NLPV, Gallipoli, Italy
June 15, 2008
Stochastic theory
of nonlinear auto-oscillator:
Spin-torque nano-oscillator
Vasil Tiberkevich
Department of Physics, Oakland University,
Rochester, MI, USA
In collaboration with:
• Andrei Slavin (Oakland University, Rochester, MI, USA)
• Joo-Von Kim (Universite Paris-Sud, Orsay , France)
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator
• Generation linewidth of a nonlinear auto-oscillator
• Theoretical results
• Comparison with experiment (STO)
• Summary
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator
• Generation linewidth of a nonlinear auto-oscillator
• Theoretical results
• Comparison with experiment (STO)
• Summary
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
Unperturbed system:
dx
 F(x )
dt
x  X 0 (0t  0 )
X 0 (  2 )  X 0 ( )
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
Unperturbed system:
dx
 F(x )
dt
x  X 0 (0t  0 )
X 0 (  2 )  X 0 ( )
Perturbed system:
dx
 F( x )  f ( t , x )
dt
x  X0 ( (t ))  y(t )
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
Unperturbed system:
dx
 F(x )
dt
x  X 0 (0t  0 )
X 0 (  2 )  X 0 ( )
Perturbed system:
dx
 F( x )  f ( t , x )
dt
x  X0 ( (t ))  y(t )
d
 0  f (t ,  )
dt
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
d
 0  f (t ,  )
dt
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
d
 0  f (t ,  )
dt
• Weakly nonlinear model
y
1.0
I  I th
0.5
0.5
0.5
0.5
0.5
I  I th
y
1.0
x
1.0
0.5
0.5
0.5
x
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
d
 0  f (t ,  )
dt
• Weakly nonlinear model
y
1.0
I  I th
0.5
0.5
I  I th
y
1.0
0.5
0.5
x
0.5
1.0
0.5
0.5
x
0.5
Complex amplitude: c  x  i  y
dc
 i (0  N | c |2 )c  [ ( I  I th )   | c |2 ]c  f (t , c)
dt
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
d
 0  f (t ,  )
dt
• Weakly nonlinear model
dc
 i (0  N | c |2 )c  [ ( I  I th )   | c |2 ]c  f (t , c)
dt
Linear and nonlinear auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
d
 0  f (t ,  )
dt
• Weakly nonlinear model
dc
 i (0  N | c |2 )c  [ ( I  I th )   | c |2 ]c  f (t , c)
dt
Linear and nonlinear auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
d
 0  f (t ,  )
dt
• Weakly nonlinear model
dc
 i (0  N | c |2 )c  [ ( I  I th )   | c |2 ]c  f (t , c)
dt
Nonlinear auto-oscillator: frequency depends on the amplitude
Linear and nonlinear auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
d
 0  f (t ,  )
dt
• Weakly nonlinear model
dc
 i (0  N | c |2 )c  [ ( I  I th )   | c |2 ]c  f (t , c)
dt
All conventional auto-oscillators are linear.
| N | 
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator
• Generation linewidth of a nonlinear auto-oscillator
• Theoretical results
• Comparison with experiment (STO)
• Summary
Spin-torque oscillator
Electric current I
Magnetic metal
Ferromagnetic metal – strong self-interaction – strong nonlinearity
Electric current – compensation of dissipation – auto-oscillatory dynamics
Spin-torque oscillator
Electric current I
p
Spin-transfer torque:
M(t)
I
 dM 
M  M  p

