.. . . . - - , 25.09.2010

Phys Rev B 40 182 (1989) ?

, .. .. 180, 3 (2010)

Lecture 1Disordered superconductors (basics)Suppression of superconductivity by disorder: 3 types of materials and 3 major mechanismsGranular superconductors and artificial arrays 1). Superconductivity in a single grain2) Granular superconductors: experiments3) Theories of SIT. Which parameter drives the S-I transition ?4) BKT transitions in 2D JJ arrays5) 2D JJ arrays with magnetic field: Bose metal ??Homogeneously disordered thin films 1) Suppression of Tc by disorder-enhanced Coulomb2) Mesoscopic fluctuations of Tc near the QPT3) Quantum S-M transition: SC islands on top of a poor metal film Experimental realization: graphen

Disordered superconductors (classical results)Potential disorder does not affect superconductive transition temperature (for s-wave) A.A. Abrikosov & L.P.Gorkov 1958 P.W.Anderson 1959In the dirty limit l > 1 or (the same in another form) G = (h/e2) d-2 >> 1What happens if G ~ 1 ?

constleads to BCS gap equation 1 = g N(0) (d/) th(/2T) Anderson theoremApprox.

Major types of disordered superconductive materialsVery strongly disordered metallic alloys with usual carrier density (~1022/cm3) bulk conductivity near Mott limit, G ~ 1Granular superconductive metals or artificially prepared arrays of islands inter-granular conductance Gt ~ 1Homogeneously disordered poor metalswith low (~< 1021/cm3) carrier density

(Second lecture)

Superconductivity v/s LocalizationGranular systems with Coulomb interaction Bosonic mechanismCoulomb-induced suppression of Tc in uniform films Fermionic mechanism

Competition of Cooper pairing and localization (no Coulomb) (2nd lecture)

I. Granular arraysReviews: I.Beloborodov et al, Rev. Mod.Phys.79, 469 (2007) R.Fazio and H. van der Zant, Phys. Rep. 355, 235 (2001)

g constExamples:Phys Rev B 1987Phys Rev B 1981

Grain radius a >> F

Basic energy scalesGrain radius a is large on atomic scale: kF a >> 1Level spacing inside grain = 1/(4a3)Coulomb energy Ec = e2/aIntra-grain Thouless energy Eth = D0/4a2

Intergrain couplingLow transmission, but largenumber of transmission modesT = (4 e2/)2

gT = T h/e2- dimensionless inter-grain conductanceA) Granular metal gT 1B) Granular insulator gT 1Narrow coherent band = gT

1) Superconductivity in a single grainWhat is the critical size of the grain ac?

What happens if a < ac ?

Assuming 0 >>a >> ac , what is the critical magnetic field ?

Critical grain size Mean-field theory gap equation: = (g/2) i /[i2 + 2]1/2Level spacing > ac = (1/ )1/3

Ultra-small grains a

Critical magnetic field for small grainac > acA. Larkin 1965

2) Granular superconductors: experiments

Very thin granular films

3D granular materials

E-beam - produced regular JJ arrays

Thin quenched-condensed filmsPb grainsSn grainsA. Frydman, O. Naaman, R. Dynes 2002

Granular v/s Amorphous filmsA.FrydmanPhysica C391, 189 (2003)

Conclusion in this paper: control parameter is the normal resistance R. Its critical value is RQ = h/4e2 = 6.5 kOhmPhys Rev B 40 182 (1989)

Bulk granular superconductorsSample thickness 200 nm

Bulk granular superconductors

Artificial regular JJ arrays

What is the parameter that controls SIT in granular superconductors ?Ratio EC/EJ ?

