 My name is Takashi Bouchi from University of Tokyo. And today I'm going to talk about VCS-VC crossovers in iron-sailing self-superconductors. And I'm going to show you our very recent unpublished results, which are quite surprising. So this work has been done in collaboration with many people who sit here. And this is the people in my group. And this is the people in the Yuji Matsuda group at Kyoto University. And we have collaboration with the KIT group and Professor Shin's group, who does the laser ARPES technique, and STM by Hanaguri-san, and theory inputs from Iriya Eremin. And the main players here is Yuta Mizukami and Ohei Tanaka in my group, and Dr. Hashimoto in Shin's lab. OK, so let me start with a very brief introduction on VCS-VC crossover. And this is a kind of general phase diagram. So this is temperature divided by the Fermi temperature. So we consider the Fermi system. And this is the strength of attraction. So you have weak coupling limit, which is here, and strong coupling limit, which is here. And in the weak coupling limit is, of course, is a VCS limit. So we have a pair size, which is actually the coherent strength. It's much larger than the inter-electron distance, which is of the order of 1 over kf. So you have a situation that kf psi is much larger than unity, right? And in the opposite side, you have so-called Bose-Einstein condensation limit. So you have a very strong interaction between these two fermions. So you can actually have a tightly bound molecules like this. So your pair size is now becoming much smaller than kf inverse. So you have a situation that kf psi is much smaller than unity. And what is important here is that there is no phase transition between these two limits. So you have just a crossover phenomena. And most interesting regime is the so-called crossover regime, where you have a strongly interacting pairs. And the pair size is of the order of 1 over kf. So you have the situation that kf psi is of the order of unity. Now here is the important parameter, which is actually the ratio between the superconducting gap delta and the fermion energy EF. And this ratio is actually is the order of 1 over psi kf. And let's suppose you have a single band. You have only whole band given by this red curve in the normal state. Then if you go into the superconducting state, what happens in the BCS limit is, as you know, you have a gap opening at the wave vector kf. So you have a minimum gap at kf, of course. And then if you do, for example, Alpe's measurements, you can see this kind of band structure in the superconducting state. But in the opposite side, what happens is that you will have a negative chemical potential. Then the gap opening occurs at k equal 0, not at kf. So you have a minimum gap here at k equal 0, at the gamma point. And in the conventional superconductors, this ratio delta over EF is very small, like 10 to the minus 4, 10 to the minus 5. So you are in this limit, of course. And in some of the under-adopt high-tc cuprace, this ratio becomes larger. And this is 10 to the minus 1 and minus 3. And so there is a strong discussion about the relevance of this BCS-BC crossover in the physics of under-adopt high-tc cuprace. So today I would like to focus on this system, iron selenium, and in this context of BCS-BC crossover. And as you know, this material is very important material. And first of all, the crystal structure is quite simple. So you have only iron selenium layers like this. And there are a number of interesting features. So the physics is very rich. First of all, you have a structural transition at 90 Kelvin, which is here in this low-t cup. You can see a kink here. And you don't see any magnetic ordering down to lowest temperatures. So here you have only in-plane anisotropy without magnetism. So people call this state as a non-magnetic nematic phase. And for example, if you look at the vortices in the superconducting state, you can see a very elongated shape, which is the evidence for the strong in-plane anisotropy in this nematic phase. And what is interesting here is that we have the so-called compensated semi-metallic electric structure, meaning that we have equal numbers of electrons and holes in the system. And most importantly, we have very small Fermi energies. And we can obtain a very beautiful single crystals by using the chemical vapor transport technique recently developed. So if you look at the STM topographic data, you can easily very hard to find any defects. And you have a very small concentration of defects. So what you can do is, for example, the quasi-particle interference by using STM techniques. So this is the data taken at the weekend by Hanaguri-San's group. And by using two different scan directions, you can actually have the electron-like Fermi surface like this and the hole-like Fermi surface like this. And so you can actually quantify the effective Fermi energies of these two bands. And this technique has a very high energy resolution. So you can have this number. And at the same time, you can take a look at the energy up by the same technique without applying magnetic field. So this data has taken 12 Tesla to suppress the superconductivity to see the electron in the hole band like this. So from these measurements, we can quantify this ratio delta over EF, which is actually very close to unity for the electron band here. And we get the number like 0.3 for the hole bands, which is here. So in both cases, these numbers are very large compared to other superconductors. So we can say that we are in the VCSVC cross-over regime. And we can do another test, which is the KF-GSI. And KF can be simply quantified by looking at this q over 2. And in-plane coherence lengths can be quantified by the HC2 measurements. So we get the number KF-GSI, which is these kind of numbers. So again, this is consistent with the VCSVC cross-over regime. So we can safely say that