Transcript Theory of self-induced ferromagnetic Josephson resonance

Nagoya University on September 5,2009
Ferromagnetic Josephson Junction
and Spin Wave Resonance
Sadamichi Maekawa
(IMR, Tohoku University)
Co-workers:
S. Hikino, M. Mori, S. Takahasi (IMR, Tohoku University)
I. Petkovic, M. Aprili (Univeriste Paris-Sud)
S. E. Barnes (University of Miami)
Reference:
I. Petkovic, M. Aprili, S.E.Barnes, F.Beuneu and S.Maekawa: to be published.
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Outline
1. Superconducting phase difference
Josephson effect, Phase difference coupled with magnetic field,
Resistively shunted junction (RSJ) model, I-V characteristic,
Ferromagnetic Josephson junction (SC/FM/SC junction)
2. Magnetization dynamics in ferromagnet (FM)
Ferromagnetic resonance (FMR)
3. Coupled superconducting phase and magnetization dynamics
Ferromagnetic Josephson junction
Differential resistance
4. Model
RSJ model + Maxwell’s equation + LLG equation
5. Differential resistance
FMR signal
2
6. Summary
DC Josephson Effect
qL
  x
B.D.Josephson Phys. Lett. 1, 251 (1962)
Phase difference between superconductors
qR
Current flow without voltage drop
Cooper pair
SC
d
I or NM
dc Josephson effect
x
SC
Josephson current
I; Insulator
NM; Normal metal
Current-Voltage characteristic
IJ = Ic sin(q )
Thickness dependence of Ic
Ic
I
Penetration depth of order
parameter to barrier
I ; I  Several tens
Josephson critical current
NM ;
Ic
0
V
0
A
 NM  A few  m
d
3
Properties of superconducting phase difference
・Gauge invariant phase
qL
y
x
z
a
Magnetic field B
B0
qR
b
q  y   
Vector potential； Ax   By
2 b

0 a
dr  A  r 
・Josephson current density
iJ  ic sin q  y 
0:Magnetic flux
quantum
IJ
B0

Fraunhofer pattern
B : Flux
4
B.D.Josephson Phys. Lett. 1, 251 (1962)
Dynamics of SC phase difference
W. C. Stewart, APL. 12, 277 (1968)
D. E. McCumber, JAP. 39, 3113 (1968)
Dynamics of superconducting phase
Current equation
qL
R
qR
I c sin q
RSJ model
Cooper pair
current
SCI X Josephson
SC
SC : superconductor
I  I c sin q 
I
V
R
dq 2e
 V
dt
Phase dynamics
V
・Phase difference
q  J t  q0
J 
2e
V :Josephson frequency
・AC Josephson current
DC voltage
AC current
IJ
I J  Ic sin J t  q0 
t
5
Josephson Effect in ferromagnetic Josephson junction (FJJ)
q  q R  q L :Phase difference between SC’s
E
2hex

NM
SC
 kF 
 FM
・Josephson current
k
SC
hex
h
FM kF  ex
vF
vF
ei ( k  hex /
vF ) x
ei (  k  hex /
vF ) x
 cos  hex x / vF 
IJ I=J =Ic Isin(
q q) )
c sin(
 c.c
Current-phase relation
Ic
0-state
SC/NM/SC junction
-state
0
d
q
FJJ junction
6
A. I. Budzin, Rev. Mod. Phys. 76, 411 (2004)
I+
Nb
Pd1-xNix
S
F
Josephson Coupling
Nb
S
Kontos et al. PRL 89, 137007 (2002)
IV-
dF1
V+
dF2
80
20
experiment at 1.5K
60
I (mA)
10
IcRn(µV)
x10
SIS
0
-10
-20
-3
SIFS
-2
-1
0
1
2
theory
0 state
40
R
Interface = 10
F = 46 Å
20
-6
W
 state
3
V (mV)
I-V characteristics
0
40
60
80
100
120
dF (Å)
140
160
180
7
Diffraction Pattern
140
120
I c ( A
100
80
60
40
+
-
20
0
/o
dF (Å)
I=-IcsinDf
-junction
60
50
I c ( A
I=IcsinDf
0-junction
40
30
20
10
0
/o
8
SC/F/SC junction
S/N/S
F
S
S/F/S junction
Ryazanov et al., PRL 86, 2427 (2001)
π-state
0-state
θ
S
F
θ
θ
S
S
θ＋π
F
S
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Magnetization dynamics in FM
Ferromagnetic resonance (FMR)
Ferromagnetic thin film
z
y
x
DC magnetic field
RF-magnetic field
M : Magnetization
Landau-Lifshitz-Gilbert (LLG) equation
dM
 
dM 
  M  H eff 
M

dt
M 
dt 
M ; Total Magnetization,  ;Gyromagnetic ratio
Heff ; Effective field,  ;Gilbert damping
Heff
M
10
SC
FM
Magnetization
dynamics
Phase dynamics
q t
Mt
Coupling
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Model
x
RSJ model : FJJ
SC
V
i  ic sin q  r , t  
R
i ; Current density
M
FM
ic ; Critical current density
d
y
Josephson current density
iJ  ic sin q  r , t  ex
SC
M is parallel to x axis.
Phase difference
q  r , t   J t 
J 
2eV
z
L
2
dxAx  r , t 
0 d / 2

d /2
d is thickness of FM
L is width of junction
Ax  r , t   By z  Bz y
Vector potential coupled with M dynamics
x
Josephson frequency
M
Bz
y
By
z
Bi : Dynamical flux density
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i=y,z
Procedure of calculation
・First step : Ac magnetic field due to ac Josephson current
Ampere’s law
Ac Josephson current
rotH  r , t   ic sin J t  ex
Magnetic field H  r , t 
Magnetic flux density B
B  r , t   0 H  r , t   M  r , t 
M
Precessional M
・Second step Using the solution of Maxwell’s and LLG equations
Phase difference
2 z y 2 y z
q r ,t  

 J t
0 L
0 L
Dynamical flux coupled with M dynamics
i  r , t   Bi  r , t  dL
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i=y,z
Analytic formula of dc Josephson current
Ic  Im
dc
IJ  
 yy

eff 
Im



 J  zz J 

1
eff
FM
y
SC
d I c
160
J 
x
2eV
Josephson frequency
M
SC

Magnetic susceptibility
Bz
By
iiIm   M J
z
i=y,z
W02  J2
2
W2   2    2W  2
J 
0 J
 0
W0    H K  M / 0 
FMR induced by ac Josephson current
HK : anisotropic field
Self-induced ferromagnetic Josephson resonance
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Voltage dependence of dV/dI
RSJ model
W. C. Stewart, APL. 12, 277 (1968)
D. E. McCumber, JAP. 39, 3113 (1968)
2
V 1 Ic2 R Ic R dc
I 

IJ , 1
R 2 V
V
 Ic R 


 V 
H K  0.6 T, M / 0  0.2 T
FJJ
CJJ
1.1
Resonance frequency
FMR
V  23 V
1
dV/dI [W]
dV/dI [W]
1.2
Im
I Jdc   yy
J    zzIm J 
0.9
10
V [V]
20
30
CJJ : Conventional Josephson junction
Wf FMR
70 GHz
FMR  11
FMR
V  23 V
V [V]
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In conclusion :
SC/FM/SC Josephson junction.
Dynamical coupling between SC phase and magnetization.
Ferromagnetic Josephson resonance on 107 Ni atoms.
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