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Fractal and Pseudopgaped Superconductors:

theoretical introduction

Based on the results obtained in collaboration withLev Ioffe and Emil Yuzbashyan Rutgers Vladimir Kravtsov ICTP Trieste

Emilio Cuevas Murcia Univ.

Mikhail Feigel’manL.D.Landau Institute, Moscow

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)

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Plan of the talk1. Introduction: why new theory is needed?2. Fractal superconductivity at the localization edge - sketch of the derivation - main features3. Superconductivity with pseudogap - origin of the psedogap - development of the superconductive correlations - qualitative features4. 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

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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 interactionK.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

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

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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 !

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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

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SC side: local tunneling conductance

Nature Physics 7, 239 (2011)

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Superconductive state near SIT is very unusual:

1. the spectral gap appears much before (with T decrease) than superconductive coherence does

2. Coherence peaks in the DoS appear together with resistance vanishing

3. Distribution of coherence peaks heights is very broad near SIT

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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 diamondBosonic 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

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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 interactionK.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

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Superconductive transition

at the mobility edge

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Theoretical model (3D) Simplest BCS attraction model, but for critical (or weakly localized) electron

eigenstates

H = H0 - g ∫ d3r Ψ↑†Ψ↓

†Ψ↓Ψ↑ Ψ = Σ cj Ψj (r) Basis of exact eigenfunctions

of free electrons in random potential

M. Ma and P. Lee (1985) :S-I transition at δL ≈ Tc

We will find that SC state is compatible with δL >> Tc

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Why do anyone may need analytical theory for S-I

transition?• Low-temperature superconductivity is the

nontrivial result of a weak 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

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Mean-Field Eq. for Tc

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163D Anderson model: γ = 0.57

d2 ≈ 1.3 in 3D

Fractality of wavefunctions

IPR: Mi = 4dr

l is the short-scale cut-off length

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3D Anderson model: long evolution from diffusive metal to the

critical pointE.Cuevas and V.Kravtsov, Phys.Rev B76 (2007)

“Box distribution”: critical disorder strength Wc = 16.5

W=5

W=2

W=10

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Modified mean-field approximation for critical temperature Tc

For small this Tc is higher than BCS value !

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Alternative method to find Tc:Virial expansion

(A.Larkin & D.Khmelnitsky 1970)

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Tc from 3 different calculations

Modified MFA equationleads to:

BCS theory: Tc = ωD exp(-1/ λ)

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Neglected so far : off-diagonal terms

Non-pair-wise terms with 3 or 4 different eigenstates were omittedTo estimate the accuracy we derived effective Ginzburg -Landau functional taking these terms into account

Parameters a, b, C and W do not contain fractal exponents

W=∫<δa(r)δa(r’)>dr’

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Can we understand increase of Tc by disorder within regularperturbative 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

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Order parameter in real space

for ξ = ξk

SC fraction =

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Tunnelling DoS

Asymmetry in local DoS:

Average DoS:

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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

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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 weakweak to strongstrong localization takes placeHypotetically the same kind of analysiswe did for 3D can be adopted for 2D case

But it was not done yet

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Superconductive state with a pseudogap

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Parity gap in ultrasmall grains K. Matveev and A. Larkin 1997

No many-body correlations

Local pairing energy

Correlations between pairs of electrons localized in the same “orbital”

-------------- EF

--↑↓---- ↓--

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Parity gap for Anderson-localized eigenstates

Energy of two single-particle excitations after depairing:

ΔP plays the role of the activation gap

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Activation energy TI from Shahar-Ovadyahu exper. (1992) and fit to the

theory

The fit was obtained with single fitting parameter

= 0.05 = 400 K

Example of consistent choice:

Similar fit with naïve exponent d=3 instead of d2 = 1.3 fails undoubtedly

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Tc versus Pseudogap

Superconductive transition exists even at δL >> Tc0

Annals of Physics 325, 1368 (2010)

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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

Here “local gap”exceeds SC gap :

only with weak coupling !

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Single-electron states suppressed by pseudogap ΔP >> Tc

Effective number of interacting neighbours

“Pseudospin” approximation

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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

T

Ktot(T)

Tc Δp

eV1 = ΔP + Δ 2eV2 = 2 ΔV2 << V1 near SIT

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Spectral weight of high-ω conductivity

constant (T-independent) in BCS

Pseudogap superconductor with ΔP >> Δ

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

Coulomb repulsion?

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Coulomb enchancement near mobility edge ??

Condition of universal screening:

Normally, Coulomb interaction is overscreened, with universal effective coupling constant ~ 1

Example of a-InOx :

Effective Coulomb potential is weak if

e2kF ~ 5 104 K 0

deeply in insulator state of InOx

i.e. for κ > 300

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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

Have been discussed in this talk:

Generalized mean-field – like theory of superconductive state for critical or weakly localized single-electron states