most important this is is entropy or Thermal mode and should be accounted for to describe experiments also control maybe controlled by cooling and But in principle, I think this provides a way to measure measure viscosity You know with qualification one needs to take thermal effects into account what I think is not known how to do is how to measure second viscosity and It's interesting because there are predictions by you know same string theory people that due to conformal Theory viscosity due to conformal symmetry viscosity is supposed to vanish second viscosity is supposed to vanish So that would be interesting to see whether it comes out like that on the planet. Thank you very much