Analysis of proximity effects
in S/N/F and F/S/F junctions
Han-Yong Choi
Na-Young Lee / SKKU
Hyeonjin Doh / Toronto
Kookrin Char / SNU
KIAS workshop
2005. 10. 25 ~ 10. 29.
Superconductivity (S) vs.
Ferromagnetism (F)
SKKU condensed-matter theory group
Proximity effect
9
8
N
F
7
Tc (K)
S
6
Nb
Nb/CoFe(10nm)
Nb/Ni(10nm)
Nb/CuNi(10nm)
5
4
0
Tc vs. d N or d F .
10
20
30
40
50
60
dNb (nm)
d S ~ S .
SKKU condensed-matter theory group
Plan
I. Introduction to proximity effect.
S/N, S/F.
II. S/N/F.
Issues of SNU data.
III. Usadel equation.
Odd triplet pairing.
Results.
IV. F/S/F.
V. Summary and outlook.
SKKU condensed-matter theory group
I. Introduction
S/N bilayers: 1960’s. [de Gennes, Rev. Mod. Phys. (’64)]
Tc/Tc,S
dPb (nm)
S
N
Y
Cu ~ 40 nm dCu (nm)
[Werthamer, Phys.Rev. (’63)]
For T  Tc ,
N 
DN
DS
1
, S 
, D  vF .
2T
2T
3
SKKU condensed-matter theory group
S/F bilayers: 1980’s & 90’s
Re{Y}
8
F
0-state
-state
Min Tc vs. dF
f=14.4nm, f=14.4cm,
7
Tc (K)
S
fit results
S=8.3nm, S=14.6cm,
TCurie=1152K
-11
b=0.28, RbA=0.6 x 10 cm
6
5
4
2
Nb(26nm)/CoFe
0
2
4
6
8
10
dCoFe (nm)
SKKU condensed-matter theory group
Origin of oscillations
S
S
F

k ,
F
U
 
k  q / 2, 
K
2h
h


 k ,
x
 
 k  q / 2, 
ix

sin( x /  m 0 )
h
 k  q / 2   k  q / 2  2h, q 
, Y  c  d cos  e k F cos 
.


k F cos 
x /  m0
h

Y  c  d cos  e

x
 cos

ix
e
h
k F cos
  x sin( x /  m 0 ) 
~ e
 ,  m 0  vF / h.
x /  m0 

 e (1i ) x /  m 3i / 8 
 ,  m   m 0 .
 Re 
2
 x m /  
dirty limit (oscillation suppressed).
SKKU condensed-matter theory group
II. S/N/F trilayers

Expectations: only one length scale in N.
S
SN
N
F
Y
Tc
SNF
0

dN
Experiments: “surprises”
two more length scales.
SKKU condensed-matter theory group
1. Short length
8
Nb(23nm)/Au
Nb(23nm)/Au/CoFe(10nm)
6
7 Nb(26nm)/Au/CoFe(10nm)
Tc (K)
Tc (K)
7
5
6
5
4
Nb(22nm)/Au/CoFe(10nm)
0
1
2
3
4
5
dAu (nm)
0
100
dAu (nm)
200
SKKU condensed-matter theory group
2. Intermediate length
Nb(23nm)/Au/CoFe(10nm)
0.0
-0.1
Tc (K)
Tc - Tc,lim (K)
0.0
-0.2
-0.1
20
-0.3
Nb(16nm)
Nb(17nm)
Nb(18nm)
40
60
80
100 120
dAu (nm)
0
100
dAu (nm)
200
SKKU condensed-matter theory group
Au & Cu
8
Nb24.3nm/Au
Nb24.3nm/Au/CoFe10nm
Tc (K)
7
6
Nb24.3nm/Cu
Nb24.3nm/Cu/CoFe10nm
5
4
0
100
200
dAu (nm)
SKKU condensed-matter theory group
Another way of looking at
the short length
superconductor
normal metal
ferromagnetic metal

dF = 10 nm
dF = 10 nm
dF = 10 nm
dN = 3 nm
dS = 23 nm
dS = 26 nm
dS = 23 nm
Which has the highest Tc?
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How to understand?


