Fractal and Pseudopgaped Superconductors: theoretical introduction Mikhail Feigel’man L.D.Landau Institute, Moscow Based on the results obtained in collaboration with Lev Ioffe and Emil Yuzbashyan Rutgers Vladimir Kravtsov ICTP Trieste Emilio Cuevas Murcia Univ. Publications relevant to this talk: Phys Rev Lett. 98, 027001(2007) (M.F.,L. Ioffe,V. Kravtsov, E.Yuzbashyan) Annals of Physics 325, 1368 (2010) (M.F., L.Ioffe, V.Kravtsov, E.Cuevas) Related publications: Phys. Rev. B 82, 184534 (2010) (M.F. L.Ioffe, M. Mezard) Nature Physics 7, 239 (2011) (B.Sacepe,T.Doubochet,C.Chapelier,M.Sanquer, M.Ovadia,D.Shahar, M. F., L..Ioffe)

Plan of the talk Introduction: why new theory is needed? Fractal superconductivity at the localization edge - sketch of the derivation - main features 3. Superconductivity with pseudogap - origin of the psedogap - development of the superconductive correlations - qualitative features For the next steps (effects of quantum fluctuations) see the talk by Lev Ioffe: - S-I transition and insulating state - quantum phase slips within pseudogap model

Superconductivity v/s Localization Coulomb-induced suppression of Tc in uniform films “Fermionic mechanism” Yu.Ovchinnikov (1973, wrong sign) Mayekawa-Fukuyama (1983) A.Finkelstein (1987) Yu.Oreg & A. Finkelstein (1999) Granular systems with Coulomb interaction K.Efetov (1980) M.P.A.Fisher et al (1990) “Bosonic mechanism” Competition of Cooper pairing and localization (no Coulomb) Imry-Strongin, Ma-Lee, Kotliar-Kapitulnik, Bulaevskii-Sadovskii(mid-80’s) Ghosal, Randeria, Trivedi 1998-2001

We consider amorphous systems with direct S-I transition Gap is NOT suppressed at the transition

Bosonic mechanism: Control parameter Ec = e2/2C 1.Grains are needed, but we don’t have 2.SIT is actually not seen in arrays in magnetic field !

Main challenges from exp. data In some materials SC survives up to very high resistivity values. No structural grains are found there. Preformed electron pairs are detected in the same materials both above Tc and at very low temp. on insulating side of SIT - by STM study in SC state - by the measurement of the activated R(T) ~ exp(T0/T) on insulating side

SC side: local tunneling conductance Nature Physics 7, 239 (2011)

Superconductive state near SIT is very unusual: the spectral gap appears much before (with T decrease) than superconductive coherence does Coherence peaks in the DoS appear together with resistance vanishing Distribution of coherence peaks heights is very broad near SIT

Class of relevant materials Amorphously disordered (no structural grains) Low carrier density ( around 1021 cm-3 at low temp.) Examples: amorphous InOx TiN thin films Possibly similar: Be (ultra thin films) NbNx B- doped diamond Bosonic v/s Fermionic scenario ? None of them is able to describe data on InOx and TiN : Both scenaria are ruled out by STM data in SC state

Superconductivity v/s Localization Coulomb-induced suppression of Tc in uniform films “Fermionic mechanism” Yu.Ovchinnikov (1973, wrong sign) Mayekawa-Fukuyama (1983) A.Finkelstein (1987) Yu.Oreg & A. Finkelstein (1999) Granular systems with Coulomb interaction K.Efetov (1980) M.P.A.Fisher et al (1990) “Bosonic mechanism” Competition of Cooper pairing and localization (no Coulomb) Imry-Strongin, Ma-Lee, Kotliar-Kapitulnik, Bulaevskii-Sadovskii(mid-80’s) Ghosal, Randeria, Trivedi 1998-2001

Superconductive transition at the mobility edge

Theoretical model (3D) H = H0 - g ∫ d3r Ψ↑†Ψ↓†Ψ↓Ψ↑ Simplest BCS attraction model, but for critical (or weakly localized) electron eigenstates H = H0 - g ∫ d3r Ψ↑†Ψ↓†Ψ↓Ψ↑ Basis of exact eigenfunctions of free electrons in random potential Ψ = Σ cj Ψj (r) S-I transition at δL ≈ Tc M. Ma and P. Lee (1985) : We will find that SC state is compatible with δL >> Tc

