Superconducting fluctuations, interactions and disorder...

Claude Chapelier, Benjamin Sacépé, Thomas DubouchetINAC-SPSMS-LaTEQS, CEA-Grenoble

Superconducting fluctuations, interactions Superconducting fluctuations, interactions Superconducting fluctuations, interactions Superconducting fluctuations, interactions

and disorder : a subtle alchemyand disorder : a subtle alchemyand disorder : a subtle alchemyand disorder : a subtle alchemy

Les défis actuels de la supraconductivité – Dautreppe 2011

Kamerlingh Onnes, H., "Further experiments with liquid helium. C. On the change of electric resistance of pure metals at very low temperatures, etc. IV. The resistance of pure mercury at helium temperatures." Comm. Phys. Lab. Univ. Leiden; No. 120b, 1911.

Superconductivity in pure metals

Kamerlingh Onnes, H., "Further experiments with liquid helium. C. On the change of electric resistance of pure metals at very low temperatures, etc. IV. The resistance of pure mercury at helium temperatures." Comm. Phys. Lab. Univ. Leiden; No. 120b, 1911.

Superconductivity in pure metals and alloys …

J.P. Burgerla supraconductivité des métaux, des alliages et des films minces (Ed.Masson)

BCS theory for clean systems

k↓

−k↑

−k↑k↓

Bloch plane waves

J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. B. 108, 1175, (1957)

J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106, 162, (1957)

ξ0

φk,σ =1√Veikr

φ−k,σ =1√Ve−ikr

Theory of dirty superconductors

En Same energy if the scatterers are non-magnetic (time-reversed symmetry)

P.W. Anderson, J. Phys. Chem. Solids. 11, 26, (1959)

A.A. Abrikosov & I.P. Gorkov, Sov. Phys. JETP 8, 1090, (1959)

Nota : if the particle size become small, the mean level spacing between different would become greater than the superconducting gap and superconductivity will be forbidden

Ht = H0 +Hdis

√ξ0l

φn,σ =∑

n

< n | k > φk,σ

φ⋆n,σ =∑

n

< n | k >⋆ φ−k,−σ

Localization and superconductivity

MetalInsulator

kF l ≈ 1kF l≪ 1 kF l≫ 1

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)

M. Feigel’man et al., Phys. Rev. Lett. 98, 027001 (2007) ; Ann.Phys. 325, 1390 (2010)

Inhomogeneous superconducting state

P.W. Anderson, Absence of diffusion in certain random lattices Phys. Rev. 109, 1492(1958)

M.P.A Fisher and G. Grinstein Anderson, Presence of quantum diffusion in two-dimensions : universal resistance at the superconductor-insulator transition Phys. Rev. Lett 64, 587(1990)

Bosoni

c

Ψ(r, T ) = ∆(r, T ) eiϕ(r,T )

Superconductor-insulator transition : two scenarios

M.P.A Fisher and G. Grinstein Anderson, Presence of quantum diffusion in two-dimensions : universal resistance at the superconductor-insulator transition Phys. Rev. Lett 64, 587(1990)

Fermionic

Bosoni

c

Ψ(r, T ) = ∆(r, T ) eiϕ(r,T )

Superconductor-insulator transition : two scenarios

H. M. Jaeger, et al. Phys.Rev.B 34, 4920 (1986)

Gallium

Granular films

D.B. Haviland, Y. Lui, A.M. Goldman, PRL 62, 2180 (1989)

d =

d =

Amorphous films

Continuous decrease of Tc

Cooper pairing suppressed at the SIT

Competition between EC and EJ

Cooper pairs localized in grains

Fermionic

Bosoni

c

Superconductor-insulator transition : the role of Coulomb interaction

Bismuth

Ψ(r, T ) = ∆(r, T ) eiϕ(r,T )

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

[kΩΩ ΩΩ

]

T [K]

TiN 1 TiN 2 TiN 3

Superconductor-insulator transition in highly disordered TiN films

B. Sacépé et al., Phys. Rev. Lett 101, 157006 (2008)


kF l 1

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

[kΩΩ ΩΩ

]

T [K]

TiN 1 TiN 2 TiN 3




STM Spectroscopy

kF l 1

( ) ( )0 ,, , ,

. .i j i ii j i

H t c c h c V nσ σ σσ σ

µ+

< >= − + + −∑ ∑

int i ii

H n nλ ↑ ↓= − ∑

A. Ghosal et al., Phys. Rev. Lett 81, 3940 (1998);

Phys. Rev. B 65, 0145001 (2001)


Beyond the mean-field BCS theory : superconducting fluctuations

Ψ(r, T ) = ∆(r, T ) eiϕ(r,T )

F [Ψ(r)] = Fn +∫dV a | Ψ(r) |2 + b

2 | Ψ(r) |4 +

4mi | ∇Ψ(r) |2

Order parameter

Ginzburg-Landau functional (zero magnetic field)

a = α(T − Tc)

