Disordered Superconductors - Ecole Meso 2016 Matiere ... · PDF fileDisordered Superconductors Cargese 2016. ... la supraconductivité des métaux, des alliages et des films minces

Claude Chapelier,INAC-PHELIQS, CEA-Grenoble

Disordered Superconductors

Cargese 2016

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. Burger

la supraconductivité des métaux, des alliages et des

films minces (Ed.Masson)

BCS theory for clean systems

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)

Theory of dirty superconductors

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)

In superconducting grains, superconductivity disappears when the mean level spacing between different electronic states

becomes greater than the superconducting gap

𝝂 𝚫 𝑳𝟑 > 𝟏Anderson criterion for superconductivity :

Localization in disordered metals

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

P.A. Lee and T.V. Ramakrishnan

Disordered electronic systems

Rev. Mod. Phys. 57, 287(1985)





Rev. Mod. Phys. 57, 287(1985)


Are Cooper pairs getting localized ?




Rev. Mod. Phys. 57, 287(1985)



Leggett’s argument : every Bose system is superfluid at T=0A.J. Legett, Topics in the theory of Helium

Physica Fennica 8, 125 (1973)




Rev. Mod. Phys. 57, 287(1985)








Rev. Mod. Phys. 57, 287(1985)





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

d =

d =




Rev. Mod. Phys. 57, 287(1985)






d =

d =

Strong localization, fluctuations and Coulomb interactions


Localization in disordered superconductors

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)

3D localization

MetalInsulator

Inhomogeneous

superconducting state

𝑘𝐹 𝑙 ≪ 1 ν Δ ξloc3 ~ 1 𝑘𝐹 𝑙 ~ 1 𝑘𝐹 𝑙 ≫ 1








3D localization 2D numerical simulation

0 ,

, , ,

. .i j i i

i j i

H t c c h c V n

int i ii

H n n

MetalInsulator

Inhomogeneous










3D localization

0 ,

, , ,

. .i j i i

i j i

H t c c h c V n

int i ii

H n n

MetalInsulator

Inhomogeneous



2D numerical simulation








3D localization 2D Monte Carlo calculation

0 ,

, , ,

. .i j i i

i j i

H t c c h c V n

int i ii

H n n

MetalInsulator

Inhomogeneous



Cooper pairing beyond the mobility edge leads to an inhomogeneous superconductor

The transition to an insulator requires quantum fluctuations

Superconducting fluctuations

Thermal fluctuations ψop = ∆ 𝑻 𝒆𝒊𝝋 𝑻

A. Larkin and A. Varlamov,

Theory of fluctuations in superconductors,

Oxford University Press (2006)


Thermal fluctuations

Amplitude fluctuationsT > Tc

ξ(T)

τ =π ђ

8 kB(T − Tc)

ψop = ∆ 𝑻 𝒆𝒊𝝋 𝑻







ξ(T)

τ =π ђ

8 kB(T − Tc)

T >∼ Tc Amplitude and phase fluctuations

ξ(T)








ξ(T)

2D : Berezinskii – Kosterlitz - Thouless

τ =π ђ

8 kB(T − Tc)


ξ(T)







Amplitude fluctuations

δTcTc

~ Gi(d) ∼1

ν 𝑘𝐵Tc 𝜉0𝑑

24−d

ν Δ ξloc3 ~ 1

T > Tc

ξ(T)

ξ0~ξloc

Gi(3)∼ 80𝑘𝐵𝑇𝑐EF

4

~ 10−12 − 10−14

Gi(2) ∼𝑒2

23 ђ𝑅⧠

Clean 3D superconductor :

Dirty 2D superconductor :

Gi~1


τ =π ђ

8 kB(T − Tc)


ξ(T)


3D localized superconductor :






Amplitude fluctuations

δTcTc

~ Gi(d) ∼1

ν 𝑘𝐵Tc 𝜉0𝑑

24−d

T > Tc

ξ(T)

Gi(3)∼ 80𝑘𝐵𝑇𝑐EF

4

~ 10−12 − 10−14

Gi(2) ∼𝑒2

23 ђ𝑅⧠

Clean 3D superconductor :

