1. 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, (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, (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)

2. 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

3. 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

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

5. 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 !

6. 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

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

8. 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

9. 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

10. 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

11. Superconductive transition
at the mobility edge

12. Theoretical model (3D) H = H0 - g ∫ d3r Ψ↑†Ψ↓†Ψ↓Ψ↑
Simplest BCS attraction model,
but for critical (or weakly
localized) electron eigenstates
H = H 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

13. Why do anyone may need analytical theory for S-I transition?
Low-temperature superconductivity is the nontrivial result of a weak interaction:
Tc ~ ( ) 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

14. Mean-Field Eq. for Tc

16. Fractality of wavefunctions4
IPR: Mi = dr
d2 ≈ in 3D
l is the short-scale
cut-off length
3D Anderson model: γ = 0.57

18. 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

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

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

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

22. 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

23. 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:
“Enhancement of superconductivity by Anderson localization”
Renormalization Group approach

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

25. Tunnelling DoS
Average DoS:
Asymmetry in local DoS:

26. 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

27. 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

28. Superconductive state with a pseudogap

29. 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”

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

31. 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

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

33. 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 :

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

35. 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

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

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

38. 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 ~ K
deeply in insulator state of InOx
Effective Coulomb potential is weak if
i.e. for κ > 300

39. 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