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