 Tak. Ah, OK. OK. Now I see that it's working. So, thank you very much to the organizers. So, today I'm going to speak about no local telemojitricity in a system which is based on a Joyce injunction in which the weak link is a two-dimensional topological insulator. So, first I would like to acknowledge the people involved. Sorry, I have to get rid of this message here. OK. No, it doesn't matter. The people involved. Yes. OK. So, the main actor in this work is John Michele Blasi, who was a PhD student in our group and then moved to as a postdoc in Geneva. Then Alessandro Braggio in PISA. Then Liliana Rachea was here from Buenos Aires in Matteo Carrega, who is now in Geneva. So, the outline of the talk is this. I will first give a very quick introduction to telemojitricity nanoscale just to set the basic notions. Then I will say a few words about topological insulators and then I will move to the main topic, which is non-local topological Joyce injunction, showing how one can get linear telemojitricity through a magnetic flux applied to the system or by applying a phase difference. And then if I have time, I will speak about non-linear effect as well. So, let me first set a few, I just need a few things just to be sure that we are on the same page. So, I'm thinking about telemojitricity in a nanoscale system. So, I'm thinking about a conductor attached to two leads where I can apply both voltage and temperature bias. I'm interested in the charge current and in the heat current flowing through here. And I want to consider the phonons. So, I'm considering very small temperature, phonons are out of a basically neglected. So, the quantity I'm looking at is thermo-power, so, which is the voltage, which appears across the conductor when a temperature difference is applied at open circuit. So, I just make here, I just put here the definition to be sure what I'm speaking about later. Also, I'm also considering thermo-conductor, just the heat flow, the heat current produced by a temperature difference. So, thermo-tricity is important, not by itself only, but also because one can use thermo-tricity effect to implement a steady-state heat engine. So, a system which uses some heat coming from outside, in this case we have a hot terminal here, a cold terminal here. Some heat is entering our system and some power is coming out. So, I'm interested in heat work conversion. So, we know for example that the maximum, the optimal performance is quantified by the Carnot efficiency, also in these steady-state engines, which just depends on the ratio between the two temperatures. So, now let me be a bit more precise on this. So, I'm considering a conductor, as I said, so the simple instance is just a two-term conductor. So, the efficiency of a heat engine is this ratio between the power produced and the input heat current. In linear response, so the point is that one can define a figure of merit, ZT, which is expressed in terms of this transport coefficient and that's why it's important to determine this coefficient because just knowing them, you will get this figure of merit. So, if you may just equal to the electric conductivity, term of power to the second and divided by the thermal conductance times temperature. So, this figure of merit is important because just knowing it, you will know, for example, the maximum efficiency of your system, which is expressed through this equation here. So, what you want is you want a large ZT. When ZT is very small, the maximum efficiency is zero. When ZT is very large, the maximum efficiency reaches the Carnot efficiency. Also, you can express the efficiency at the maximum power of the same ZT. So, now, when dealing with nanoscopic systems, one can ask some questions to ask how quantum effects affect the performance of a heat engine. So, for example, what is the role of confinement or quantization or what is the role of quantum interference. This is what you can ask when dealing with the nanoscopic system where quantum effects can be important. So, actually, in the 90s it was first realized that in order to enhance the heat work conversion, so in order to enhance the ZT figure of merit, what is important is somehow to get, to reach some energy filtering, so to have some transport coefficients, which has sharp features in the energy. So, for example, the generalize conductivity. And here, for example, in the first paper, in this paper here, they studied quantum wells or superlattices, realizing that if you decrease the spacing between the layers, you can increase ZT. In this other paper here, they realized that the optimal performance is reached when this kernel here is a delta function. So, a quantum dot, in principle, can give you the optimal, it's an optimal heat engine. So, nowadays, there's a vast literature about all this here. I just mentioned a few review papers on that. Now, what I'm going to, in this talk, to focus on, is topology and superconductivity. Topology, actually, in what we consider, is two-dimensional topological insulators and how we can use to look at thermal electricity. So, let me first, I've got one slide just explaining what a two-dimensional topological insulator is. So, basically, it's a two-dimensional electronic system, which has a gap in the density of states, so it's insulating in the gap, but presents pairs