 Okay. So, let's start the last talk of this session, which is going to be given by Aaron Kapitulnik from Stanford University. Thank you very much. I would like to thank the organizers for the invitation to come and talk to you about this work. I'm going to talk about also transport, but I will concentrate on thermal diffusivity, basically diffusivity, but mostly thermal diffusivity, which most of it, I mean, in fact, almost all the work was done by J.C. Young. The new student is Eric and Sean has been around and samples come from all over. I'll talk today about hold-op and electron-dop cooperates. So, just a reminder, I'm sure you all seen it, especially today, a few times. But let me start from the Ohm's Law. Ohm's Law is a constitutive relation between the electric field and the current, which if you start from the electric field, writing it as a gradient of the potential, you can work it back to see that it gives you this relation between the electric field and now n is the number density, which means that if you now divide the current in Ohm's Law by the charge, you get the number current density, which then give you a relation between the number current density and the gradient of the density. And this is a diffusion relation and the diffusion constant, if you just work out these relations, is nothing but the diffusion constant that appears in the Einstein relation. Well, you can do the same thing with Fourier's Law, as we saw in Cameron Bernier's talk today. You start with heat current, the coefficient that related to the gradient in the temperature is the thermal conductivity. You work out through the first law of thermodynamics that this current, which is just the change in heat with time, leads, in fact, to another diffusion equation, now with respect to the temperature in the sample, with the assumption, of course, that the temperature is well defined, and this relation here appears the specific heat, and here the thermal conductivity gives you another type of Einstein relation, because this is a diffusion equation, so this is going to be the thermal diffusivity, so Kappa, the thermal conductivity, is the specific heat times the thermal diffusivity. Now, we know that if you divide these two Einstein relations, you get the so called Lorentz number times the temperature, that is, the Wiedemann-Ferns Law, which works very well for most metals. Most metals, I mean, there are some exceptions. We heard a little bit about tungsten diphosphide today. Tungsten itself is a very interesting material with such a deviation from Wiedemann-Ferns law. Now, if I start now from the electrical expression, the Einstein relation, then just simple relations, as we heard for experimentalist writing density of states, and the diffusivity, which is one over the dimensionality times the velocity that is V Fermi squared times the relaxation time, we can rewrite it in terms of epsilon Fermi tau, or in terms of K Fermi L, then the conductivity is through the formula, which can then rewritten in terms of K Fermi and the mean free path. And we see that, in fact, there is this combination of K Fermi times L, that we don't expect it to be larger than one. Now, if you look back at the expression of the diffusivity, you see that this expression of K Fermi L not larger than one also give you a bound on the diffusivity, which is just plug it in here, the K Fermi L equal one, and you get one over dimensionality times h bar over m, or m star if we are talking about some transport in a general material. So diffusivity bound is just h bar over m. Now, I want to remind you that kind of the relation, this is very simple to simple Drude or Boltzmann transport, which is that in order to define transport in momentum, with well defined momentum, you need to create a wave packet. The wave packet will have a width around some momentum, say the Fermi momentum. And then if you look at the energy around this momentum, there is the fundamental p square over 2m, then there is the variation in energy, and then there is the way by which the wave packet is going to spread in time, which means that if you write now that time you have an observer at some x naught, and you ask when is that wave packet going through that person, it's going to be that delta x over the fundamental velocity, that's h bar over delta p, you can write delta x, you can write in terms of h bar over delta p, and you get the familiar uncertainty relation for energy and time. But then there is another hidden thing in this kind of relation, which is you do not allow the wave packet to spread too much. And the way the wave packet spreads is that delta t of spread is h bar over this term, which needs to be much larger than the delta t that appears in the uncertainty relation. So this again leads through some very simple algebra to the fact that the diffusivity always needs to be larger now in terms of this wave packet larger than h bar over m. So this kind of brings us back to that relation that appeared in the Drude formula that when k fermiel needs to be much larger than one for it to work, it also means that d needs to be much larger than h bar over m. Both are equivalent in talking about transport. Obviously the limit at which k fermiel is of order one is the so-called multi-offering limit, and which now we have that, or as was proposed by others, if you want to discuss well-defined quasi-particles, then what you want is that the mean free path will be larger than the Fermi wavelengths. This introduces a factor of 2 pi. And I think this factor of 2 pi appears everywhere, including in Planckian relaxation time, whether it's h bar over kt or it's h bar over 2 pi kt, I'm not going to discuss or negotiate this. For me, this is the same. Anyway, the whole issue about exceeding the multi-offer regular