Transcript Spin-wave chaos in the coincidence regime of nonlinear

Model for spin-wave chaos in the
coincidence regime of nonlinear
ferromagnetic resonance
1
A. Krawiecki , A. Sukiennicki
1,2
1
Faculty of Physics, Warsaw University of Technology,
Koszykowa 75, 00-662 Warsaw, Poland
2
Department of Solid State Physics, University of Łódź,
Pomorska 149/153, 90-283 Łódź, Poland
Nonlinear ferromagnetic resonance
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Ferromagnetic sample is placed in perpendicular dc
and rf (with frequencies in the GHz range) magnetic
fields.
The uniform precession of magnetization (uniform
mode) is excited in the sample by the rf field. In the
coincidence regime the rf field frequency wp is close to
the uniform mode frequency wo.
If the rf field amplitude hT exceeds a certain threshold
hthr, the uniform mode decays into spin-wave pairs.
The measured quantity is usually absorption in the
sample, which is proportional to the uniform mode
amplitude.
As the rf field amplitude is increased, periodic (with
frequencies in the range of kHz) and then chaotic
oscillations of absorption appear.
The 1st-order Suhl instability (coincidence regime)
Decay of the uniform
mode (pumped in
resonance) into spinwave pairs with half the
pumping frequency and
opposite wave vectors
Theoretical description
The magnetic Hamiltonian
  
 
2
1
D
H   M  dV  M  H ext  2 M  H dip  2 M 0  M i  
i  x, y , z
V


2
0
M 0  saturationmagnetization;



H ext  H 0 ez  hT ex cosw p t  externalmagneticfield;
 ' 


M
r
H dip      dV '   '   dipolar field;
 V

r

r


D  exchangeconstant,V  sample volume.
 
The Hamiltonian contains the Zeeman energy, the energy of magnetic dipolar
interactions, and the exchange energy; all magnetizations and fields are
normalized to the saturation magnetization.
The Holstein-Primakoff canonical transformation

M   M x  iM y  s 2M 0   2 ss 
  2M 
12
12
0



s1 
ss  
 4M 0 
M z  M 0  ss  ;   the gyromagnetic ratio
The Fourier expansion
s V
1 2
s e

k

k

ik r
;

s V
1 2
s

k


k
e

ik r
The Bogolyubov transformation
sk  k ak   k a k ,
where k   Ak  w k  2w k  ,  k  Bk  Ak  w k  2w k 
12

A   H0

k
 Dk   4M 
2

0

k ,0
12
N  N z

Bk  4M 0  k , 0 N   1   k , 0 kT2 2k 2 ,
 1    k

k ,0
T
2
Bk ,

2k 2 ,
w k2   2 H 0  4N z M 0  Dk 2 H 0  4N z M 0  Dk 2  4M 0 sin 2 k ,
N z , N   N x  N y  2  demagnet ization coefficients, kT  k x  ik y .
The Hamiltonian in the canonical form
H  hT cosw p t  I k ak  c.c.   w k ak ak  H 3  higher order termsin ak , ak .

k
H3 
 V
  
k1 , k 2 , k3
  
k1 , k 2 , k3

k

ak1 ak2 ak3  c.c.  U k1 ,k2 ,k3 ak1 ak2 ak3  c.c.
I k , Vk1 ,k2 ,k3 , U k1 ,k2 ,k3  interact ion coefficients.
 The above Hamiltonian contains non-resonant three-mode interaction terms,
e.g., U k1 ,k2 ,0ak1 ak2 a0 . Such terms should be removed by another canonical
transformation, which, however, should leave the resonant terms
(e.g., Vk,k,0ak ak a0 ) intact.
 The removal of all non-resonant terms from H3 influences the higher-order
terms in the Hamiltonian.
 However, since the basic nonlinear process in the case under study is the 1storder Suhl instability (the resonant three-mode process), the higher-order
terms in the Hamiltonian can be subsequently neglected.
The second quasi-canonical transformation
Let us assume that only the uniform mode (denoted by zero) is directly excited
by the rf field and has frequency close to wp, and the spin-wave pairs have
frequencies close to wp/2. Then the following transformation removes the nonresonant terms from H3:




 Vk1 ,k2 ,k ak1 ak2
Vk,k1 ,k2  Vk1 ,k ,k2 ak1 ak2 U k ,k1 ,k2  U k1 ,k ,k2  U k1 ,k2 ,k ak1 ak2
 k  ak   









w k  w k1  w k2
w k  w k1  w k2
k1 , k 2  w k  w k1  w k 2

Vk1 ,k2 ,k ak1 ak2
w w w

k

k1

k2


k ,0
1   1  

k1 , 0

k2 ,0

V


  
k , k1 , k 2

 Vk1 ,k ,k2 ak1 ak2
w w w

k

k1

k2
[ A similar transformation is well known in the case of parallel pumping:
V.S. Zakharov et al., Usp. Fiz. Nauk 114, 609 (1974)]


k2 ,0




1   1   

k ,0

k1 , 0
Equations of motion for the spin-wave amplitudes
The Hamiltonian and canonical equations (with damping)



