Stochastic theory of nonlinear auto-oscillator: Spin-torque nano-oscillator

Stochastic theory of nonlinear auto-oscillator:Spin-torque nano-oscillator

Vasil TiberkevichDepartment of Physics, Oakland University,

Rochester, MI, USA

NLPV, Gallipoli, Italy June 15, 2008

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

Models of (weakly perturbed) auto-oscillators

Auto-oscillator – autonomous dynamical system with stable limit cycle


Auto-oscillator – autonomous dynamical system with stable limit cycle• Phase model

Unperturbed system:

)(xFx

dt

d)( 000 tXx )()2( 00 XX



)(xFx

dt

d)( 000 tXx )()2( 00 XX

),()( xfxFx

tdt

d )())((0 tt yXx

Unperturbed system:

Perturbed system:



Unperturbed system:

Perturbed system:

)(xFx

dt

d)( 000 tXx )()2( 00 XX

),()( xfxFx

tdt

d )())((0 tt yXx



• Weakly nonlinear model

0 .5 0 .5x

0 .5

0 .5

1 .0y

1 .0 0 .5 0 .5x

0 .5

0 .5

1 .0y

thII thII




0 .5 0 .5x

0 .5

0 .5

1 .0y

1 .0 0 .5 0 .5x

0 .5

0 .5

1 .0y

thII

yixc Complex amplitude:

thII

Linear and nonlinear auto-oscillators






Nonlinear auto-oscillator: frequency depends on the amplitude




All conventional auto-oscillators are linear.

|| 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

Spin-transfer torque:

p M(t)

Electric current I

J. Slonczewski, J. Magn. Magn. Mat. 159, L1 (1996)L. Berger, Phys. Rev. B. 54, 9353 (1996)

pMMTM

0S

S M

I

dt

d

VMe

g

0

B 1

2

Equation for magnetization

Landau-Lifshits-Gilbert-Slonczewski equation:

MSGeff ][ TTMH

M

dt

d][ eff MH GT ST



– conservative torque (precession)

precession

MSGeff ][ TTMH

M

dt

d][ eff MH GT ST

][ eff MH




– dissipative torque (positive damping)

precession

M

damping

SGeff ][ TTMHM

dt

d][ eff MH GT ST

][ eff MH

]][[ eff0

GG HMMT

M


SGeff ][ TTMHM

dt

d



– dissipative torque (positive damping)

– spin-transfer torque (negative damping)

precession

M

damping

spin torque][ eff MH GT ST

][ eff MH

]][[ eff0

GG HMMT

M

]][[0

S pMMT M

I



precession

M

damping

spin torqueSGeff ][ TTMH

M

dt

d][ eff MH GT ST

When negative current-induced damping compensates natural magnetic damping,

self-sustained precession of magnetization starts.

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






• Strong influence of thermal noise (spin-torque NANO-oscillator)




• Strong influence of thermal noise (spin-torque NANO-oscillator)

• Strong frequency nonlinearity

Outline

• Introduction







• Summary

Nonlinear oscillator model

2|| cp – oscillation power

)arg(c – oscillation phase

0)()()( cpcpcpidt

dc cp)( cp)(cpi )(

precession

M

damping

spin torque


0)()()( cpcpcpidt

dc cp)( cp)(cpi )(

precession



precession

M

damping

spin torque


0)()()( cpcpcpidt

dc cp)( cp)(cpi )(

precession positive damping



precession

M

damping

spin torque

0)()()( cpcpcpidt

dc cp)( cp)(cpi )(


negative damping

positive dampingprecession



precession

M

damping

spin torque

How to find parameters of the model?

0)()()( cpcpcpidt

dc cp)( cp)(cpi )(

How to find parameters of the model?

0)()()( cpcpcpidt

dc cp)( cp)(cpi )(

• 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

0)()()( cpcpcpidt

dc

Stationary solution: )exp()( 0g0 itiptc


Stationary power is determined from the condition of vanishing total damping:

)()( 00 pp

Stationary frequency is determined by the oscillation power:

)( 0g p

Deterministic theory: Spin-torque oscillator

Stationary power is determined from the condition of vanishing total damping:

)()( 00 pp

Qp

1

0

Stationary frequency is determined by the oscillation power:

)( 0g p

thII – supercriticality parameter

0th I – threshold current

QNNp

1

000g

0 – FMR frequency

N – nonlinear frequency shift

Generated frequency

0 2 4 6 8 100

5

10

15

20

25

30

35

Ge

ne

ratio

n f

req

ue

ncy

g /2 (

GH

z)

Bias magnetic field H0 (kOe)

0 30 60 900

5

10

15

20

25

30

35

H0 = 5 kOe

Ge

nera

tion

fre

quen

cy

g /2 (

GH

z)

Magnetization angle 0 (deg)

H0 = 8 kOe

Frequency as function of applied field

Frequency as function of magnetization angle

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

– thermal equilibrium power of oscillations:

Random thermal noise:

Stochastic Langevin equation:

– scale factor that relates energy and power: dpppE )()(

)(2)()( nn tttftf

)()()()()( B

n p

TkppppD

00

2|| pc

)(nn tfD

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.