limit appeared when people measured resistivity of many of these newly discovered at a time high-temperature superconductors, hold-opt superconductors, and found that resistivity continues to increase whether linear or different than linear, but increase much beyond this multi-offer regular limit, and as if nothing happens at that limit, at that limit of k fermiel equal one or L mean free path is of order of the Fermi wavelengths. So there was no saturation. The people who noticed it at the beginning were Emory and Kievelson, they called these metals bed metals and even tried, they noticed the fact that many times rho is proportional to t. They even tried to make the connection to the high-temperature superconductivity at the time. I think that this still remains an unclear issue whether there is or not any relations to this. Now the linear resistivity was then highlighted and we saw it several times today by Brunenau where they showed that if you then rewrite the Drude formula and in terms of such numbers as a function of one over v fermi, there is a linear line with a proportionality constant alpha and the previous talk discussed in details whether what is the value of alpha which in general has been shown especially if you plot it in a log-log plot to be very close to one. Now the general transport considerations whether you exceed or not the multi-offer regular limit can be discussed as I did at the beginning using Ohm's law and using Fourier's law just using thermodynamics and when you do that you don't need to discuss quasi-particles and whether they exist or not. So if you do that you can still ask yourself about whether this kind of transport can continue in a metallic fashion beyond the multi-offer regular limit and an approach that was taken by Sean Hartnell a few years ago was to say that the idea that transport in such strongly coupled systems is governed by the smallest possible relaxation time and then using a simple uncertainty relation as I used it earlier this leads to if you allow for delta E the energy to be not smaller than is bound by a KT then you get a relaxation time that is bound by H bar over KT which is what people nowadays call Planckian relaxation time. If you then plug this into the simple formula of diffusivity you get that the diffusivity needs to be larger than H bar times a velocity which for electrons is going to be the Fermi velocity square over KT. Well all this is a very simple picture which then allowed Sean to draw this diagram resistivity which is at low temperature maybe if you want to talk about good versus or coherent versus incoherent particles will be limited by K to 1 here without any temperature dependence but then as you increase the temperature there is going to be a diffusivity bound that goes like 1 over T because of this Planckian relaxation time. Copper and high Tc are going to be therefore in different regimes but I think all this is still something that people need to discuss. Anyway I want to mention something that is very important since I'm going to talk about thermal diffusivity which is phonons phonons and insulators. Now for electrons we saw that there is a multi-offer regular limit. We are dealing with materials all the materials we are dealing with will have phonons and in general the electron-phonon interaction is going to be of order 1 whether it's 0.5 or 1.2 it doesn't matter it's going to be of order 1 so we need to pay attention to phonons and it turned that phonons also have their own multi-offer regular limit which is the mean free path should not exceed the minimum of the minimum wavelengths of the phonons that you excite or the lattice constant whichever comes first and if you do then there is a limit for the phonon diffusivity and phonon transport and one can then argue what what it should be the consequence of that this is not going to be the main topic of this of this talk. In any event if you take this minimum mean free path for the phonons you can then discuss a minimum diffusivity for the phonons which is going to be of order of the sound velocity times this L minimum as I defined here. In turn this also will give you a thermal conductivity a minimum thermal conductivity that you can find is approached in many insulators especially insulators which are perovskite like and and there have been a lot of recent studies of this including common banyan studies of Stonzium titanate. So if we now assume that phonons also if they have a problem being defined as good quasi particles will be limited by this Planckian relaxation time then there is going to be therefore a phonon bound for the phonon diffusivity also of order of h bar but now the velocity that we need to put in is of order of the sound velocity square over kt. The interesting thing is that when both electrons and phonons are going to be bound by the same relaxation time then you can put together their velocities and this is going to be discussed soon. In any event what I want to mention is that for many of the materials that have been discussed today and I'm going to discuss in this talk it turned that both the based on measurements the electron diffusivity and the phonon diffusivity are getting close to the more their respective limit of temperatures of order of say 150 or 200 of 200 Kelvin again this factor of 2 pi can shift you from 200 to 300 but I'm not going to well I mean not going to be very specific on that. So we'll show that actually neither electrons nor phonons are well defined quasi particles and see where it take us. Now the measurements we are doing are measurements of thermal diffusivity and the reason is that if you start from the charge equation of Einstein equation which is the charge susceptibility times the charge diffusivity that it involves the density of states