Stochastic theory of nonlinear auto-oscillator: Spin-torque nano-oscillator · 2008. 6. 15. ·...
Transcript of Stochastic theory of nonlinear auto-oscillator: Spin-torque nano-oscillator · 2008. 6. 15. ·...
Stochastic theory
of nonlinear auto-oscillator:
Spin-torque nano-oscillator
Vasil Tiberkevich
NLPV, Gallipoli, Italy June 15, 2008
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• 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• 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:
)(xFx
=dt
d)( 000 φω += tXx )()2( 00 φπφ XX =+
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
)(xFx
=dt
d)( 000 φω += tXx )()2( 00 φπφ XX =+
Unperturbed system:
Perturbed system:
),()( xfxFx
tdt
d+= )())((0 tt yXx += φ
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
Unperturbed system:
Perturbed system:
)(xFx
=dt
d)( 000 φω += tXx )()2( 00 φπφ XX =+
),(0 φωφ
tfdt
d=−
),()( xfxFx
tdt
d+= )())((0 tt yXx += φ
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
),(0 φωφ
tfdt
d=−
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
• Weakly nonlinear model
1.0y
1.0y
thII <thII >
),(0 φωφ
tfdt
d=−
-0.5 0.5x
-0.5
0.5
-1.0 -0.5 0.5x
-0.5
0.5
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
• Weakly nonlinear model
1.0y
1.0y
thII <
),(0 φωφ
tfdt
d=−
thII >
-0.5 0.5x
-0.5
0.5
-1.0 -0.5 0.5x
-0.5
0.5
yixc α+=
),(]||)([)||( 2
th
2
0 ctfccIIccNidt
dc=−−+++ βαω
Complex amplitude:
Models of (weakly perturbed) auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
),(0 φωφ
tfdt
d=−
• Weakly nonlinear model
),(]||)([)||( 2
th
2
0 ctfccIIccNidt
dc=−−+++ βαω
dt
Linear and nonlinear auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
• Weakly nonlinear model
),(]||)([)||( 2
th
2
0 ctfccIIccNidt
dc=−−+++ βαω
),(0 φωφ
tfdt
d=−
dt
Linear and nonlinear auto-oscillators
Auto-oscillator – autonomous dynamical system with stable limit cycle
• Phase model
• Weakly nonlinear model
),(]||)([)||( 2
th
2
0 ctfccIIccNidt
dc=−−+++ βαω
),(0 φωφ
tfdt
d=−
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
• Weakly nonlinear model
),(]||)([)||( 2
th
2
0 ctfccIIccNidt
dc=−−+++ βαω
),(0 φωφ
tfdt
d=−
dt
All conventional auto-oscillators are linear.
β<<|| N
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator• 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
p M(t)
Electric current I
Spin-transfer torque:
J. Slonczewski, J. Magn. Magn. Mat. 159, L1 (1996)
L. Berger, Phys. Rev. B. 54, 9353 (1996)
[ ][ ]pMMTM
××==
0
S
S M
I
dt
d σ
VMe
g
0
B 1
2
µεσ =
Equation for magnetization
Landau-Lifshits-Gilbert-Slonczewski equation:
MSGeff ][ TTMH
M++×= γ
dt
d][ eff MH ×γ
GT ST
Equation for magnetization
Landau-Lifshits-Gilbert-Slonczewski equation:
precession
MSGeff ][ TTMH
M++×= γ
dt
d][ eff MH ×γ
GT ST
– conservative torque (precession)
precession
][ eff MH ×γ
Equation for magnetization
Landau-Lifshits-Gilbert-Slonczewski equation:
precession
M
damping
SGeff ][ TTMHM
++×= γdt
d][ eff MH ×γ
GT ST
– conservative torque (precession)
– dissipative torque
(positive damping)
precession
][ eff MH ×γ
]][[ eff
0
GG HMMT ××−=
M
γα
Equation for magnetization
SGeff ][ TTMHM
++×= γdt
d
Landau-Lifshits-Gilbert-Slonczewski equation:
precession
M
damping
spin
torque][ eff MH ×γGT ST
– conservative torque (precession)
– dissipative torque
(positive damping)
– spin-transfer torque
(negative damping)
precession
][ eff MH ×γ
]][[ eff
0
GG HMMT ××−=
M
γα
]][[0
S pMMT ××+=M
Iσ
Equation for magnetization
Landau-Lifshits-Gilbert-Slonczewski equation:
precession
M
damping
spin
torqueSGeff ][ TTMHM
++×= γdt
d][ eff MH ×γ
GT ST
precession
When negative current-induced damping
compensates natural magnetic damping,
self-sustained precession of magnetization starts.
