Superconducting proximity effect in a diffusive...

A. Cottet and W. BelzigUniversity of Basel

(now: Orsay and Konstanz)

Superconducting proximity effect in a diffusive ferromagnet with spin-active interfaces

S F

EexΔ

Phys. Rev. B 72, 180503R (2005)

Manifestations of the oscillations of the order parameter

S F SI0 sin(ϕ) d

V. V. Ryazanov et al., PRL. 86, 2427–2430 (2001)

I 0 (

µA

)

T. Kontos et al.,PRL 86, 304 (2001)

d/ξF

-+

d (A)

I 0 R

n (µ

V) T. Kontos et al.,

PRL 89, 137007 (2002)

d=75 A

A. Bauer et al., PRL 92, 217001 (2004)

A. Bauer et al., PRL 92, 217001 (2004)

d=75 A Φ0 /2

π

S F S

d

S F Ie-

]ˆ,ˆˆ]ˆ,ˆˆ[

]ˆ,ˆ[2

ˆ ]ˆ,ˆ[ˆ

ˆ2

[

GGDiGGDGiG

GDG

DiGGGGx

GGg

FFFS

FMR

FFSTF

FF

!!

++

++

++="

"

#$

%

{ }

ˆ,ˆ.ˆ

]ˆ,ˆ.[ˆ

GmD

GmD

S

S

3

3

!"

!"

rr

rr

=

=

+

#

!!

"

#

$$

%

&'= ( )*

n

S

n

S

nttGG

*

,,ImQ+

Phase-shifting conductance

Boundary conditions for a diffusive S/F interface

D. Huertas-Hernando et al.

!=

n

nT TGG Q

V. V. Kuprianov et Lukichev (1988)

Magnetoresistance term

Tunnel conductance

1<<nT

F is weakly polarized

Following D. Huertas-Hernando, Y. Nazarov and W. Belzig (2002)

!!

"

#

$$

%

&+= '('() /4Q )(Im

*

,,

*

,,S

n

S

n

n

F

n

F

nn ttrrTGG*

!

!

"#

$%&

'(=

""

#

$

%%

&

'(=

)*

)*)*

n

F

n

F

nMR

n

S

n

S

n

n

F

n

F

n

ttGG

TttrrGG

22

Q

Q 42

,,

*

,,

*

,,/)(Im+

Josephson current in a SFS junction

eQTkgI n

n

BFS /)(

,

!"#

#$=

!

"(#$ ) = arg(i % (1+ #$ ))

#$ =G$ /Gt

F

tt

d

F

RLe

d

G

eI !

"

"

#

#$!

%#

&

++

+

=' 2

0

)1(1

))(sin(

d<<ξF , ⇒

0x

S F

d

L

!" R

t!

Asymmetric case :

S

2

S!

2

S!"L

t!

)()()(

)()()(

xxx

xxx

SFISFS

SFISFS

!!!

!!!

"###

"$$$

+=

+=

L! R!T=0

!

"t

R >> "

t

L

Josephson current in a SFS junctionvery asymmetric case

1 2 3 4 5

-0.2

0.0

0.2

0.4

2 4

-3

0

3

15.0/ =!

!=!=!

TkB

RL

L

t

R

t !!

Fd !/

γφR = -1

R

t

L

tG! "

oeI

Fd !/

γφRγ φR = - 0

.5γφR = +2

0

π0

60 80 100 120 1400

20

40

60

80

!" =

-1.3

!" =

0

#F=36 A

e I0

/G

tR

(µe

V)

d(A)

60 80 100 120 140

0.98

0.99

1.00

!" =

0

d(A)

!" =

-1.6

#F=50 A

N(0

)/N0

0

ta =0.4

+

-+ -+

-

Theoretical interpretation ofProximity effect measurements in PdNi

0

0

0 0

0

0

CONCLUSIONS

A new framework for describing the superconductingproximity effect in diffusive ferromagnets:

The interfaces are characterized by GT and Gφ

Gφ modifies the phase and amplitude of the spatialoscillations of physical signals

New interpretation of proximity effect measurements in PdNi

Identification of experimental signatures of spin-dependentinterfacial phase shifting

exp(-x/ξN)

Superconducting proximity effectat a superconducting/normal interface (S/N)

E

EFk

-k

x

!"##

NSN

N(E)

Tk

D

B

N

h=!