  TS 
M0
 dt S
 g B 1
 
2 e M0 V
J. Slonczewski, J. Magn. Magn. Mat. 159, L1 (1996)
L. Berger, Phys. Rev. B. 54, 9353 (1996)
Equation for magnetization
Landau-Lifshits-Gilbert-Slonczewski equation:
dM
  [Heff  M ]  TG  TS
dt
M
Equation for magnetization
Landau-Lifshits-Gilbert-Slonczewski equation:
dM
  [Heff  M ]  TG  TS
dt
M
precession
 [Heff  M] – conservative torque (precession)
Equation for magnetization
Landau-Lifshits-Gilbert-Slonczewski equation:
dM
  [Heff  M ]  TG  TS
dt
damping
M
precession
 [Heff  M] – conservative torque (precession)
TG  
G
M0
[M  [M  Heff ]] – dissipative torque
(positive damping)
Equation for magnetization
Landau-Lifshits-Gilbert-Slonczewski equation:
dM
  [Heff  M ]  TG  TS
dt
damping
M
precession
 [Heff  M] – conservative torque (precession)
TG  
TS  
G
M0
I
M0
[M  [M  Heff ]] – dissipative torque
(positive damping)
[M  [M  p]] – spin-transfer torque
(negative damping)
spin
torque
Equation for magnetization
Landau-Lifshits-Gilbert-Slonczewski equation:
dM
  [Heff  M ]  TG  TS
dt
damping
M
precession
When negative current-induced damping
compensates natural magnetic damping,
self-sustained precession of magnetization starts.
spin
torque
Microwave power
Experimental studies of spin-torque oscillators
S.I. Kiselev et al., Nature 425, 380 (2003)
Experimental studies of spin-torque oscillators
M.R. Pufall et al., Appl. Phys. Lett. 86, 082506 (2005)
Specific features of spin-torque oscillators
S.I. Kiselev et al., Nature 425, 380 (2003)
M.R. Pufall et al., Appl. Phys. Lett. 86, 082506
(2005)
Specific features of spin-torque oscillators
• Strong influence of thermal noise
(spin-torque NANO-oscillator)
S.I. Kiselev et al., Nature 425, 380 (2003)
M.R. Pufall et al., Appl. Phys. Lett. 86, 082506
(2005)
Specific features of spin-torque oscillators
• Strong influence of thermal noise
(spin-torque NANO-oscillator)
S.I. Kiselev et al., Nature 425, 380 (2003)
• Strong frequency nonlinearity
M.R. Pufall et al., Appl. Phys. Lett. 86, 082506
(2005)
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator
• Generation linewidth of a nonlinear auto-oscillator
• Theoretical results
• Comparison with experiment (STO)
• Summary
Nonlinear oscillator model
dc
 i ( p )c   ( pp))c   ( pp))c  0
dt
p | c | – oscillation power
2
damping
  arg( c) – oscillation phase
M
precession
spin
torque
Nonlinear oscillator model
precession
dc
 i ( p )c   ( pp))c   ( pp))c  0
dt
p | c | – oscillation power
2
damping
  arg( c) – oscillation phase
M
precession
spin
torque
Nonlinear oscillator model
precession
positive
damping
dc
 i ( p )c   ( pp))c   ( pp))c  0
dt
p | c | – oscillation power
2
damping
  arg( c) – oscillation phase
M
precession
spin
torque
Nonlinear oscillator model
precession
positive
damping
negative
damping
dc
 i ( p )c   ( pp))c   ( pp))c  0
dt
p | c | – oscillation power
2
damping
  arg( c) – oscillation phase
M
precession
spin
torque
How to find parameters of the model?
dc
 i ( p )c   ( pp))c   ( pp))c  0
dt
How to find parameters of the model?
dc
 i ( p )c   ( pp))c   ( pp))c  0
dt
• Start from the Landau-Lifshits-Gilbert-Slonczewski equation
• Rewrite it in canonical coordinates
• Perform weakly-nonlinear expansion
• Diagonalize linear part of the Hamiltonian
• Perform renormalization of non-resonant three-wave processes
• Take into account only resonant four-wave processes
• Average dissipative terms over “fast” conservative motion
• and obtain analytical results for
( p)
 ( p)
 ( p)
Deterministic theory
dc
 i ( p )c   ( p )c   ( p )c  0
dt
2
p | c | – oscillation power
Stationary solution:
c( t ) 
p0 exp( igt  i0 )
Stationary power is determined from the condition of vanishing total damping:
 ( p0 )   ( p0 )
Stationary frequency is determined by the oscillation power:
g   ( p0 )
Deterministic theory: Spin-torque oscillator
Stationary power is determined from the condition of vanishing total damping:
 ( p0 )   ( p0 )
 1
p0 
 Q
  I I th – supercriticality parameter
I th  0 
– threshold current
Stationary frequency is determined by the oscillation power:
g   ( p0 )
 1
g  0  Np0  0  N
 Q
0
– FMR frequency
N
– nonlinear frequency shift
Generated frequency
Frequency as function of
applied field
Frequency as function of
magnetization angle
35
Generation frequency g /2 (GHz)
Generation frequency g /2 (GHz)
35
30
25
20
15
10
5
H0 = 8 kOe
30
25
H0 = 5 kOe
20
15
10
5
0
0
0
2
4
6
8
Bias magnetic field H0 (kOe)
Experiment: W.H. Rippard et al., Phys.
Rev. Lett. 92, 027201 (2004)
10
0
30
60
Magnetization angle 0 (deg)
Experiment: W.H. Rippard et al., Phys.
Rev. B 70, 100406 (2004)
90
Stochastic nonlinear oscillator model
Stochastic Langevin equation:
dc
 i ( p )c   ( p )c   ( p )c  Dn f n (t )
dt
Random thermal noise:
f n (t ) f n (t )  2 (t  t )
kBT
Dn ( p )   ( p ) ( p )   ( p )
l( p)
2
|
c
|
 – thermal equilibrium power of oscillations:
 0
 p
 0