Experimental studies of spin-torque oscillators
Mic
row
ave
po
we
r
S.I. Kiselev et al., Nature 425, 380 (2003)
Mic
row
ave
po
we
r
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
S.I. Kiselev et al., Nature 425, 380 (2003)
• Strong influence of thermal noise
(spin-torque NANO-oscillator)
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)
• Strong influence of thermal noise
(spin-torque NANO-oscillator)
M.R. Pufall et al., Appl. Phys. Lett. 86, 082506
(2005)
• Strong frequency nonlinearity
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator• Stochastic model of a nonlinear auto-oscillator
• Generation linewidth of a nonlinear auto-oscillator
• Theoretical results
• Comparison with experiment (STO)
• Summary
Nonlinear oscillator model
0)()()( =Γ−Γ++ −+ cpcpcpidt
dcω cp)(+Γ cp)(−Γcpi )(ω
2|| cp = – oscillation power
)arg(c=φ – oscillation phase
0)()()( =Γ−Γ++ −+ cpcpcpidt
ω )(+ )(−)(
precession
M
damping
spin
torque
Nonlinear oscillator model
0)()()( =Γ−Γ++ −+ cpcpcpidt
dcω cp)(+Γ cp)(−Γcpi )(ω
precession
0)()()( =Γ−Γ++ −+ cpcpcpidt
ω )(+ )(−)(
2|| cp = – oscillation power
)arg(c=φ – oscillation phase
precession
M
damping
spin
torque
Nonlinear oscillator model
0)()()( =Γ−Γ++ −+ cpcpcpidt
dcω cp)(+Γ cp)(−Γcpi )(ω
precessionpositive
damping
0)()()( =Γ−Γ++ −+ cpcpcpidt
ω )(+ )(−)(
2|| cp = – oscillation power
)arg(c=φ – oscillation phase
precession
M
damping
spin
torque
0)()()( =Γ−Γ++ −+ cpcpcpidt
dcω cp)(+Γ cp)(−Γcpi )(ω
Nonlinear oscillator model
negative
damping
positive
dampingprecession
0)()()( =Γ−Γ++ −+ cpcpcpidt
ω )(+ )(−)(
2|| cp = – oscillation power
)arg(c=φ – oscillation phase
precession
M
damping
spin
torque
How to find parameters of the model?
0)()()( =Γ−Γ++ −+ cpcpcpidt
dcω cp)(+Γ cp)(−Γcpi )(ω
How to find parameters of the model?
0)()()( =Γ−Γ++ −+ cpcpcpidt
dcω cp)(+Γ cp)(−Γcpi )(ω
• Start from the Landau-Lifshits-Gilbert-Slonczewski equation
• Rewrite it in canonical coordinates
• Perform weakly-nonlinear expansion
Diagonalize linear part of the Hamiltonian• 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+Γ )( p−Γ)( pω
Deterministic theory
0)()()( =Γ−Γ++ −+ cpcpcpidt
dcω
Stationary solution: )exp()( 0g0 φω itiptc +−=
2|| cp = – oscillation power
Stationary power is determined from the condition of vanishing total damping:
)()( 00 pp −+ Γ=Γ
Stationary frequency is determined by the oscillation power:
)( 0g pωω =
Deterministic theory: Spin-torque oscillator
Stationary power is determined from the condition of vanishing total damping:
)()( 00 pp −+ Γ=Γ
Qp
+−
=ζζ 1
0
thII=ζ – supercriticality parameter
σ0th Γ=I – threshold current
Stationary frequency is determined by the oscillation power:
)( 0g pωω =
QNNp
+−
+=+=ζζ
ωωω1
000g
0ω – FMR frequency
N – nonlinear frequency shift
Generated frequency
25
30
35
Generation frequency ω
g /2
π (GHz)
20
25
30
35
H0 = 5 kOe
Generation frequency ω
g /2
π (GHz) H
0 = 8 kOe
Frequency as function of
applied field
Frequency as function of
magnetization angle
0 2 4 6 8 100
5
10
15
20
Generation frequency
Bias magnetic field H0 (kOe)
0 30 60 900
5
10
15
20
Generation frequency
Magnetization angle θ0 (deg)
Experiment: W.H. Rippard et al., Phys.
Rev. Lett. 92, 027201 (2004)
Experiment: W.H. Rippard et al., Phys.
