Introduction
Disorder & superconductivity : milestones
Kamerlingh Onnes, H., "On the change of electric resistance of pure metals at very
low temperatures, etc. IV. The resistance of pure mercury at helium temperatures." Fermionic
A.M. Finkel’stein
JETP Lett. 45, 46 (1987)
BCS theory
Superconductivity
scenario
Bosonic scenario
M.P.A. Fisher, PRL 65, 923 (1990)
Abrikosov, Gorkov
Anderson theorem
1911
1957
1958 - 1959
Time
1987
1990
2000 - 2010
Localization vs.Superconductivity
Anderson localization
A. Kapitulnik, G. Kotliar, Phys. Rev. Lett. 54, 473, (1985)
M. Ma, P.A. Lee, Phys. Rev. B 32, 5658, (1985)
G. Kotliar, A. Kapitulnik, Phys. Rev. B 33, 3146 (1986)
M.V. Sadowskii, Phys. Rep., 282, 225 (1997)
A.
Ghosal et al., PRL 81, 3940 (1998) ; PRB 65, 014501 (2001)
B.
K. Bouadim, Y. L. Loh, M. Randeria, N. Trivedi, Nature Physics (2011)
M. Feigel’man et al., Phys. Rev. Lett. 98, 027001 (2007) ; Ann.Phys. 325, 1390 (2010)
TIN Superconductor-Insulator transition
8
7
6
R [k]
5
4
3
2
TiN 1
TiN 2
TiN 3
1
0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
T [K]
1,0
 = 154 µeV
1,5
1,0
0,5
-0,5
 = 225 µeV
1,5
1,0
0,5
Teff = 0,32 K
Teff = 0,35 K
0,0
-1,0
G(V), normalized
G(V), normalized
G(V), normalized
1,5
0,5
2,0
2,0
2,0
0,0
V [mV]
0,5
1,0
0,0
-1,0
-0,5
 = 260 µeV
Teff = 0,25 K
0,0
0,5
1,0
V [mV]
Increasing disorder
0,0
-1,0
-0,5
0,0
0,5
V [mV]
Sacépé et al., PRL 101, 157006 (2008)
1,0
TIN Superconductor-Insulator transition
1,0
 = 154 µeV
1,5
1,0
0,5
-0,5
 = 225 µeV
1,5
1,0
0,5
Teff = 0,32 K
Teff = 0,35 K
0,0
-1,0
G(V), normalized
G(V), normalized
G(V), normalized
1,5
0,5
2,0
2,0
2,0
0,0
V [mV]
0,5
1,0
0,0
-1,0
-0,5
 = 260 µeV
Teff = 0,25 K
0,0
0,5
1,0
V [mV]
Increasing disorder
0,0
-1,0
-0,5
0,0
0,5
V [mV]
Sacépé et al., PRL 101, 157006 (2008)
1,0
Pseudogap
Superconducting fluctuations correction to the DOS
8
7
6
5
R [k]
Preformed Cooper pairs above Tc
B. Sacépé, et al., Nature Communications (2010)
4
3
2
TiN 1
TiN 2
TiN 3
1
0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
T [K]
A. Varlamov and V. Dorin, Sov. Phys. JETP 57, 1089, (1983)
Meso2012
Localization of preformed Cooper pairs in
highly disordered superconducting films
Thomas Dubouchet, Benjamin Sacépé, Marc Sanquer, Claude Chapelier
INAC-SPSMS-LaTEQS, CEA-Grenoble
T. Baturina, Institute of semiconductor Physics - Novosibirsk