of conducting edge states. So, for example, here, you have an insulator in contact with this two-dimensional topology insulator. The system is insulating in the gap, but you have these two channels, edge states. This edge state has got a helicality, so they are helical, in the sense that they spin up, for example, in this case, propagate to the right, or spin down, propagate to the left. So, the special relation looks like this. We have a conduction band and valence band, and then we have these two lines here, which represents the left-going propagating spin-up modes, sorry, right-going propagating spin-up modes, and left-going propagating spin-down modes. So, this effect was predictive for graphene with spin-up interaction, or make-and-tell-your-ride, but nowadays there are many different system, which displays this property. So, the helical edge states can be explained, can be described by a low-energy effective Hamiltonian of this kind, where we have the Fermi velocity and the Z-pauli matrix. Just to say that there are, as I said, many different material systems. We have also indium arsenide, gallium antimonide, quantum wells, and also atomically thin crystal of tungsten-telluride more recently. Now, people, as already studied JSON junction with this two-dimensional topology insulator inside, it is interesting because it gives rise to a strange current phase relation, so, which is 4 pi periodic. So, here I just mentioned it, the first few papers now, in the first years, now there are, of course, much more, and these, at least, I don't think is complete of experiments on this system here. So, it's something which people are looking at nowadays. So, now let me come to the actual system we are looking at. So, we are considering two-dimensional topological insulator, for example, here spin-up going, propagating right, spin-up down, propagating left. We are, we have two supergranate electrodes on the left and on the right, which are proximizing the two-dimensional topological insulator, so, we have a phase, phi left, phi right, on the left and the right, subgundating phase. In addition, so, this is a non-local setup in the sense that we are putting a normal probe somewhere along the edge, which is coupled to the two edge states, something like a CM tip. So, this probe is connected, this probe, we can have a bias voltage, vn, applied to it. The superconductors are grounded, the position is X0 between 0 and L. And what we are interested in is in the charge current flowing through the probe and the heat current flowing in, for example, the left superconductor. We are looking at the effect of a magnetic flux threading the junction and also of a pure phase, phase difference. But I will spend more time on the effect of a magnetic flux. The temperature of, so, T plus delta T half is the temperature of the left reservoir, of the left superconductor, T minus delta T half of the ratchets, and the normal probe is a temperature T in the middle. But we see that it is not very important in this temperature here. So, I'm considering just half of the two-dimensional, of the two-dimensional Ti. So, I'm assuming that the width of the Ti is very large, so the edge state on the upper side and the lower side are at the top. And the lower side are decoupled. I'm assuming that the probe can be coupled to the edges with an arbitrary transmission, T squared, which is pin-independent and energy-independent. With T squared, we can go from zero to one. So, we can range, we can go from the tundering regime to the very clean regime. Very clean contact. So, the system can be modeled through a budo di gen hamiltonian, where we have the particle-like excitations up here, the whole-like excitations down here. And they are coupled through the order parameter delta. So, I'm assuming an S-wave superconductor. So, I'm coupling spin-up and spin-down electrons. Delta X is the gap in the system, which is actually approximating induced gap. It's the gap which is induced by the superconducting electrodes put on top of the... to the Ti. So, the emiltonian H is described by the lu energy effectively emiltonian, the one I showed before, plus in case we are applying a magnetic flux, plus a Doppler shift energy, which is due to this magnetic flux, which takes this form here, which is basically a shift in the momentum of the condensate. So, now... So, we are going to see first is how this presence of this Doppler shift energy will impact the... the transport. So, this is the dispersion curve, where for in the in the proximized region, so delta is finite, with zero Doppler shift, so in the absence of a... of a magnetic flux. So, what we see is that there's a gap, delta opening between spin down particle and spin down holes. So, these are spin down particle, spin up particles. So, a gap opens here, as it should do. Then, what's the effect of the Doppler shift? So, shift vertically, in two branches in opposite directions. In such a way that on the left branch we have a gap, which is reduced by the amount epsilon delta s. So, this gap is delta minus epsilon delta s. On the right branch the gap is increased by the same amount epsilon delta s. Ds, sorry. And this comes in the fact that when you just look at the special curve in the superconductivity, the two branches are shifted down and up branches are shifted