Rev. B 70, 100406 (2004)
Stochastic nonlinear oscillator model
)()()()( nn tfDcpcpcpidt
dc=Γ−Γ++ −+ω
Random thermal noise:
Stochastic Langevin equation:
)(2)()( nn tttftf ′−=′∗ δ
)(nn tfD
η – thermal equilibrium power of oscillations:
λ – scale factor that relates energy and power: ∫= dpppE )()( ωλ
)(2)()( nn tttftf ′−=′ δ
)()()()()( B
np
TkppppD
λωη ++ Γ=Γ=
η===Γ=Γ −−00
2|| pc
Fokker-Planck equation for a nonlinear oscillator
Probability distribution function (PDF) P(t, p, φ) describes probability
that oscillator has the power p = |c|2 and the phase φ = arg(c) at the
moment of time t.
[ ] 2
2
nn
2
)()(2)()()(2
φφω
∂∂
+
∂∂
∂∂
=∂∂
−Γ−Γ∂∂
− −+
P
p
pD
p
PppD
p
PpPppp
pdt
dP [ ] φ
ω∂∂
−Γ−Γ∂∂
− −+
PpPppp
p)()()(2
2
2
nn
2
)()(2
φ∂∂
+
∂∂
∂∂ P
p
pD
p
PppD
p[ ]
2n2
)(2)()()(2φφ
ω∂
+
∂∂
=∂
−Γ−Γ∂
− −+pp
ppDp
pPppppdt
Deterministic terms
describe noise-free
dynamics of the oscillator
Stochastic terms describe
thermal diffusion in the
phase space
[ ] φ
ω∂
−Γ−Γ∂
− −+ pPpppp
)()()(22n
2)(2
φ∂+
∂∂ pp
ppDp
)()()()()( B
np
TkppppD
λωη ++ Γ=Γ=
Stationary PDF
Stationary probability distribution function P0:
• does not depend on the time t (by definition);
• does not depend on the phase φ (by phase-invariance of FP equation).
[ ]
=Γ−Γ− −+
dp
dPppD
dp
dPppp
dp
d 0n0 )(2)()(2
Ordinary differential equation for P0(p):
dpdpdp
Stationary PDF for an arbitrary auto-oscillator:
Constant Z0 is determined by the normalization condition 1)(0
0 =∫∞
dppP
′
′Γ
′Γ−′−= ∫
+
−p
pdp
pp
TkZpP
0B
00)(
)(1)(exp)( ω
λ
Analytical results for STO
“Material functions” for the case of STO:
)1()( G Qpp +Γ=Γ+ )1()( pIp −=Γ− σ Gth Γ== III σζ
Analytical expression for the stationary PDF:
[ ]( )ηζ
ηζ
ββ 2
2
0)(
)1)((exp
)1()(
QQE
QQpQ
Qp
QpP
+++−
+=
Analytical expression for the mean oscillation power
Here ηζβ 2)1( QQ+=
∫∞
−−=1
)( dttexE xt ββ – exponential integral function
( )( ) QQQE
Q
Qp
+−
+
++−
++
=ζζ
ηζηζ
ζη
β
1
)(
)(exp1
2
2
Power in the near-threshold region
2
3
Oscillator power p (a.u.)
4 5 6 7 8 90
1
Oscillator power
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• 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
C L
+R0 –Rs
Classical result (see, e.g., A. Blaquiere, Nonlinear
System Analysis (Acad. Press, N.Y., 1966)):
Full linewidth of an electrical oscillator
2
0
2
00B2U
RTk ωω =∆
~e(t)
Linewidth of a conventional oscillator
C L
+R0 –Rs
Classical result (see, e.g., A. Blaquiere, Nonlinear
System Analysis (Acad. Press, N.Y., 1966)):
Full linewidth of an electrical oscillator
2
0
2
00B2U
RTk ωω =∆
~e(t)
Physical interpretation of the classical result
L
R
2
00 =Γ 2
002
1CUE =
LC
10 =ω
Frequency Equilibrium half-linewidth Oscillation energy
0
B02E
TkΓ=∆ω
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)]
out
2
00
22
P
Γ=∆
ωω
h
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)]
out
2
00
22
P
Γ=∆
ωω
h
Physical interpretation of the classical resultPhysical interpretation of the classical result
0
B02E
TkΓ=∆ω
0out0 Γ= PE2
0B
ωh=Tk
Effective thermal energy Oscillation energy
Linewidth of a spin-torque oscillator
0.1
0.2
0.3 Q(V)=2650
Q(W)=4235 f=35.89 GHz
2∆f = 13.5 MHz
Amplitude (nV/Hz1/2)
Experiment by W.H. Rippard et al.,
DARPA Review (2004).