2

2n

n 2

)()(2)()()(2

P

p

pD

p

PppD

p

PpPppp

pdt

dP

Deterministic terms describe noise-free

dynamics of the oscillator

Stochastic terms describe thermal diffusion in the

phase space

P

pPpppp

)()()(2 2

2n

n 2

)()(2

P

p

pD

p

PppD

p

)()()()()( B

n p

TkppppD

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).

dp

dPppD

dp

dPppp

dp

d 0n0 )(2)()(2

Ordinary differential equation for P0(p):

Stationary PDF for an arbitrary auto-oscillator:

Constant Z0 is determined by the normalization condition 1)(0

0

dppP

p

pdp

pp

TkZpP

0B00 )(

)(1)(exp)(

Analytical results for STO

“Material functions” for the case of STO:

)1()( G Qpp )1()( pIp Gth III

Analytical expression for the stationary PDF:

Analytical expression for the mean oscillation power

Here 2)1( QQ

1

)( dttexE xt – exponential integral function

2

2

0 )(

)1)((exp

)1()(

QQE

QQpQ

Qp

QpP

QQQE

QQ

Q

Qp

1

)(

)(exp1

2

2

Power in the near-threshold region

4 5 6 7 8 90

1

2

3

Osc

illat

or p

ower

p (

a.u.

)

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

~e(t)

Classical result (see, e.g., A. Blaquiere, Nonlinear System Analysis (Acad. Press, N.Y., 1966)):Full linewidth of an electrical oscillator

20

200B2

U

RTk

Linewidth of a conventional oscillator

C L

+R0 –Rs

~e(t)

Classical result (see, e.g., A. Blaquiere, Nonlinear System Analysis (Acad. Press, N.Y., 1966)):Full linewidth of an electrical oscillator

20

200B2

U

RTk

Physical interpretation of the classical result

L

R

20

0 200 2

1CUE

LC

10

Frequency Equilibrium half-linewidth Oscillation energy

0

B02E

Tk

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

200

22

P

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

200

22

P

Physical interpretation of the classical result

0

B02E

Tk

0out0 PE2

0B

Tk

Effective thermal energy Oscillation energy

Linewidth of a spin-torque oscillator

35.850 35.875 35.900 35.925

0.0

0.1

0.2

0.3 Q(V)=2650Q(W)=4235 f=35.89 GHz

f = 13.5 MHz

Am

plit

ud

e (

nV

/Hz1/

2 )

Frequency (GHz)

Experiment by W.H. Rippard et al., DARPA Review (2004).

0eff0

B0G2

pVM

Tk

H

Theoretical result [J.-V. Kim, PRB 73, 174412 (2006)]:


35.850 35.875 35.900 35.925

0.0

0.1

0.2

0.3 Q(V)=2650Q(W)=4235 f=35.89 GHz

f = 13.5 MHz

Am

plit

ud

e (

nV

/Hz1/

2 )

Frequency (GHz)


0eff0

B0G2

pVM

Tk

H

Physical interpretation:

0

B02E

Tk



35.850 35.875 35.900 35.925

0.0

0.1

0.2

0.3 Q(V)=2650Q(W)=4235 f=35.89 GHz

f = 13.5 MHz

Am

plit

ud

e (

nV

/Hz1/

2 )

Frequency (GHz)


0eff0

B0G2

pVM

Tk

H


MHz5.13222 exp f MHz35.0222 theor f


0

B02E

Tk


35.850 35.875 35.900 35.925

0.0

0.1

0.2

0.3 Q(V)=2650Q(W)=4235 f=35.89 GHz

f = 13.5 MHz

Am

plit

ud

e (

nV

/Hz1/

2 )

Frequency (GHz)


0eff0

B0G2

pVM

Tk

H


This theory does assumes that the frequency is constant

const)( 0 p

MHz5.13222 exp f MHz35.0222 theor f


0

B02E

Tk

How to calculate linewidth?

Power spectrum S() is the Fourier transform of the autocorrelation function K():

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.

deKS i)()( )()()( tctcK


dc

Power and phase fluctuations

1)( 0

B

0

pE

Tk

p

p

)(0

)()()( titi epetptc

Two-dimensional plot of stationary PDF

)Re(c

)Im(c

|| cp

)arg(c

Power fluctuations are weak

Oscillations – phase-modulated process:

Autocorrelation function:

)0()(exp)( 0 iipK

Linewidth of any oscillator is determined by the phase fluctuations

Linearized power-phase equations


dc Stochastic Langevin equation:



dc

)(0 )()( tietpptc


Power-phase ansatz: 0|| pp



dc

)(0 )()( tietpptc

pNetfp

Dp

dt

d

etfpDpdt

pd

i

i

)(Im)(

)(Re22

n0

n0

n0neff

pN


Power-phase ansatz: 0|| pp

Stochastic power-phase equations:

0eff pGG Effective damping: dppdG )(

Nonlinear frequency shift coefficient: dppdN )(

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

20

eff

)()()(

epE

Tkptptp

Power fluctuations


||2

0

B

eff

20

eff

)()()(

epE

Tkptptp

Power fluctuations are small

)( 0

B

pE

Tk

Power fluctuations


||2

0

B

eff

20

eff

)()()(

epE

Tkptptp ||2 eff e

)( 0

B

pE

Tk

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:

)()()( 0ptt

Phase fluctuations


)()()( 0ptt Determines mean oscillation frequency

Phase fluctuations



Variance of the phase difference:

GG

NNp

eff

0 – dimensionless nonlinear frequency shift

eff

||22

0

B2

2

1||||

)()(

eff

e

pE

Tk

GG

NNp

eff

0

Phase fluctuations



Variance of the phase difference:

eff

||22

0

B2

2

1||||

)()(

eff

e

pE

Tk Determines oscillation linewidth

GG

NNp

eff

0 – dimensionless nonlinear frequency shift

Auto-correlation function:

)(

2

1exp)( 2)(

00 piepK

GG

NNp

eff

0

Low-temperature asymptote

||)1()(

)( 2

0

B2 pE

Tk||

“Random walk” of the phase


Lorentzian lineshape with the full linewidth

||)1()(

)( 2

0

B2 pE

Tk

Frequency nonlinearity broadens linewidth by the factor2

2 11

GG

N

lin2

0

B2 2)1()(

)1(2 pE

Tk

||



Lorentzian lineshape with the full linewidth

lin2

0

B2 2)1()(

)1(2 pE

Tk

||)1()(

)( 2

0

B2 pE

Tk||

Frequency nonlinearity broadens linewidth by the factor2

2 11

GG

N

Region of validity K300~1

)(2

0effB

pETk


High-temperature asymptote

2eff

2

0

B2

)()( pE

Tk 2

“Inhomogeneous broadening” of the frequencies


Gaussian lineshape with the full linewidth

Different dependence of the linewidth on the temperature:

T~2

2eff

2

0

B2

)()( pE

Tk 2

)(||22

0

Beff pE

Tk



Gaussian lineshape with the full linewidth

Different dependence of the linewidth on the temperature:

T~2

Region of validity

2eff

2

0

B2

)()( pE

Tk 2

)(||22

0

Beff pE

Tk

K300~)(

20eff

B pE

Tk


Outline

• Introduction







• Summary

Temperature dependence of STO linewidth

0 50 100 150 2000

100

200

300

400

500

600

700

low-temperatureasymptote

Lin

ew

idth

(M

Hz)

Temperature (K)

device #1second harmonic

high-temperatureasymptote

0 50 100 150 200 250 3000

100

200

300

400

500

600

700

low-temperatureasymptote

Lin

ew

idth

(M

Hz)

Temperature (K)

device #2first harmonic

high-temperatureasymptote

Low-temperature asymptote gives correct order-of-magnitude estimation of the linewidth.

Experiment: J. Sankey et al., Phys. Rev. B 72, 224427 (2005)

Angular dependence of the linewidth

)(1

)()1(2

0

B

2

0

B2

pE

Tk

GG

N

pE

Tk

Nonlinear frequency shift coefficient N strongly depends on the orientation of the bias magnetic field

0 30 60 90-20

0

20

40

No

nlin

ea

r fr

eq

ue

ncy

sh

ift N

/2 (

GH

z)

Out-of-plane magnetization angle 0 (deg)

0 30 60 90-10

-8

-6

-4

-2

0

No

nlin

ea

r fr

eq

ue

ncy

sh

ift N

/2 (

GH

z)

In-plane magnetization angle 0 (deg)

Isotropic film,out-of-plane magnetization

Anisotropic film,in-plane magnetization

N


45 50 55 60 65 70 75 80 85 900

50

100

150

200

250

Experiment

10 x (|N| = 0)

Ge

ne

ratio

n li

ne

wid

th

(M

Hz)


(|N| > 0)

Experiment: W.H. Rippard et al., Phys. Rev. B 74, 224409 (2006)

Out-of-plane magnetization

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

Nonlinear frequency shift coefficient N strongly depends on the orientation of the bias magnetic field

0 30 60 90-20

0

20

40

No

nlin

ea

r fr

eq

ue

ncy

sh

ift N

/2 (

GH

z)


0 30 60 90-10

-8

-6

-4

-2

0

No

nlin

ea

r fr

eq

ue

ncy

sh

ift N

/2 (

GH

z)

In-plane magnetization angle 0 (deg)

Isotropic film,out-of-plane magnetization

Anisotropic film,in-plane magnetization

N


0 30 60 90 1200

1000

2000

3000

4000

|N| = 0(x 10)

Lin

ew

idth

(M

Hz)

Field angle (deg)

|N| > 0

0 30 60 900

500

1000

1500

2000

2500

|N| = 0(x 10)

Lin

ew

idth

(M

Hz)

Field angle (deg)

|N| > 0

Experiment: K. V. Thadani et al., arXiv: 0803.2871 (2008)

In-plane magnetization

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 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

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