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Superconductivity
Introduction
Disorder & superconductivity : milestones
BCS theory
Anderson localization
Abrikosov, GorkovAnderson theorem
1911 1957 1958 - 1959 1987 1990
Time
2000 - 2010
Bosonic scenario
Fermionic scenario
Localization vs.Superconductivity
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)
A.M. Finkel’steinJETP Lett. 45, 46 (1987) M.P.A. Fisher, PRL 65, 923 (1990)
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."
TIN Superconductor-Insulator transition
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,00
1
2
3
4
5
6
7
8
R [
k]
T [K]
TiN 1 TiN 2 TiN 3
-1,0 -0,5 0,0 0,5 1,00,0
0,5
1,0
1,5
2,0
G(V
), n
orm
ali
zed
V [mV]
= 260 µeVT
eff = 0,25 K
-1,0 -0,5 0,0 0,5 1,00,0
0,5
1,0
1,5
2,0
= 225 µeVT
eff = 0,32 K
G(V
), n
orm
ali
zed
V [mV]
-1,0 -0,5 0,0 0,5 1,00,0
0,5
1,0
1,5
2,0
= 154 µeVT
eff = 0,35 K
G(V
), n
orm
ali
zed
V [mV]
Increasing disorder Sacépé et al., PRL 101, 157006 (2008)
TIN Superconductor-Insulator transition
-1,0 -0,5 0,0 0,5 1,00,0
0,5
1,0
1,5
2,0
G(V
), n
orm
ali
zed
V [mV]
= 260 µeVT
eff = 0,25 K
-1,0 -0,5 0,0 0,5 1,00,0
0,5
1,0
1,5
2,0
= 225 µeVT
eff = 0,32 K
G(V
), n
orm
ali
zed
V [mV]
-1,0 -0,5 0,0 0,5 1,00,0
0,5
1,0
1,5
2,0
= 154 µeVT
eff = 0,35 K
G(V
), n
orm
ali
zed
V [mV]
Increasing disorder Sacépé et al., PRL 101, 157006 (2008)
Superconducting fluctuations correction to the DOS
Preformed Cooper pairs above Tc
A. Varlamov and V. Dorin, Sov. Phys. JETP 57, 1089, (1983)
Pseudogap
B. Sacépé, et al., Nature Communications (2010)
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,00
1
2
3
4
5
6
7
8
R [
k]
T [K]
TiN 1 TiN 2 TiN 3
Localization of preformed Cooper pairs in highly disordered superconducting films
Meso2012
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
Samples : e-gun evaporation of high purity In2O3
onto Si/SiO2 substrate under O2 pressure
D. Shahar, Weizmann Institute of Science
InOx
Amorphous Indium Oxide
Thickness : 15 nm (red & grey) and 30 nm (bleu) – 3D regime
InO#1
InO#3
1 mm
Nearly critical samples
High disorder (red) and low disorder (bleu)
•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
InO#1
InO#2InO#1
InO#3
D. Shahar and Z. Ovadyahu, Phys. Rev. B 46, 10917 (1992)
InOx
V. F. Gantmakher et al., JETP 82, 951 (1996)
Fit : s-wave BCS density of states
Typical spectrum measured at 50 mKInO#3
STM and transport measurements
Absence of quasi-particle excitations at low energies
Spatial fluctuations of the spectral gap Δ(r)
II.1 Superconducting inhomogeneities Superconducting inhomogeneities
Map of the spectral gap
Gaussian distribution
InO#3
II.1 Superconducting inhomogeneities Superconducting inhomogeneities
Spectra measured at different locations (T=50mK)
Spatial fluctuations of the spectral gap Δ(r)G
, N
orm
aliz
edG
, N
orm
aliz
ed
Gaussian distribution
InO#3
II.1 Superconducting inhomogeneities Superconducting inhomogeneities
Fluctuations of Δ(r) and superconducting transition
5.5)(
3
CBTk
r
Spatial fluctuations of ∆ / (1.76kB)
Spatial fluctuations of thecoherence peaks height
Coherence peaksG
, N
orm
aliz
edG
, N
orm
aliz
ed
Spatial fluctuations of thecoherence peaks height
Coherence peaks
Statistical study
G,
Nor
mal
ized
InO#3
G,
Nor
mal
ized
Incoherent spectra
Statistical study
Full spectral gap without coherence peaks
G,
Nor
mal
ized
G, N
orm
aliz
ed
InO#3
Incoherent spectra
Statistical study
G,
Nor
mal
ized
G,
Nor
mal
ized
High disorder
Full spectral gap without coherence peaks
InO#1
λ λ
Superconducting
Insulating
A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) & PRB 65, 014501 (2001)
Signature of localized Cooper pairs
0 ,, , ,
. .i j i ii j i
H t c c h c V n
int i ii
H n n
λ
« Insulating » gap Egap
Localized Cooper pairs