0eff0
B0G2
pVM
Tk
H∂∂
≈∆ω
αω
Theoretical result [J.-V. Kim, PRB 73, 174412 (2006)]:
35.850 35.875 35.900 35.925
0.0Amplitude (nV/Hz
Frequency (GHz)
Linewidth of a spin-torque oscillator
0.1
0.2
0.3 Q(V)=2650
Q(W)=4235 f=35.89 GHz
2∆f = 13.5 MHz
Amplitude (nV/Hz1/2)
Experiment by W.H. Rippard et al.,
DARPA Review (2004).
0eff0
B0G2
pVM
Tk
H∂∂
≈∆ω
αω
Physical interpretation:
B02E
TkΓ=∆ω
Theoretical result [J.-V. Kim, PRB 73, 174412 (2006)]:
35.850 35.875 35.900 35.925
0.0Amplitude (nV/Hz
Frequency (GHz)
0
02E
Γ=∆ω
Linewidth of a spin-torque oscillator
0.1
0.2
0.3 Q(V)=2650
Q(W)=4235 f=35.89 GHz
2∆f = 13.5 MHz
Amplitude (nV/Hz1/2)
Experiment by W.H. Rippard et al.,
DARPA Review (2004).
0eff0
B0G2
pVM
Tk
H∂∂
≈∆ω
αω
Physical interpretation:
Theoretical result [J.-V. Kim, PRB 73, 174412 (2006)]:
B02E
TkΓ=∆ω
35.850 35.875 35.900 35.925
0.0Amplitude (nV/Hz
Frequency (GHz)
MHz5.13222 exp =∆=∆ πωf MHz35.0222 theor =∆=∆ πωf
0
02E
Γ=∆ω
Linewidth of a spin-torque oscillator
0.1
0.2
0.3 Q(V)=2650
Q(W)=4235 f=35.89 GHz
2∆f = 13.5 MHz
Amplitude (nV/Hz1/2)
Experiment by W.H. Rippard et al.,
DARPA Review (2004).
0eff0
B0G2
pVM
Tk
H∂∂
≈∆ω
αω
Physical interpretation:
Theoretical result [J.-V. Kim, PRB 73, 174412 (2006)]:
B02E
TkΓ=∆ω
35.850 35.875 35.900 35.925
0.0Amplitude (nV/Hz
Frequency (GHz)
This theory does assumes that the frequency is constant
const)( 0 == ωω p
MHz5.13222 exp =∆=∆ πωf MHz35.0222 theor =∆=∆ πωf
0
02E
Γ=∆ω
How to calculate linewidth?
Power spectrum S(Ω) is the Fourier transform of the autocorrelation
function K(τ):
two-time function
∫ Ω=Ω ττ τdeKS i)()( )()()( tctcK ∗+= ττ
Autocorrelation function K(τ) 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.
)()()()( nn tfDcpcpcpidt
dc=Γ−Γ++ −+ω
Power and phase fluctuations
1)( 0
B
0
<<≈∆
pE
Tk
p
p
)()()()( titi epetptc φφ ≈=
Two-dimensional plot of stationary PDF
)Im(c
|| cp =
)arg(c=φ
Power fluctuations are weak
Oscillations – phase-modulated process:
)(
0
)()()( titi epetptc φφ ≈=)Re(c
Autocorrelation function:
[ ])0()(exp)( 0 φτφτ iipK −≈
Linewidth of any oscillator is determined by the phase fluctuations
Linearized power-phase equations
)()()()( nn tfDcpcpcpidt
dc=Γ−Γ++ −+ωStochastic Langevin equation:
Linearized power-phase equations
)()()()( nn tfDcpcpcpidt
dc=Γ−Γ++ −+ω
)(
0 )()( tietpptc φδ+=
Stochastic Langevin equation:
Power-phase ansatz: 0|| pp <<δ
Linearized power-phase equations
)()()()( nn tfDcpcpcpidt
dc=Γ−Γ++ −+ω
)(
0 )()( tietpptc φδ+=
[ ]etfpDppd iδ
δ φ=Γ+ −)(Re22
Stochastic Langevin equation:
Power-phase ansatz: 0|| pp <<δ
Stochastic power-phase equations:
[ ]
[ ] pNetfp
Dp
dt
d
etfpDpdt
i
i
δωφ
δ
φ
φ
+=+
=Γ+
−
−
)(Im)(
)(Re22
n
0
n0
n0neff
pN δ
( ) 0eff pGG −+ −=ΓEffective damping: dppdG )(±± Γ=
Nonlinear frequency shift coefficient: dppdN )(ω=
Nonlinear frequency shift creates additional source of the phase noise
Linear system of equations.