H  hT cos(w pt )(I 0  c.c.)  w k  k   V0,k 0 k k  c.c.
k
 k


0
 k
H
 k k  i 
t
 k
 0  iI 0

k
 k
H
 k k  i
t
 k
hT
cos(w pt )  0  iw0  0  i V0,k k k
2
k
 k  k  iw k  k  iV0,k k 0
I0 - interaction coefficient between the uniform mode and the rf field,
0, k - complex amplitudes of the uniform mode and spin waves,
0, k  phenomenological damping of the uniform mode and spin waves,
V0,k - coefficients of nonlinear interactions between the uniform mode and
spin-wave pairs.
Separation of the fast time dependence
 k   k exp(iqk ), qk  const;
u0   0 exp(iw pt ), uk    k exp(iw pt 2).
u0  iI 0
hT
 0  iw 0 u0  i V0,k uk2
2
k
uk  0  iw k uk  iV u u

0, k

k 0
w 0  w p  w 0  0
w k  w p 2  w k  0
The 1st-order Suhl instability threshold
hthr  min
k
20  iw0 k  iw k
I 0 V0,k
Just above the threshold only one (critical) spin-wave pair is excited; if hT
exceeds much the threshold, other pairs with frequency close to w p/2 can be
excited. However, experimental results (low correlation dimension of chaotic
attractors, etc.) suggest that even deeply in the chaotic regime the oscillations
of absorption appear due to interactions of a small number of spin-wave pairs
with the uniform mode.
Model with two spin-wave pairs
 0  0  iw 0 ,  k  k  iw k
  hT hthr ,V  V0, 2 V0,1 ,
 0  arg I 0 ,  1  argV0,1 ,
b0  V0,1 11 exp(i 0 )u0 ,
bk 
V0,1
1
expi  0  1  2uk .
Equations of motion in a dimensionless form
b0  i  0  1   0  iw 0 b0  ib12  iVb22
b1  1  iw1 b1  ib1b0
b2  2  iw 2 b2  iVb2b0
where   d d 1t 
• The model with one spin-wave pair ( with a2=0) shows transition to chaos
via period-doubling,
•Inclusion of a second spin-wave pair, with higher Suhl instability threshold,
can lead to quasiperiodicity, Pomeau-Maneville type-III intermittency, etc.
•The chaotic behavior of the models with one or two spin-wave pairs is in
qualitative agreement with experiments on spin-wave chaos in the
coincidence regime.
Example: route to chaos via period-doubling
Model with one spin-wave pair,
left column: time series of absorption,
right column: chaotic attractor.
Parameters:
0  1.0; w 0  1.5; 1  1.0; w1  3.0,
(a, b)   1.75; (c, d )   1.84;
(e, f )   1.86; ( g , h)   2.0.
Example: route to chaos via quasiperiodicity
Model with two spin-wave pairs,
left column: time series of absorption,
right column: power spectrum of
absorption.
P arameters:
0  1.0; w 0  1.0; 1  1.0; w1  3.0,
2  1.0; w 2  3.0; V  0.952.
(a, b)   1.3, periodic motion,
(c, d )   1.32, quasiperiodic motion,
(e, f )   1.34, chaoticmotion.
Example: type-III Pomeau-Maneville intermittency
Model with two spin-wave pairs,
(a) time series of absorption,
(b) mean duration of laminar phases
vs. the control parameter,
(c) probability distribution of
durations of laminar phases.
Parameters:
0  1.67; w 0  1.67; 1  1.0; w1  3.33,
2  1.0; w 2  4.67; V  1.048.
(a )   8.03,
(b)  c  7.9908, the straight line has
slope  0.8  1,
(c)   7.992, the straight line has
slope  1.52  1.5.
Conclusions
• Systematic derivation of the equations of motion for spinwave amplitudes in the coincidence regime of nonlinear
ferromagnetic resonance above the 1-st order Suhl instability
threshold was presented,
• The non-resonant three-mode interaction terms can be
removed by means of the quasi-canonical transformation,
which leaves only resonant three-mode terms in the
Hamiltonian, and the higher-order terms can be neglected,
• The model equations with one or two parametric spin-wave
pairs show transition to chaos via, e.g., period doubling,
quasi-periodicity, Pomeau-Maneville intermittency, etc., and
the results of simulations are in qualitative agreement with
experimental results.
Thank you for your
attention