Dimensionless conductance g = (h/4e2) R-1 ? (for 2D case)

3) Theoretical approaches to SIT K.Efetov ZhETF 78, 2017 (1980) [Sov.Phys.-JETP 52, 568 (1980)] Hamiltonian for charge-phase variablesM.P.A.Fisher, Phys.Rev.Lett. 65, 923 (1990) General duality Cooper pairs Vortices in 2D

R.Fazio and G.Schn, Phys. Rev. B43, 5307 (1991) Effective action for 2D arrays

K.Efetovs microscopic HamiltonianControl parameterEc = e2/2C

Artificial arrays:major term in capacitance matrix is n-n capacitance Cqi and i are canonically conjugated

Logarithmic Coulomb interaction Artificial arrays with dominating capacitance of junctions: C/C0 > 100U(R) = Coulomb interaction of elementary chargesFor Cooper pairs, x by factor 4

Competition between Coulomb repulsion and Cooper pair hopping:Duality charge-vortex: both charge-charge and vortex-vortex interaction are Log(R) in 2D. Vortex motion generates voltage: V=0 jV Charge motion generates current: I=2e jcAt the self-dual point the currents are equal RQ=V/I=h/(2e)2=6.5k.RQM.P.A.Fishers duality argumentsInsulator is a superfluid of vorticesIn favor of this idea: usually SIT in films occurs at R near RQProblems: i) how to derive that duality ?What about capacitance matrix in granular films ?iii) Critical R(T) is not flat usually

Can we reconcile Efetovs theory and result of duality approach ? We need to account for capacitance renormalization to due to virtual electron tunneling via AES [PRB 30, 6419 (1984)] action functionalChargingSts = gVirtual tunnelingJosephson (if d/dt

Mean-field estimate with renormalized actionSC transition atStrong renormalization of C:T=0:J = g/2

Can one disentangle g and EC/EJ effects ? JETP Lett. 85(10), 513 (2007) This model allows exact duality transformationControl parameterExperimentally, it allows study of SIT in a broad range of g and/or EJ/EC

4) Charge BKT transition in 2D JJ arraysLogarithmic interaction of Cooper pairs 2eU(R) =8 ECTemperature of BKT transition is T2 = EC/R.Fazio and G.Schn, Phys. Rev. B43, 5307 (1991)

Not observed !The reason: usually T2 is above parity temperature Interaction of pairs is screened by quasiparticlesCharge BKT is at T1 = EC/4

(unless T* is above T2 )

5) Bose metal in JJ array ?At non-zero field simple Josephson arrays show temperature-independent resistance with values that change by orders of magnitude.

Dice array (E.Serret and B.Pannetier 2002; E.Serret thesis, CNRS-Grenoble)Foto from arxiv:0811.4675At non-zero field Josephson arrays of more complex (dice) geometry show temperature independent resistance in a wide range of EJ/Ec

The origin of Bose metal is unknown Hypothesis: it might be related to charge offset noise

Elementary building block

Al2O3+Elementary building blockAlAl-+-+

Conclusion for granular superconductive arrays:

1) basic features of SIT are understood

2) really quantitative SIT theory is not constructed yet

II. Homogeneously disordered SC films

1) Suppression of Tc by disorder-enhanced Coulomb2) Mesoscopic fluctuations of Tc near the QPT3) Quantum S-M transition: SC islands on top of a poor metal film4) Experimental realization: graphen

1) Suppression of Tc in amorphous thin films by disorder-enhanced Coulomb interactionTheoryExperimentReview:Generalization to quasi-1D stripes:Phys. Rev. Lett. 83, 191 (1999) Yu. Oreg and A. M. Finkel'stein Similar approach for 3D poor conductornear Anderson transition:P. W. Anderson, K. A. Muttalib, and T. V. Ramakrishnan, Phys. Rev. B 28, 117 (1983)

Amorphous v/s Granular films

Suppression of Tc in amorphous thin films: experimental data and fits to theoryJ.Graybeal and M.BeasleyPbfilms

Suppression of Tc in amorphous thin films: qualitative pictureDisorder increases Coulomb interaction and thus decreases the pairing interaction (sum of Coulomb and phonon attraction). In perturbation theory:Roughly,But in fact BCS-type problem withenergy-dependent coupling must be solvedReturn probability in 2DThe origin of the correction term ???????g = h/e2 It is revival of strong Coulomb repulsion, due to slow diffusion at g ~ 1

