 We are going to start the last session of this wonderful conference, and we start with Elisabetta Paladino from Catania, so we're ahead. Thank you very much. So first of all, let me thank the organizers very much for the kind invitation for the wonderful conference of these days. And as I said, I'm based in Catania, which is the town you see here. It's behind the Hetna, which is one of the highest active volcanoes of Europe, and there is close to the sea, so it's a nice place also to work. And the work I'm going to present has been done together with Francesco Pellegrino, who is actually an expert in graphene and empinofalci. Okay, so motivation and outline of this presentation. There's been some interest on graphene-josephson junctions in recent years, and there are several reasons for that. I don't need to convince you that graphene-josephson junctions are a very interesting heterostructure to study fundamental phenomena related to proximitized superconductivity in a two-dimensional material. And that graphene-based gate months are very promising from the point of view of, of course, gate-tuneability, control by microwave characterization, resilience to strong magnetic fields, and the possible extensibility. Now, the topic I'm going to tell you about today is on these systems, and in particular what we did was to investigate the effect of a dilute ensemble of microscopic impurities, which we model with the Andersson model, and using the Dirac-Bogolubov-Degen model for the graphene. And what we found is how the current phase relation is modified by the presence of these microscopic impurities, which we model quantum mechanically, and how their presence influence the characteristics of the graphene-current phase relation, in particular the skewnets and critical current, which have a peculiar temperature dependence due to these impurities. This is a type of transport analysis. I will also show how the current noise spectrum may give a very precise spectroscopic characterization of these impurities. If time will allow, I will also pass in a second part of this presentation to discuss another possible mechanism of current fluctuations in short ballistic graphene-josephson junctions, in particular critical current fluctuations, due to a phenomenological model in this case, which is often used starting from semi-conducting devices, which is called McWhorter model. It's a typical model which leads to one over half noise, and we found that it can give rise also to one over half noise in the critical current and in this condition with peculiar features related to the statistics of the fluctuations. But to start with, let's summarize briefly, we already seen this picture a number of times during this conference, what ballistic graphene-josephson junctions are. We consider, as I said, the short junction regime, where the supercurrent flows because of proximity effect on graphene, and the formation of coherent superposition of electrons and all due to under-effect processes at the constructive interference of under-effect processes at the normal superconducting interfaces on the two sides. This coherent superposition gives rise to what we know as under-effect bound states. The supercurrent, which flows, depends on the phase difference of the energies of these under bound states, which is written down here, which is related to the microscopic characteristics of the junction, in particular to the transmittance of the junction, which is given by these two k-factors. Let me just anticipate that due to the fact that these k-factors depend on the density of carriers in graphene, this carrier density in graphene may give rise to fluctuations of the critical current, which will be the last topic of this presentation. Now, this type of devices have been implemented by using very clean interfaces between graphene and superconductors via encapsulating graphene in hexagonal and boron nitride, which allows to have one dimension at edge contact. And this type of setup has allowed the measurement of the current phase relation, which, as you see, has a characteristic non-sinusoidal behavior, as a difference of what we are used to in Josephson tunnel junction. This cuteness is tunable by the gate voltage, because this is embedded in a circuit, and in addition, it depends on temperature. This is the sketch of how the cuteness, which is a measure of the non-sinusoidality of the current phase relation as a function of temperature, looks like in this specific experiment, and you see that whereas the overall qualitative trend is predicted by the Tito-Benakar-Bogolubov-Degen approach, it's quantitatively not imperfect agreement. Then, let me just mention a couple of very recent experiments, which pointed out the possible presence of microscopic impurities in this setup. This is an experiment done by the Hollywood group. Here they have a gate one, and the gate one with aluminum electrodes. They managed to perform coherent control of this gate one, and in particular here I showed the Ramsey-Fringes experiment, which they did. They get a rather short coherence time in this experiment, but what matters for what I'm going to tell you is that when they perform the time discrete Fourier transform of this signal, which is what you see here, on top of the frequencies which they expect from the frequency mismatch due to the Ramsey procedure, they also observe some unexplained additional frequencies, which could have different sources. What they say here is that they might be associated with coupling to spurious, to leveled system, or in principle also excitations or to photo transitions in the hetero structure. So it's open what the origin is. On a different setup with a different superconductor, these are experiments at MIT. What they observe also are effects possibly due to microscopic quantum dots, they say. And this is, for instance, a measure which they report on the conductance. In principle this should reveal the band stratography, but these big spikes point to the possible presence of spurious effect. So these are one of the reasons why we looked at this problem. So let's come to the model we use. So the sketch of the setup is here. We have a graphene layer in