V. Vinokur, Material Science Division, Argonne National Laboratory - Argonne
M. Baklanov, A Satta, IMEC - Leuven
Maoz Ovadia, Dan Shahar, The Weizmann Institute of Science
Mikhail Feigel'man, L. D. Landau Institute for Theoretical Physics
Lev Ioffe, Serin Physics laboratory, Rutgers University
InOx
Amorphous Indium Oxide
Samples: e-gun evaporation of high purity In2O3
onto Si/SiO2 substrate under O2 pressure
D. Shahar, Weizmann Institute of Science
Thickness : 15 nm (red & grey) and 30 nm (bleu) – 3D regime
1 mm
InO#1
InO#3
InOx
Nearly critical samples
High disorder (red) and low disorder (bleu)
InO#1
InO#2
V. F. Gantmakher et al., JETP 82, 951 (1996)
InO#3
InO#1
•kFle ~ 0.4-0.5 < 1  localized regime (Ioffe-Regel criterion)
•Tc comprised between 1K and 2K
•Carrier density : N = 3.5 x 1021 cm-3
D. Shahar and Z. Ovadyahu, Phys. Rev. B 46, 10917 (1992)
STM and transport measurements
InO#3
Typical spectrum measured at 50 mK
Fit : s-wave BCS density of states
 Absence of quasi-particle excitations at low energies
II.1 Superconducting
Superconducting
inhomogeneities
inhomogeneities
Spatial fluctuations of the spectral gap Δ(r)
Map of the spectral gap
InO#3
Gaussian distribution
II.1 Superconducting
Superconducting
inhomogeneities
inhomogeneities
Spatial fluctuations of the spectral gap Δ(r)
InO#3
G, Normalized
G, Normalized
Spectra measured at different locations (T=50mK)
Gaussian distribution
II.1 Superconducting
Superconducting
inhomogeneities
inhomogeneities
Fluctuations of Δ(r) and superconducting transition
Spatial fluctuations of ∆ / (1.76kB)
( r )
3
 5.5
k BTC
Coherence peaks
G, Normalized
G, Normalized
Spatial fluctuations of the
coherence peaks height
R
G()  G(eV  )
100
G (eV  )
Coherence peaks
Spatial fluctuations of the
coherence peaks height
G, Normalized
G, Normalized
Statistical study
R
G()  G(eV  )
100
G (eV  )
InO#3
Incoherent spectra
Full spectral gap
without coherence peaks
Normalized
G,G,
Normalized
Statistical study
R
G()  G(eV  )
100
G (eV  )
InO#3
Incoherent spectra
Full spectral gap
without coherence peaks
G, Normalized
G, Normalized
Statistical study
R
G()  G(eV  )
100
G (eV  )
InO#1
High disorder
Signature of localized Cooper pairs
A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) & PRB 65, 014501 (2001)
H 0  t
  c  c   h.c.   V    n 
i , j  ,

