Thin-film Giaever transformer Vortex drag in a Thin-film Giaever transformer Yue (Rick) Zou (Caltech) Gil Refael (Caltech) Jongsoo Yoon (UVA) Past collaboration: Victor Galitski (UMD) Matthew Fisher (Caltech) T. Senthil (MIT) The superconductor-insulator transition in thin films has drawn a lot of attention. One reason is that the experiments seem to expose electrons at the height of their bipolar disorder – with a mani-depressive tendency towards superconductivity on one hand, insultor on the other, and maybe even a new kind of metal. The electrons are both strongly correlated and in a highly disordered environmnet. Many theories have been put forth, and two directions in particular stick in my mind: a percolation picture, and a vortex condensation picture. They give similar predications, and are still far from being verified with no clear evidence. Here I would like to describe this two paradigms – including my contributions to the vortex paradigm, and then suggest and discuss an experiment that may decide qualitatively between the two. $$$: Research corporation, Packard Foundation, Sloan Foundation

Outline Experimental Motivation – SC-metal-insulator in InO, TiN, Ta and MoGe. Two paradigms: - Vortex condensation: Vortex metal theory. - Percolation paradigm Thin film Giaever transformer – amorphous thin-film bilayer. Predictions for the no-tunneling regime of a thin-film bilayer Conclusions I’ll start with a very quick base line of the experiments, move to discuss the percolation and vortex paradigms, then describe the suggest bilayer drag experiments – an echo of the Giaever transformer. And conclude.

Quantum vortex physics SC-insulator transition Thin films: B tunes a SC-Insulator transition. V I InO Ins B B (Hebard, Paalanen, PRL 1990) The story of supreconductor insulator transition started with this measurement of the resistance vs. temperature of an InO Film in a perpendicular magnetic field. As the magnetic field is raised, it seems that the film goes from wanting to be SC at the lowest temperatures, to wanting to be insulating. SC

Observation of Superconductor-insulator transition Thin amorphous films: B tunes a SC-Insulator transition. InO: More recent studies of InO showed saturation at about 100mK, and a set of distinctly different insulators – very weak, peaking at a few kOhms, intermediate – 100kOhm, and huge – GOhm. (Sambandamurthy, Engel, Johansson, Shahar, PRL 2004) (Steiner, Kapitulnik) Saturation as T 0 Insulating peak different from sample to sample, scaling different – log, activated.

Observation of a metallic phase MoGe: Apart from the intriguing variety of insulating behavior, other weak insulators also showed saturation – here is MoGe, and Ta. (Mason, Kapitulnik, PRB 1999)

Observation of a metallic phase Is the saturation in the intermediate field range indicating a new phase? Quite possibly, although some object and think of it as a saturation of electron temperatures? (Qin, Vicente, Yoon, 2006) Saturation at ~100mK: New metalic phase? (or saturation of electrons temperature)

Vortex Paradigm How do we think about this SIT, possibly SMIT in terms of the vortex paradigm?

Correlated states in higher dimensions: Superconductivity Pairs of electrons form Cooper pairs: (c’s are creation operators for electrons) Cooper pairs are bosons, and can Bose-Einstein condense: Meissner effect H. K. Onnes, Commun. Phys. Lab.12,120, (1911) Gap in tunneling density of states www.egglescliffe.org.uk http://dept.phy.bme.hu/research/sollab_nano.html

Free energy of a superconductor – Landau-Ginzburg theory Most interesting, phase dynamics . Supercurrent: Vortices: (Josephson I) V Vortex motion leads to a voltage drop: (Josephson II)