this material, iron selenite, is located deep inside this VCSVC cross-over regime. And then what is quite important in this phase diagram is that we have another kind of temperature scale other than Tc, which is actually a pairing temperature. So we have an extended temperature regime between the pairing temperature and actual Tc, where you have a so-called preformed cooper pairs, or you may have the so-called pseudo gap. So first, we want to look at what is the superconducting fluctuations in this system. And we have evidence for very large superconducting fluctuations. So this is, for example, a diamagnetic response. We have a temperature dependence of the magnetization. And this is the diamagnetic component. And you can see it starts from fairly high temperature than Tc, which is about 9 Kelvin. And here is the resistivity curve. And if you look carefully, you can see downturn curvature starting at around 20 Kelvin, which is here. And you can see the derivative. The rho dt shows an upturn like this. But it is quite difficult to quantify this superconducting fluctuation only from this data. So we decided to look at the magnetic torque. And because that we can actually measure magnetic torque in a very high sensitive way by using the so-called micro cantilever technique like this. And we can have sensitivity much higher than the commercial script magnetometer. And essentially, what we are doing is that we scan the magnetic field like this from the c-axis and measure the angular dependence of the magnetic torque. So we see this kind of sine curves like this. And the torque is given by this formula, of course. And in this kind of setup, the amplitude of this torque curve is proportional to the difference between the c-axis susceptibility and a-axis susceptibility. So we can quantify this delta chi in a very high resolution. So this is the temperature dependence of delta chi. And as you expected, we see some anomaly at structural phase transition or nematic phase transition at Ts. Now, we can do this kind of measurements at different fields. So this is a set of data showing the temperature dependence of the magnitude of delta chi, which is the difference between chi c and chi a. And as you can see, below some temperature, we see a very strong field dependence. And this field dependence of delta chi is coming from the diamagnetic susceptibility due to the superconducting fluctuations. So we can quantify this number as a function of temperature by subtracting high field data. And these are the data for 0.5 and 1 tesla, which can be compared with the theoretical prediction of superconducting fluctuations in the conventional mechanism, meaning that it's an athero-masof-rackin-type Gaussian fluctuations, which is given by this formula. So this theory prediction is for the zero field limit. And we can kind of calculate this delta chi a l as a function of temperature, which is shown by this blue curve. And as you can see, we have a kind of tendency that if you go to a lower field, this magnetism becomes larger and larger. And even for the 0.5 tesla, we have strongly enhancement from this theoretical prediction in the zero field limit. So from this, we conclude that we have observed giant superconducting fluctuations in the iron selenium up to T star, which is of the order of 20 Kelvin, which is actually twice high than the actual Tc. So these giant superconducting fluctuations can be considered as a signature of BCSBC crossover. And the recent theory paper also concluded that this is actually the signature of BCSBC crossover. Yes. So you are saying that the quantitative value is much larger than the mass of large in prediction. But the temperature dependence is also different. Or did you check if the temperature dependence is still square root of T minus Tc or not? OK. So because of the large error, it's kind of difficult to distinguish. But within an error, you can actually fit if you have a. So the discrepancy is due to the fact that you can estimate x i from upper critical field. And then the pre-facto that you get is much larger than what you expect. OK. So far, so good. I mean, this experiment is consistent with BCSBC crossover. Then we pose a question. What about the specific heat? Because this is the simplest thermodynamic quantity. And in the case of BCS, we have a very simple temperature dependence of C over T, for example. And we have actually jump at Tc. Of course, we have a second phase transition. And the temperature dependence in the superconducting state depends on the structure of the superconducting gap. But if you have a conventional S wave superconductor, you will see this exponential decay. However, in the case of BEC, the temperature dependence of the specific heat looks completely different. And this is the simplest case for 3D both gas for BEC. And if you look at C over T as a function of T, you actually have the gradual increase of C over T towards Tc. And you don't have a jump, but you have a kink. So it's not like a typical second phase transition. And then in the superconducting state, you have a power law dependence. And in this simplest case, you will get square root T dependence of C over T. So we have decided to look at this specific heat very carefully. So we developed the high resolution setup for the specific heat measurement for particularly designed for measuring the small samples. So we actually use this thermometer as a heater as well. And we have a grass fiber here to see what's the change of the temperature if you make a pulse of the heat. And we record this increase and decreasing branches. And then we can actually obtain the specific heat as a function of temperature. Yes? Well, for example, OK, this one. Yeah, this is just the simplest one. So I'm not saying this. So the advantage of this kind of new setup is that we can reduce the so-called addenda component, which is the background contribution