1. Obvious/mundane explanation.
Bad interfaces. higher interface resistance higher Tc.
But, interface resistance bet metals are similar.
Oscillations in Tc vs. dF.
2. More exotic explanation.
From new physics like triplet pairing?
Inhomogeneous exchange fields are predicted to induce
enhanced superconductivity by spin triplet excitations.
[Rusanov et al, PRL (2004), Bergeret et al, PRL (2001), …].
SKKU condensed-matter theory group
Nb/Au/Co60Fe40
1.00
7.6
Nb(24nm)/Au(10nm)/CoFe(d nm)
Nb(24nm)/Au/CoFe(d nm)
7.4
Tc (K)
7.2
Tc / Tc(dCoFe=0)
Au = 5 nm
Au = 10 nm
Au = 30 nm
7.0
6.8
6.6
6.4
0
1
2
3
4
5
dCoFe (nm)
6
7
8
9
bNF=0.5
0.95
0.90
0
1
2
3
4
5
6
7
8
dCoFe (nm)
SKKU condensed-matter theory group
Two options to understand the
short length scale (~ 2 nm)
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Triplet?
8
Nb(23nm)/Au
Nb(23nm)/Au/CoFe(10nm)
6
7 Nb(26nm)/Au/CoFe(10nm)
Tc (K)
Tc (K)
7
5
6
5
4
Nb(22nm)/Au/CoFe(10nm)
0
1
2
3
4
5
dAu (nm)
0
100
200
dAu (nm)
SKKU condensed-matter theory group
III. Usadel formalism
 


F ( r1 , r2 , t1  t2 )    ( r1 , t1 ) ( r2 , t2 ) .

 
 
 
F ( r1 , r2 , t )  F ( R, r , t )  F ( R, k , in )  f ( x, in )  
F ( x, k , in ).

z
k
Usadel equation
 

2 
ˆ
 Tc 2 F ( x, i )   F  ( x)  i sgn(  ) HF ,
x
2
 
  f tx  if ty
ˆ
F  f s  f   i y  
  f s  f tz


f s  f tz 

f tx  if ty 
O
1
 f  f  , f tz  1  f   f  , ftx   1  f   f  , f ty 
2
2
2
 f s ( x, i ) 
 0
 ( x ) 







f
(
x
,
i

)
D
 tz

 0  ˆ   hz
2
 
, F ( x, i )  
, ( x )  
, H 
f tx ( x, i ) 
h
0 
2Tc


 x


 f ( x, i ) 
 h
 0 
 ty

 y
fs 
1
 f  f .
2i 
 hz hx h y 

0
0 0
.
0
0 0

0
0 0 
x
SNF
SKKU condensed-matter theory group
Boundary conditions
Self-consistency relation
z
 ( x)

 Tc 0 

( x) ln    2T  
 f s ( x, in ) .
T 
n  0  n


Boundary modeled by

Boundary conditions.
V ( x )  V0  Vx x  V y y  Vz z  ( x ).
x
 
 
FS ( x, i )   N
FN ( x, i )  0,
x
x


 
SN
ˆ
2. FS  FN    N
FN .
x
  bNF
0 i mNF
0 


NF
b
0
0 
 0
ˆ SN   bSN , ˆ NF   NF
NF
.
i m
0

0
b


NF 
 0
0
0
b 

1.  S
O
S N F
SKKU condensed-matter theory group
Odd triplet pairing?
 


F ( r1 , r2 , t1  t2 )    ( r1 , t1 ) ( r2 , t2 ) .

 
 
 
F ( r1 , r2 , t )  F ( R, r , t )  F ( R, k , in )  f ( x, in )  
F ( x, k , in ).

k
 Antisymmetry
requirement (at t1=t2): F changes sign


under r1,  r2 ,  .
For    , F ( x, r ,  0)   F ( x,r ,  0).  F ( x, r  0,  0)  0.
F ( x, r  0,  0) 
1


1
F
(
x
,
k
,
i

)

 
n
n
k
f



( x, in )  0.
n
f  ( x,in )   f  ( x, in ). Odd frequency triplet pairing.
SKKU condensed-matter theory group
Solution: by extending the Green’s
function method of Fominov et al, PRB 2002.
SKKU condensed-matter theory group
Solution



The basic idea is to solve the homogeneous equations
with appropriate boundary conditions to obtain a single
equation for the singlet pairing component f s ( x, i ) ,
and the boundary conditions in terms of

f s ( x, i ) and f s ( x, i )
x
within the S region. 0  x  d S .
The obtained differential equation is then solved by
constructing Green’s function following standard
procedure, say, in Arfken.
SKKU condensed-matter theory group
Triplet pairing in S/N/F




S = conventional s-wave singlet superconductor.
Tc determined by the singlet pairing component.
Triplet pairing components are induced in addition to
the singlet component (via spin-flip scatterings).
Triplet components are s-wave (even in k), and odd in
frequency. Long length scale.
Triplet components change Tc indirectly by changing
singlet component via boundary conditions.
SKKU condensed-matter theory group
Procedures for understanding
Tc vs. dN of Nb/Au/CoFe.
 