X-Y model for superconducting film: Cooper pairs as Bosons When the superconducting order is strong – ignore electronic excitations. Standard model for bosonic SF-Ins transition – “Bose-Hubbard model”: Hopping: Charging energy: Large U - Mott insulator (no charge fluctuations) Large - Superfluid (intense charge fluctuations, no phase fluctuations) The vortex paradigm thinks about the films as having bosons – cooper pairs – in them. These are described by a Bose hubbard model. Phi is the SC phase variable and this is the Josephson term, giving hopping. N is the local particle density, which appears in the charging energy. Large U – we get an insulator with suppressed charged fluctuations. Large rho_S – the superfluid wins, and global phase coherence arises. SIT. superfluid insulator

Vortex description of the SF-insulator transition (Fisher, 1990) Vortex: Vortex hopping: (result of charging effects) Vortex-vortex interactions: superfluid insulator Cooper-pairs: Following Matthew Fisher, we can dualize this to discuss the model in terms of vortices. When the phases indicate a winding – this implies a vortex. Naively, we would think of vortices as bosonic, and therefore can have their own phase angle – theta. With a hopping t_V – which will determined by the disorder and coulomb interactions in the films, a local density conjugate to theta. Also, the vortices in thin films will have a log interaction. At large rho_S the vortices are localized because of interactions, giving a vortex-insulator, which is a phase coherent CP condensate. But weak rho_S or strong vortex hopping – V-superfluid which implies destruction of phase coherence. –CP Insulator. Condensed vortices = insulating CP’s V-superfluid V- insulator Vortices:

Universal (?) resistance at SF-insulator transition Assume that vortices and Cooper-pairs are self dual at transition point. Current due to CP hopping: 2e EMF due to vortex hopping: Resistance: In reality superconducting films are not self dual: vortices interact logarithmically, Cooper-pairs interact at most with power law. Samples are very disordered and the disorder is different for cooper-pairs and vortices.

Disorder localized Electrons: Magnetically tuned Superconductor-insulator transition Net vortex density: Disorder pins vortices for small field – superconducting phase. Large fields some free vortices appear and condense – insulating phase. Larger fields superconductivity is destroyed – normal (unpaired) phase. Let’s concentrate on the thin films in a magnetic field and see if we can explain the generic Magnetoreisstance curve. The net vortex density is given by the magnetic field in this familiar fashion. At low fields disorder can pin vortices, and we have a V-insulator, which is a standard superconductor. At larger fields some free vortices appear – and if they are bosonic, they condense. – Vortex superfluid, which is the insulator – this peak. The physics of this peak is then that of an incoherent superconductor where cooper pairs are localized, and threfore infinite resistance. Then, at larger fields, we get a normal – unpaired phase, of mildly localized electrons. Disorder pins vortices: Free vortices: Vortex SF insulator Disorder localized Electrons: Normal (unpaired) Phase coherent SC

Disorder localized Electrons: Magnetically tuned Superconductor-insulator transition Net vortex density: Problems Saturation of the resistance – ‘metallic phase’ Non-universal insulating peak – completely different depending on disorder. But problems always come up – the resistance saturates at finite temperature in this region, and the peak is not universal. Metallic phase? Disorder localized Electrons: Normal (unpaired)

Two-fluid model for the SC-Metal-Insulator transition (Galitski, Refael, Fisher, Senthil, 2005) Uncondensed vortices: Cooper-pair channel Disorder induced Gapless QP’s (electron channel) (delocalized core states?) Finite conductivity: To tackle this problem, together with Matthew Fisher, Victor Galitski, and T. Senthil,we suggested a phenomenological modification of the theory. We clearly have vortices in the film, but for some reason, they don’t want to condense. Therefore we can treat them within RPA, as if they were fermionic – i.e., like a classical diffusive gas. They have a finite conductivity – and therefore flow in the direction of the force – this is vortex current – and if we try to put a current of cooper pairs through, the vortices will move normal to it. We posit that in addition, another fermionic gas is present – gapless Bogolubov quasi particles. Possibly there due to the disorder , or due the the vortex core states. This presents an electronic channel which also has a finite conductivity. Two channels in parallel:

Transport properties of the vortex-metal Effective conductivity: Assume: - grows from zero to . - grows from zero to infinity. B B B Vortex metal This is the effective conductivity. We have no microscopic calculation of the conductivities. But there is really only one generic possibiltiy. Upto a certain field both vortices and gapless electrons are localized. B sub e, some density of states of gapless QP’s appears, and its conductance grows up to the normal-state conductance – in the absence of pairing, that is the maximal value that diffusive electron states can give. The vortex conductivity is also zero upto some field, and then it increases gently, until eventually it diverges at B_V. in this range we can discuss the vortices within RPA. They always have some finite – albeit large – conductance. Let’s put these two together. There are two possibilities – in the weak insulators, the B_e is smaller than B_V – spinons appear before vortices condense. Without spinons, the resistance of the film is the conductance of the vortices. After vortices are condensed – all we have is spinons- who carry all the current – they becom electrons. In the middle we have a resistance which is the reisstance of the spinons parallel to the conductance of the vortices. Never diverging. This middle regime is the vortex metal. This should remind you of the weak InO. Weak insulators: Ta, MoGe Weak InO Normal (unpaired)

Transport properties of the vortex-metal Effective conductivity: Chargless spinons contribute to conductivity! Assume: - grows from zero to . - grows from zero to infinity. B B B Vortex metal The case of strong insulators is covered by the case of vortices condensing before the gapless electrons appear. This gives a good insulator sliver in the middle. Strong insulators: TiN, InO Insulator Normal (unpaired)

More physical properties of the vortex metal Cooper pair tunneling (Naaman, Tyzer, Dynes, 2001). A superconducting STM can tunnel Cooper pairs to the film: We can also probe the film with a superconducting tip, as was tried in Dynes group, in which case, we expect the tunneling conductance to be a combinatoin of 2e tunneling, and a cooper pair tunneling.

More physical properties of the vortex metal Cooper pair tunneling A superconducting STM can tunnel Cooper pairs to the film: Vortex metal phase: Normal phase: CP CP Vortex metal Normal (unpaired) B Insulator In the normal phae, we expect the tunneling to just be proportional to the electorn conductance which is here sigma _S and since the electrons are in a localized phase, we expect this to diverge as ln T^2, or if in the coulomb glass, as e^1/T^0.5. The dependence of the conductance in the vortex metal phase, we calculate to be determined by the vortex conductivity, and it goesto zero relly fast – T^lnT. This dependence will dominate in this area. And the more mild one will dominate in the right hand side. e strongly T dependent

Percolation Paradigm (Trivedi, Dubi, Meir, Avishai, Spivak, Kivelson, et al.)

Pardigm II: superconducting vs. Normal regions percolation Strong disorder breaks the film into superconducting and normal regions. SC NOR NOR B R Moving on to the percolation paradigm, let’s see if we understand this. Here, in zero field the film is a superconductor. But as the field increases some regions become normal, and the resistance increases a bit.

Pardigm II: superconducting vs. Normal regions percolation Strong disorder breaks the film into superconducting and normal regions. NOR SC SC SC SC SC R But as the field increases the SC no longer percolates, conduction happens through electrons, and the SC is confined to islands, which are inaccessible to single electrons, and have too much charging energy for cooper pairs. Electrons now have to move through narrow localized channels, giving rise to the large resistance. B Near percolation – thin channels of the disorder-localized normal phase.

Pardigm II: superconducting vs. Normal regions percolation Strong disorder breaks the film into superconducting and normal regions. NOR R Finally, at large field, all electrons become normal, and the the conduction is through the electorns which are somewhat localized by the disorder B Near percolation – thin channels of the disorder-localized normal phase. Far from percolation – normal electrons with disorder.