of the sample holder. And for example, if you use a typical PPMS, which is a commercial kind of measurement setup, you get very large kind of background contributions. But we can actually reduce by a factor of like 1,000. And then we can actually measure very precisely the temperature dependence of small single crystals with a mass of 10 to the 10 or 100 micrograms. And this is a kind of typical example for measured for iron selenium. So we kind of subtracted the form term to extract the electric contribution, C e over T, as a function of temperature. So we use this 81 microgram sample for these typical measurements, which can be compared with other previously published papers. And here you can actually see essentially consistent results. And as you can see, we see a clear jump at Tc. And we see kind of T linear behavior in the low temperature limit. And this T linear behavior is considered as evidence for the nodal-like superconducting gap in this system, or very small gap minima. But the fact that we see a clear jump at Tc is more consistent with the BCS-like state in this material. So what's happening here? So one important contribution from theory side is given by the paper by our chairman. And they considered actually two-band system. Of course, iron selenium is the march-band superconductor, so you should consider this, anyways. But so far, BCS-BC crossover physics has been only considered for a single band system, particularly focusing on the ultra-cold atomic systems. And for example, if you have one electron band shown by red-dashed curve in the normal state, then if you put the large delta over EF ratio, then you get the BC-like state. And in the superconducting state, you should see this kind of band structure. And the minimum gap opens at k called 0, of course. However, they actually considered two bands, one whole band and one electron band like this. And then they put strong inter-band interactions between these two. Of course, you can expect that in iron selenium. Then what they found is that at least one of the band has this kind of BCS-like structure in the superconducting state, even if you have a large delta over EF values. So this suggests that the march-band effect complicates the physics of BCS-BC crossover. And our results of specific heat may be explained by this kind of march-band effect for iron selenium. So we actually further continuous our research to this sulfur substituted system. So this is the isoelectronic substitutions. And we can obtain high-quality single crystals up to 20% of sulfur content. And this is the temperature dependence of resistivity. And as you can see, we can see the anomalies associated with the pneumatic transition. And the transition temperature is shifting down as you increase the sulfur content. And this is the phase diagram we obtained from these kind of measurements. And the structural transition goes down. And it disappears at about 17%. Now, if you look at STM, then you can actually see some features associated with the sulfur position. And you can actually count the number of sulfur atoms in this area. And this is a wider field of view. And you can actually see that there is no evidence for any segregations in this kind of high substitution concentration samples. Also, we can observe the phantom oscillations up to a very large number of x, meaning that we can actually have very clean crystals, clean and homogeneous. So this means that this is a very good system to kind of study the intrinsic physics. And then the question is, what's going on about the BCSBC crossover in this sulfur substituted systems? So let's look at the specific heat as a function of temperature at different composition. So this is, again, the iron selenium. So you have seen already. So we see a clear jump at Tc for iron selenium. So if you go to 10% sulfur substitution, here you get a pneumatic temperature about 70 Kelvin. So you are still deep in the pneumatic phase. Then you see, again, a clear jump at Tc in the C over T as a function of T. And you also see a T linear-like behavior in the low temperature limit. Now, what is surprising is that if you go to 20%, which is outside the pneumatic phase, so we are actually in the tetragonal phase, then this C over T look completely different. So as you can see, this is going up gradually, and we have no jump at Tc, and we have a kink. And below Tc, we have this kind of convex curvature like this, so which is more like a power row. So this absence of the clear jump at Tc is not consistent with BCS-like state. And it's more consistent with BC-like kind of features as I showed. So first thing, I'm an experimenter. So first thing I was worried is that maybe we should check the Tc distribution, first of all, in this particular sample. As I mentioned, we have the very clean and homogeneous samples, so it is unlikely that this comes from the Tc distribution, first of all. And we use the tiny crystals, and we made this setup for the tiny crystals measurements. However, there is a chance that we may have Tc distribution. So what we have done is that we actually looked right at this critical concentration around 16%. And here, surprisingly, we see a kind of double transition like this. And one transition is at around 8 Kelvin, and the other is at 4 Kelvin. So even if we have a very clean and homogeneous sample, we have a finite size of the crystal, so we may see some kind of Kevin Carlin homogeneity. That is inevitable, right? But what we found is that these two transitions can be reproduced by using two different data sets. One is that specific heat at 13% percent, which is deep inside the pneumatic phase. And another is the 20% data, which is outside the pneumatic phase. Then if you actually add these two, then you can essentially reproduce the data at 16%. So this strongly suggests that we have a kind of rapid jump in Tc as a function of x like this. And in