Tuesday 28 July 2015 – Goetz Seibold - Intrinsic spin Hall effect in systems with striped spin-orbit coupling

After introducing the spin Hall effect, Götz described the basics in terms of the skew scattering mechanism and the side jump, and how in the stationary state, in a homogeneous system, the intrinsic conduction is cancelled by a diffusive contribution countering it.

Against this background he proposed to look at spatially non-homogeneous systems, and proposed a periodic modulation of the Rashba spin-orbit coupling for the 2-dimensional electron liquid (2DEL) at the interface between a thin lanthanum aluminate film and a strontium titanate substrate (LAO/STO). The modulation proposed could be achieved, e.g., putting periodically spaced metallic stripes as top electrodes to dope in a one-dimensionally

varying manner. The spin-orbit  coupling would be thus modulated, and the carrier concentration would also oscillate with maximal density for maximal coupling. The theory  is, however, much more general, quite applicable to very different systems, while keeping the periodic stripe geometry on a 2D system.

In this scenario, it seems that even the ground-state would display non-zero spin currents. Considering now the system in the presence of an applied electric field, but in the stationary situation, one finds suitable conditions for which one can get a non-zero spin-hall sigma, while keeping a zero “stationarity parameter” gamma  (i.e. the one which identifies the spin-hall conductivity of the equivalent homogeneous case).

Such a behaviour would be related to localised states at the stripes in the 2DEL. This Rashba spin-orbit coupling characteristics should be robust against disorder.

The charge transport would be strongly suppressed in the direction of the applied field, giving rise to large spin-hall angles.

The presentation was on the technical side, but offering good insights into the physics of the problem. As normal with last lectures in a dense session that is running late, the discussion was brief.  There was a question on the range of density where this effect is robust. The answer was that it corresponds (in the homogeneous- Rashba-coupling case) to the regime where a single chiral band is occupied. However, for the inhomogenenous case such an identification is not so simple. A second question was on the possibility to achieve the same effect with different inhomogeneous profiles for the Rahba coupling, and the answer was that in a previous work it was shown that if the Rashba coupling oscillates randomly aorund a fixed value it cannot be seen.

Blogged by Emilio Artacho

Tuesday 28 July 2015 - Stefano Gariglio - Study of Superconductivity at LaAlO3/SrTiO3 Interfaces by Field Effect

Stefano starts his talk by a general introduction on the fascinating properties of the LaAlO3/SrTiO3 (LAO/STO) interface, namely superconductivity, strong spin-orbit coupling and magnetism. In this interface between a non-polar material (STO) and a polar material (LAO), an electronic reconstruction takes place to avoid the divergence of the electrostatic potential as the LAO thickness increases. As a result, electrons are transferred from the surface of the LAO to the STO where Ti ions can have a mixed-valence ionic state. A 2DEG is formed at the interface and extends in the STO on a typical thickness of 10 nm. We will get back to this point latter. One key question that Stefano addressed is whether or not this electronic reconstruction is specific to the LAO/STO interface. The answer is...no and several other oxides interfaces have been found to be conducting and even superconducting with similar transition temperatures than the one of LAO/STO. But of course, one always needs the same ingredient : a polar discontinuity. Let's look at a few examples. If you replace the LaAlO3 layer by a LaGaO3 layer, another wide gap and polar material, you also obtain a 2DEG which is superconducting below 300mK. Not very surprising since Ga has the same valence state than Al (just one row below in the periodic table). A 2DEG is also observed with LaTiO3 and LaVO3, two Mott insulators where the transition metal ion (Ti or V) can take many different valence states. The LaTiO3/SrTiO3 interface is also superconducting but so far there is no report of superconductivity in the LaVO3/SrTiO3 one. Good, but these are all STO based heterostructures.  Can we use another non-polar material to replace STO ? The answer is yes. H. Hwang and collaborators have shown that under proper atomic boundary conditions, the TiO2/LaAlO3 interface is conducting with a high-mobility.

Then Stefano reported a measurement of both the perpendicular and the parallel critical magnetic field of superconducting LAO/STO interfaces over the entire phase diagram (i.e. as a function of electrostatic back gating). By checking carefully the temperature dependence of these two critical fields, it is in principle possible to discriminate between single-gap and two-gap superconductivity. That's an important point for the LAO/STO interface since, back in the early days, two gaps have been measured in bulk doped STO by tunneling spectroscopy. As far as we can say from the experimental data of Stefano, it seems that there is only one superconducting gap in the LAO/STO interface.