Magneto-resistance curves in the percolation picture (Dubi, Meir, Avishai, 2006) Simulate film as a resistor network: NOR island SC island Normal links: Nor-SC links: SC-SC links: The magneto resistance is then calculated by solving a resistor network. The film is divided into sites on a square lattice. The sites are either normal or superconducting. Between normal sites we model conduction as in a disordered glass – with radnom onsite energies, and this activated form. Between superconducting island, we have very little reisstance, and from normal to SC links, the resistance is inhibitavely high – activated with a charging gap. Solving this network gives indeed graphs as this, by choosing parameters correctly. Reuslting MR:

Drag in a bilayer system Two theories, very similar magneto resistance is obtained generically. Can probably be fine tuned to get any more refined measurements such as nonlinear and AC conductivity. But here is a an experiment that will qualitatively should be able to tell between the two approaches.

Giaever transformer – Vortex drag Two type-II bulk superconductors: B The idea to measure drag in a bilayer comes from the Giaever transformer idea. Put two type || superconudctors one on top of the other, put it in a field – vortices with flux tubbes penetrate both layers. If a current flows, the vortices will move together in both layers. Consider the drag resistance – what is the voltage drop on layer 2 due to a current in layer 1. The voltage is only due to vortex motion, and therefore it is the same for both. The drag resistance is then just the resistance of the bottom layer. The strong binding between the vortices in the two layer arises essentially because of current-current coupling of the demagnetizing curretnly around the vortices. Notice where the curret goes. This can only happen for vortices though. Vortices tightly bound:

2DEG bilayers – Coulomb drag Two thin electron gases: Coulomb force creates friction between the layers. We can consider the same thing in 2d electron systems. In this case it is the coulomb interaction between electrons that may produce drag. This will be significant only if the 2degs are dilute – proportional to 1/n^2. To see why that is, consider the potential on layer 2 due to a dilute gas of electrons in layer 1. the potential will have deep wells, and when you move the electrons in layer 1, this potential moves, and takes the electrons with it. But if the electrons in layer 2 are dense, the potential has more undulations and less amplitude, and the drag is much much smaller. Note that the direction of the current is completely opposite to the one induced by the vortex motion. Inversely proportional to density squared: Opposite sign to Giaever’s vortex drag.

2DEG bilayers – Coulomb drag Two thin electron gases: “Excitonic condensate” Example: The drag measurement was successfully used in 2deg bilayers, to probe the excitonic condensate there. The signal is clearly visible (Kellogg, Eisenstein, Pfeiffer, West, 2002)

? Thin film Giaever transformer Drag suppressed Significant Drag Insulating layer, Josephson tunneling: Amorphous (SC) thin films or Percolation paradigm Vortex condensation paradigm Drag is due to coulomb interaction. Drag is due to inductive current interactions, and Josephson coupling. Electron density: Such an experiment can also be used to illuminate the SIT field. Let’s conisder a device which has two amorphous thin films with an insulating layer in between – if possible 5nm? The insulating layer could be either completely tunnel proof, or allow some small josephson tunneling – so that the interaction between vortices is not only the current-inductive coupling. Within the percolative paradigm, drag would only come through coloumb interaction. If vortices are responsible for the resistance in each layer, the drag would come from vortex drag, and will be due to the inductive or josephson coupling between the layers. But the electron density Is huge compared to the 2degs – more than three orders of manitude bigger. Vortex density is pretty small though. So it seems plausible that the drag is suppressed in the percolation paradigm, while it should be signigicant within the vortex paradigm. Vortex density: ? ( QH bilayers: ) Drag suppressed Significant Drag

?

r Vortex drag in thin films bilayers: interlayer interaction Vortex current suppressed. e.g., Pearl penetration length: r Vortex attraction=interlayer induction. Also suppressed due to thinness. So far we considered the drag without josephson coupling between the layers. Let me briefly review what we got. First, we expected that inductive coupling alone will not be enough to induce significant drag – since the layers are so thin, the amount of demagnetizing currents are so small that for instance, the london penetration length stretches out to become the pearl length – 2 lambda london square over the thickness of the superconductor. We calculated the inductive coupling between two vortices at distance r assuming their stationary current pattern, and we get this interaction. It is hard to read it, but first notice – phi_0 is the unit of flux, and from lambda_eff^2 we get indeed d^2 in fromt as we expect – since the layers are thin. Now at long range, we recover the log interaction, and at short range, we get a restoring force, again proportional to thickness squared.

