Benjamin SacépéBenjamin SacépéInstitut Néel, CNRS & Université Joseph Fourier, Grenoble

Localization of preformed Cooper-pairs Localization of preformed Cooper-pairs

in disordered superconductorsin disordered superconductors

Lorentz Center, Leiden 2011

Benjamin SacépéBenjamin SacépéInstitut Néel, CNRS & Université Joseph Fourier, Grenoble

Localization of preformed Cooper-pairs Localization of preformed Cooper-pairs

in disordered superconductorsin disordered superconductors

Thomas Dubouchet, Claude Chapelier, Marc SanquerThomas Dubouchet, Claude Chapelier, Marc SanquerCEA Grenoble

Maoz Ovadia, Dan ShaharMaoz Ovadia, Dan ShaharWeizmann Institute of Science, Rehovot

M. Feigel’manM. Feigel’manL.D. Landau Institut for Theoretical Phyiscs, Moscow

L. IoffeL. IoffeRutgers University, Piscataway

Lorentz Center, Leiden 2011

Superconductor-Insulator Transition (SIT)Superconductor-Insulator Transition (SIT)

Main ingredients :

1. Disorder localization

2. Attractive pairing superconducting phase

3. Coulomb interaction competes with pairing

4. Reduced dimensionality affects 1,2 and 3

d =

d =

Quench condensed Bismuth

D.B. Haviland, Y. Lui, A.M. Goldman, PRL (‘89)

Amorphous indium oxideAmorphous indium oxide

G. Sambandamurthy, et al. PRL 94, 017003, (2005)

Magnetic field-tuned SITMagnetic field-tuned SIT

For similar results in TiN films see : T. Baturina, et al. PRL (2007)

V. F. Gantmakher et al., JETP 82, 951 (1996)

0.20K

SuperconductorSuperconductor

Magnetic field-tuned SITMagnetic field-tuned SIT

0.20K

InsulatorInsulator

Positive magnetoresistance at low field :

Superconducting correlations in insulators !?

SuperconductorSuperconductor

Magnetic field-tuned SITMagnetic field-tuned SIT

0.20K

InsulatorInsulator

Positive magnetoresistance at low field :

Superconducting correlations in insulators !?

« Insulating correlations » in superconductors ???

SuperconductorSuperconductor

Magnetic field-tuned SITMagnetic field-tuned SIT

E

µL

µR

NR(E)NL(E)

TipTip SampleSample

2ΔeV

E

µL

µR

NR(E)NL(E)

TipTip SampleSample

2ΔeV

mK-STM setup : tunneling spectroscopymK-STM setup : tunneling spectroscopy

Cryostat : inverted dilution

50mK< T< 6K

C. Chapelier’s setup, CEA Grenoble

Combined transport and spectroscopy measurements

Thickness : 15 nm (blue) and 30 nm (red)

3D regime

Samples : e-gun evaporation onto Si/SiO2 substrate of high purity In2O3 under O2 pressure

1 mm

InO#1

InO#2

AmorphousAmorphous Indium Oxyde (a-InO Indium Oxyde (a-InOxx))

Thickness : 15 nm (blue) and 30 nm (red)

3D regime

1 mm

InO#1

InO#2

AmorphousAmorphous Indium Oxyde (a-InO Indium Oxyde (a-InOxx))

V. F. Gantmakher et al., JETP 82, 951 (1996)

Samples : e-gun evaporation onto Si/SiO2 substrate of high purity In2O3 under O2 pressure

Fit : s-wave BCS density-of-states

Typical spectrum measured at 50 mKTypical spectrum measured at 50 mKInO#1

Tunneling spectroscopy of amorphous indium oxydeTunneling spectroscopy of amorphous indium oxyde

Inhomogeneities of Inhomogeneities of ΔΔ(r)(r)

Map of the spectral gapMap of the spectral gap

8%

Gaussian distribution

For similar results in TiN and NbN films see :

B. Sacépé, et al. PRL 101, 157006 (2008)

M. Mondal, et al. PRL 106, 047001 (2011)

Spectra measured at different locations Spectra measured at different locations (T=50mK)(T=50mK)

G,

Nor

mal

ized

G,

Nor

mal

ized

Inhomogeneities of Inhomogeneities of ΔΔ(r)(r)

Gaussian distribution

8%

For similar results in TiN and NbN films see :

B. Sacépé, et al. PRL 101, 157006 (2008)

M. Mondal, et al. PRL 106, 047001 (2011)

BCS ratio BCS ratio ΔΔ/T/Tcc =1.76 =1.76 ? ?

