Discussion Session: Inhomogeneous electronic states in superconductors

Suggested question : How to disentangle the unavoidable atomic level inhomogeneity of real materials from the electronic inhomogeneity ?

The first debated question was the role of fractality in the spontaneously emergent spatial inhomogeneities observed in highly disordered superconductors. According to the results of numerical simulations reported by Prof. Trivedi, this fractality holds only close to the metal-insulator transition in the non-interacting particles Anderson localization transition. However, deeper in the insulating state where the superconductor-insulator transition (SIT) takes place, once the pairing interaction is switched on, this fractal character of the wave functions does not show up in the u and v BCS operators, possibly because the wave functions are now too strongly localized. Dr. Feigel’man remarked that this could also be a side effect of the non perturbative implementation of the pairing potential in these calculationsand stressed that, in the fractal theory of the SIT, the spatial variations of the order parameter are expected to show the fractal behaviour only in the limited range of distances because at large scales local pairing amplitude relies on the participation of several localized wave-functions while no fractal structure is expected at distances less than a mean free path of electrons. This is also true for any other physical observable which varies in real space. This range of distances significant for realistic physical systems with small T_c/E_F ratio and short mean free path but shrinks as T_c/E_F →1 as is the case for numerical simulations. Moreover, it has been emphasized that the fractal character of the wave-functions at the mobility edge does not lead to the SIT, but rather to an enhancement of the pairing interaction. The superconducting properties disappear for higher disorder when the mean level spacing is much higher than the superconducting gap. The main role of fractality is to give broad spatial distribution of the pairing amplitude of electrons in a single localized state. In conclusion, no exotic physical length scale appears in the theory, beside the usual superconducting coherence length and the localization length. There seems to be no contradiction between numerical results of Prof. Trivedi and analytical theory.

The second discussed issue was the possibility that inhomogeneities might explain the puzzles observed by Prof. Raychaudhuri on NbN films. Namely, for several disordered films, the transport and Hall effect measurements at high temperatures give the value of (k_Fl) that is much smaller than unity. In this situation one expects that these materials become insulators at low temperatures. However, in case of NbN films, the resistance remains finite when extrapolated to zero temperature. One possible explanation of this disagreement is that the measured conductivity is dominated by a small percolating volume of the film. Another explanation attributes it to a significant thermal dependence of the electronic density which is only known at room temperature.

Although disorder driven electronic segregations in nominally homogeneous materials seem to become an established concept supported by theoretical calculations and experimental observations, the usual dichotomy between granular and homogeneously disordered systems is still extensively emphasized in most of the presentations which show the famous sets of R(T) curves in Ga and Bi. Although this systematic opposition might be misleading, the general opinion is that it should be continued to be discussed because, despite being an oversimplification, it helps non-specialists to clarify a complex SIT landscape.

Prepared by Lev Ioffe and Claude Chapelier

Vladimir Manucharyan: Superinductance: engineering and characterization.

He reports about large inductances realized with series arrays of Josephson-junctions and means to characterize them. The experimental techniques used may very well be useful to characterize large inductances realized with highly disordered superconductors.

He starts by showing a catalogue of commercially available inductors (Coilcraft) and shows that they always have a failure mode, which implies that the impedance will never exceed the quantum resistance. This is due to the finestructure constant, which he blames to be too small.

Large inductance circuits are useful for 1; enhanced coupling in QED systems 2; electrical current metrology with Bloch oscillations in JJ’s. 3; Topologically protected quantum states in network of JJ’s. 4; eliminate flux noise in flux qubits (cf. Kerman’s poster).

One could use disordered superconductors, which requires a careful choice of controllable material and of which as of today the electrodynamics is poorly understood. Instead the ‘workhorse’ of superconducting quantum-engineering is used, the Al tunneljunction. A chain of JJ’s can act as an inductor, even tunable because of its dependence on the bias current.

In designing the most useful inductor one needs to consider a long list of possible failures, which are not a priori trivially known. Therefore it is very important to be able to measure and characterize the inductor thoroughly.

He describes 2 methods, of which one of them is easy to implement for highly disordered superconductors. The other one requires integration in a high-Q qubit structure, which is less readily available. The outcome is that coherent quantum phase slip is resolved down to 100 kHz, which connects his presentation to the first two of the workshop by Hans Mooij and Oleg Astafiev, although the QPS is realized in a series array of JJ’s (like by Pop, .., Guichard et al from Grenoble. )

The speaker believes that the highly disordered superconductors are potentially very useful for compact high-Q resonators coupled to semiconductor quantum dots, Rydberg atoms on a chip, polar molecules, or anything with a small dipole moment and a long life time.