ξ2 = 24mαTc ξ(T ) = ξ0

√Tc

T−Tc

Gi ≈ e2

23 R

A. Larkin & A. Varlamov, in theory of superconducting fluctuations (Oxford)

Gi ≪ ǫ = ln( TTc)≪ 1

τGL = π8kB(T−Tc)

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

[kΩΩ ΩΩ

]

T [K]

TiN 1 TiN 2 TiN 3

superconducting fluctuations in highly disordered TiN films


W. Escoffier et al. , Phys. Rev. Lett. 93, 217005, (2004)

B. Sacépé et al., Nat. Comm., (2010)


B. Sacépé et al., Nat. Comm., (2010)

Superconductor-insulator transition in highly disordered InO films

High disorder (red) and low disorder (blue)

•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

kF l < 1

Spectra measured at different locations (T=50mK)

G, N

orm

aliz

edG

, Nor

mal

ized

Gaussian distribution

Superconductor-insulator transition in highly disordered InO films

Map of the spectral gap

Spatial fluctuations of thecoherence peaks height

Coherence peaks

G, N

orm

aliz

edG

, Nor

mal

ized

( ) ( )

( )

G G eVR

G eV

∆ − > ∆= ×> ∆

Statistical study

G, N

orm

aliz

edG

, Nor

mal

ized


Coherence peaks


Coherence peaks

( ) ( )

( )

G G eVR

G eV

∆ − > ∆= ×> ∆

Statistical study

G, N

orm

aliz

edG

, Nor

mal

ized

Incoherent spectra

( ) ( )

( )

G G eVR

G eV

∆ − > ∆= ×> ∆

Statistical study

Full spectral gap without coherence peaks

Incoherent spectra

Statistical study

( ) ( )

( )

G G eVR

G eV

∆ − > ∆= ×> ∆

G, N

orm

aliz

edG

, Nor

mal

ized

High disorder

Full spectral gap without coherence peaks

resistivity ×××× 2

InO#3Tc ~ 1.7 K

InO#1Tc ~ 1.2 K

Proliferation of spectra without coherence peaks

Increase of disorder

Proliferation of incoherent spectra at the superconductor-insulator transition

8%σ∆

~

16%σ∆

~

B. Sacépé et al., Nat. Phys. (2011)

Superconducting

Insulating

A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) & PRB 65, 014501 (2001)

Numerical calculations: superconductivity with disorder

With increasing disorder:• Superconductivity becomes « granular-like »

• Spectral gap is not the SC order parameter

Signature of localized Cooper pairs

Pseudogap state above Tc

BCS peaks appear along with superconducting phase coherence

Macroscopic quantum phase coherenceprobed 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 coherenceprobed at a local scale

Fractal superconductivity near the mobility edge

Egap = p + BCS

“parity gap”

“BCS gap”

•Tunneling spectroscopy(single-particle DOS)

Egap = p + BCS

“BCS gap”

•Point-contact spectroscopy(Andreev reflection = transfer of pairs)

Tunnel barrier

Transparent interface

Egap = p + BCS

• p “parity gap”: pairing of 2 electrons in localized wave functions

• BCS “BCS gap”: long-range SC order between localized pairs

∆p

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 « 1

T = 300 mK

Z value

Tunnel regime

Contact regime

• Single-particle transfers ~ T• Two-particles transfers ~ T 2

Blonder, G. E., Tinkham, M., and Klapwijk T.M. Phys. Rev. B 25, 7 4515 (1982)


From tunnel to contact regime

Agrait, N. et al., Phys. Rev. B 46, 9 5814 (1992)

Tunnel regime

Contact regime

Control parameter :

Contact resistance Rc = VBias / It


From tunnel to contact in a-InOx

Tunnelregime

Contact regime

T=50mK Rc=65k

Rc=6k

Tunnelregime

Contact regime

Distinct energy scales for pairing and coherence

From tunnel to contact in a-InOx

. Contact : additional peaks at eV ±±±± 200 µeV

. Energy scale BCS independent of Rc

BCS

Egap = p + BCS

Egap

Distinct energy scales for pairing and coherence

Two distinct energy scales in a-InOx

Distinct energy scales for pairing and coherencein disordered a-InOx

Egap(r)= p(r)+ BCS

• BCS probed by AR remains uniform

• Egap probed by STS fluctuates

BC

S

[10

-4eV

]

Egap [10-4 eV]

11 evolutions on 3 samples

BCS=Egap

Conclusion

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

• “Partial” condensation of pairs below TcRectangular spectra at 50mK = localized Cooper pairs

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

• Distinct energy scales for pairing and coherenceSTS measures Egap and Andreev reflection measures BCS

• Interactions : CoulombTc decreases continuously in homogeneously disordered systems

• Inhomogeneous superconducting stateSpatial fluctuations of the gap : a precursor to Cooper pair localization

Superconducting fluctuations, interactions and disorder...

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