and in general it's very difficult to separate diffusivity from the susceptibility. Now in do the formula they are mixed together and then when you analyze like Brunet all analyzed you cannot see the difference but in principle there is no reason why these two should be lumped together and therefore what we are interested in is to separate them and measure the diffusivity. Now it's as I said it's hard to measure the diffusivity of the electric part but it's easier to do it for the heat diffusivity and then I will argue that if you are measuring the heat diffusivity and you are in the limit of being bound then it doesn't matter which relaxation you are talking about they should be bound together. So we expect that in the regime I'm going to talk about these two are going to be similar. So I'm going to discuss now measurements of the thermal diffusivity which we remember is the ratio between the thermal conductivity and the specific heat. A word of caution people measure sometimes thermal conductivity as the quantity to be measured and then they measure specific heat and they divide the two to obtain the diffusivity. In principle this should not work because thermal conductivity tells you only about the excitations that will carry entropy from one side of the sample to the other. Specific heat measures all the excitations if you have for example a non dispersing optical mode it will contribute to see it will not contribute to the thermal conductivity. Just a remark. In any event for simple kinetic approach for simple materials it has been used simply that every channel that contributes heat conduction should be added and therefore I have the channel for the electrons and the channel for the phonons you add them together and then to obtain the full thermal conductivity which if you measure the full specific heat you can define some effective diffusivity. But this is not going to be neither of the electron or the phonon it's going to be some kind of effective diffusivity. So we are going to measure this effective diffusivity we are going to see what we get and then we go back and see what we can say about this combination of diffusivities for the electrons for the phonons or for whatever there is in the material. So the way we do it is using a method by which we measure directly the diffusivity. We don't measure thermal conductivity we don't measure specific heat. We only measure the way by which heat is transported in the material and we extract out the thermal diffusivity. So we start with a heat equation and we introduce heat at a certain point these are two laser points at a certain point by using a time modulated heat source and then a few microns away we measure the change in reflectivity of the material only the change in reflectivity we don't care about the reflectivity itself and in fact we don't even care about the change in reflectivity because the only thing we care about is to measure the phase shift between the heat source and where we measure because this is going to tell me what is the diffusivity because if I heat and at a time t equals 0 let's say at the peak of heat then it's going to take some time for the peak of heat to reach the measurement point and I'm going to measure this time by measuring the phase shift of the wave that comes to the measuring point and this gives me the diffusivity in fact you can solve the heat equation and in the infinite limit you see that there is an exponential term that goes like square root of the frequency by which I modulate the heat over twice the diffusivity of the heat times the distance I can change the distance I can change the frequency and by that I can get the diffusivity in length scales of order of a few microns which is very good because many of the materials we are going to talk about may have domains may have twin boundaries may have a lot of defects with this we can avoid these defects we can average the diffusivity over many parts of the material and in particular we can measure anisotropy we can measure anisotropy very easily as I'll show you in a minute so as I said this is the phase shift that we measure so we can then plot it as a function of frequency typically we measure between 200 hertz and 30 kilohertz and when we get such a good fit we know that there is only one type of fluid that transports the heat in time scales that with respect to the size that I'm using and here you can see we measured over 18 microns distance then then I mean and here this was measured between 500 hertz and 20 kilohertz then there is within this time scale there is just one type of fluid that is transporting the heat and this is very important okay as I mentioned it's very easy to use this method to measure anisotropy of the diffusivity because I can place the heating spot at one point and then go around with the measuring spot and this is some YBCO sample in which we measure the full diffusivity and then I can find the maximum the minimum and yes the laser beam you can see from in fact from here so it's it's about one micron it's it's focused at around one micron and the typical distance we will use is between a few microns to about 20 microns depending on the material yes is there any instance of the third direction that the heat diffuses also in the sample not only yeah so so in in in principle I mean what we do is we solve the the full heat equation but if if there is a so anyway to cut it short the answer is the answer is yes the third direction is is coming in and then you need to distinguish whether you have a material in which the third direction is much lower diffusivity or not but this is modeled in three in three dimensions and and but the diffusivity because I'm measuring it only to within one optical penetration depths is is in the in the plane but the