grey superposed to electrodes. So this is grey is the junction and then there is a substrate. We consider the short junction regime. I'm going to tell later on which is the precise condition. We describe graphene with the usual Dirac-Bogolub of the gene approach. It's written down here. This Z corresponds to the valet index and we have the usual fermionic field operators. We assume from the very beginning that both the gap and the electrostatic potential have a step-like behaviour in the different part of the setup. And the alpha is the phase difference on the two sides of the junction. Now in the short junction limit is written around here. We know that we have a series of discrete under bound states and we also have a continuum, but in this regime the continuum does not depend on the phase difference. So continuum states do not carry supercurrent. So we're only interested in the under bound states which are the ones which carry supercurrent and we write the effective Hamiltonian of the under bound states in this simple form. Here we have the under levels here. Z again is the valet index and K is the momentum, the transverse momentum. Then we have to include impurities and we model impurities with the under some model, which is a model which has been used to describe defects in graphene. It's written down here. This is the case where we suppose there is a single energy level for these impurities, but we'll pass later on to a case where we consider a distribution of energies. The interaction between the graphene and impurities is expressed in terms of a tunneling Hamiltonian, which is written down here. So it describes a single electron tunneling between the Anderson level described in the impurities and the under bound states. And you see it is expressed here in terms of a four by four matrix. A and B are the two graphene sublattices, whereas plus, minus are already defined. And we assume a short range interaction potential which comes simply if you take a tight binding type of description for the graphene and you assume that the impurity sits at one of the carbon side, but a random place. This results in a potential which depends on position. It depends on a tunneling amplitude, down or to. It depends on the area of the unit cell. M and Md are some coefficients related to the two graphene sublattices. And this is a phase factor coming from the hexagonal symmetry of the graphene. Overall, we get the total Hamiltonian, which has a block stretch. On the diagonal, we have under bound states impurities on the other side and then we have off diagonal terms which describe, of course, the tunneling. Our goal is to find the Green's function corresponding to this Hamiltonian which is itself a quite big matrix. Since we are only interested in super current carrying states, we consider only the block which is related to the under bound states and we call it under bound states Green function which is just the projection of the total Green function on this subspace. And we can express this Green function in terms of an effective Hamiltonian which is written down here. It consists of the under part and then the projection of the interaction part. For the case of a disordered ensemble of impurities, so we assume that these impurities are randomly distributed in the graphene, so it means a random distribution of the positions where this impurity sits and also of these other parameters which describe the short range interaction potential. And then we get an effective Hamiltonian which looks like this. It's expressed again in terms of the fermionic operators for the under bound states which are written down here, the tunneling amplitude is here and the energy so far localized energy of the impurity comes here. Of course, we do not know how these impurities are distributed in energy so we consider a road distribution or not necessarily, by the way, a distribution of energies and we describe it like this and the result in the effective Hamiltonian is having a pre-factor which depends on the impurities energies in this way. Next step is to evaluate the equilibrium supercurrent. This is a quite tricky calculation I have to say which is mainly responsible of Francesco Pellegrino but the sketch of the result is very simple and nice. So we start from the current operator. This is not our original result, it was known. The current operator consists of a part which is diagonal in the under bound states Hamiltonian and a part which is off diagonal and comes in whenever the transmission is not unitary. Then we average this operator over the equilibrium distribution of this hybridized state and we get an expression for the current which is similar in a structure as the one you know in the absence of impurities but of course we have an effect of impurity which is included in a spectral function this is quite standard as a result, let's say the spectral function has this structure of course it depends on the under bound states in the absence of impurities and the impurities effect is written down here. To specify to get some explicit result we decided to take a distribution, an orange distribution of impurities centered around some finite frequency epsilon knot and with some width which is this parameter gamma which gives also a Laurentian distribution for this term which enters the spectral function. The result is the following. For simplicity let me just look at the case where the distribution of impurities in energy is centered at the Fermi energy so epsilon knot is equal to zero and we get this doublet structure where these are the under bound states the width of the distribution of energies enters here as an imaginary contribution and the reminder of the tunneling interaction is in this factorial. Now the physical result which comes out can be understood in a very simple sketch which is given here. So let's look at the structure of the under bound state in the clean limit so without any impurity. We know we have pairs of under bound states these pairs give rise to super current flowing in opposite directions so you see here that the lowest energy states corresponds to the current flowing along this direction to the left side so to speak whereas the upper