l – scale factor that relates energy and power: E ( p )  l   ( p )dp
Fokker-Planck equation for a nonlinear oscillator
Probability distribution function (PDF) P(t, p, ) describes probability
that oscillator has the power p = |c|2 and the phase  = arg(c) at the
moment of time t.
dP 
P  
P  Dn ( p)  2 P
 2 p ( p)   ( p)P ( p)

2 pDn ( p)  

dt p
 p 
p 
2 p  2
Deterministic terms
describe noise-free
dynamics of the oscillator
Stochastic terms describe
thermal diffusion in the
phase space
kBT
Dn ( p )   ( p ) ( p )   ( p )
l( p)
Stationary PDF
Stationary probability distribution function P0:
• does not depend on the time t (by definition);
• does not depend on the phase  (by phase-invariance of FP equation).
Ordinary differential equation for P0(p):

d
2 p ( p)   ( p)P0   d 2 pDn ( p) dP0 
dp
dp 
dp 
Stationary PDF for an arbitrary auto-oscillator:
 l p
  ( p)  
dp
P0 ( p )  Z 0 exp 
 ( p)1 

  ( p)  
 k BT 0

Constant Z0 is determined by the normalization condition
 P ( p)dp  1
0
0
Analytical results for STO
“Material functions” for the case of STO:
 ( p)  G (1  Qp )
 ( p)  I (1  p)
  I I th  I G
Analytical expression for the stationary PDF:

Q
exp  (  Q )(1  Qp ) Q 2
P0 ( p ) 
(1  Qp ) 
E (  Q ) Q 2 
Analytical expression for the mean oscillation power
Q  exp  (  Q ) Q 2    1
p
1 