Blogged by Teun Klapwijk

Thierry Klein: Superconducting properties of boron-doped silicon

Thierry Klein started off with review of superconductivity in covalent bonded materials and row IV systems. He gave example of fullerenes and Si-Clathrates. He then discussed the reasonably recent example of superconductivity in diamond. He motivated that it exected that the el-phonon coupling constant could be as big as 280 meV. He gave a naive estimate forecast of a Tc based on this el-phonon coupling for this sample of 200K.

Was discovered by Ekimov et al. in 2004. It is believed that superconductivity occurs in an sp3 band.

It appears with B doping at the onset of the metal-insulator transition. Tc is NOT 200K in this compound; it is limited to ~10K presumably due to the low charge carrier density.

Diamond superconducting films can also be grown by plasma assisted CVD, which appeared to this blogger to have a max Tc in these films of about 3K. It has also been discovered intercalated graphite has Tc around 11K in 2005. In 2007 they discovered superconductivity in B doped Si.

Gas immersion laser doping is done to increase B concentration of Si. Apparently it is not possible to doped to high enough concentrations in a melt. The necessary concentrations of B are above the solubility limit of B in Si. To this blogger they blast and melt the surface of a Si with a laser. The surface recrystallize under some atmosphere of B which is incident. They can make highly doped B doped Si layer.

At 1% Boron concentrations the transition starts and rises linear. This is difference than diamond, where Tc comes up with an exponent of ~ 0.5. The superconducting transition doesn't coincide with the MIT. That is at much lower concentrations. The max Tc looks to be about 0.5K. The superconductivity in this system is extreme type II. Unlike the case of diamond where k_Fl is about 1, k_Fl in this system is about 10.

Calculations can postdict the superconductivity reasonably, but there is uncertainty about mu* (as there is always). But with a mu* of about 0.14 (typical for metals), they can describe Tc.

Blogged by Peter Armitage

Michael Gershenson: Magnetic-field driven phase transitions in unconventional Josephson arrays

Michael started off by reminding us of the Bosonic model of superconductor-insulator transition (SIT).

In this model as applied to Josephson Junction arrays, there is charge-flux duality. Can be described by a Hamiltonian with Josephson couplings between scing island and charging energies for the islands.

In the regime where Ec
There are a number of possible complications in the simple arrays including random charge, and Josephson energies, and flux noise.

They have endeavored to overcome these issues by making JJ arrays with a large number of nearest-neighbor islands. These are arrays where there is a supercell of many tightly connected neighbors. This supercell is connected to its neighboring supercells, by cells that have less interconnects. With these array they can explore of a wide range of JJ parameters and an effective Ej/Ec created which is enhanced over the bare value by factor of N^2.

Michael 1st presented data for array without ground planes and then for arrays conducting ground plane. They found multiple SIT (due to commensurate effects) over a wide range of critical resistances R ~ 3-20 k were observed. "Metallic" phases with very low (typically < 100 mK) characteristic energies were found.

SIT observed at low “critical” Rcr ~ few ohms resemble the “dirty boson” SIT however the duality is lacking for the transitions observed at larger Rcr . On the “insulating” side of the SIT, the R(T) dependences can be fitted with the Arrhenius law R(T)~exp(T0/T), where kBT0 is close to the “Coulomb” gap 2eV* (V* is the offset voltage across the whole array). Michael speculated that this may be a signature of some collective process and/or macroscopic inhomogeneity. The threshold for quasiparticle generation at high bias currents is surprisingly universal for samples with vastly different zero-bias resistances and that this power scales with the array area.

Blogged by Peter Armitage

Wei Liu : Dynamical study of phase fluctuations and their critical slowing down in amorphous superconducting films

Wei Liu introduced her talk by a description of the superconducting transition as seen in the thermal dependance of the resistance. She divided the curve into three main parts: a normal state behavior at high temperature, a superconducting fluctuations region when the curve bends down at the transition and a superconducting state characterized by a zero resistance. She emphazised that the critical region of the transition is divided into an amplitude fluctuation region at the beginning of the downturn of the curve and a phase fluctuation region at lower temperature. In the phase fluctuation region, free vortices are thermally generated down to the Berezinsky–Kosterlitz–Thouless (BKT) transition where these fluctuations are frozen out. The speaker described different signatures of this BKT transition: a universal resistance curve [P. Minnhagen (1987)] and the non linear I-V characteristic [K. Epstein (1982)].The critical temperature of the BKT transition is theoretically signalled by a jump in the exponent of this charactreristic (not seen in transport measurements as far as the blogger knows). A related universal jump has indeed been seen in the damping rate of a torsion oscillator immersed inHe3/He4 mixtures. The method used by Wei Liu is based on the frequency dependance of the superfluid stifness. Although it was not the premature end of her talk, she immediately anticipated her conclusions: the microwave complex conductivity can characterize two-dimensionnal quantum systems by providing an experimental signature of the superfluid stifness. She observed that the dynamic slows down at the transition.

