Post on 19-Sep-2020
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+
]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
Theoretical interpretation ofProximity effect measurements in PdNi
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
Theoretical interpretation ofProximity effect measurements in PdNi
Usadel Equations+ spin degenerate boundary conditions of V. V. Kuprianov et Lukichev (1988)
0
+ -
0
Theoretical interpretation ofProximity effect measurements in PdNi
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
Theoretical interpretation ofProximity effect measurements in PdNi
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
Theoretical interpretation ofProximity effect measurements in PdNi
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