Localization of preformed Cooper-pairs
in disordered superconductors
Benjamin Sacépé
Institut Néel, CNRS & Université Joseph Fourier, Grenoble
Lorentz Center, Leiden 2011
Localization of preformed Cooper-pairs
in disordered superconductors
Benjamin Sacépé
Institut Néel, CNRS & Université Joseph Fourier, Grenoble
Thomas Dubouchet, Claude Chapelier, Marc Sanquer
CEA Grenoble
Maoz Ovadia, Dan Shahar
Weizmann Institute of Science, Rehovot
M. Feigel’man
L.D. Landau Institut for Theoretical Phyiscs, Moscow
L. Ioffe
Rutgers University, Piscataway
Lorentz Center, Leiden 2011
Superconductor-Insulator Transition (SIT)
Quench condensed Bismuth
Main ingredients :
d=
d=
D.B. Haviland, Y. Lui, A.M. Goldman, PRL (‘89)
1.
Disorder
localization
2.
Attractive pairing
3.
Coulomb interaction
4.
Reduced dimensionality
superconducting phase
competes with pairing
affects 1,2 and 3
Magnetic field-tuned SIT
Amorphous indium oxide
V. F. Gantmakher et al.,
JETP 82, 951 (1996)
G. Sambandamurthy, et al. PRL 94, 017003, (2005)
For similar results in TiN films see :
T. Baturina, et al. PRL (2007)
Magnetic field-tuned SIT
Superconductor
0.20K
Magnetic field-tuned SIT
Superconductor
0.20K
Positive magnetoresistance at low field :
Superconducting correlations in insulators !?
Insulator
Magnetic field-tuned SIT
Superconductor
0.20K
Positive magnetoresistance at low field :
Superconducting correlations in insulators !?
« Insulating correlations » in superconductors ???
Insulator
mK-STM setup : tunneling spectroscopy
C. Chapelier’s setup, CEA Grenoble
50mK< T< 6K
E
Tip
Sample
µL
eV
NL(E)
Cryostat : inverted dilution
 Combined transport and spectroscopy measurements
2Δ
µR
NR(E)
Amorphous Indium Oxyde (a-InOx)
Thickness : 15 nm (blue) and 30 nm (red)
 3D regime
1 mm
InO#2
InO#1
Samples : e-gun evaporation onto Si/SiO2
substrate of high purity In2O3 under O2 pressure
Amorphous Indium Oxyde (a-InOx)
Thickness : 15 nm (blue) and 30 nm (red)
 3D regime
1 mm
InO#2
InO#1
Samples : e-gun evaporation onto Si/SiO2
substrate of high purity In2O3 under O2 pressure
V. F. Gantmakher et al.,
JETP 82, 951 (1996)
Tunneling spectroscopy of amorphous indium oxyde
InO#1
Typical spectrum measured at 50 mK
Fit : s-wave BCS density-of-states
Inhomogeneities of Δ(r)
Map of the spectral gap
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)
Inhomogeneities of Δ(r)
Spectra measured at different locations
G, Normalized
G, Normalized
(T=50mK)
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)
Fluctuations of Δ(r) and superconducting transition
BCS ratio Δ/Tc =1.76 ?
Fluctuations of Δ(r) and superconducting transition
 Definition of Tc : zero-resistance state (macroscopic phase coherence)
6
2 ( r )
 11 !?
Tc
Fluctuations of the BCS peaks
G, Normalized
G, Normalized
Fluctuations of peak heights !
Fluctuations of the BCS peaks
R
G()  G(eV  )
G(eV  )
Fluctuations of the BCS peaks
R
G()  G(eV  )
G(eV  )
Fluctuations of the BCS peaks
R  0 !?
Extreme case : « Insulating » gap
Spectra measured at different locations
(T=50mK)
R  0 !?
Let’s approach the SIT
InO#2
InO#1
G, Normalized
G, Normalized
Sample InO#2 : disorder  2
High disorder
sample
R
G()  G(eV  )
G(eV  )
Let’s approach the SIT
InO#2
InO#1
Increase of disorder