Why do anyone may need analytical theory for S-I transition? Low-temperature superconductivity is the nontrivial result of a weak interaction: Tc ~ (10-4 - 10-3) EF It leads to relatively long coherence length ξ >> lattice constant Thus straightforward computer simulation of interacting problem in relevant parameter range is impossible Combination of analytical theory and numerical results might be very useful

Mean-Field Eq. for Tc

Fractality of wavefunctions 4 IPR: Mi = dr d2 ≈ 1.3 in 3D l is the short-scale cut-off length 3D Anderson model: γ = 0.57

3D Anderson model: long evolution from diffusive metal to the critical point E.Cuevas and V.Kravtsov, Phys.Rev B76 (2007) “Box distribution”: critical disorder strength Wc = 16.5 W=10 W=5 W=2

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

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

Tc from 3 different calculations Modified MFA equation leads to: BCS theory: Tc = ωD exp(-1/ λ)

Neglected so far : off-diagonal terms Non-pair-wise terms with 3 or 4 different eigenstates were omitted To estimate the accuracy we derived effective Ginzburg -Landau functional taking these terms into account W=∫<δa(r)δa(r’)>dr’ Parameters a, b, C and W do not contain fractal exponents

Can we understand increase of Tc by disorder within regular perturbative approach ? Yes: - for 2D case without Coulomb interaction (only Cooper int.) Talk by Vladimir Kravtsov at KITP, Santa Barbara, 13 Sept.2010 “Can disorder increase superconducting Tc?” for 2D case with short-range repulsion and Cooper interaction I.Burmistrov, I. Gornyi and A. Mirlin arXiv: 1102.3323 “Enhancement of superconductivity by Anderson localization” Renormalization Group approach

Order parameter in real space for ξ = ξk SC fraction =

Tunnelling DoS Average DoS: Asymmetry in local DoS:

Superconductivity at the Mobility Edge: major features Critical temperature Tc is well-defined through the whole system in spite of strong Δ(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

What to do with really thin films ? Where are no Anderson transition in 2D But localization length Lloc ~ exp(π g) g = h/e2Rsqr varies very sharply in the region g ~ 1 where crossover from weak to strong localization takes place Hypotetically the same kind of analysis we did for 3D can be adopted for 2D case But it was not done yet

Superconductive state with a pseudogap

Parity gap in ultrasmall grains Local pairing energy ------- ------- EF --↑↓-- --  ↓-- Parity gap in ultrasmall grains K. Matveev and A. Larkin 1997 No many-body correlations Correlations between pairs of electrons localized in the same “orbital”

Parity gap for Anderson-localized eigenstates Energy of two single-particle excitations after depairing: ΔP plays the role of the activation gap

Activation energy TI from Shahar-Ovadyahu exper Activation energy TI from Shahar-Ovadyahu exper. (1992) and fit to the theory The fit was obtained with single fitting parameter Example of consistent choice: = 400 K = 0.05 Similar fit with naïve exponent d=3 instead of d2 = 1.3 fails undoubtedly

Annals of Physics 325, 1368 (2010) Tc versus Pseudogap Superconductive transition exists even at δL >> Tc0

Correlation function M(ω) No saturation at ω < δL : M(ω) ~ ln2 (δL / ω) (Cuevas & Kravtsov PRB,2007) Superconductivity with Tc << δL is possible This region was not noticed previously only with weak coupling ! Here “local gap” exceeds SC gap :

Single-electron states suppressed by pseudogap ΔP >> Tc “Pseudospin” approximation Effective number of interacting neighbours

Qualitative features of “Pseudogaped Superconductivity”: STM DoS evolution with T Double-peak structure in point-contact conductance Nonconservation of the full spectral weight across Tc eV1 = ΔP + Δ 2eV2 = 2 Δ V2 << V1 near SIT Ktot(T) Tc Δp T

Spectral weight of high-ω conductivity constant (T-independent) in BCS Pseudogap superconductor with ΔP >> Δ

Major unresolved theoretical problem with the developed approach: what happens to Coulomb repulsion?

Coulomb enchancement near mobility edge ?? Normally, Coulomb interaction is overscreened, with universal effective coupling constant ~ 1 Condition of universal screening: Example of a-InOx : e2kF ~ 5 104 K deeply in insulator state of InOx Effective Coulomb potential is weak if i.e. for κ > 300

Have been discussed in this talk: Generalized mean-field – like theory of superconductive state for critical or weakly localized single-electron states To be explained: Activated R(T) on the insulating side close to SIT Strong fluctuations of coherence peak heights on the superconducting side Nature of the SIT within pseudogap model See next talk for the results beyond MFA