under bound state corresponds to a current flowing in the opposite direction and this mathematically comes from this J here. If you see on the left-hand side the Fermi distribution so it's clear that if we are at zero temperature or a very small temperature the underf current comes from mainly the effect due to the lowest under bound state. Now when we have interaction this picture changes and in the case of sharp resonances we get a structure which is intuitive in a sense of this type of level structure so we have the repulsion of the levels so the upper lowest states are a little bit below and a little bit above the amplitude state but also we get a formation of a doublet close to the Fermi energy. Now what's peculiar because of the symmetry of this Hamiltonian these levels are such that the hybridized level can be directly traced either to the lower or to the upper under bound states so here you see the same color that you have on the left-hand side gray, gray, black, black and the gray and black lines correspond to super current flowing in opposite direction. Then before showing the result let me anticipate what we can expect from this picture. If we are at zero temperature let's assume we are at zero temperature under this condition we will have current flowing due to this lowest energy state as we are in the clean limit but we also get a contribution from the hybridized level with the upper under bound states. This state carries a current which is smaller which is in the opposite direction so the overall current will be smaller and I have to say that this type of hybridization this is a sketch of course this level depends on K and this type of hybridization is such that the splitings are smaller the closer K is to the closer we are to the total transmission to tau K equal to 1 but these states or larger transmission are also the ones which are mainly responsible for the skewness of the current phase relation. So I hope that I convinced you that the qualitative picture which comes out is the following here you have the current phase relation in this color you have the clean limit and you can see that it's clearly skewned then the different colors correspond to have a presence of impurities if we look at the gamma going to zero limit so let's assume we have really sharp resonances coming from these impurities you get the dashed black dashed line here which is clearly with a reduced skewns with respect to the clean case and the origin is precisely the mechanism I emphasized. Now we can also look at what happens if we increase the temperature and here we plot the skewness which is defined as 2 minus the maximum the value of the phase where the current phase relation is the maximum over pi minus 1 so for a very small temperature let's look at the clean limit first for a very small temperature we have the zero temperature case and then we have a monotonic decrease of the skewness the monotonic decrease of the skewness when we have a bare system simply comes from the fact that we only have a pair of under bound states and as soon as the higher energy bound state is populated it gives a current flowing in the opposite direction so it reduces the skewness and this takes place monotonically but in this case we have this type of structure so what happens is that there is an intermediate frequency scale in which we have first a reduced skewness let's look at the black line so initially for a very small temperature we have the reduced skewness with respect to the clean case and this mainly comes from the presence of these two or current flowing because of these two levels then if we increase a little bit the current sorry, the temperature this level comes into play but this gives an opposite contribution to the super current with respect to the other state so we approach a behavior similar to the one we had in the absence of impurities and this gives rise to an increase of the skewness for a while as the temperature increases further this further level comes into play and we have again the analogous picture that we had in the case of the clean system so a decrease do not manage to point in the right direction a decrease of the skewness as temperature is increasing okay, this is, I didn't say these were results okay, these were results in case of a finite bias sorry, a finite doping and this effect also occurs in the critical current so the critical current reveals basically the same physics so it has this non-monotonic behavior as the skewness then we pass to the equilibrium super current why? because in the current phase relation you have an overall effect with these impurities with this peculiar temperature dependence but if we want really to make spectroscopy of the presence of these impurities more convenient to look at the super current noise spectrum, the equilibrium spectrum which we can evaluate is written down here it has a structure which is fully understandable and we have that the spectral function enters with these two terms with the typical dependence on energy now in this case the processes which come into play are related to the possible transitions between the states and the transitions are here indicated with the colored arrows we have a transition at higher energy which is the same or most of the same order of magnitude of the transition you had in the clean limit then you have transition at intermediate energies the blue and the green ones but the green ones if you look at the Fermi distribution on the left-hand side are basically forbidden by the Pauli exclusion principle if you wish or Pauli blocking because there's no possibility to tunnel between these pair of levels and then we have an additional very low frequency component coming from transition between these two doublets now we plot on the left-hand side the current spectrum the gray area is the area where we expect to have current noise without any impurities in the clean limit and this takes place in a finite frequency range which is all the order of 2 delta not roughly the clean limit is again these Cian lines which you barely see here what happens if we have also impurity is that we get features as smaller frequencies precisely those related to this process