Now let's go back to the extension of the 2DEG in the STO substrate. From the perpendicular critical magnetic field, Stefano extracted the GL coherence length has a function of  back gate voltage or equivalently as a function of the sheet conductance.  As expected, the coherence length takes a minimum value when the Tc is maximum (for a conductance of approximately 1 mS) and follows an inversed dome shape. From the parallel critical field, Stefano extracted the thickness of the superconducting 2DEG using again a simple GL picture. For low conductance (negative gate voltage), the extension of the 2DEG in the STO is about 10 nm and it increases up to 30 nm for the highest conductance (positive gate voltage). This result is consistent with the electrostatic filling of the highest subbands in the interfacial quantum well, which delocalize deeper in the STO substrate. This is a beautiful result, which should help us to better understand the superconducting properties of these interfaces as a function of electrostatic gating. However, it also raises a very fundamental question. This analysis being entirely based on a simple GL picture, how can we explain that the parallel critical field exceeds by far the Pauli limit? Everybody agrees that the strong Rashba spin-orbit coupling probably plays a role into this but this key point still needs to be clarified.

Blogged by Nicolas Bergeal

Tuesday 28 July 2015 - Marco Grilli - Nanoscopic inhomogeneity, intrinsic charge instability, and novel metal-to-superconductor quantum criticality in oxide heterostructures

Marco reviewed a number of experimental results on LAO/STO and argued that they indicat a high degree of inhomogeneity in the superconducting state. He emphasized two key observations:

1. The temperature range where resistance starts deviating from the metallic value -- but without vanishing completely -- is much larger than T_c itself.  This behavior is typical for granular superconducting materials.   

2. The resistance curves exhibit long low-temperature tails which are characteristic for systems exhibiting a percolation transition.

Based on these observation, Marco and collaborators proposed that in the “underdoped” side of the superconducting dome, LAO/STO is in a phase separated state. Here, nanoscale superconducting puddles are embedded in a metallic background. As a consequence, below the percolation threshold there is no long-range phase coherence, which is suggested as an explanation for the experimentally observed pseudogap behavior.

Marco then explained that the phase separated state may originate from an intrinsic mechanism. The charge carriers at the interface (electrons) and the outer surface (holes) arise from electronic reconstruction due to the polarity of the system. If the system forms puddles of higher and lower charge carrier concentration,  then it is electrostatically favorable for the electrons to mirror the hole puddles. Moreover, the hole density determines the depth of the potential well experienced by the electrons, and correspondingly the strength of the spin-orbit coupling. The key result was that such a phase separated state may be energetically favorable over a homogeneous state, as indicated by a negative compressibility.

Marco predicted that within this scenario the superconducting fluctuations exhibit an anomalous dynamical critical exponent. This affects a number of observables and may offer a route to experimentally testing the theory.

Blogged by Karen Michaeli

Tuesday 28 July 2015- Harold Hwang - Superconductivity in STO heterostructures

The discoverer of conductivity at LAO/STO interfaces himself is giving us an update on superconductivity in STO heterostructures, how neat is that!

STO is of course so interesting because it is the lowest density bulk superconductor. The delta doped systems under consideration here have a density of just a few percent of an electron per unit cell. The technical trick to get these type of samples is to use very high growth temperatures at low oxygen pressure. In this way as many as possible oxygen vacancies appear, which enhances the Sr stoichiometry. Oxygen can then be refilled later. By varying the thickness of a 1% Nb doped STO interlayer, a 2D superconductor can be realized. For thin layers, the magnetic field analysis of the superconductivity shows 2D behavior. If this layer is thicker than about 100 nm a transition to 3D arises, consistent with a coherence length of that order.

Decreasing the thickness of the dopant plane greatly enhances the electron mobility. That is very special! The more 2D the sample is, the cleaner the system, clean enough to see beautiful quantum oscillations. Intriguingly, at low densities, superconductivity disappears while the mobility stays large (2D metal).

Following a Japanese prediction, Harold has been trying to pursue topological superconductivity in a bilayer Rashba system by using two dopant layers. Experimentally, it looks like a twoband superconductor, where the two bands arise form quantum confinement. The coupling between bands can be played with.

To probe the superconductivity in the low density doped STO layer with a tunnel probe, you run into the issue of having a large Schottky barrier. Harold resolved this challenge by including a LAO unit cell that acts as a compensating dipole. Conductance increases by putting in an insulator! Isn’t this field great… A superconducting tunnel spectrum with clear coherence peaks was obtained and could be fitted very well with just thermal broadening.