modeling is a three dimensional okay so let me first show you some results on underdoped YBCO not yet optimally doped underdoped and samples are here they are between 6.5 and 6.75 oxygen doping and here is the result first I told you I can measure the anisotropy here is the anisotropy in the diffusivity as a function of angle going around one point then the other one goes around and then we locate the highest and the lowest and measure it as a function of temperature this is from room temperature down to about 20 Kelvin along the B direction that's the chain direction and then the a direction that's perpendicular to the chain direction and these are actually aligned very very precisely with the two principle axis of the material so this is for 6.6 this is point 6.75 I have others as well now the the interesting thing is which you don't see it from here is if you divide now the two directions in order to look at the anisotropy itself and surprisingly you find that this is just these two but we have 6.67 falling on top of it as well is that the anisotropy first of all is very large if you don't remember there for example anisotropy in the sound velocity of this material whether you go along the chain or perpendicular is just a few percent this is the anisotropy that people observe in resistivity which is an electrical type of measurement moreover this is the onset that people find in charge for charge order in this material again this is something that people will observe in in electrical measurements and it's not supposed to appear so strongly in the in say phonon measurements I want to mention that in the past people in fact dismissed any possibility that heat transport in this temperature range is associated with electron simply by measuring the resistivity using Wiedemann Franz multiplying getting a very low thermal conductivity and then assuming well that's it nothing there is no electronic thermal conductivity whatsoever it's all about about phonons okay now you can see from these results that this cannot be I mean here is for example for the 6.75 measured on similar I mean similar batch crystals the thermal diffusivity that's the squares and this is the resistivity measurement I mean they follow each other how can it be if the thermal diffusivity is just a lattice thermal diffusivity it's the same side there is no scaling here this is there is no scaling of the of the anisotropy it's that whatever is measured on this material moreover if you look now especially in the A direction at the inverse diffusivity I plot inverse diffusivity I'm going to plot lots of inverse diffusivity because this is supposed to appear in resistivity and you look at the resistivity of the 6.75 they have very similar temperature dependence I don't know how to scale it because they are all kind of ends and stars and and other numbers but you can see that the temperature dependence is very similar in fact there is an attempt to do some scaling with the dotted line is the temperature dependence of the diffusivity so again we see that diffusivity that was supposed to be only phonons has an electronic character okay let me now go to the electron doped cuprates okay these are samples from regreen that we measured recently show you the data here is NCCO near optimal doping as grown as grown is barely superconducting this is Sumerium Serium Copper Oxide as grown that's the thermal this is the thermal diffusivity then these are different samples but annealed I mean we got two different samples from from week two different samples from for the now demium we have also PCCO data which I we just appeared in the past couple of days so it's not in the talk but it looks very similar okay so you can then for these materials from all the data you can see that this is first of all it's isotropic in the plane okay unlike the YBCO this is isotropic in the plane second you can see the numbers they are very similar to the YBCO numbers sound velocities are of the same of the same order typically of order 6 10 to the 5 centimeter per second you can then extract this minimum phonon mean free path from from our diffusivity data at room temperature and you get that well maybe it's one maybe it's two lattice constants but it is of the order of the lattice constant so phonons should have a problem as well as the electrons which is something that people notice long time ago for this material there's a paper of the bucket that's the Green's group that showed that so I plot now the inverse diffusivity inverse thermal diffusivity as we measured it for some we measured it up to 630 degrees okay from about 77 so showing here is a safe from 100 degrees to I mean this one to about 400 600 600 600 and the interesting thing is to notice is that the inverse diffusivity for all these systems is linear it's linear through wherever should be the multi-offering limit both for the electrons and for the phonons in in in this material this is despite the fact that the resistivity is not always linear the SCCO near optimal topic is sometimes linear but NCCO in general goes like T-square PCCO goes like T-square all the way to very high temperatures which is also something that is not well understood now if we then try to fit this line of inverse thermal diffusivity it's easier to do here because it's perfect perfect linear we find that the inverse diffusivity is fit with AT plus B it's a straight line now in the same spirit as in the YBCO I'm going to assume now a relaxation time which is Planckian relaxation time that is limited by H bar over KT and therefore this AT is the inverse of H bar times velocity squared over dimensionality times KT okay to the to the minus one because this is inverse diffusivity okay and then I have a constant B now if I do that this means that I assume that