InO#1
Tc ~ 1.7 K
8%
resistivity × 2


16%
InO#2
Tc ~ 1.3 K
 Proliferation of gaps without peaks
Superconductivity and disorder
Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave Superconductor
A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 65, 014501 (2001)
λ
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
Superconductivity and disorder
Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave Superconductor
A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 65, 014501 (2001)
λ
λ
 Insulating gap induced by pairing interaction
Egap ~   loc

3
loc
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)
Insulating 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 65, 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)
Reduced BCS Hamiltonian built on
eigenstates of the Anderson problem
H    j c†j c j 
j
E gap ~

D
 loc
with
2

M jk c†j c†j ck ck 

 jk
M jk   dr  2j k2
In the high-disorder regime when
M jk   jk  dr  4j (r )   jk
In the lowest order:
Egap ~   loc

3
loc
 loc  
1
3
loc
Superconductivity and disorder
Role of Spatial Amplitude Fluctuations in Highly Disordered s-Wave Superconductor
A. Ghosal, M. Randeria, N. Trivedi, PRL 81, 3940, (1998) and PRB 65, 014501 (2001)
λ
λ
 Insulating gap induced by pairing interaction
Egap ~   loc

3
loc
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)
Superconductivity and disorder
Recent QMC simulations
Disorder
K. Bouadim, Y. Loh, M. Randeria, N. Trivedi, arXiv:1011.3275
Insulator
SC
Pairing gap in the insulator
 Proliferation of incoherent –localized— Cooper-pairs
when approaching the SIT
Pairing gap in the insulator
Simulations on the Bethe lattice
Lev 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)
 Proliferation of incoherent –localized— Cooper-pairs
when approaching the SIT
Pseudogap above Tc : preformed pairs
T-dependence of the local DOS
Tc
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 superconductors
Superconducting fluctuations in quasi-2D TiN films ( thickness < 5nm )
B. Sacépé, et al. Nature Commun. 1:140 (2010)
A. Varlamov and V. Dorin, Sov. Phys. JETP 57, 1089, (1983)
Local versus macroscopic phase coherence
BCS peaks appears at Tc
independently of gap inhomogeneities
Tpeak : Temperature below which peaks start to grow
 BCS peaks give a local signature of the superconducting phase coherence
Condensation versus localization of preformed Cooper pairs
Formation of a pseudogap without BCS peaks at T>Tc
Local pairing without phase coherence at T > Tc
Preformed Cooper-pairs
« Insulating » gap at T<<Tc
Local pairing without phase coherence at T << Tc
Spectral signature of
localized Cooper pairs
Conclusions
•
Preformed Cooper-Pairs above Tc
Pseudogap 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 pairs
Gap in the DOS remains & coherence peaks disappear
Localization of preformed Cooper pairs in disordered superconductors
Nature Physics 7, 239 (2011)
Coulomb interaction : Zero-Bias Anomaly
Disorder-enhanced Coulomb interaction
B. Altshuler, et al., Phys. Rev. Lett. 44, 1288, (1980)
Quantum corrections to the DOS
Soft coulomb gap
TiN 1
Superconducting fluctuations

 nT 
l
G i ln  Tc 
We need a global theory !
Quantum corrections to the conductivity
Disorder-enhanced Coulomb interaction
(2D weak-localization/Aronov-Altshuler corrections)
  0 
A
  
ln  
 RQ  T 
Superconducting fluctuations
(2D Aslamasov-Larkin correction …)
Dynamical Coulomb blockade
P. Joyez and D. Estève, PRB 56, 1848, (1997)
Magnetic field-tuned SIT
Amorphous indium oxyde
d = 15 nm
Huge magnetoresistance peak : superconductivity-related ?
Spectral fluctuations
Spectral gap map
Coherence peak map
Disorder-induced inhomogeneities in TiN
Disorder-tuned SIT in ultra-thin films of TiN
Titanium nitride
T. Baturina (Novosibirsk)
V. Vinokur (Argonne National Lab.)
Gaped insulator
made of localized
Cooper pairs ?
B. Sacépé, et al. PRL 101, 157006 (2008)