Superconducting gap Δ delocalized Cooper pairs
G,
No
rmal
ized
G,
No
rmal
ized
G,
No
rmal
ized
G,
No
rmal
ized
Signature of localized Cooper pairs
resistivity × 2
InO#3Tc ~ 1.7 K
InO#1Tc ~ 1.2 K
Proliferation of spectra without coherence peaks
Proliferation of incoherent spectra at the superconductor-insulator transition
8%
~
16%
~
B. Sacépé, et al., Nature Physics (2011)
resistivity × 2
InO#3Tc ~ 1.7 K
InO#1Tc ~ 1.2 K
Proliferation of spectra without coherence peaks
Proliferation of incoherent spectra at the superconductor-insulator transition
8%
~
16%
~
High disorder
Low disorder
B. Sacépé, et al., Nature Physics (2011)
M. Feigel’man et al., Phys. Rev. Lett. 98, 027001 (2007) ; Ann.Phys. 325, 1390 (2010)
Pseudogap state above Tc
Tc
Pseudogap state above Tc
Tc
Pseudogap state above Tc
BCS peaks appear along with superconducting phase coherence
Macroscopic quantum phase coherence probed at a local scale
Tpeak ~ Tc
Definition of Tc
Phase coherence signaled at Tc independently of ∆(r) Justification of Tc
Definition of Tc
Macroscopic quantum phase coherence probed at a local scale
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
11)(2
6 CB
gap
Tk
rE
Superconductivity beyond the mobility edgeM. Feigel’man et al., Phys. Rev. Lett. 98, 027001,
(2007)M. Feigel’man et al., Ann. Phys. 325, 1390 (2010)
Egap = Δp + ΔBCS
•∆p “parity gap”: pairing of 2 electrons in localized wave functions
•∆BCS “BCS gap”: long-range SC order between localized pairs
BCS model built on fractal eigenfunctions of the Anderson problem
Two contributions to the spectral gap
Superconductivity beyond the mobility edgeM. Feigel’man et al., Phys. Rev. Lett. 98, 027001,
(2007)M. Feigel’man et al., Ann. Phys. 325, 1390 (2010)
Egap = Δp + ΔBCS
•∆p “parity gap”: pairing of 2 electrons in localized wave functions
•∆BCS “BCS gap”: long-range SC order between localized pairs
BCS model built on fractal eigenfunctions of the Anderson problem
Two contributions to the spectral gap
Egap = Δp + ΔBCS
“parity gap”
“BCS gap”
•Tunneling spectroscopy(single-particle DOS)
Egap = Δp + ΔBCS
“BCS gap”
•Point-contact spectroscopy (Andreev reflection = transfer of pairs)
How to measure the SC order parameter ?
Tunnel barrier
Transparent interface
Point-Contact Andreev Spectroscopy
Conductance of a N/S contact
Sample
Tip
Sample
Tip
Sample
Tip
Normal metal Superconductor
Barrier : parameter Z
Transmission : T = 1 / (1+ Z2)
Z » 1
Z ~ 1
Z « 1T = 300 mK
Z value
Tunnel regime
Contact regime
• Single-particle transfer ~ T • Two-particles transfer ~ T 2
Blonder, G. E., Tinkham, M., and Klapwijk T.M. Phys. Rev. B 25, 7 4515 (1982)
Point-Contact Andreev Spectroscopy
From tunnel to contact in a-InOx
Tunnel regime
Contact regime
T=50mK Rc=65kΩ
Rc=6kΩ
Egap
∆BCS
Egap = Δp + ΔBCS
Tc
Andreev signal : evolution with T
Egap(T) = Δp + ΔBCS(T)
Egap and ∆BCS evolve on distinct temperature range ∆BCS : local signature of SC phase coherence
. ∆BCS evolves between 0 and ~ Tc
. Egap evolves between 0 and ~ 3-4
Tc
T-evolution of Andreev signal
Distinct energy scales for pairing and coherence
Two distinct energy scales in a-InOx
Distinct energy scales for pairing and coherence in disordered a-InOx
Egap (r) = Δp(r) + ΔBCS
• ∆BCS probed by AR remains uniform
• Egap probed by STS fluctuates
∆B
CS
[1
0-4 e
V]
Egap [10-4 eV]
11 evolutions on 3 samples
∆BCS =
Egap
Conclusion : Fluctuation and localization of preformed Cooper pairs
• Fluctuating Cooper-Pairs above Tc
• “Partial” condensation of these preformed pairs below Tc
• SIT occurs through the localization of Cooper-pairs
• But inhomogeneous superconducting state at T<Tc
• Homogeneously disordered system with a continuous decrease of Tc
• Distinct energy scales for pairing and coherence
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