Fluctuations of Fluctuations of ΔΔ(r) and superconducting transition(r) and superconducting transition

Fluctuations of Fluctuations of ΔΔ(r) and superconducting transition(r) and superconducting transition

Definition of Tc : zero-resistance state (macroscopic phase coherence)

2 ( )6 11 !?

c

r

T

G, N

orm

aliz

edG

, Nor

mal

ized

Fluctuations of peak heights !Fluctuations of peak heights !

Fluctuations of the BCS peaksFluctuations of the BCS peaks

( ) ( )

( )

G G eVR

G eV

Fluctuations of the BCS peaksFluctuations of the BCS peaks

( ) ( )

( )

G G eVR

G eV

Fluctuations of the BCS peaksFluctuations of the BCS peaks

Fluctuations of the BCS peaksFluctuations of the BCS peaks

0!?R

Extreme case : « Insulating » gap Extreme case : « Insulating » gap

Spectra measured at different locations Spectra measured at different locations (T=50mK)(T=50mK)

0!?R

G,

Nor

mal

ized

G,

Nor

mal

ized

High disordersample

Let’s approach the SITLet’s approach the SIT

( ) ( )

( )

G G eVR

G eV

Sample InO#2 : disorder Sample InO#2 : disorder 2 2

InO#1

InO#2

resistivity × 2

8%

InO#1Tc ~ 1.7 K

InO#2Tc ~ 1.3 K

16%

Proliferation of gaps without peaksProliferation of gaps without peaks

Let’s approach the SITLet’s approach the SIT

Increase of disorderIncrease of disorder

InO#1

InO#2

λ

Superconductivity and disorderSuperconductivity and disorder

With increasing disorder : Superconductivity becomes « granular-like » Spectral gap remains finite even at large disorder Spectral gap is NOT anymore the SC order parameter

λ

Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave SuperconductorRole of Spatial Amplitude Fluctuations in Highly Disordered s-Wave SuperconductorA. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 6565, 014501 (2001)

λ

Insulating gap induced by pairing interactionInsulating gap induced by pairing interaction

λ

Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave SuperconductorRole of Spatial Amplitude Fluctuations in Highly Disordered s-Wave SuperconductorA. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 6565, 014501 (2001)

Superconductivity and disorderSuperconductivity and disorder

M. Ma, and P. A. Lee, PRB 32, 5658, (1985)M. Feigel’man, et al., PRL 98, 027001, (2007)M. Feigel’man, et al, Ann. Phys. 325, 1390 (2010)M. Feigel’man, et al, PRB (2010)

3~gap loc

loc

E

Insulating gap due to pairingInsulating gap due to pairing

P. W. Anderson, J. Phys. (Paris) Colloq. 37, C4-339 (1976)

M. Ma, and P. A. Lee, PRB 32, 5658, (1985)

K. A. Matveev and A. Larkin, PRL. 78, 3749, (1997)

A. Ghosal, et al. PRL 81, 3940, (1998) and PRB 6565, 014501 (2001)

M. Feigel’man, et al. PRL 98, 027001, (2007)

M. Feigel’man, et al. Ann. Phys. 325, 1390 (2010)

M. Feigel’man, et al. PRB 82, 184534 (2010)

In the lowest order:

2~ D

locgapE

3~gap loc

loc

E

† † †j j j jk j j k k

j jk

H c c M c c c c

Reduced BCS Hamiltonian built on eigenstates of the Anderson problem

2 2jk j kM dr with

In the high-disorder regime when

43

1( )jk jk j jk

loc

M dr r

loc

λ

Insulating gap induced by pairing interactionInsulating gap induced by pairing interaction

λ

Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave SuperconductorRole of Spatial Amplitude Fluctuations in Highly Disordered s-Wave SuperconductorA. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 6565, 014501 (2001)

Superconductivity and disorderSuperconductivity and disorder

M. Ma, and P. A. Lee, PRB 32, 5658, (1985)M. Feigel’man, et al., PRL 98, 027001, (2007)M. Feigel’man, et al, Ann. Phys. 325, 1390 (2010)M. Feigel’man, et al, PRB (2010)

3~gap loc

loc

E

Superconductivity and disorderSuperconductivity and disorder

Recent QMC simulationsRecent QMC simulationsK. Bouadim, Y. Loh, M. Randeria, N. Trivedi, arXiv:1011.3275