down and up. And so, this makes this movement, this makes this shift of the branches. So, now let's look at what do we expect in telemetricity in this system. So, here we are applying a positive Doppler shift and energy. And the branch on the left is going down, on the right is going up. This is the distance between zero and so, this is the gap for the left branch. And what we see is that we can have a thermoelectric effect so, just applying a temperature difference so, with Vn equal to zero applying a temperature difference we can have a thermoelectric effect coming from the fact that we have holes, so this is the hole branch we have holes hot holes coming from from the left and the cold particles coming from the right described by this particle dispersion. This can happen when the temperature is of the order of this gap, of course. So, when the temperature is of the order this branch starts to be active in telemetricity in some current, we expect some current flowing through the normal through the normal probe. So, we described this using the scattering approach so, the system is actually three terminals, so contains three terminals, but we have just two relevant affinities, so biases which is first, first is Vn the bias here the voltage bias, the probe and the second is the temperature difference between the two superconductors delta T. So, what we are going to use is a partial let's say on sagam matrix, it's a non-local sagam matrix 2 by 2 which connects the two which relates the two the two relevant currents so the charge current in the probe and the heat current in the superconductor to the two affinities Vn and delta T. So, we are getting rid of some of some question, but these are the important one the one we are interested in. So, here I'm assuming that we are in the linear regime delta T small delta T is much smaller than T. So, in this way we can define this non-local on sagam matrix which is a bit weird because for example what we find is that L21 is equal to minus L12 which is the usual symmetry one expects but this comes from the fact that this is incomplete. The whole on sagam matrix will have the proper the proper symmetries. Ok, now we can we can see the first the effect one calculates for example L12 L12 is this terms which gives the the magnetic effect so charge current due to a temperature difference no, temperature difference between the two superconductors what you find is that this is the plot of L12 normalized in some way as a function of the Doppler shift energy normalized with delta and as a function here of the transmission in the coupling with the probe so what one sees is that one has an effect when epsilon ds divided by delta is over the one Ok, and this makes sense because it's when the first branch the branch on the left or on the right goes down and close to the origin for example in this case so this was positive delta s and one get a positive a positive signal here and for negative delta s the branches move in an opposite way and one gets these value and negative and one sees that the effect increases by increases increasing the coupling so the tunneling there is a little effect so in this case I took a length which is of the order of the coherence length so one just to understand a bit more one can see what happens to L11 so for example on the electrical conductance electrical conductance looks like this again as a function of the energy of the Doppler shift and one sees that there are two peaks here two peaks which basically reflects the presence of under the bound states within the the rejoice the weak link so we have two bound states which then of course goes with open up when increasing the the coupling with probe so basically the coupling is giving width to these bound states up to here where everything is open for example when you increase the length what you will find is a figure like that so there are going to be many under bound states so these actually under bound states I am saying this because there are the effects on the on the thermoelectricity so the conductance is less interesting maybe because there is nothing between up to energy or epsilon ds of course because in order to have a heat current in superconductor all the heat current simply due to excitation so you have to have energies above above delta so there are no features coming from under bound states because under bound states don't play any role here in the in thermal conductivity ok so here there is one can switch on some phase difference and this is the effect we see that there is more effect when also a magnetic flux plus a pure phase difference is switched on but we will see later the effect of a phase difference so now we calculate something which is usually measured which is the thermo power the cyber coefficient defined as defined a few slides ago which is simply given by this ratio here of matrix of on saga coefficients and we will find that so this is thermo power for different temperatures we will find that this thermo power is larger for smaller temperature reaching quite high values so in this case is something like 50 micro volt per Kelvin and has got this kind of dependence with respect to eds so there is a peak which is less broader for small temperature here is the same plot but again is also some presence of some phase phase