2 
ˆ
 Tc 2 F ( x, i )   F  ( x)  i sgn(  ) HF ,
x
2
Parameters of Usadel equation:
i ,  i , (for i = S, N, F), Tc0.
hex,  bSN ,  bNF ,  mNF . (interface)
1. Fit S/F (Nb/CoFe): hex, Tc0.
2. Fit S/N (Nb/Au):  bSN .
3. Fit S/N/F (Nb/Au/CoFe) to determine  bNF ,  mNF .
SKKU condensed-matter theory group
Nb/CoFe
SF
From S/F,  b  0.34.
SKKU condensed-matter theory group
Nb/Au
8.0
Nb~7.0nm, Nb=15.2cm
Au~85nm, Au=2.3cm,
Tc (K)
7.5
-11
2
b~1.15, RbA~2.24 x 10 cm
7.0
Nb(23nm)/Au
6.5
0
100
200
dAu (nm)
SN
From S/N,  b  1.15.
SKKU condensed-matter theory group
Quantitative analysis S/N/F
From S/N/F,  bSN  1.15,  bNF  0.4.
NF
No need to introduce  m .
SKKU condensed-matter theory group
Usadel calculations.

By solving the Usadel equation, lim S/N/F  S/F
d N 0
because S/N/F still has two interfaces
(mathematically) in the limit dN 0.

Short length scale of ~ 2-3 nm:
The length scale over which electrons feel the interface.
Not the physical material length.
SKKU condensed-matter theory group
Pairing amplitudes
F
N
S
SKKU condensed-matter theory group
Triplet components
F
N
S
SKKU condensed-matter theory group
2. Intermediate length
Nb(23nm)/Au/CoFe(10nm)
0.0
-0.1
Tc (K)
Tc - Tc,lim (K)
0.0
-0.2
-0.1
20
-0.3
Nb(16nm)
Nb(17nm)
Nb(18nm)
40
60
80
100 120
dAu (nm)
0
100
dAu (nm)
200
Could never match the experimental observations of
more than one length scales.
Intermediate length not understood.
SKKU condensed-matter theory group
Yamazaki et al.: Nb/Au/Fe (MBE)
Length scale of 2.1 nm.
SKKU condensed-matter theory group
Nb/Au/Co60Fe40
1.00
7.6
Nb(24nm)/Au(10nm)/CoFe(d nm)
Nb(24nm)/Au/CoFe(d nm)
7.4
Tc (K)
7.2
Tc / Tc(dCoFe=0)
Au = 5 nm
Au = 10 nm
Au = 30 nm
7.0
6.8
6.6
6.4
0
1
2
3
4
5
dCoFe (nm)
6
7
8
9
bNF=0.5
0.95
0.90
0
1
2
3
4
5
6
7
8
dCoFe (nm)
SKKU condensed-matter theory group
Results for S/N/F





It seems that it is the interface resistance  b
that caused the Tc jump (short length scale) on Tc vs. dN
for Nb/Au/CoFe.
S/F :  bSF  0.34.
S/N/F :  bSN  1.15,  bNF  0.4.
 bSF   bSN NF   bNF for continuity.
Intermediate length of ~ 20 nm not understood.
Oscillations in Tc vs. dF not understood.
SKKU condensed-matter theory group
IV. F/S/F

Parallel & antiparallel
F S
F
F S
F
TcP  TcAP because the F effect is canceled in
antiparallel junctions.
Proximity switch
device.
SKKU condensed-matter theory group
Gu et al., PRL 2002
You et al., PRB 2004


TcAP  TcP is much smaller in experiment
compared with theoretical calculation.
Why?
SKKU condensed-matter theory group
Why?


Two F’s are not identical.
Triplet components (induced by spin flip scatterings at
S/F interfaces).
SKKU condensed-matter theory group
Triplet pairing components.


Tunneling conductance for FSF.
Effects of triplet pairing components.
F
F
S
SKKU condensed-matter theory group
Nb/SrRuO3
S
F
S
M
F
M
M
M
SKKU condensed-matter theory group
V. Summary & Outlook





No need for triplet pairing components for
Nb/Au/CoFe.
It is the interface resistance that caused the Tc
jump. Short length scale of ~ 2 nm: the length
scale over which electrons feel the interface.
Not the physical material length.
Not understood: intermediate length of ~ 20 nm,
Tc vs. dF of S/N/F.
Tc difference between parallel and antiparallel F’s
of F/S/F is reduced by triplet components.
Search for the odd-frequency triplet pairing in
artificial junctions of S, N, and F.
SKKU condensed-matter theory group