Dis

orde

rD

isor

der

SC

Insulator

Pairing gap in the insulatorPairing gap in the insulator

resis

tivity

× 2

Proliferation of incoherent –localized— Cooper-pairs Proliferation of incoherent –localized— Cooper-pairs

when approaching the SIT when approaching the SIT

Pairing gap in the insulatorPairing gap in the insulator

resis

tivity

× 2

Proliferation of incoherent –localized— Cooper-pairs Proliferation of incoherent –localized— Cooper-pairs

when approaching the SIT when approaching the SIT

Simulations on the Bethe latticeSimulations on the Bethe latticeLev Ioffe, Misha Feigel’man

M. Feigel’man, et al., PRL (2007)M. Feigel’man, et al, Ann. Phys. (2010)M. Feigel’man, et al, PRB (2010)

Tc

Pseudogap above TPseudogap above Tcc : preformed pairs : preformed pairs

T-dependence of the local DOST-dependence of the local DOS

Pseudogap in TiN and NbN films :

B. Sacépé, et al. Nature Commun. 1:140 (2010)

M. Mondal, et al. PRL 106, 047001 (2011)

Pseudogap in quasi-2D conventional superconductorsPseudogap in quasi-2D conventional superconductors

A. Varlamov and V. Dorin, Sov. Phys. JETP 57, 1089, (1983)

B. Sacépé, et al. Nature Commun. 1:140 (2010)

Superconducting fluctuations in quasi-2D TiN films ( thickness < 5nm )

Local versus macroscopic phase coherenceLocal versus macroscopic phase coherence

Tpeak : Temperature below which peaks start to grow

BCS peaks give a local signature of the superconducting phase coherence BCS peaks give a local signature of the superconducting phase coherence 

BCS peaks appears at Tc BCS peaks appears at Tc

independently of gap inhomogeneitiesindependently of gap inhomogeneities

« Insulating » gap at T<<T« Insulating » gap at T<<Tcc

Local pairing without phase coherence at T << Tc

Formation of a pseudogap without BCS peaks at T>TFormation of a pseudogap without BCS peaks at T>Tcc

Local pairing without phase coherence at T > Tc

Spectral signature of localized Cooper pairs

Preformed Cooper-pairs

Condensation versus localization of preformed Cooper pairsCondensation versus localization of preformed Cooper pairs

ConclusionsConclusions

Localization of preformed Cooper pairs in disordered superconductors

Nature Physics 7, 239 (2011)

• Preformed Cooper-Pairs above TcPseudogap in the DOS between Tc and ~ 3-4 Tc

• “Partial” condensation of pairs below Tc

Rectangular spectra at 50mK = localized Cooper pairs

• SIT occurs through the localization of Cooper pairsGap in the DOS remains & coherence peaks disappear

Disorder-enhanced Coulomb interactionDisorder-enhanced Coulomb interaction

B. Altshuler, et al., Phys. Rev. Lett. 44, 1288, (1980)

Coulomb interaction : Zero-Bias AnomalyCoulomb interaction : Zero-Bias Anomaly

Soft coulomb gapSoft coulomb gap

Superconducting fluctuationsSuperconducting fluctuations

We need a global theoryWe need a global theory !

ln lni c

T

G T

TiN 1

Quantum corrections to the DOSQuantum corrections to the DOS

TR

A

Q

ln0

Superconducting fluctuationsSuperconducting fluctuations (2D Aslamasov-Larkin correction (2D Aslamasov-Larkin correction

…)…)

Disorder-enhanced Coulomb interactionDisorder-enhanced Coulomb interaction(2D weak-localization/Aronov-Altshuler (2D weak-localization/Aronov-Altshuler

corrections)corrections)

Quantum corrections to the conductivityQuantum corrections to the conductivity

Dynamical Coulomb blockadeDynamical Coulomb blockade

P. Joyez and D. Estève, PRB 56, 1848, (1997)

Magnetic field-tuned SITMagnetic field-tuned SIT

Huge magnetoresistance peak : superconductivity-related ?

Amorphous indium oxydeAmorphous indium oxyded = 15 nmd = 15 nm

Spectral gap map Coherence peak map

Spectral fluctuationsSpectral fluctuations

Disorder-induced inhomogeneities in TiNDisorder-induced inhomogeneities in TiN

B. Sacépé, et al. PRL 101, 157006 (2008)

Titanium nitrideT. Baturina (Novosibirsk)V. Vinokur (Argonne National Lab.)

Gaped insulator Gaped insulator made of localized made of localized Cooper pairs ?Cooper pairs ?

Disorder-tuned SIT in ultra-thin films of TiN Disorder-tuned SIT in ultra-thin films of TiN