Why Coulomb repulsion does not kill phonon attraction (sometimes)?Coulomb pseudo-potential () = 0 / [1+ 0 ln (EF/)]Normally, 0 ~ 1 and ln (EF/) >> 1 ~ D ~ 0.05 eV whereas EF ~ 5 eVTolmachev Logarithm At the energy scale D relevant for phonon process, fullamplitude in the Cooper channel can be attractive at small = 0 - 1/ln (EF/) > 0

Why disorder leads to Coulomb revival ?Uij = i 2(r) j 2(r) U(r-r) dr drMatrix element of Coulomb repulsion:U(r) is short-range = V-1 [1 + (1/g) ln (1/ij ) ]Enhancement due to disorderUniform term subject toTolmachev Log

Diagrams for Coulomb suppression of the Cooperon

Fluctuations of the local Density of States= 1 (T-invariant) or = 2 (no T-invariance)Review: A.Mirlin, F.Evers, Rev. Mod. Phys. (2008)

Renormalization Group: summation of major diagrams

In the form developed in M.F., A.Larkin & M.Skvortsov (Phys.Rev.B 2000)

Coulomb suppression of Tc(wrong sign)At ln(1/Tc0) > 5gc > 4 i.e. Rc < RQ Conclusion:Superconductor MetalQuantum phase transition

Fluctuations near Tc2) Mesoscopic fluctuations of Tc near the Finkelsteins Quantum Phase TransitionM. Feigelman and M. Skvortsov, Phys. Rev. Lett. 95 057002 (2005)

Thermal fluctuations: disorder and dimensionality

Fluctuations due to disorder

The goal:To study mesoscopic fluctuation near the Coulomb-suppressed TcWay to go:

1) Derive the Ginzburg-Landau expansion2) Include mesoscopic fluctuations of K(r,r)3) Droplets of superconductive phase above Tc

Ginzburg-Landau expansion: result

Mesoscopic fluctuations of the kernel K(r,r): step 1

Mesoscopic fluctuations of the kernel K(r,r): step 2

Superconductor with fluctuating TcSov.Phys.JETP54, 378 (1981)Linearized GL equation is similar to a Schrdinger eq. with Gaussian random potential Localized tail states I.Lifshitz, Zittarz & Langer, Halperin & Lax (mid-60s)(formally equivalent to instanton solutions in some effective field theory)

Mesoscopic vs. thermal fluctuations

Experiment on TiN filmsB.Sacepe, C.Chapelier,T.Baturina et alPhys. Rev.Lett. 101,157006 (2008)

Conclusion of the part 2):Quantitative theory of this transition is not developed yetMajor open problem: Density of States smearing near critical conductance

3) Quantum S-M transition: SC islands on the top of a poor metal film

M. V. Feigelman and A. I. Larkin Chem. Phys. Lett. 235, 107 (1998)

B. Spivak, A. Zyuzin, and M. Hruska Phys. Rev. B 64, 132502 (2001)

M. V. Feigelman, A. I. Larkin and M.A.Skvortsov Phys. Rev. Lett. 86, 1869 (2001)

Critical conductance for T=0 SC state Ideal contacts between islands and film:can be >> 1bdb >> d

Experimental realization: graphenSuggested in M. Feigelman, M. Skvortsov and K. Tikhonov JETP Letters 88(11), 747-751 (2008); arXiv:0810.0109] First experiment: B.M. Kessler, C.O. Girit, A. Zettl, V. Bouchiat arxiv: 0907.3661

1st ExperimentNew experiments are on the way:Bilayer graphen, STM probe,Low-fraction covering

Graphene + superconducting islands

The End of First Lecture

Lecture 2. Superconductivity in amorphous poor conductors: pseudo-gap and new SIT scenario1) Motivation from experiments2) BCS-like theory for critical eigenstates3) Superconductivity with pseudo-gap4) Quantum phase transition: Cayley tree model

M.Feigelman, L.Ioffe,V.Kravtsov,E.Yuzbashyan, Phys Rev Lett.98, 027001, 2007 M.Feigelman, L.Ioffe,V.Kravtsov,E.Cuevas, Ann.Phys. 325, 1390 (2010)L.Ioffe and M. Mezard Phys.Rev.Lett. 105, 037001 (2010) M.Feigelman, L.Ioffe and M.Mezard, arXiv:1006.5767

Direct evidence for the gap above the transition (Chapelier, Sacepe 2007). Activation behavior does not show gap suppression at the critical point as a function of the disorder (Shahar & Ovaduyahu, 1992)!