proportional to MR slope. Vortex drag in thin films within vortex metal theory Perturbatively: (Kamenev, Oreg) Drag generically proportional to MR slope. Expect: Looking more from the side, we want to know drag, and we use the vortex metal theory for it. Perturbatively, we need a commutator of the current in one layer with that in the other, which corresponds to this diagram, with two interaction lines connecting the two polarization bubbles. Here comes a big surprise – we would expect naively that the vortex drag conductance depends on the individual conductance over the vortex density, but following von Oppen Simon and Stern, who thought about a similar problem in 2degs, this should actually be dependent on d sigma d n, but since sigma v is the resistance, and n is the vortex density – the magnetic field, we get proportionality to the slope of the magneto resistance squared. This is a general result for drag calculations. Good for strong insulators. The final answer we get is written here. Note that we took the screened interaction between the vortices. Chi is the denisty response function of the vortices, calculated within RPA. (Following von Oppen, Simon, Stern, PRL 2001) Answer: U – screened inter-layer potential. - Density response function (diffusive FL)

Vortex drag in thin films: Results Our best chance (with no J tunneling) is the highly insulating InO: The dependence on the slope of the magento resistance shows that the most hope to see anything without Josephson coupling is by using the most insulating InO sample. In that case we calculated the vortex metal parameters from the experimental observatoin, put them in this result, and this is what we got: at the region of high slope, at the lowest Temperature of 70mK, we expect to see 0.1mOhm, not too small, possibly barely measurable? As we go further away , it fades into the sidelines. I won’t tell you the calculation, but we also calculated in the condensed vortex regime, which gives significantly lower drag as expected. We smoothend the result with a gaussian of width ½ Tesla – which suppressed the maximum here quite a bit. The conservative maximum is then – 0.1mOhm. I would like to say that I think it is likely that the drag resistance should diverge at the transition to the insulator, but we can’t support this so much. maximum drag: Note: similar analysis for SC-metal ‘bilayer’ using a ground plane. Experiment: Mason, Kapitulnik (2001) Theory: Michaeli, Finkel’stein, (2006)

Percolation picture: Coulomb drag Solve a 2-layer resistor network with drag. - Normal - SC Can neglect drag with the SC islands: And now the competing theory. If we get something completely different, then we are happy, since this experiment will hold promise to decide. What we did is solve a two-layer resistnace network, with drag appearing between links that sit directly on top of each other. It turns out that for this calculation we can neglect any drag to do with the SC regions, and the drag of parallel nromal links we model as the drag for a localized electron glass, which has the expression here – it is a result modified from Efrat Shimshony’s paper for the coulomb glass for our purposes. Normal-Normal drag – use results for disorder localized electron glass: (Shimshoni, PRL 1987)

? Percolation picture: Results Drag resistances: Solution of the random resistor network: Compare to vortex drag: ? Armed with these results, we calculated the drag in the percolation picture, and it also has a peak – but look – it peaks at 10^-11 ohms. Much

Conclusions Vortex picture and the puddle picture: similar single layer predictions. Giaever transformer bilayer geometry may qualitatively distinguish: Large drag for vortices, small drag for electrons, with opposite signs. Drag in the limit of zero interlayer tunneling: vs. Intelayer Josephson should increase both values, and enhance the effect. (future theoretical work) Amorphous thin-film bilayers will yield interesting complementary information about the SIT.

Conclusions What induces the gigantic resistance and the SC-insulating transition? What is the nature of the insulating state? Exotic vortex physics? Phenomenology: Vortex picture and the puddle picture: similar single layer predictions. Experimental suggestion: Giaever transformer bilayer geometry may qualitatively distinguish: Large drag for vortices, small drag for electrons.