Can be solved in general case.
Power fluctuations
Correlation function for power fluctuations:
||2
0
B
eff
2
0eff
)()()(
τδτδ Γ−+
ΓΓ
=+ epE
Tkptptp
Power fluctuations
Correlation function for power fluctuations:
||2
0
B
eff
2
0eff
)()()(
τδτδ Γ−+
ΓΓ
=+ epE
Tkptptp
Power fluctuations are small
)( 0
B
pE
Tk
Power fluctuations are small
Power fluctuations
Correlation function for power fluctuations:
||2
0
B
eff
2
0eff
)()()(
τδτδ Γ−+
ΓΓ
=+ epE
Tkptptp
||2 eff τΓ−e
)( 0
B
pE
Tk
Power fluctuations are smallPower fluctuations are small
Non-zero correlation time of power fluctuations
Additional noise source –N δp for the phase fluctuations is
Gaussian, but not “white”
Phase fluctuations
Averaged value of the phase difference:
τωφτφ )()()( 0ptt −=−+
Phase fluctuations
Averaged value of the phase difference:
τωφτφ )()()( 0ptt −=−+ Determines mean
oscillation frequency
Phase fluctuations
Averaged value of the phase difference:
τωφτφ )()()( 0ptt −=−+ Determines mean
oscillation frequency
Variance of the phase difference:
−+ −=
Γ=
GG
NNp
eff
0ν – dimensionless nonlinear frequency shift
Γ−
−+Γ=∆Γ−
+eff
||22
0
B2
2
1||||
)()(
eff τ
τνττφe
pE
Tk
−+ −=
Γ=
GG
NNp
eff
0ν
Phase fluctuations
Averaged value of the phase difference:
τωφτφ )()()( 0ptt −=−+ Determines mean
oscillation frequency
Variance of the phase difference:
Γ−
−+Γ=∆Γ−
+eff
||22
0
B2
2
1||||
)()(
eff τ
τνττφe
pE
Tk Determines oscillation
linewidth
−+ −=
Γ=
GG
NNp
eff
0ν – dimensionless nonlinear frequency shift
Auto-correlation function:
∆−= −
)(2
1exp)( 2)(
00 τφτ τω pi
epK
−+ −=
Γ=
GG
NNp
eff
0ν
Low-temperature asymptote
||)1()(
)( 2
0
B2 τντφ +Γ≈∆ +pE
Tk||τ
“Random walk” of the phase
Low-temperature asymptote
Lorentzian lineshape with the full linewidth
||)1()(
)( 2
0
B2 τντφ +Γ≈∆ +pE
Tk
2B2 2)1()1(2 ωννω ∆+=Γ+=∆ +
Tk
||τ
“Random walk” of the phase
Frequency nonlinearity broadens linewidth by the factor
2
2 11
−+=+
−+ GG
Nν
lin
2
0
B2 2)1()(
)1(2 ωννω ∆+=Γ+=∆ +pE
Low-temperature asymptote
Lorentzian lineshape with the full linewidth
2B2 2)1()1(2 ωννω ∆+=Γ+=∆ +
Tk
||)1()(
)( 2
0
B2 τντφ +Γ≈∆ +pE
Tk||τ
“Random walk” of the phase
lin
2
0
B2 2)1()(
)1(2 ωννω ∆+=Γ+=∆ +pE
Frequency nonlinearity broadens linewidth by the factor
2
2 11
−+=+
−+ GG
Nν
Region of validity K300~1
)(2
0effB ν+
ΓΓ
<<+
pETk
High-temperature asymptote
2
eff
2
0
B2
)()( τντφ ΓΓ≈∆ +
pE
Tk 2τ
“Inhomogeneous broadening” of the frequencies
High-temperature asymptote
Gaussian lineshape with the full linewidth
2
eff
2
0
B2
)()( τντφ ΓΓ≈∆ +
pE
Tk 2τ
||22 BTkΓΓ=∆ νω
“Inhomogeneous broadening” of the frequencies
Different dependence of the linewidth on the temperature:
T~2 ∗∆ω
)(||22
0
Beff
pE
TkΓΓ=∆ +∗ νω
High-temperature asymptote
Gaussian lineshape with the full linewidth
2
eff
2
0
B2
)()( τντφ ΓΓ≈∆ +
pE
Tk 2τ
||22 BTkΓΓ=∆ νω
“Inhomogeneous broadening” of the frequencies
Different dependence of the linewidth on the temperature:
T~2 ∗∆ω
Region of validity
)(||22
0
Beff
pE
TkΓΓ=∆ +∗ νω
K300~)(
2
0effB ν
pETk
ΓΓ
>>+
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator• 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
400
500
600
700
Linewidth (MHz)
device #1
second harmonic
high-temperature
asymptote 400
500
600
700
low-temperature
Linewidth (MHz)
device #2
first harmonic
high-temperature
asymptote
Experiment: J. Sankey et al., Phys. Rev. B 72, 224427 (2005)
0 50 100 150 2000
100
200
300
low-temperature
asymptote
Linewidth (MHz)
Temperature (K)
0 50 100 150 200 250 3000
100
200
300 low-temperature
asymptote
Linewidth (MHz)
Temperature (K)
Low-temperature asymptote gives correct order-of-magnitude estimation of the
linewidth.