difference ok so maybe I've got till 5 minutes so what we saw is that the helical nature which gives rise to this thermo this is basically this fact here the fact that you can have holes hot holes in this case so the helical nature is crucial for this effect to occur there is no dependence in the linear response there is no dependence on this Tm also beyond the response there is no dependence on the temperature of the number of probes and the difference in temperature of the two superconductors delta T also there is no dependence on the position of the probe so this is true in an ideal system so I probably won't have time but when I introduce some asymmetries between left and right this is not true anymore so one can ask just switch off completely the double shift effect but one somehow imposes a phase difference between the two superconductors and in this case also one finds thermoelectric effect but in this case due to under the interference so to resonant processes occurring and so the interesting thing is that one can find a very easy it's very simple to write the current in the probe because it's simply given by a contribution which is given by which is controlled by the distribution function in the lead so this is relative to particles this is relative to holes and another contribution controlled by the distribution superconductors and when so when there is no voltage applied to the probe this term here goes away and one remains with this term here only which is given by which is controlled by the different temperature difference between the superconductor which is basically controlled by this quantity Q which gives this is the probability for a particle to go from the left superconductor to the normal probe and a particle to go from the superconductor so actually a Andrijev process for a particle to go into a hole in any case they are describing basically resonant processes occurring within the weak link and also this Q is very simple form which is the length of the weak link enters through this cosine function as well as of course of course phi everything is controlled also by the transmission of the so the coupling of the probe and for example when phi in this generalized this Onsager symmetry relation so this is not important now ok so if one plots L12 so determinatrik coefficient so this L12 against the length and against the phase difference one find of course some periodic dependence just given by the formula I just showed you so maybe what is interesting here is the fact that still L12 so the determinatrik response is maximized for transmission so for coupling which are in the middle of the range so not for small so they should be somehow in the middle of the middle of the range so the response is very much characterized by this oscillation in phi but also in length ok so here just plotting again the c-back coefficient so s the measurable thermal power against transmission here phase difference and also temperature so temperature between around half of the of critical temperature are the optimal one to see large value of this c-back coefficient now I'm just comparing the two effects where we have pure double shift or pure under the interferometric effect and with double shift the effect is much much much much larger with respect to with respect to the interferometry so I still have a few minutes or ok so no so just what was the effect of more realistic effect and some estimates about the some estimates about the heat generation and efficiency of any tangents of this kind so the main message I want to convey is a critical effect induced by eminiti flux which is very much dependent on the helical nature of these states so all what I told you is basically containing this paper here so thank you very much thank you I already see a raised hand thank you very much for the very city talk I was wondering if phase slips would look different in such a system because the phase slip would correspond to voltage and they also correspond to heat so if the helical nature of the insulator would correspond to somehow make the phase slip different any effect I think so yes we never looked at it but it is definitely interesting to look at I have no idea what thank you I had a question because you were only looking at kinetic energy of the charge careers regarding the heat transport I am considering the heat current as you the complete heat current just to particles but it is just kinetic you neglect any plasmonic effect in the edge channel it is only kinetic do you have an idea of the edge reconstruction of topological edge states I am not familiar with them in quantum hole effect it can dominate in several regions so you have a lot of electrostatic content in the energy of the 1D wave propagating in the edges here it is only the kinetical part so if the two channels are overlapping on the same position they will be very strongly coulomb interacting so I guess it will change the chirality I mean we change many things but I don't think people have studied this it is even a mess in the quantum hole because no this thank you are there any other questions we still have time for one quick question in the room or in the chat it is always good to wait a moment until people in the chat figure out what to click on they are old if not let's thank the speaker again thank you