superconducting fluctuations, interactions and disorder...
TRANSCRIPT
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)
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
Superconductor-insulator transition in highly disordered TiN films
B. Sacépé et al., Phys. Rev. Lett 101, 157006 (2008)
B. Sacépé et al., Phys. Rev. Lett 101, 157006 (2008)
STM Spectroscopy
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
Superconductor-insulator transition in highly disordered TiN films
B. Sacépé et al., Phys. Rev. Lett 101, 157006 (2008)
B. Sacépé et al., Phys. Rev. Lett 101, 157006 (2008)
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)
Superconductor-insulator transition in highly disordered TiN films
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
superconducting fluctuations in highly disordered TiN films
W. Escoffier et al. , Phys. Rev. Lett. 93, 217005, (2004)
B. Sacépé et al., Nat. Comm., (2010)
superconducting fluctuations in highly disordered TiN films
B. Sacépé et al., Nat. Comm., (2010)
superconducting fluctuations in highly disordered TiN films
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
Spatial fluctuations of thecoherence peaks height
Coherence peaks
Spatial fluctuations of thecoherence peaks height
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)
Point-Contact Andreev Spectroscopy
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
Point-Contact Andreev Spectroscopy
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