Andreev reflection

D diffusion constant in N

PROGRAM:

1) Proximity effect in diffusive S/F circuits

2) New boundary conditions for describing S/F interfaces

3) Predictions for S/F, S/F/I and S/F/S geometries

4) Interpretation for the proximity effect measurements in Nb/PdNi

5) Conclusions

S F

EexΔ

x

FS

exp(-x/ξF)

N(E)

E

EF

Eex

F

k+Q

-k+Q

ex

FE

Dh=!

Q =Eex

Fvh

Superconducting proximity effectat a superconducting/ferromagnetic interface (S/F)

D diffusion constant in F

!"##

Theoretical description of proximity effect from Gorkov to Usadel equations

Usadel Eqs.

Diffusive limit

]ˆ),(ˆˆ)ˆ[( 3 GriEexn

r!++ "#$

( )GGD rrˆˆ!!

=

),(ˆ nrG !r

vF vF

Tkn Bn )12( += !"

3/lvDF

=

P

r

1/kF

Gorkov Eqs.

BCS description

),,( ErPrr

Γ̂

2/)(

2/)(

21

21

rrR

rrr

rrr

rrr

!=

+=

1x

Eilenberger Eqs.

Quasi-classical limit

),,

(ˆ Erg

r

vF

vF

ξS

1x

Spin degenerate boundary conditionsfor a diffusive S/N interface

Ballistic zoneIsotropisation zoneDiffusive zone

!=

n

nT TGG Q

12

, <<=!" "nn tT ] ˆ,ˆ [ˆ

ˆ2 GGGx

GGg

NSTN

NN =!

!⇒

V. V. Kuprianov et Lukichev (1988)

!!

"

#

$$

%

&=

NS

NS

nn

nn

rt

trS

''

''

'

,,

,,!"

= SS

Tunnel limit:

S S

n

t!,

S

n

r!,

N

n

t!,

N

n

r!,

GNˆ

GSˆ

le ξS(N)le

h/2eG =Q

Barrier described in a scattering approach

N

: NgConductance per unit lenght of N

60 80 100 120 140

0.98

0.99

1.00

d(A)

!F=50 A

N(0

)/N0

60 80 100 120 1400

20

40

60

80

!F=36 A

e I0

/G

tR

(µe

V)

d(A)

+ -

0


0.4

small d

==F

Lt

tg

dGa0

2/1)/( exF EDh=!

0

Data of T. Kontos et al. (2001, 2002)

60 80 100 120 1400

20

40

60

80

!F=36 A

e I0

/G

tR

(µe

V)

d(A)

60 80 100 120 140

0.98

0.99

1.00

d(A)

!F=50 A

N(0

)/N0

+

-

0


Usadel Equations+ spin degenerate boundary conditions of V. V. Kuprianov et Lukichev (1988)

0

+ -

0


2/1)/( exF EDh=!

0

Data of T. Kontos et al. (2001, 2002)

0.4

small d

==F

Lt

tg

dGa0

Spin dependent interfacial scattering

1 2

2

!,nt

1

!,nr

1

!,nt

2

!,nr

)2(1,

,,

)2(1,

,,

!

!!

!

!!

"

"

r

nn

t

nn

i

i

err

ett

1(2)1(2)

1(2)1(2)

=

=

Spin-dependence of the phases !