2
  Q 
E (  Q ) Q      Q
Here
  (1  Q) Q 2

E ( x )   e  xt t   dt
1
– exponential integral function

Oscillator power p (a.u.)
Power in the near-threshold region
3
2
1
0
4
5
6
7
8
9
Bias current I (mA)
Experiment: Q. Mistral et al., Appl. Phys. Lett. 88, 192507 (2006).
Theory: V. Tiberkevich et al., Appl. Phys. Lett. 91, 192506 (2007).
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator
• Generation linewidth of a nonlinear auto-oscillator
• Theoretical results
• Comparison with experiment (STO)
• Summary
Linewidth of a conventional oscillator
Classical result (see, e.g., A. Blaquiere, Nonlinear
System Analysis (Acad. Press, N.Y., 1966)):
Full linewidth of an electrical oscillator
C
k BT R002
2 
U 02
+R0
e(t)
L
–Rs
~
Linewidth of a conventional oscillator
Classical result (see, e.g., A. Blaquiere, Nonlinear
System Analysis (Acad. Press, N.Y., 1966)):
Full linewidth of an electrical oscillator
C
k BT R002
2 
U 02
+R0
L
–Rs
Physical interpretation of the classical result
k BT
2  0
E0
0 
1
LC
Frequency
0 
R0
2L
Equilibrium half-linewidth
e(t)
E0 
~
1
CU 02
2
Oscillation energy
Linewidth of a laser
Quantum limit for the linewidth of a single-mode laser (Schawlow-Townes limit)
[A.L. Schawlow and C.H. Townes, Phys. Rev. 112, 1940 (1958)]
0 02
2 
2 Pout
Linewidth of a laser
Quantum limit for the linewidth of a single-mode laser (Schawlow-Townes limit)
[A.L. Schawlow and C.H. Townes, Phys. Rev. 112, 1940 (1958)]
0 02
2 
2 Pout
Physical interpretation of the classical result
k BT
2  0
E0
k BT 
0
2
Effective thermal energy
E0  Pout 0
Oscillation energy
Linewidth of a spin-torque oscillator
Experiment by W.H. Rippard et al., Theoretical result [J.-V. Kim, PRB 73, 174412 (2006)]:
DARPA Review (2004).
f=35.89 GHz
f = 13.5 MHz
1/2
Amplitude (nV/Hz )
0.3
Q(V)=2650
Q(W)=4235
0.2
0.1
0.0
35.850
35.875
35.900
Frequency (GHz)
35.925
2    G
0 k BT
H M 0Veff p0
Linewidth of a spin-torque oscillator
Experiment by W.H. Rippard et al., Theoretical result [J.-V. Kim, PRB 73, 174412 (2006)]:
DARPA Review (2004).
f=35.89 GHz
f = 13.5 MHz
1/2
Amplitude (nV/Hz )
0.3
Q(V)=2650
Q(W)=4235
0 k BT
H M 0Veff p0
Physical interpretation:
0.2
k BT
2  0
E0
0.1
0.0
35.850
2    G
35.875
35.900
Frequency (GHz)
35.925
Linewidth of a spin-torque oscillator
Experiment by W.H. Rippard et al., Theoretical result [J.-V. Kim, PRB 73, 174412 (2006)]:
DARPA Review (2004).
f=35.89 GHz
f = 13.5 MHz
1/2
Amplitude (nV/Hz )
0.3
2    G
Q(V)=2650
Q(W)=4235
Physical interpretation:
0.2
k BT
2  0
E0
0.1
0.0
35.850
0 k BT
H M 0Veff p0
35.875
35.900
35.925
Frequency (GHz)
2f exp  2 2  13.5 MHz
2f theor  2 2  0.35 MHz
Linewidth of a spin-torque oscillator
Experiment by W.H. Rippard et al., Theoretical result [J.-V. Kim, PRB 73, 174412 (2006)]:
DARPA Review (2004).
f=35.89 GHz
f = 13.5 MHz
1/2
Amplitude (nV/Hz )
0.3
2    G
Q(V)=2650
Q(W)=4235
Physical interpretation:
0.2
k BT
2  0
E0
0.1
0.0
35.850
0 k BT
H M 0Veff p0
35.875
35.900
35.925
Frequency (GHz)
2f exp  2 2  13.5 MHz
2f theor  2 2  0.35 MHz
This theory does assumes that the frequency is constant
( p)  0  const
How to calculate linewidth?
Power spectrum S() is the Fourier transform of the autocorrelation
function K():
S ()   K ( )ei d
K ( )  c(t   )c (t )
two-time function
Autocorrelation function K() can not be found from the stationary PDF.
Ways to proceed:
• Solve non-stationary Fokker-Planck equation
[J.-V. Kim et al., Phys. Rev. Lett. 100, 167201 (2008)];
• Analyze stochastic Langevin equation.
dc
 i ( p )c   ( p )c   ( p )c  Dn f n (t )
dt
Power and phase fluctuations
Two-dimensional plot of stationary PDF
Im( c)
Power fluctuations are weak
p

p0
p | c |
k BT
 1
E ( p0 )
Oscillations – phase-modulated process:
  arg( c)
c( t ) 
Re(c )
p(t )ei ( t ) 
p0 ei ( t )
Autocorrelation function:
K ( )  p0 exp i ( )  i (0)
Linewidth of any oscillator is determined by the phase fluctuations
Linearized power-phase equations
dc
Stochastic Langevin equation:  i ( p )c   ( p )c   ( p )c  Dn f n (t )
dt
Linearized power-phase equations
dc
Stochastic Langevin equation:  i ( p )c   ( p )c   ( p )c  Dn f n (t )
dt
Power-phase ansatz:
c( t ) 
p0  p(t ) ei ( t )
| p | p0
Linearized power-phase equations
dc
Stochastic Langevin equation:  i ( p )c   ( p )c   ( p )c  Dn f n (t )
dt
Power-phase ansatz:
c( t ) 
p0  p(t ) ei ( t )
Stochastic power-phase equations:
dp
 2eff p  2 Dn p0 Re f n (t )e i
dt
d
Dn
  ( p0 ) 
Im f n (t )e i  N p
dt
p0