can be anticipated and if we look again to the let's say sharp regime where we have resonances you see that we get this is a square root of divergence just because we are in the gamma equal to zero limit and this the value of this frequency can be easily estimated considering how these processes enter the spectral function and it's quite easy to understand that because of the way they enter the spectral function we expect these square root divergences in correspondence to maximum and minima of the differences between the possible transitions these maximum and minima are related to the extremes of the tunneling amplitude so for instance here we can precisely understand where this first transition takes place this peaks takes place of course if we have a broader distribution of impurity we won't have square root divergences but we have finite width peaks this is this picture applies to the case of absence of any doping but we can also consider what happens in the doping the system is involved but still the physical phenomenon is the one I introduce and one can directly count how many transitions one expect in addition one more qualitative feature which characterizes the presence of this impurity is the fact that we have a finite noise also a zero frequency and we can evaluate analytically this finite zero frequency the current spectrum it takes this form it has a linear temperature dependence which is plausible in the presence of this order and it's nice to see how this amplitude the slope in this case depends on the parameters of the impurity states also on the tunneling amplitude of the density and of course again on the transmissivity of the junction this is the behavior depending on the chemical potential and we also observe a peculiar non-monotonic behavior with increasing the width of the energy distribution this is basically related to the spectral weight at the Fermi energy of these processes okay how much time do I have? 5 okay fine so in the last 5 minutes I will just give a flash, a flavor or the other possible processes giving rise to current noise in Grafingiososon junction actually Grafingiososon junctions are a unique two-dimensional structure to study noise and in particular one ORF noise and there is this nice radio by Alexander Ballanding giving a really nice overview on these results there are two experimental results which are closely related to the process I am going to describe briefly one is obtained by this group and what they consider is the effect of they consider in this Grafingioson junction and what they can see is that they obtain one ORF noise and then they consider the effect of boron nitride defects in this hetero structure and charge in homogeneity and they find some anomalous peak in the amplitude of the noise these are not reported here which they attribute to impurity states which originate from the carbon atom which may replace the nitrogen site in the HBN crystal so a model not that far from the one we are considering whereas these other results are from the experiment of Petyakon's group these are on a Grafingioson junction in the close to the diffusive limit in this case they also obtain one ORF noise in the critical current but the mechanism seems to originate from a different effect which is variation of the proximity induced gap in the Grafing junction so I would say that the problem is still not totally solved let me just give an hint of how we suggest that critical current noise may originate the idea is the following the critical current of the Grafingioson junction depends on the chemical potential and it has this characteristic shape so that fluctuations classical fluctuations of the critical current may depend linearly or quadratically on fluctuations of the chemical potential depending on what we call the second point in a sense in a qubit type of language so it means that depending on the doping we can pass from one regime to the other but how can we have fluctuations of the chemical potential basically due to the two-dimensional structure of Grafing the chemical potential is the rate to the square root of the density of carriers so it's plausible to expect that if we have a mechanism leading to fluctuations in the final we will have fluctuations in the chemical potential and therefore fluctuations in the critical current so this is in a few words the mechanism we consider the reason for fluctuations of the carrier density we consider is given by as I said the so-called MAC quarter model the idea is sketched here in the case of Grafing so Grafing is here, black this is a substrate and these spots here that in the substrate there may be some charge traps which induce some trapping and the trapping of charges from the Grafing channel to these traps with a rate with a tunneling rate which depends on the distance of these traps from the Grafing layer if we consider a proper distribution of these trap rates we get basically the MAC quarter model applied to this specific setup in the case of the Grafing Josephson junction here in this work we assume that there is a gate voltage which fixes the voltage of the Grafing so this implies a constraint between the fluctuations of the chemical potential and of the carrier density in Grafing which are described here and in particular you see the fluctuations of the carrier density in Grafing is related to fluctuations of the chemical potential in different orders through coefficients which are the quantum capacitance and its derivative which are parameters which are to a certain extent tunable via the chemical potential and depends also on temperature this gives rise to fluctuations of the critical current and you see that we have fluctuations coming from correlators of different orders and the prefectors which are tunable depends on the critical current derivative of the carrier order first and higher order derivatives they enter the power spectrum actually in this form so the spectrum is 1 over F you see the frequency here but the prefector depends on the combination of correlators of different orders which prefector which we can really predict depending on the doping which we have in the system a similar structure the current density noise