Class of relevant materialsAmorphously disordered (no structural grains)Low carrier density ( around 1021 cm-3 at low temp.)Examples: InOx NbNx thick films or bulk (+ B-doped Diamond?) TiN thin films Be (ultra thin films)

T0 = 15 KR0 = 20 kWOn insulating side (far enough):Kowal-Ovadyahu 1994D.Shahar & Z.Ovadyahuamorphous InO 1992

Disorder-controlled SIT: nonzero gap at the transitionRed curve: same sample after 4 days annealing at 300K but not presented in the following slidesNear-critial InOxB.Sacepe M.OvadiaD.Shahar (2009)T0 = 1.9 KActivation gap

Example: Disorder-driven S-I transition in TiN thin filmsT.I.Baturina et al Phys.Rev.Lett 99 257003 2007 Specific Features of Direct SIT:

Insulating behaviour of the R(T) separatrix

On insulating side of SIT, low-temperature resistivity is activated: R(T) ~ exp(T0/T)

Crossover to VRH at higher temperatures

Seen in TiN, InO, Be (extra thin) all areamorphous, with low electron density

Baturina et al 2007 T0 = 0.38 K R0 = 20 kWTiN film 5 nm thickness

Magnetic-field induced SIT and giant magnetoresistanceInOx D.Shahar et al (2004)Transport by pairs

Gap vanishes at Bc

Conclusion 1:Resistivity follows activation law

R ~ exp(T0/T)

both near SIT and far in the insulating state

SC side: local tunneling conductance

Local tunneling conductance-2Gap widthsPeak heightsMore disorderLess disorder

Conclusion 2 Superconductive state near SIT is very unusual: the spectral gap appears much before (with T decrease) than superconductive coherence doesCoherence peaks in the DoS appear together with resistance vanishing

Bosonic v/s Fermionic scenario ? None of them is able to describe data on InOx and TiN3-d scenario: competition between Cooper pairing and localization (without any role of Coulomb interaction)

Theoretical model Simplest BCS attraction model, but for critical (or weakly localized) electron eigenstatesH = H0 - g d3r = cj j (r)Basis of localized eigenfunctionsM. Ma and P. Lee (1985) :S-I transition at L TcWe will see that in fact SC state survives far into the region L >> Tc

Superconductivity at the Localization Threshold: L 0Consider Fermi energy very close to the mobility edge:single-electron states are extended but fractal and populate small fraction of the whole volumeHow BCS theory should be modified to account for eigenstates fractality ?Method: combination of analitic theory and numerical data for Anderson mobility edge model

Mean-Field Eq. for Tc

3D Anderson model: = 0.57 D2 1.3 in 3DFractality of wavefunctions IPR: Mi = 4drl is the short-scale cut-off length

Modified mean-field approximation for critical temperature TcFor small this Tc is higher than BCS value !

Order parameter in real spacefor = kSC fraction =

Tunnelling DoSAsymmetry in local DoS:Average DoS:

Superconductivity at the Mobility Edge: major featuresCritical temperature Tc is well-defined through the whole system in spite of strong (r) fluctuations Local DoS strongly fluctuates in real space; it results in asymmetric tunnel conductance G(V,r) G(-V,r)Both thermal (Gi) and mesoscopic (Gid) fluctuational parameters of the GL functional are of order unity

Superconductivity with Pseudogap Now we move Fermi- level into the range of localized eigenstates