Effective damping:


eff  G  G  p0
| p | p0
Linear system of equations.
Can be solved in general case.
G  d ( p) dp
Nonlinear frequency shift coefficient: N  d( p) dp
Nonlinear frequency shift creates additional source of the phase noise
Power fluctuations
Correlation function for power fluctuations:
 
p(t   )p(t )  p 
 eff
2
0
 kBT 2 eff | |

e
 E ( p0 )
Power fluctuations
Correlation function for power fluctuations:
 
p(t   )p(t )  p 
 eff
2
0
Power fluctuations are small
 kBT 2 eff | |

e
 E ( p0 )
Power fluctuations
Correlation function for power fluctuations:
 
p(t   )p(t )  p 
 eff
2
0
 kBBT 2 effeff ||||

e
 E ( p00)
Power fluctuations are small
Non-zero correlation time of power fluctuations
Additional noise source –N p for the phase fluctuations is
Gaussian, but not “white”
Phase fluctuations
Averaged value of the phase difference:
 (t   )   (t )  ( p0 )
Phase fluctuations
Averaged value of the phase difference:
 (t   )   (t )  ( p0 )
Determines mean
oscillation frequency
Phase fluctuations
Averaged value of the phase difference:
 (t   )   (t )  ( p0 )
Determines mean
oscillation frequency
Variance of the phase difference:
2 eff | |



k
T
1

e
2
2
B

 ( )  
|  |   |  | 
E ( p0 ) 
2eff 


Np0
N

– dimensionless nonlinear frequency shift
eff
G  G
Phase fluctuations
Averaged value of the phase difference:
 (t   )   (t )  ( p0 )
Determines mean
oscillation frequency
Variance of the phase difference:
2 eff | |



k
T
1

e
2
2
B

 ( )  
|  |   |  | 
E ( p0 ) 
2eff 


Determines oscillation
linewidth
Np0
N

– dimensionless nonlinear frequency shift
eff
G  G
Auto-correlation function:
 1

K ( )  p0e i ( p0 ) exp   2 ( )
 2

Low-temperature asymptote
 2 ( )  
k BT
(1   2 ) |  |
E ( p0 )
“Random walk” of the phase
Low-temperature asymptote
 2 ( )  
k BT
(1   2 ) |  |
E ( p0 )
“Random walk” of the phase
Lorentzian lineshape with the full linewidth
k BT
2  (1   )
 (1   2 ) 2lin
E ( p0 )
2
Frequency nonlinearity broadens linewidth by the factor
 N 

1    1  
 G  G 
2
2
Low-temperature asymptote
 2 ( )  
k BT
(1   2 ) |  |
E ( p0 )
“Random walk” of the phase
Lorentzian lineshape with the full linewidth
k BT
2  (1   )
 (1   2 ) 2lin
E ( p0 )
2
Frequency nonlinearity broadens linewidth by the factor
 N 

1    1  
 G  G 
2
2
 eff
Region of validity kBT  
 
 E ( p0 )

~ 300 K
2
 1 
High-temperature asymptote
k BT 2
 eff  2
E ( p0 )
“Inhomogeneous broadening” of the frequencies
 2 ( )  
High-temperature asymptote
k BT 2
 eff  2
E ( p0 )
“Inhomogeneous broadening” of the frequencies
 2 ( )  
Gaussian lineshape with the full linewidth
2  2 |  |  eff
kBT
E ( p0 )
Different dependence of the linewidth on the temperature:
2  ~ T
High-temperature asymptote
k BT 2
 eff  2
E ( p0 )
“Inhomogeneous broadening” of the frequencies
 2 ( )  
Gaussian lineshape with the full linewidth
2  2 |  |  eff
kBT
E ( p0 )
Different dependence of the linewidth on the temperature:
2  ~ T
   E ( p0 )
Region of validity kBT   eff 
    2 ~ 300 K
  