Dirty 2D superconductor :

Disorder drastically enhances thermal fluctuations


τ =π ђ

8 kB(T − Tc)

ν Δ ξloc3 ~ 1 ξ0~ξloc Gi~1


ξ(T)

3D localized superconductor :





TIN Superconductor-Insulator transition

T. I. Baturina, et al.PRL 99, 257003 (2007)Sacépé et al., PRL 101, 157006 (2008)

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

TiN


Sacépé et al., PRL 101, 157006 (2008)

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


Sacépé et al., PRL 101, 157006 (2008)


Sacépé et al., PRL 101, 157006 (2008)


Quantum fluctuations

K.B. Efetov,

Phase transition in granulated superconductors,

JETP 51, 1016 (1980)

K.A. Matveev and A.I. Larkin

Parity effect in ground state energies of ultrasmall superconducting grains,

Phys. Rev. Lett. 78, 3749(1997)

ψop = ∆ 𝒆𝒊𝝋



∆N ∆φ >1

2

K.B. Efetov,


JETP 51, 1016 (1980)



Phys. Rev. Lett. 78, 3749(1997)

N,𝜑

D

EJ =h π Δ

8 e2 Rtth(

Δ

2T)

Ec = e2

4 π ϵ0D

δ =1

ν 𝐷3


δ =1

ν 𝜉𝑙𝑜𝑐3



∆N ∆φ >1

2

K.B. Efetov,


JETP 51, 1016 (1980)



Phys. Rev. Lett. 78, 3749(1997)

N,𝜑

D

EJ =h π Δ

8 e2 Rtth(

Δ

2T)

Ec = e2

4 π ϵ0D

Gallium

H. M. Jaeger, et al.

Phys.Rev.B 34, 4920 (1986)


δ =1

ν 𝐷3

δ =1




∆N ∆φ >1

2

K.B. Efetov,


JETP 51, 1016 (1980)



Phys. Rev. Lett. 78, 3749(1997)

N,𝜑

D

EJ =h π Δ

8 e2 Rtth(

Δ

2T)

Ec = e2

4 π ϵ0D

Gallium


Phys.Rev.B 34, 4920 (1986)

2D Monte Carlo calculation

0 ,

, , ,

. .i j i i

i j i

H t c c h c V n

int i ii

H n n



Phys. Rev. Lett. 78, 3749(1997)


δ =1

ν 𝐷3

δ =1




∆N ∆φ >1

2

K.B. Efetov,


JETP 51, 1016 (1980)



Phys. Rev. Lett. 78, 3749(1997)

N,𝜑

D

EJ =h π Δ

8 e2 Rtth(

Δ

2T)

Ec = e2

4 π ϵ0D

Gallium


Phys.Rev.B 34, 4920 (1986)

2D Monte Carlo calculation

0 ,

, , ,

. .i j i i

i j i

H t c c h c V n

int i ii

H n n

Δp = δ

2 lnδ

Δ



Phys. Rev. Lett. 78, 3749(1997)

∆ = 1.76 kB Tc

Eg = ∆ + ∆p

Parity gap


δ =1

ν 𝐷3

δ =1


Coulomb depairing in disordered superconductors

A.M Finkelstein

Pis’sma Zh. Esk. Theor. Fiz., 45, 46 (1987)


A.M Finkelstein

Pis’sma Zh. Esk. Theor. Fiz., 45, 46 (1987)R. A. Smith, M.Y. Reizer, and J. W. WIlkins

Phys. Rev. B 51, 6470(1995)


A.M Finkelstein

Pis’sma Zh. Esk. Theor. Fiz., 45, 46 (1987)R. A. Smith, M.Y. Reizer, and J. W. WIlkins

Phys. Rev. B 51, 6470(1995)

Short range Coulomb interaction continuously decreases Tc and Δ in the same proportion