Local pairing in addition tocollective pairing

Parity gap in ultrasmall grains

K. Matveev and A. Larkin 1997 No many-body correlations Local pairing energyCorrelations between pairs of electrons localized in the same orbital-------------- EF------ --(Lecture 1)

Parity gap for Anderson-localized eigenstatesEnergy of two single-particle excitations after depairing:P plays the role of the activation gap

P(M) distribution

Insulating state: parity gap scaling near mobility edge

Activation energy TI from Shahar-Ovadyahu exp. and fit to the theoryThe fit was obtained with single fitting parameter = 0.05= 400 KExample of consistent choice:

Local (on-site) attraction + disorderPhys. Rev.B 65, 014501 (2001)

Critical temperature in the pseudogap regimeHere we use M() specific for localized statesMFA is OK as long as MFA:is large

Correlation function M()No saturation at < L :M() ~ ln2 (L / )(Cuevas & Kravtsov PRB,2007)

Superconductivity with Tc < L is possible

This region was not found previously

Here local gapexceeds SC gap :

Critical temperature in the pseudogap regimeWe need to estimate MFA:It is nearly constant in a very broad range of ~ ()3

Tc versus PseudogapTransition exists even at L >> Tc0

Low-energy effective Hamiltonian for pseudogaped SC stateP >> Tc

Single-electron states suppressed by pseudogapEffective number of interacting neighboursPseudospin approximation

Qualitative features of Pseudogaped Superconductivity: STM DoS evolution with T

Double-peak structure in point-contact conductance

Nonconservation of full spectral weight across Tc

VTKtot(T)Tcp

Andreev point-contact spectroscopy2eV1 = 2eV2 = + P

T.Dubouchet,thesis, Grenoble(11 Oct. 2010)

Spectral weight of high- conductivityis constant (T-independent) in BCS Pseudogap superconductor with P >>

S-I-T: Third ScenarioBosonic mechanism: preformed Cooper pairs + competition Josephson v/s Coulomb S I T in arraysFermionic mechanism: suppressed Cooper attraction, no paring S M T

Pseudospin mechanism: individually localized pairs - S I T in amorphous media

SIT occurs at small Z and lead to paired insulator How to describe this quantum phase transition ? Cayley tree model is solved (M.F.,L.Ioffe & M.Mezard)

Superconductor-Insulator Transition Simplified model of competition between random local energies (iSiz term) and XY coupling

Distribution function for the order parameterLinear recursion (T=Tc)Solution in the RBS phase:T=0Diverging 1st moment

Phase diagramSuperconductorHopping insulator

g TemperatureEnergyRSB stateFull localization:Insulator withDiscrete levelsMFA linegcMajor feature: green and red line meet at zero energyThis is NOT the case in presense of magnetic field

Order parameter: scaling near transitionTypical value near the critical point:

Threshold for activated transport Nonzero line-width appears above threshold frequency only:

Nonzero activation energy for transport of pairs is due to the absence of thermal bath at low

This is T = 0 result !Nonzero but low temperatures:

Experimental phase diagram

Our interpretation

Conclusions for Lecture 2

Pairing on nearly-critical states produces fractal superconductivity with relatively high Tc but very small superconductive density

Pairing of electrons on localized states leads to hard gap and Arrhenius resistivity for 1e transport

Pseudogap behaviour is generic near S-I transition, with insulating gap above Tc

New type of S-I phase transition is described On insulating side activation of pair transport is due to ManyBodyLocalization threshold

Superconductivity is extremely inhomogeneous near SIT, for 2 different reasons: i) fractality, ii) lack of self-averaging

The End

Alternative method to find Tc:Virial expansion (A.Larkin & D.Khmelnitsky 1970)

Coulomb enchancement near mobility edge ?? Condition of universal screening:Normally, Coulomb interaction is overscreened, with universal effective coupling constant ~ 1Example of a-InOx :Effective Coulomb potential is weak:

Ginzburg-Landau expansion: step 1

Ginzburg-Landau expansion: step 2

Andreev contact spectroscopyT.Dubouchet,thesis, Grenoble(11 Oct. 2010)