Transcript Document
PseudoGap Superconductivity and
Superconductor-Insulator transition
Mikhail Feigel’man
L.D.Landau Institute, Moscow
In collaboration with:
Vladimir Kravtsov
Emilio Cuevas
Lev Ioffe
Marc Mezard
ICTP Trieste
University of Murcia
Rutgers University
Orsay University
Short publications: Phys Rev Lett. 98, 027001 (2007)
arXiv:0909.2263
Plan of the talk
1. Motivation from experiments
2. BCS-like theory for critical eigenstates
- transition temperature
- local order parameter
3. Superconductivity with pseudogap
- transition temperature v/s pseudogap
4. Quantum phase transition: Cayley tree
5. Conclusions
Superconductivity v/s Localization
• Coulomb-induced suppression of Tc in uniform
films
“Fermionic mechanism”
A.Finkelstein (1987) et al
• 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, Bulaevsky-Sadovsky(mid-80’s)
Ghosal, Randeria, Trivedi 1998-2001
There will be no grains in this talk !
We consider amorphous systems with direct S-I transition
SUPERCONDUCTOR-INSULATOR:
EXPERIMENTAL EVIDENCE
Direct evidence for the gap above the transition (Chapelier,
Sacepe). Activation behavior does not show gap suppression at
the critical point as a function of the disorder (Sahar,
Ovaduyahu, 1992)!
First look: critical behavior as predicted by boson
duality (Haviland, Liu, Goldman 1989, 1991)
Class of relevant materials
• Amorphously 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)
Special example: nanostructured Bi films
(J.Valles et al)
Bosonic mechanism:
Control parameter
Ec = e2/2C
BOSE MODEL (PREFORMED COOPER
PAIRS)
•
R
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=φ0 jV
Charge motion generates current: I=2e jc
At the self-dual point the currents are equal →
RQ=V/I=h/(2e)2=6.5kΩ.
M.P.A.Fisher 1990
Insulator
EJ / E C 1
EJ / E C 1
RQ
EJ / E C 1
cos(i ) 0
T
In superconducting films have cores → friction
→ vortex motion is not similar to charges
→ duality is much less likely in the films
→ intermediate normal metal is less likely in
Josephson arrays (no vortex cores)
Duality approach to SIT in films
Model of
Josephson
array (SC state)
Duality transformation
(approximate)
Insulating state of
Josepshon array
?????
Duality transformation
(non-existing)
Continous amorphous
disordered film
(SC state)
????????
Insulating state of
disordered film
SUPERCONDUCTOR-INSULATOR:
EXPERIMENTAL EVIDENCE
If duality arguments are correct, the same behavior
should be observed in Josephson arrays…
First look: critical behavior as predicted by
boson duality (Haviland, Liu, Goldman 1989,
1991)
At zero field simple Josephson
arrays show roughly the critical
behavior. However, the critical R is
not universal. (Zant and Mooji,
1996) and critical value of EJ/Ec
differs.
SUPERCONDUCTOR-INSULATOR:
EXPERIMENTAL EVIDENCE
If Josephson/Coulomb model is correct, the same
behavior should be observed in Josephson arrays…
At non-zero field simple Josephson
arrays
show
temperature
independent resistance with values
that change by orders of magnitude.
(H. van der Zant et al, 1996)
SUPERCONDUCTOR-INSULATOR:
EXPERIMENTAL EVIDENCE
If Josephson/Coulomb model is correct, the same
behavior should be observed in Josephson arrays…
At non-zero field simple Josephson
arrays show temperature independent
resistance with values that change by
orders of magnitude.
(H. van der Zant et al, 1996)
SUPERCONDUCTOR-INSULATOR:
EXPERIMENTAL EVIDENCE
If Josephson/Coulomb model is correct, the same
behavior should be observed in Josephson arrays…
BUT IT IS NOT
At non-zero field Josephson arrays of
more complex (dice) geometry show
temperature independent resistance
in a wide range of EJ/Ec. (B.Pannetier
and E.Serret 2002)
Bosonic v/s Fermionic
scenario ?