The experiment relies on the Corbino geometry of the sample which allows a broadband spectrometric study (100 MHz-40 GHz with 1Hz resolution). The measurement consists in recording, with a network analyser, the microwave reflexion of a line ended by the sample cooled down to 300 mK. Both the real and imaginary components of the complex impedance can thus be obtained.

The sample was a 30 nm thick amorphous Indium oxide film with a sheet resistance below 1500 Ohms. An Azlamazov-Larkin paraconductivity analysis of the thermal dependance of the resistance gives a superconducting temperature sligthly below 3 K.

Weil Liu showed the real and imaginary part of the conductance of this film as a function of frequency for different temperatures. In the real conductance, the spectral weight moves to lower frequency as the temperature is decreased. In the supercondsucting state, a gap in the frequency opens and a delta function peak appears at zero frequency as expected for a superconductor with zero DC resistance. The plot of the logarithm of the imaginary conductance shows a linear behavior whose slope continuously changes from a positive value at high temperature to a negative one at low temperature. This imaginary component times the frequency is a direct measure of the energy scale of the superfluid stiffness. The latter becomes frequency dependent above T_BKT (approximately 2.3K) and while its value should be four times T_BKT at this temperature (according to BKT theory), the speaker rather estimates it around 12 K. However, the audience found that there was not such a discrepancy between theory and measurements, taking into account the width of the experimental curves.

The real part of the conductance was also shown for different temperature and a peak above T_BKT could be observed that shifts to even higher temperature when the AC frequency is increased. This peak unveils the superfluid density.

The last part of the talk consisted in a scaling analysis of the phase and the amplitude of the complex impedance. Both quantities could be rescaled on universal curves for a significant range of temperature above the transition. This analysis allowed Wei Liu to extract a characteristic relaxation frequency which decreases rapidly above T_BKT. Such an observation is consistent with a vortex activation scenario. However the blogger did not understand whether the extracted vortex core energy was in agreement or not with BCS theory.

During the discussion, the validity of the BKT interpretation of the data in such a disordered superconductor was questionned. Indeed, one expects the vortex to be pinned. It was recognized that edge pinning was irrelevant because of the Corbino geometry. However, internal pinning could be detrimental. Whether the short length scales probed by these high frequency measurements allows disregarding pinning remains an open question.

Blogged by Claude Chapelier

Discussion group 3: Superconductor-insulator transition: new and old systems

Suggested question: identify experimental tests for the latest developments in the theory of the SIT

Experiments suggested to check recent theories of S-I transition:

1) RF stimulation of activated conductance. Can be used to probe the nature of insulating state which contains (according to theory of Feigelman, Ioffe, Mezard) energy threshold for delocalized excitation, ω_d(g) which vanishes when coupling constant g approaches the critical point g_c RF excitation with frequency ω which is a bit above ω_d may lead to strong increase of conductivity due to specific excitation

of mobile excitations without much heating of the sample.

2) Mapping of supercurrent distribution. According to the theory, close to SIT supercurrent is distributed very inhomogeneously along the sample. It would be important to measure distribution of supercurrents experimentally. Possible methods could include, for example, magnetic AFM or magnetic-field-sensitive NV-centers in diamond, both methods being able to measure magnetic field produced by super-current with a good spatial resolution.

3) Spatial correlation of peak heights and gap widths.

More extensive STM studies are needed to investigate correlation functions of the spatial inhomogeneities of the superconducting state. The spatial evolution of the gap and coherence peak height correlation functions as a function of disorder would be of great interest for understanding how disorder transforms superconductors into insulators.

4) Fate of pseudogap while moving away from SIT ?

According to theoretical description of strongly disordered superconductors, the size of pseudogap is expected to decrease while switching to less disordered samples. Thus it would be important to study if pseudogap features (e.g. seen in InOx near S-I transition) are diminished for less disordered samples, with higher Tc values (like ones used in the phase slip experiment by Oleg Astafiev, or in high-frequency measurements reported Peter Armitage ).

Prepared by B.Sacepe and M. Feigel’man