Angular dependence of the linewidth
)(1
)()1(2
0
B
2
0
B2
pE
Tk
GG
N
pE
Tk+
−++ Γ
−+=Γ+=∆ νω ν
Nonlinear frequency shift coefficient N strongly depends on the
orientation of the bias magnetic field
Isotropic film, Anisotropic film,
N
0 30 60 90-20
0
20
40
Nonlinear frequency shift N/2
π (GHz)
Out-of-plane magnetization angle θ0 (deg)
0 30 60 90-10
-8
-6
-4
-2
0
Nonlinear frequency shift N/2
π (GHz)
In-plane magnetization angle φ0 (deg)
Isotropic film,
out-of-plane magnetization
Anisotropic film,
in-plane magnetization
Angular dependence of the linewidth
)(1
)()1(2
0
B
2
0
B2
pE
Tk
GG
N
pE
Tk+
−++ Γ
−+=Γ+=∆ νω
Nonlinear frequency shift coefficient N strongly depends on the
orientation of the bias magnetic field
Isotropic film, Anisotropic film,
νN
0 30 60 90-20
0
20
40
Nonlinear frequency shift N/2
π (GHz)
Out-of-plane magnetization angle θ0 (deg)
0 30 60 90-10
-8
-6
-4
-2
0
Nonlinear frequency shift N/2
π (GHz)
In-plane magnetization angle φ0 (deg)
Isotropic film,
out-of-plane magnetization
Anisotropic film,
in-plane magnetization
Angular dependence of the linewidth
150
200
250
Experiment
Generation linewidth 2
∆ω
/2π (M
Hz)
2∆ω (|N| > 0)
Out-of-plane magnetization
45 50 55 60 65 70 75 80 85 900
50
100
Experiment
10 x 2∆ω (|N| = 0)
Generation linewidth
Out-of-plane magnetization angle θ0 (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
)(1
)()1(2
0
B
2
0
B2
pE
Tk
GG
N
pE
Tk+
−++ Γ
−+=Γ+=∆ νω
Nonlinear frequency shift coefficient N strongly depends on the
orientation of the bias magnetic field
Isotropic film, Anisotropic film,
νN
0 30 60 90-20
0
20
40
Nonlinear frequency shift N/2
π (GHz)
Out-of-plane magnetization angle θ0 (deg)
0 30 60 90-10
-8
-6
-4
-2
0
Nonlinear frequency shift N/2
π (GHz)
In-plane magnetization angle φ0 (deg)
Isotropic film,
out-of-plane magnetization
Anisotropic film,
in-plane magnetization
Angular dependence of the linewidth
2000
3000
4000
Linewidth (MHz)
|N| > 0
1500
2000
2500
Linewidth (MHz)
|N| > 0
In-plane magnetization
0 30 60 90 1200
1000|N| = 0
(x 10)
Linewidth (MHz)
Field angle (deg)
0 30 60 900
500
1000
|N| = 0
(x 10)
Linewidth (MHz)
Field angle (deg)
Experiment: K. V. Thadani et al., arXiv: 0803.2871 (2008)
Outline
• Introduction
• Linear and nonlinear auto-oscillators
• Spin-torque oscillator (STO)
• Stochastic model of a nonlinear auto-oscillator• 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 • 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.