None of them is able to describe
data on InOx and TiN
3-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 electrons
H = H0 - g ∫ d3r Ψ↑†Ψ↓†Ψ↓Ψ↑
Ψ = Σ cj Ψj (r)
M. Ma and P. Lee (1985) :
Basis of localized eigenfunctions
S-I transition at δL ≈ Tc
Superconductivity at the
Localization Threshold: δL → 0
Consider Fermi energy very close
to the mobility edge:
single-electron states are extended
but fractal
and populate small fraction of the
whole volume
How BCS theory should be modified to account
for eigenstate’s fractality ?
Method: combination of analitic theory and numerical
data for Anderson mobility edge model
Mean-Field Eq. for Tc
Fractality of wavefunctions
IPR: Mi =
D2 ≈ 1.3
3D Anderson model: γ = 0.57
4
dr
in 3D
Modified mean-field approximation
for critical temperature Tc
For small
this Tc is higher than BCS value !
Order parameter in real space
for ξ = ξk
SC fraction =
Superconductivity with Pseudogap
Now we move
Fermi- level into the
range of localized eigenstates
Local pairing
in addition to
collective pairing
Local pairing energy
1. Parity gap in ultrasmall grains
K. Matveev and A. Larkin 1997
------------- EF
--↑↓--- ↓--
No many-body correlations
Correlations between pairs of electrons localized in the same “orbital”
2. Parity gap for Andersonlocalized eigenstates
Energy of two single-particle excitations after depairing:
P(M) distribution
Critical temperature in the
pseudogap regime
MFA:
Here we use M(ω) specific for localized states
MFA
is OK as long as
is large
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 found
previously
Here “local gap”
exceeds SC gap :
MFA:
Critical temperature in the
pseudogap regime
We need to estimate
~
(
3
)
It is nearly constant in a
very broad range of
Tc versus Pseudogap
Transition exists even at δL >> Tc0
Single-electron states suppressed by pseudogap
“Pseudospin” approximation
Effective number of interacting neighbours
Qualitative features of
“Pseudogaped Superconductivity”:
•
STM DoS evolution with T
• Double-peak structure in point-contact
conductance
V
• Nonconservation of full spectral weight
across Tc
Ktot(T)
Tc
Δp
T
Third Scenario
• Bosonic mechanism: preformed Cooper pairs +
competition Josephson v/s Coulomb – S I T in arrays
• Fermionic 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 (L.Ioffe & M.Mezard)
Superconductor-Insulator
Transition
Simplified model of competition
between random local energies
(ξiSiz term) and XY coupling
Phase diagram
Temperature
Energy
Hopping insulator
Full localization:
Insulator with
Discrete levels
Superconductor
MFA line
RSB state
gc
g
Fixed activation energy is due to the absence of thermal bath at low ω
Conclusions
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 Cayley tree, at least). On insulating side activation of
pair transport is due to ManyBodyLocalization threshold
Coulomb enchancement near mobility edge ??
Normally, Coulomb interaction is overscreened,
with universal effective coupling constant ~ 1
Condition of universal screening:
Example of a-InOx
Effective Couloomb potential is weak:
Alternative method to find Tc:
Virial expansion
(A.Larkin & D.Khmelnitsky 1970)
Neglected : off-diagonal terms
Non-pair-wise terms with 3 or 4 different eigenstates were omitted
To estimate the accuracy we derived effective GinzburgLandau functional taking these terms into account
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
Tc from 3 different calculations
Modified MFA equation
leads to:
BCS theory: Tc = ωD exp(-1/ λ)
Activation energy TI from ShaharOvadyahu exp. and fit to theory
The fit was obtained with
single fitting parameter
Example of consistent choice:
= 0.05
= 400 K