Following D. Huertas-Hernando, Y. Nazarov and W. Belzig (2002)

θ

V

P1 P2

I

FF

N

Spin dependence of the interfacial diffusion phase shiftsEffects in the F/N/F diffusive case

S. Urazdin et al.,Phys. Rev. B 71, 100401 (2005)

A. Braatas et al., (2001)

! "#$

%&' (= )*

n

N

n

N

nmixrrGG

*

,,1Q

Im(G

mix )= 0

Im(Gmix) increases

Spin accumulationsupressed by interfacial

precession effects for θ≠0, π

N

r

N

rmixG

!"==

,,0 ## if )Im(

Py/Cu/Py

S/F/N ballistic point contact

S

r

S

rD

!"#=

,,$$

$

E. Zhao et al. PRB 70, 134510 (2004)

][2 0 2 !"# # =$ DA

)/( arccos2 != eVA

"

Andreev resonances:

1.0 symbols

05.0 lines

=

=

n

n

T

T

!"=D

!"=D

0=!D

2

A!

A!

NSS

r !" #,-

S

r !" ,

h,-σ

e,σ

e,σ

h,-σ

e,σ

NFS

point contact

Angular parametrisation of the problem

)(n

!"#

: Pairing angle

: Superconducting phase)(n

!"#

Usadel equations in the limit of a weak proximity effect :

2

3

2

2

2

2

2

0

!"

!!

#

!

$$

$

$

$$

$$

xQ

Qk

x F

%

%=

=&&%

% !!"

#$$%

&+=

ex

n

n

Eik

''(( )sgn( 2

with

2/1)/(exFEDh=!

Bulk solutions

0)(

)(

=

=

n

n

!"

#!"

$

$

: F In

2 / : S In

Boundary conditions in F for a S/F interface:

)sin()sin( !!""!

###

SSTF Gx

g $=%

%

( ) !"##!!!!

"$""" iGGx

g SSSTF )cos()sin()cos( +%%=&

&

Description of S/F diffusive hybrid circuits in the limit of weak proximity effect

)arctan(n

S!

"#

=Rigid boundary conditions:

)sgn(1 nik !"" +=Δ Eex ⇒

In this seminar:

Semi-infinite S/F structure

F

xk

n

St

SFe

ikx

!

"#

##

$#%

&%&

'

+=

)sgn(

)sin()(

F

Ftt

g

G !"

##

)(

)( =

0

S

!"

t!

F

x!

-10 -5 0 5 100.0

0.5

1.0

-10 -5 0 5 10-0.5

0.0

0.5

t

SF x

!

" # )0( =

!"!"

!

" )]0(arg[ =#xSF

0=!Q

Experimental parameters for Nb/PdNi structures

0 40 80 1200

50

100

150

200

!(µ"

.cm

)

d (A)

mean free path led<dsat : le ~ d

d<dsat : le constant

S/F/I = Nb/PdNi/Alox/Al

S/F/S = Nb/PdNi/Alox/Al/Nb L

t

R

t !! / 1

Eex ~ d

ΔNb=1.35 meV, ΔAl/Nb=0.6 meV Eex ⇒ )sgn(1 nik !"" +=~10 meV

T=1.5K Tc / 6~

⇒

3 fitting parameters:

satF

satLt

tg

dGa

,

0=

L

!"L

t! F!

2/1

02

)/(

3/

2

exF

eF

F

ED

lvD

DNe

h=

=

=

!

"

!"

#$

%%

F

Ft

t

G )(

)( =

!LG"

d<dsat

d>dsat

000,, Fta !"#

d

a Ft

00! 0

!" 0

F!

satd

d0

!"

dd

a

sat

Ft

00!d

dsatF

0!

0

taNb/Pd : = 0.2

EFEex

dEex !

exE

Experimental parameters for Nb/PdNi structures

0

F!

constraints:

~ 50 A


60 80 100 120 1400

20

40

60

80

!F=36 A

e I0

/G

tR

(µe

V)

d(A)

60 80 100 120 140

0.98

0.99

1.00

d(A)

!F=50 A

N(0

)/N0

0

ta =0.4

+-

0 0


60 80 100 120 1400

20

40

60

80

!" =

0

#F=36 A

e I0

/G

tR

(µe

V)

d(A)

60 80 100 120 140

0.98

0.99

1.00

!" =

0

d(A)

#F=50 A

N(0

)/N0

0

ta =0.4

+

-+

-

0

0 0

0

Superconducting proximity effect in a diffusive...

Documents

Transcript of Superconducting proximity effect in a diffusive...