there is a single velocity but at least to within the frequencies that we measured which is the time scale of these measurements we see only one type of fluid that is taking the heat so it makes sense to use just one velocity I'll call it VB don't ask me why now I want to mention that that many insulators this was pointing pointed out to me by Kaman Benya recently especially perovskites have diffusivity and here is a paper I forgot to to cite it I can give it to whoever wants but that they all have diffusivities which are just a few times larger than what we measure but with slopes that are much larger for similar sound velocities the fact that our slope is lower means that the velocity associated with it is higher if it's higher it means that it's higher than sound velocity because all these actually are quite consistent with sound velocity being the proportionality okay I'll come to it in a minute so let's try and understand this data in some possible interpretation so first of all we already discussed the fact that probably neither electrons nor phonons have very good quasi particles we also show that especially it's easy to see for the electron doped cuprates less so for the YBCO because there was some curvature which we are not sure where it comes from it could very well be because YBCO has very high the bi-temperature so there are still changes in the specific heat but again we are not sure but certainly for the electron doped it's very very linear and we measured quite a few samples now from this family and they all show this behavior so we see that this is consistent with this Planckian relaxation time now with some velocity because once you assume a Planckian relaxation time you need to assume a velocity but this velocity is not the Fermi velocity that one would expect in fact we find that it is between the sound velocity and the Fermi velocity somewhere in between the two larger than the sound velocity by a few smaller than the Fermi velocity by a few there is not that big difference in this in these materials so so that's that's for the for the temperature dependence term but what about the free constant well it turned that the free constant which then appears in the limit of t goes to zero because if I have AT plus B I take T to zero I get B is some diffusion constant to the minus one it's just a number so what is this B so I'm going to conjecture that similar to the Madison rule that appears in the good quasi particle regime where when you take the t goes to zero of all scattering times you are left with the impurity scattering and the way you add the time you add rates of scattering I claim that the same thing should happen here I should add the I should head I should add the above the multi-offering limit I should add rates or inverse diffusivity it's the same however I should not be able to see quote unquote beyond the multi-offering limit because if I'm above the multi-offering limit and I don't the transport is of in in incoherent transport I cannot say what is going to happen below the multi-offering limit when good quasi particles exist and therefore I conjecture that this limit that is the limit of t goes to zero that constant B should be one over the H bar over m just like what I started with which is the diffusivity at the multi-offering limit which happens to be the quantum of diffusion for these particles with effective mass m so that's the cartoon I want to draw that if I'm here I'm I have the good Madison rule I can look from any temperature and see what is going to be the t equal to zero impurity scattering that is I can take the limit of t goes to zero of good quasi particles but if I'm here and I look beyond towards t equal to zero there is a screen here and the screen is the diffusivity at the multi-offering limit which is H bar over m star so here are the numbers if I assume that indeed that constant B that we measure is H bar over m star this is what I get for m star for NCCO for SCCO they are all of order of what people measure for these materials so this seems to be very very suggestive that indeed I have the speed gives me a velocity that is in between the sound and Fermi velocity and the free term gives me the diffusivity at the multi-offering limit okay I have a chair and that wants to kill me yes oh no okay so if you if you try to do the same for the resistivity again this is the bucket paper which they show that for many of these electron doped cup rates it goes like the resistivity at high temperature above the multi-offering limit goes like t square so this means that if you write the resistivity this way and you can then if the inverse diffusivity is proportional to temperature then the compressibility or the charge susceptibility should have temperature dependence and in fact there is no reason why it shouldn't I mean it can have temperature dependence especially at high temperatures and there have been some theories for that as well if I do that I take for example in SCCO there is a slope because for SCCO annealed SCCO it's very linear then I can get and I assume again using a relation similar to brewing et al I get also an effective mass which is very close to what people measure okay so these are the conclusions and the last slide before I thank you we find that thermal diffusivity for many of these strongly correlated systems at high temperatures above the multi-offering limit thermal diffusivity cannot be explained only by phonons has to be explained by some combination of electrons and phonons we call it a soup of electrons and phonons with each which is characterized by a unique velocity call it VB and once you are above the multi-offering limit and you look down in your extrapolation of the thermal diffusivity you cannot see beyond the quantum of diffusion thank you