i
j
i
i,
λ
i,
H int    ni ni
i
λ
Superconducting
Insulating
Superconducting gap Δ
 delocalized Cooper pairs
λ
G, Normalized
G, Normalized
G, Normalized
G, Normalized
Signature of localized Cooper pairs
« Insulating » gap Egap
 Localized Cooper pairs
Proliferation of incoherent spectra at the superconductor-insulator transition


~
8%
InO#3
Tc ~ 1.7 K
resistivity × 2

~

16%
B. Sacépé, et al., Nature Physics (2011)
 Proliferation of spectra without coherence peaks
InO#1
Tc ~ 1.2 K
Proliferation of incoherent spectra at the superconductor-insulator transition


~
8%
InO#3
Tc ~ 1.7 K
resistivity × 2

~

16%
B. Sacépé, et al., Nature Physics (2011)
 Proliferation of spectra without coherence peaks
M. Feigel’man et al., Phys. Rev. Lett. 98, 027001 (2007) ; Ann.Phys. 325, 1390 (2010)
InO#1
Tc ~ 1.2 K
High disorder
Low disorder
Pseudogap state above Tc
Tc
Pseudogap state above Tc
Tc
Pseudogap state above Tc
Definition of Tc
Macroscopic quantum phase coherence
probed at a local scale
Tpeak ~ Tc
 BCS peaks appear along with superconducting phase coherence
Definition of Tc
Macroscopic quantum phase coherence
probed at a local scale
 Phase coherence signaled at Tc independently of ∆(r)
 Justification of Tc
Signatures of the localization of Cooper pairs
A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) & PRB 65, 014501 (2001)
K. Bouadim, Y. L. Loh, M. Randeria, N. Trivedi, Nature Physics (2011)
M. Feigel’man et al., Phys. Rev. Lett. 98, 027001 (2007) ; Ann.Phys. 325, 1390 (2010)
•Spatial fluctuations of the spectral gap
•rectangular-shaped” gap at T«Tc
•Statistic distribution of coherence peaks height
•Pseudogap regime above Tc
•Anomalously large spectral gap
6
2 Egap (r )
k BTC
 11
Two contributions to the spectral gap
Superconductivity beyond the mobility edge
M. Feigel’man et al., Phys. Rev. Lett. 98, 027001, (2007)
M. Feigel’man et al., Ann. Phys. 325, 1390 (2010)
BCS model built on fractal eigenfunctions of the Anderson problem
 Egap = Δp + ΔBCS
•∆p “parity gap”: pairing of 2 electrons in localized wave functions
•∆BCS “BCS gap”: long-range SC order between localized pairs
Two contributions to the spectral gap
Superconductivity beyond the mobility edge
M. Feigel’man et al., Phys. Rev. Lett. 98, 027001, (2007)
M. Feigel’man et al., Ann. Phys. 325, 1390 (2010)
BCS model built on fractal eigenfunctions of the Anderson problem
 Egap = Δp + ΔBCS
•∆p “parity gap”: pairing of 2 electrons in localized wave functions
•∆BCS “BCS gap”: long-range SC order between localized pairs
How to measure the SC order parameter ?
“BCS gap”
Egap = Δp + ΔBCS
•Tunneling spectroscopy
(single-particle DOS)
Tunnel barrier
“parity gap”
“BCS gap”
•Point-contact spectroscopy
(Andreev reflection = transfer of pairs)
Transparent interface
Egap = Δp + ΔBCS
Point-Contact Andreev Spectroscopy
Conductance of a N/S contact
Blonder, G. E., Tinkham, M., and Klapwijk T.M. Phys. Rev. B 25, 7 4515 (1982)
Barrier : parameter Z
Normal
metal
Superconductor
• Single-particle transfer ~ T
• Two-particles transfer ~ T 2
Transmission : T = 1 / (1+ Z2)
Tunnel regime
Z»1
Tip
Sample
Z value
Z~1
Tip
Sample
T = 300
mK
Z«1
Contact regime
Tip
Sample
Point-Contact Andreev Spectroscopy
From tunnel to contact in a-InOx
Rc=6kΩ
∆BCS
Contact
regime
Egap = Δp + ΔBCS
Egap
T=50mK
Rc=65kΩ
Tunnel
regime
Andreev signal : evolution with T
T-evolution of Andreev signal
Egap(T)= Δp + ΔBCS(T)
. ∆BCS evolves between 0 and ~ Tc
. Egap evolves between 0 and ~3-4Tc
Tc
 Egap and ∆BCS evolve on distinct temperature range
 ∆BCS : local signature of SC phase coherence
Distinct energy scales for pairing and coherence
Two distinct energy scales in a-InOx
Egap (r)= Δp(r) + ΔBCS
∆BCS [10-4 eV]
11 evolutions on 3 samples
• ∆BCS probed by AR remains uniform
∆BCS =Egap
• Egap probed by STS fluctuates
Egap [10-4 eV]
 Distinct energy scales for pairing and coherence
in disordered a-InOx
Conclusion : Fluctuation and localization of preformed Cooper pairs
•
Homogeneously disordered system
with a continuous decrease of Tc
•
But inhomogeneous superconducting state at T<Tc
•
Fluctuating Cooper-Pairs above Tc
•
“Partial” condensation of these preformed
pairs below Tc
•
SIT occurs through the localization of Cooper-pairs
•
Distinct energy scales for pairing and coherence