Written by: Alexander Brinkman

Tuesday 28 July 2015- Uwe Pracht - Pairing enhancement versus phase disordering in coupled nanograins of a conventional superconductor

Uwe Pracht takes us back to the longest-standing problem discussed in this workshop. Granular aluminum exhibits dome-like superconductivity, where the critical temperature increases with resistivity, contrary to what has been seen on much of the materials discussed in this workshop. The experimental facts are known already for a long time, but have never been satisfyingly explained.

Actually, as Uwe shows, many of the exotic phenomena that are discussed this week, such as a superconducting dome in the phase diagram, but also evidence of a pseudogap, have already been shown in this system. In his talk, Uwe tries to answer two main questions: how do the relevant energy scales (Delta, superfluid stiffness, Tc) evolve in the phase diagram, and what is the origin of the superconducting dome.

Uwe has studied these materials by looking at the electromagnetic response of various granular aluminum films. From the frequency-dependent response functions \sigma_1 and \sigma_2, he extracts the energy gap \Delta and the superfluid stiffness, respectively, using a conventional Mattis-Bardeen fit of the response.

In this way, he has been able to identify different regions in the phase diagram of granular aluminum. At low disorder, Tc increases, together with \Delta. This is attributed to a decoupling of the different superconducting grains, which enhances the shell effect in the individual grains. At the same time, the superfluid density is continuously decreasing, which has a detrimental effect on the superconductor. After a cross-over regime, the phase fluctuations in the system take over. Here, the spectral gap stays more or less constant, whereas Tc is going down.

In this region exactly, it seems that a pseudo-gap like feature seems to appear in the data. The analysis of this feature is based on the fact that individual grains that remain superconducting, but loose long-range phase coherence. However, the data in this regime is noisy, and the errorbars do not exclude a gapless state above Tc. It is definitely a very interesting system that deserves being studied in much more detail.

Blogged by Eduard Driessen

Tuesday 28 July 2015 - Misha Skvortsov - Density of states and Inhomogeneity in disordered superconductors

Misha starts with the general motivation: we have experimental (and theoretical) evidence than in disordered systems the local gap becomes inhomogeneous. How to describe this? Which are the consequences in the physical observables?

First of all, Misha clarifies the difference between “disorder” and “inhomogeneity”. When one talks about disorder one usually compares the superconducting correlation length xi with the mean-free path l, so that clean limit is for l/xi>>1, while dirty limit is the opposite. Inhomogeneity is something different, it has to do with the ratio between l and Fermi wavelength, and it implies the absence of self-averaging. Question by Dan Shahar: it can occur in principle also in a clean, correlated system. Answer by Misha: yes, but we try to make our life simpler first, and we stick on the case of disorder-induced inhomogeneity. In addition we want a small parameter to deal with, so we consider disordered systems but away from the SIT. Misha wants to focus on the effects of inhomogeneity on two measurable quantities: 1) the (tunneling) DOS and 2) the superfluid density.

For the DOS Misha first introduces the phenomenological Random-Coupling Constant (RCC) model by Larkin-Ovchinnikov: a space-dependent coupling constant can have the same effect as Abrikosov-Gorgov like magnetic impurities (as implemented by Usadel equations), i.e. smearing of the BCS coherence peak in the DOS plus a reduction of the hard gap. What’s new here is that the RCC model is derived starting from mesoscopic fluctuations of the order parameter. This allows one to connect the famous eta_DOS depairing parameter in Usadel equations to the SC order-parameter fluctuations. At the end one obtains both the smearing of the peaks and a tail below the hard gap. However, even using the kind of mesoscopic inhomogeneity that one expects in the model elaborated by Misha e coworkers for the Coulomb suppression of Tc, the overall effect on the DOS is 100 times smaller than the smearing observed in TiN films. Why? This remains an open interesting question (but see also comment at the end of the talk).

For what concerns the superfluid density Misha explains that an outcome of the RCC model is that the Cooperon and the diffuson get coupled in the presence of the gauge field. This implies that the superfluid density (i.e. the response to the gauge field) is not only suppressed by disorder in the usual Mattis-Bardeen sense (i.e. only a fraction of the total carries condense in the superfluid state) but it gets an additional suppression due to the mesoscopic order-parameter fluctuations. Once again, this can be mapped into the result of Usadel equations, BUT the depairing parameter eta_EM turns out to be different from the one eta_DOS that accounts for the DOS smearing: this is a quite relevant take-home message for experimentalist trying to fit data with the Usadel equations.