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

Gallium

Granular films


d =

d =

Amorphous films

Continuous decrease of Tc

Cooper pairing suppressed at the SIT

Competition between EC

and EJ

Cooper pairs localized in grains

Superconductor-insulator transition : two scenarios

Bismuth


Frydman, A., Physica C : Superconductivity 391, 189 (2003) Hsu, S.-Y., and Valles, J. M. Phys. Rev. B 48, 4164 (1993)

Continuous decrease of Tc

Cooper pairing suppressed at the SIT

SIT due to phase fluctuations

Cooper pairs localized in grains

Granular filmsAmorphous films

Superconductor-insulator transition : two scenarios


Sacépé et al., PRL 101, 157006 (2008)

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

TiN


Sacépé et al., PRL 101, 157006 (2008)

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

TiN


-1,0 -0,5 0,0 0,5 1,00,0

0,5

1,0

1,5

2,0

G(V

), n

orm

ali

ze

d

V [mV]

= 260 µeV

Teff

= 0,25 K

-1,0 -0,5 0,0 0,5 1,00,0

0,5

1,0

1,5

2,0

= 225 µeV

Teff

= 0,32 K

G(V

), n

orm

ali

ze

d

V [mV]

-1,0 -0,5 0,0 0,5 1,00,0

0,5

1,0

1,5

2,0

= 154 µeV

Teff

= 0,35 K

G(V

), n

orm

ali

ze

d

V [mV]

Increasing disorder Sacépé et al., PRL 101, 157006 (2008)

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

TiN

-1,0 -0,5 0,0 0,5 1,00,0

0,5

1,0

1,5

2,0

G(V

), n

orm

ali

ze

d

V [mV]

= 260 µeV

Teff

= 0,25 K

-1,0 -0,5 0,0 0,5 1,00,0

0,5

1,0

1,5

2,0

= 225 µeV

Teff

= 0,32 K

G(V

), n

orm

ali

ze

d

V [mV]

-1,0 -0,5 0,0 0,5 1,00,0

0,5

1,0

1,5

2,0

= 154 µeV

Teff

= 0,35 K

G(V

), n

orm

ali

ze

d

V [mV]



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

TiN

-1,0 -0,5 0,0 0,5 1,00,0

0,5

1,0

1,5

2,0

G(V

), n

orm

ali

ze

d

V [mV]

= 260 µeV

Teff

= 0,25 K

-1,0 -0,5 0,0 0,5 1,00,0

0,5

1,0

1,5

2,0

= 225 µeV

Teff

= 0,32 K

G(V

), n

orm

ali

ze

d

V [mV]

-1,0 -0,5 0,0 0,5 1,00,0

0,5

1,0

1,5

2,0

= 154 µeV

Teff

= 0,35 K

G(V

), n

orm

ali

ze

d

V [mV]



Tc [K] Δ/Tc

4.7 1.8

1.3 2.3

1 2.6

0.45 4

TiN


Tc [K] Δ/Tc

4.7 1.8

1.3 2.3

1 2.6

0.45 4

Sacépé et al., PRL 101, 157006 (2008)


Tc [K] Δ/Tc

4.7 1.8

1.3 2.3

1 2.6

0.45 4

Sacépé et al., PRL 101, 157006 (2008)

Spatial fluctuations of Tc

M. A. Skvortsov and M. V. Feigel’man, Phys. Rev. Lett. 95, 057002, (2005)

Superconductivity and Coulomb interaction

Spatial fluctuations of Tc

M. A. Skvortsov and M. V. Feigel’man, Phys. Rev. Lett. 95, 057002, (2005)

Superconductivity and Coulomb interaction

M.V. Feigelman and M.A. Skvortsov, Phys. Rev. Lett. 109, 147002 (2012)

A.I. Larkin and Yu. N. Ovchinnikov, Sov. JETP 34, 1144 (1972)