it is the same the same structure of different prefectors however and here I just give a snapshot of how the amplitude of the 1 over F noise looks like in this case for the critical current and for the density of carriers in Grafing and the take on message is that we may have slightly no completely different temperature dependencies linear for the critical current and cubic or then passing to quadratic for the carrier density and this effect is entirely due to how the prefectors in the amplitude of this noise behave as a function of the working point so of the doping and depend also on the type of correlator which enters the the noise which is dominant if you wish so without commenting that much let me say that the message is that for this second part is that of this second part ok we can get critical current noise due to under level broadening in this case we can also describe this process as broadening under level and it allows really make a sense of the correlations which are more relevant for these fluctuations for the other part of the of the talk the model was microscopic and again the other take on message is that not only we may have a modification and explain how the current phase relation is modified but this impurity but we also may have a detailed spectroscopic evidence of these impurities thank you very much thank you for the nice talk so the session is open for questions anybody so this seems like a quite involved calculation and I wonder if it's possible to do it also for magnetic impurities in case there are some on graphene I'm not sure ok it's complicated as you said it's quite involved I try to keep it as simple as possible we are doing that we are trying to see what happens for magnetic impurities right thank you let's turn it off up ok thank you more technical points as you pointed out at the very beginning there is this work by Benakka which is basically a book where you have the gene theory for the backscattering using impurities so would it be possible to within this approach also to get information about the statistics or is this something which is something that is due to the description I mean this green's function type of calculation that you are doing ok the calculation is in the same spirit I would say because the starting point is above the gene approach statistics what do you mean statistics of impurity state or in which sense yeah yeah exactly statistical properties you mean ok I think about it I don't think it's exactly what we did ok or at least these are fermionic impurities let's say in this sense what we obtain comes from considering the electrons transition but in a way what you showed is that you have these due to the impurities you develop further in-gap states basically and due to time reversal symmetry they always appear in pairs and questions whether just from the properties of these levels whether this already is sufficient to derive most of what you have this was not clear to me how much because you could somehow was the argument that there is some occupation by these levels and basically what you see when you consider this queerness as a function of temperature is that this place basically reflects the structure this level structure yeah so you could say well let's go to Benaka and as you know I mean to get this sub-level structure you have to solve these nonlinear equations for the energies and then do some say naive statistics where you firmly populate things and then to derive this from this analysis is this something that is fair or is this we didn't think about it but I understand what you mean probably one could try to look at it in this way what we wanted to do was to start from a microscopic model maybe it can be another way of re-deriving this result probably yes thank you for the nice talk as far as I understand you considered wide junctions right because of that you shortened I was wondering and if you in these wide junctions you see any signature of the band structure of graphene itself if there is something special because you have graphene and not any other two-dimensional thing I would say that the behaviors close to the charge neutrality point precisely come from the band structure of graphene these are peculiar but the same type of calculation could be done also with a different spectral density yeah is there any photo I have also so you you go first good so you started from a ballistic graphene and then you can see the conductance goes down or the critical current goes down and in the end do you will you go towards a diffusive limit where the graphene just becomes a diffusive conductor I don't think with a dilute ensemble of impurities I do not expect that we get the regime we try to to maintain the ballistic nature although we did not make any check but the working assumption is like that okay I have a short question related to the 1 over f noise you see you have on the soft straight you have charge but I mean why 1 over f when we have 1 over f noise I mean 2 levels we are considering some fluctuations why not why noise I mean okay yes could also be white noise now the idea was that we consider as far as the 1 over f model we use the standard McWater so the McWater tells you that you have let's say a substrate which is some irregular type of material where you have drop distributed randomly typically you have a huge number of these traps and they are located randomly so with the distance which is different and if you consider the typical structure that tunneling amplitude may have if you simply think to sort of double or multiple well potential you get naturally a distribution of distances and then of amplitude which is such to give 1 over f noise and since actually 1 over f noise is we are looking really for this type of model since 1 over f noise is one of the main that's been to certain standard we come out again as a problem for also this gate months of the future probably and we wanted to see how the critical current is modified by this possible which probably is not the only one effects related probably other features but you know 1 over f noise it's still a mystery already in the convention of Josephson junction so thank you very much thank you to all of you