the tetragonal phase, we have a kind of flat x dependence of Tc. And in the pneumatic phase, we also have essentially x independent Tc like this. So you have a jump. So this means that this kind of gradual increase cannot be attributed to the Tc distribution because in the tetragonal phase, we have a fairly constant Tc as a function of x. So we believe that this is an interesting feature of this material. And as I said, in the superconducting state, we have a power law dependence at least down to below 1 Kelvin, which we can measure in this technique. And we get like a square T dependence. And this is kind of surprisingly similar to what is expected in the BEC case in the simplest 3D Bose gas without any interactions. So I'm not saying that we are in this kind of state, but it is surprising that it's an experimental fact that we see this kind of anomalous temperature dependence of the specific heat. So this is, again, the comparison between the iron selenium results in the pneumatic phase. And this is 20% sulfur adopt sample in the tetragonal phase. You see a clear difference between these two. Now we can also take a look at the tunneling conductance curve at different composition. So this is the sulfur content. And the blue curves in the pneumatic phase. And the red curves in the tetragonal phase. And as you can see, they look quite different between these two phases. And in the pneumatic phase, you see clear quasi-particle peaks. But in the tetragonal phase, this is strongly dumped. And it's not clear from this figure, but in the tetragonal phase, we see a finite zero energy density of states, very large one, which is present. And this finite zero energy state is quite consistent with this specific heat data, because this measurement has been taken at 0.445 Kelvin, which is about here. And in the specific heat, we see a very large density of states, even at this kind of low temperature. So in this sense, these two measurements show a quite consistent picture. And we believe that we have obtained thermodynamic evidence for possible BEC-like state in tetragonal iron selenium sulphur. And this is quite amazing, because there is no any solid sofa to have this kind of BEC-like state. Only ultra-cold atomic systems can talk about BEC-sofa. So if this is correct, it's like a revolution in the solid state physics. So we can also take a look at the torque magnetometry in the 20% sample. And we did a similar analysis as iron selenium. And again, we got a large enhancement of this delta chi below T star, which is about 20 Kelvin again. But you should know that the temperature scale and the field scale should be quite different between these two systems, because here we have Tc as about 9 Kelvin, but here we have Tc as 4 Kelvin. And so associated Hc2 is also suppressed compared with this. So we see a large enhancement, even at 1 Tesla. This means that the superconducting fluctuations are more pronounced in the self-adopt tetragonal phase. Now I have asked the Professor Shin's group to look at the laser-alpes measurement in this system. And here is a set of data taken at 2 Kelvin. And here is the ortho-rombic phase, and here is the tetragonal phase. And as you can see, in all of these systems, they see a quite flat band like this, which is consistent with the vicinity of VCS-VC crossover regime. But if you look carefully, if you zoom up this part, then you can have convex down curvature in the ortho-rombic phase. But you have a convex up curvature in the tetragonal phase. So this is, again, consistent with the VEC-like state in this tetragonal side. Now the question is, what about delta over Ef? Then from quantum oscillation measurements, we can look at the size of the film surface, which is related to the film energy. And most of the bands show the increasing trend of the frequency, which means that Ef is going up in energy if you go to the tetragonal side. And at the same time, the energy gap is shrinking if you go to the tetragonal side. So this means that delta over Ef is actually smaller in the tetragonal phase. So this is kind of opposite to what you expect in the VEC case. So the conclusion is that the observed VEC-like state in tetragonal side is quite unexpected. So I think that we should think about another key parameter that determines the VCS-VC crossover in this kind of March band case. So this is the final kind of slides. And I talked to Ilia Eremin and who kind of suggested the importance of the two-band effect. And he considered, again, this kind of two-band system, as Professor Chubko did. And they considered the interband interaction and also the intraband interactions. And what they found is that if you change the ratio between the intraband versus interband interactions, then if you have more stronger interband interactions, you actually get VCS-like states. So this is the calculation of pairing temperature and TC. But if you increase the intraband interactions, then you get kind of deviations between these two, meaning that you're likely to have the VEC-like state. So this suggests that actually another parameter is this ratio between the intraband and the interband interactions. So I think the time is up, so I can come to the conclusion. So we have studied the kind of thermodynamic study of this VCS-VEC crossover in this iron-celled new sulfur superconductors. So we have obtained the clean and homogeneous single crystals. And we observed the unexpected VEC-like behaviors in this tetragonal site, which has a smaller delta over EF. And so this is still, we have several open questions. But one possible scenario is that maybe interband versus intraband interactions, the ratio between these two may be an important parameter to describe this system. And finally, this is a very good bulk system for studying march band VCS-VEC crossover in any series. Thank you very much for your attention. OK, great. Questions and comments?