W. Escoffier, et al., PRL 93, 217005, (2004)

Thermal dependence of the Density of States

W. Escoffier, et al., PRL 93, 217005, (2004)


B. Sacépé et al., Nat. Comm., (2010)


Pseudogap above Tc



Pseudogap above Tc



Pseudogap above Tc

Pseudogap is due to pre-formed Cooper pairs

InO#1

InO#2

D. Shahar and Z. Ovadyahu, Phys. Rev. B 46, 10917 (1992)

InOx Superconductor-Insulator transition

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

InOx

InO#1

InO#2

D. Shahar and Z. Ovadyahu, Phys. Rev. B 46, 10917 (1992)

InOx Superconductor-Insulator transition

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

InOx


Pseudogap above TcInOx

B. Sacépé et al., Nat. Phys. (2011)


Pseudogap above TcInOx

B. Sacépé et al., Nat. Phys. (2011)

Spectra without coherence peaks are the signature of localized pre-formed Cooper pairs

Eg = ∆ + ∆p

Parity gap

Point-Contact Andreev Spectroscopy

How to measure the order parameter ?

Normal

metal Superconductor

T =1

1 + Z2

Eg = ∆ + ∆p

Parity gap



Eg = ∆ + ∆p

Parity gap



Normal


T =1

1 + Z2

T = 300

mK

Z valueZ » 1

Z « 1

Tunnel regime

Contact regime

Z ~ 1

Blonder, G. E., Tinkham, M., and Klapwijk T.M.

Phys. Rev. B 25, 7 4515 (1982)

Eg = ∆ + ∆p

Parity gap



Normal


T =1

1 + Z2

Sample

Tip

Sample

Tip

Sample

Tip

T = 300

mK

Z valueZ » 1

Z « 1

Tunnel regime

Contact regime

Z ~ 1


Phys. Rev. B 25, 7 4515 (1982)

Eg = ∆ + ∆p

Parity gap



Normal


T =1

1 + Z2

Sample

Tip

Sample

Tip

Sample

Tip

Eg = ∆ + ∆p

Eg = ∆ + ∆p

T = 300

mK

Z valueZ » 1

Z « 1

Tunnel regime

Contact regime

Z ~ 1


Phys. Rev. B 25, 7 4515 (1982)


InOx film far from the Superconductor-Insulator Transition : Tc = 3.5K

InOx

• Homogeneous

• No pseudogap

Eg = ∆ + ∆p


Sample

Tip

Sample

Tip

Sample

Tip

Eg = ∆ + ∆p

Eg = ∆ + ∆p

T = 300

mK

Z valueZ » 1

Z « 1

Tunnel regime

Contact regime

Z ~ 1

Close to the superconductor-insulator transition

Eg = ∆ + ∆p


Sample

Tip

Sample

Tip

Sample

Tip

Eg = ∆ + ∆p

Eg = ∆ + ∆p

T = 300

mK

Z valueZ » 1

Z « 1

Tunnel regime

Contact regime

Z ~ 1

Close to the superconductor-insulator transition

∆

Eg = ∆ + ∆p

Superconductivity , disorder, Coulomb interaction and localization

Disorder & Localization :

Tc decreases faster than with disorder : huge /Tc ratio Parity gap Strong spatial fluctuations of

Localized Cooper pairs characterized by spectra without coherence peaks

Disorder & Coulomb interaction :

Continuous decrease of Tc and with disorder Keeps /Tc ratio constant Spatial mesoscopic fluctuations of Tc

Disorder :

Stong superconducting fluctuations above Tc Pseudogap due to preformed Cooper pairs

Magnetic field studies through the SIT

G. Sambandamurthy et al., Phys. Rev. Lett. 92, 107005, (2004)

G. Kopnov et al., Phys. Rev. Lett. 109, 167002, (2012)

InOx




Stewart, Jr. et al. , Science 318, 1273, (2007)

H.Q. Nguyen et al., Phys. Rev. Lett. 103, 157001 (2009)

ϕ =h

2e

InOx Bi




Stewart, Jr. et al. , Science 318, 1273, (2007)

H.Q. Nguyen et al., Phys. Rev. Lett. 103, 157001 (2009)

ϕ =h

2e

Cooper pair insulator ?

InOx Bi

D. Sherman et al., Phys. Rev. Lett. 108, 177006, (2012)


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Disordered Superconductors - Ecole Meso 2016 Matiere ... · PDF fileDisordered Superconductors Cargese 2016. ... la supraconductivité des métaux, des alliages et des films minces