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

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 =+



• Phase model

)(xFx

=dt

d)( 000 φω += tXx )()2( 00 φπφ XX =+

Unperturbed system:

Perturbed system:

),()( xfxFx

tdt

d+= )())((0 tt yXx += φ



• 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



• Phase model

),(0 φωφ

tfdt

d=−



• 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



• Phase 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:



• Phase model

),(0 φωφ

tfdt

d=−


),(]||)([)||( 2

th

2

0 ctfccIIccNidt

dc=−−+++ βαω

dt

Linear and nonlinear auto-oscillators


• Phase model


),(]||)([)||( 2

th

2

0 ctfccIIccNidt

dc=−−+++ βαω

),(0 φωφ

tfdt

d=−

dt



• Phase model


),(]||)([)||( 2

th

2

0 ctfccIIccNidt

dc=−−+++ βαω

),(0 φωφ

tfdt

d=−

dt

Nonlinear auto-oscillator: frequency depends on the amplitude



• Phase model


),(]||)([)||( 2

th

2

0 ctfccIIccNidt

dc=−−+++ βαω

),(0 φωφ

tfdt

d=−

dt

All conventional auto-oscillators are linear.

β<<|| N

Outline

• Introduction







• 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



precession

MSGeff ][ TTMH

M++×= γ

dt

d][ eff MH ×γ

GT ST

– conservative torque (precession)

precession

][ eff MH ×γ



precession

M

damping

SGeff ][ TTMHM

++×= γdt

d][ eff MH ×γ

GT ST


– dissipative torque

(positive damping)

precession

][ eff MH ×γ

]][[ eff

0

GG HMMT ××−=

M

γα


SGeff ][ TTMHM

++×= γdt

d


precession

M

damping

spin

torque][ eff MH ×γGT ST


– dissipative torque

(positive damping)

– spin-transfer torque

(negative damping)

precession

][ eff MH ×γ

]][[ eff

0

GG HMMT ××−=

M

γα

]][[0

S pMMT ××+=M

Iσ



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


M.R. Pufall et al., Appl. Phys. Lett. 86, 082506

(2005)



• Strong influence of thermal noise

(spin-torque NANO-oscillator)


(2005)



• Strong influence of thermal noise

(spin-torque NANO-oscillator)


(2005)

• Strong frequency nonlinearity

Outline

• Introduction







• Summary

Nonlinear oscillator model

0)()()( =Γ−Γ++ −+ cpcpcpidt

dcω cp)(+Γ cp)(−Γcpi )(ω

2|| cp = – oscillation power

)arg(c=φ – oscillation phase


ω )(+ )(−)(

precession

M

damping

spin

torque




precession


ω )(+ )(−)(



precession

M

damping

spin

torque




precessionpositive

damping


ω )(+ )(−)(



precession

M

damping

spin

torque




negative

damping

positive

dampingprecession


ω )(+ )(−)(



precession

M

damping

spin

torque

How to find parameters of the model?



How to find parameters of the model?



• 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


dcω

Stationary solution: )exp()( 0g0 φω itiptc +−=


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

QQ

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







• 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)


0.1

0.2

0.3 Q(V)=2650

Q(W)=4235 f=35.89 GHz

2∆f = 13.5 MHz




0eff0

B0G2

pVM

Tk

H∂∂

≈∆ω

αω

Physical interpretation:

B02E

TkΓ=∆ω


35.850 35.875 35.900 35.925

0.0Amplitude (nV/Hz

Frequency (GHz)

0

02E

Γ=∆ω


0.1

0.2

0.3 Q(V)=2650

Q(W)=4235 f=35.89 GHz

2∆f = 13.5 MHz




0eff0

B0G2

pVM

Tk

H∂∂

≈∆ω

αω



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


0.1

0.2

0.3 Q(V)=2650

Q(W)=4235 f=35.89 GHz

2∆f = 13.5 MHz




0eff0

B0G2

pVM

Tk

H∂∂

≈∆ω

αω



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.


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


dc=Γ−Γ++ −+ωStochastic Langevin equation:



dc=Γ−Γ++ −+ω

)(

0 )()( tietpptc φδ+=


Power-phase ansatz: 0|| pp <<δ



dc=Γ−Γ++ −+ω

)(

0 )()( tietpptc φδ+=

[ ]etfpDppd iδ

δ φ=Γ+ −)(Re22


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


||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


||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


τωφτφ )()()( 0ptt −=−+ Determines mean

oscillation frequency

Phase fluctuations




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




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


Lorentzian lineshape with the full linewidth

||)1()(

)( 2

0


Tk

2B2 2)1()1(2 ωννω ∆+=Γ+=∆ +

Tk

||τ


Frequency nonlinearity broadens linewidth by the factor

2

2 11

−+=+

−+ GG

Nν

lin

2

0

B2 2)1()(

)1(2 ωννω ∆+=Γ+=∆ +pE


Lorentzian lineshape with the full linewidth

2B2 2)1()1(2 ωννω ∆+=Γ+=∆ +

Tk

||)1()(

)( 2

0


Tk||τ


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


Gaussian lineshape with the full linewidth

2

eff

2

0

B2

)()( τντφ ΓΓ≈∆ +

pE

Tk 2τ

||22 BTkΓΓ=∆ νω


Different dependence of the linewidth on the temperature:

T~2 ∗∆ω

)(||22

0

Beff

pE

TkΓΓ=∆ +∗ νω


Gaussian lineshape with the full linewidth

2

eff

2

0

B2

)()( τντφ ΓΓ≈∆ +

pE

Tk 2τ

||22 BTkΓΓ=∆ νω


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







• 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


π (GHz)

In-plane magnetization angle φ0 (deg)

Isotropic film,

out-of-plane magnetization

Anisotropic film,

in-plane magnetization


)(1

)()1(2

0

B

2

0

B2

pE

Tk

GG

N

pE

Tk+

−++ Γ

−+=Γ+=∆ νω




νN

0 30 60 90-20

0

20

40


π (GHz)


0 30 60 90-10

-8

-6

-4

-2

0


π (GHz)


Isotropic film,


Anisotropic film,



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


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)


)(1

)()1(2

0

B

2

0

B2

pE

Tk

GG

N

pE

Tk+

−++ Γ

−+=Γ+=∆ νω




νN

0 30 60 90-20

0

20

40


π (GHz)


0 30 60 90-10

-8

-6

-4

-2

0


π (GHz)


Isotropic film,


Anisotropic film,



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







• 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.

stochastic theory of nonlinear auto-oscillator: spin-torque nano-oscillator · 2008. 6. 15. ·...

Documents