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator
• Generation linewidth of a nonlinear auto-oscillator
• Theoretical results
• Comparison with experiment (STO)
• Summary
Temperature dependence of STO linewidth
Experiment: J. Sankey et al., Phys. Rev. B 72, 224427 (2005)
700
700
device #1
second harmonic
600
500
high-temperature
asymptote
400
Linewidth (MHz)
Linewidth (MHz)
500
300
200
100
0
50
100
Temperature (K)
150
high-temperature
asymptote
400
low-temperature
asymptote
300
200
100
low-temperature
asymptote
0
device #2
first harmonic
600
0
200
0
50
100
150
200
250
300
Temperature (K)
Low-temperature asymptote gives correct order-of-magnitude estimation of the
linewidth.
Angular dependence of the linewidth
2




kBT
N
kBT
2




2  (1   )
 1 


E ( p0 )   G  G   E ( p0 )


Isotropic film,
out-of-plane magnetization
40
20
0
-20
0
30
60
90
Out-of-plane magnetization angle 0 (deg)
Nonlinear frequency shift N/2 (GHz)
Nonlinear frequency shift N/2 (GHz)
Nonlinear frequency shift coefficient N strongly depends on the
orientation of the bias magnetic field
Anisotropic film,
in-plane magnetization
0
-2
-4
-6
-8
-10
0
30
60
90
In-plane magnetization angle 0 (deg)
Angular dependence of the linewidth
2




kBT
N
kBT
2




2  (1   )
 1 


E ( p0 )   G  G   E ( p0 )


Isotropic film,
out-of-plane magnetization
40
20
0
-20
0
30
60
90
Out-of-plane magnetization angle 0 (deg)
Nonlinear frequency shift N/2 (GHz)
Nonlinear frequency shift N/2 (GHz)
Nonlinear frequency shift coefficient N strongly depends on the
orientation of the bias magnetic field
Anisotropic film,
in-plane magnetization
0
-2
-4
-6
-8
-10
0
30
60
90
In-plane magnetization angle 0 (deg)
Angular dependence of the linewidth
Generation linewidth  (MHz)
Out-of-plane magnetization
250
 (|N| > 0)
200
150
Experiment
100
10 x  (|N| = 0)
50
0
45
50
55
60
65
70
75
80
85
90
Out-of-plane magnetization angle 0 (deg)
Experiment: W.H. Rippard et al., Phys. Rev. B 74, 224409 (2006)
Theory: J.-V. Kim, V. Tiberkevich and A. Slavin, Phys. Rev. Lett. 100, 017207 (2008)
Angular dependence of the linewidth
2




kBT
N
kBT
2




2  (1   )
 1 


E ( p0 )   G  G   E ( p0 )


Isotropic film,
out-of-plane magnetization
40
20
0
-20
0
30
60
90
Out-of-plane magnetization angle 0 (deg)
Nonlinear frequency shift N/2 (GHz)
Nonlinear frequency shift N/2 (GHz)
Nonlinear frequency shift coefficient N strongly depends on the
orientation of the bias magnetic field
Anisotropic film,
in-plane magnetization
0
-2
-4
-6
-8
-10
0
30
60
90
In-plane magnetization angle 0 (deg)
Angular dependence of the linewidth
In-plane magnetization
4000
2500
2000
3000
Linewidth (MHz)
Linewidth (MHz)
|N| > 0
2000
|N| = 0
(x 10)
1000
0
|N| > 0
1500
1000
|N| = 0
(x 10)
500
0
0
30
60
90
Field angle (deg)
120
0
30
60
Field angle (deg)
Experiment: K. V. Thadani et al., arXiv: 0803.2871 (2008)
90
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator
• Generation linewidth of a nonlinear auto-oscillator
• Theoretical results
• Comparison with experiment (STO)
• Summary
Summary
• “Nonlinear” auto-oscillators (oscillators with power-dependent
frequency) are simple dynamical systems that can be described by
universal model, but which have not been studied previously
• There are a number of qualitative differences in the dynamics of
“linear” and “nonlinear” oscillators:
• Different temperature regimes of generation linewidth (low- and
high-temperature asymptotes)
• Different mechanism of phase-locking of “nonlinear” oscillators
[A. Slavin and V. Tiberkevich, Phys. Rev. B 72, 092407 (2005); ibid., 74, 104401 (2006)]
• Possibility of chaotic regime of mutual phase-locking [preliminary results]
• Spin-torque oscillator (STO) is the first experimental realization of a
“nonlinear” oscillator. Analytical nonlinear oscillator model correctly
describes both deterministic and stochastic properties of STOs.