Finally, Misha makes an interesting comment on the fact that the different kinds of disorder (dense weak scatterers, or diluted strong scatterers, or grain boundaries, etc.) introduce different short-scale effects that can be finally relevant to explain the real mechanisms of Tc suppression and all the non-universal aspects of the experimental findings (including the quantitative discrepancies between theory and experiments). Indeed, even if on average only the diffusion coefficient matters, the mesoscopic fluctuations of the SC order parameters are very sensitive to short-scale structure of disorder. This question opens an interesting perspective for future theoretical and experimental work.

Blogged by Lara Benfatto

Tuesday 28 July 2015 – Claudio Castellani - Intra-gap optical absorption in disordered superconductors

Claudio Castellani talks about collective excitations in strongly disordered superconductors. 

He starts his talk by reviewing the two possible mechanisms, fermionic or bosonic, to induce a superconductor insulator transition. In the latter Cooper pairs break down and then single fermions undergo Anderson localization while in the former Cooper pairs becomes localized directly. Recent experimental results in Bi films, InO, TiN suggest that the bosonic mechanism is at work. An important consequence of the bosonic mechanism is that a finite gap still exists above Tc.  It is becoming broadly accepted this feature, once believed to be exclusive of cuprates, is indeed generic in any strongly disordered superconductor.

He continues his presentation discussing the Ioffe and Mezard proposal regarding the existence of a  glassy phase in strongly disordered superconductors that is controlled by a rather large length scale that is not related to the coherence length. Ioffe and Mezard model, a spin chain related to the disordered superconductor, by Anderson pseudo-spin representation, is defined on a Cayley tree. The loop-less structure of the Cayley tree suppresses localization effects and reduces substantially the effort to obtain results. Claudio points out that the price to pay is that the results cannot be extrapolated to any material or realistic model. This is a further motivation to study this problem in a more realistic setting: the disordered attractive Hubbard model in two dimensions.

After introducing the model and briefly reviewing previously relevant literature such as the well known papers by Trivedi and co-workers, he then presents results for the normalized spatial distribution function of the order parameter. It was computed numerically by solving the associated BdG equations, namely, in the mean-field limit. Small deviations from mean field were computed in the random phase approximation.  I believe that technically this approximation should be fine provided that disorder or Coulomb interactions are not strong enough. More quantitatively I suspect that this approximation breaks down in the insulator side when the bulk gap is of the order of the main level spacing in a localization volume.

The resulting spatial distribution clearly illustrates the difference in disordered superconductors between the spectral gap, that can be measured by tunnelling (STM), from the amplitude of the order parameter. Claudio stresses that this will be important for the rest of the talk. As in the Ioffe-Mezard model the distribution has fat tails though the details are substantially different. Claudio argues that the distribution is Tracy-Widom, that it is known to be relevant in the context of 2d polymers. Using his words it is also universal in the sense that qualitatively it does not depends on the disorder or interaction strength of the Hubbard model. It was not clear to the exact extension of the universality and the reason why the superconducting problem is so closely related with the 2d polymer one.

In the last part of the talk, based on previous results, he addresses the transport properties, more specifically current response, in the same disordered Hubbard model. 

From a technical point of view the main difference with respect to the clean or weakly disordered case is the need to include vertex corrections (we recall that the treatment of deviations from mean field is perturbative). The numerical result clearly show that the current flows through rather filamentary structures. Only some parts of sample contribute but still long-range order is preserved. After a lively exchange with the audience it is argued that this behaviour could be confirmed experimentally by studying the coherence peak and the tunnelling gap by STM techniques.

Finally he presents results for the optical conductivity. Disorder induces a novel coupling to the vector potential. Clear deviations from the BCS results (Mathis-Bardeen) due to disorder are observed. There is a missing spectral weight, originated in SC islands, it is claimed to be transferred to high frequencies. No evidence at all of Higgs mode. The conductivity seems to be dominated by phase, no amplitude, fluctuations.

I asked whether these results are robust to Coulomb interactions that are not taken into account. It is well known that in clean superconductors no phase collective excitations are observed because Coulomb interactions pushes the frequency of the excitation above the gap. I am told that preliminary results suggests that disorder prevent this mixing and therefore these results maybe be observed experimentally.

He finishes by commenting recent results on an insulating peak by Ovadia et. al. that might be interpreted as a signature of either Many-Body localization or glassy physics. I mention that I do not see a clear contradictions between the two interpretation since Many-body localization is also related to extremely slow dynamics.

Blogged by Antonio Garcia-Garcia