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University of Messina – Giovanni Finocchio

Modeling of Spin PolarizedCurrent-Driven Magnetization

Dynamic in Nanostructures

G. Finocchio1, L. Torres2, B. Azzerboni1

1Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate. University of Messina. Salita Sperone 31, 98166, Messina Italy.2Departamento de Fisica Aplicada. Universidad de Salamanca. Plaza de la Merced s/n 37008. Salamanca. Spain.

2006 NATO ADVANCED STUDY INSTITUTE Magnetic Nanostructures for Micro-Electromechanical

Systems and Spintronic Applications


Outline

• Introduction

• Spin-Transfer Torque in Micromagnetics

• Reversal Processes– Combined Action of AC field and SPC– Magnetic Tunnel Junction– Exchanged Biased Samples

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Introduction

Standard Micromagnetic Hypotheses

x

z

y

dVj

Mj

V

µi

a

V = Uj dVj

Mj =

Mj+1 = Mj + δθj (δθj <<1)

Micromagnetic Scale

Atomic Scale

∑i

µi /dVj

10-910-10 10-8 10-7 10-6 10-5 10-4 10-3 ... d (m)

Micromagnetics Domain Theory Macroscopic models

Quantum Mechanics

M(r) is a continuous function

Domains and domain walls are

clearly differentiated Statistic models

1 nm 1 µm 1 mm

Individual spins

-. M(r,t) is a continuos function of position (state)-.M(r,t) modulus is considered constant in each cell


( )∫ +++=V

demagzeemananisexch dvG εεεεTotal Free Energy

Exchange ( )2exch mA ∇=εiS

jSIt likes neighboring cells to be aligned

Anisotropy ( )2Kanis umK ⋅=ε It tries to align magnetization

cells along easy axis

External Field ext0zeeman HM ⋅−= µεN S

extHIt is minimum when the cells present magnetization along the field

Demagnetizing Field

Magnetostatic Contribution d0

demag HM2

⋅−= µε+ + + + + +

- - - - - - - -

dH

It tries to align the magnetization of the boundary cells tangent to the boundary so that no magnetic charges are present

Introduction

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Landau-Lifshitz Equation

( )effHMtd

Md ×= γ

M×Heff

Precession

Heff

M

( )effs

HMMM

××− γα

Dissipation

M×(M ×Heff)

Gilbert Equation

effdtd

HMM ×−= 0γ

Precession

Heff

M

dM /dt

dtd

M Seffeff

MHH

0γα−→

⎟⎠⎞

⎜⎝⎛ ×+×−=

dt

d

Mdt

d

Seff

MMHM

M αγ 0

Precession + Dissipation

MdM/dt

Heff M x dM/dt

M

G1dvH

0eff δ

δµ

−= 00 2 e

g e

m

µγ =

Effective Field

Introduction


Start

Stop Criterium

Parameters Input

End

Effective Field Calculation

Time Integration

Output: Mag. & Energy

NOT

YES

-Public codes: OOMMF

-Commercial codes: micromagus (Berkov), LLG (Scheinfein), etc

Salamanca-Messina Code, Features:-Fast: Demagnetizing interaction (97% of Comp. Time) treated by FFT techniques using Fortran subroutines optimized in assembler. Ten times faster than Numerical Recipes’ implementations.-Inclusion of temperature effect:Gauss-Seidel-Projection-Method used for time integration leading to cell-size independent results.-Inclusion of eddy currents:Coupled 3D solution of LLG and Maxwell equations.-Inclusion of Spin-Transfer Torque:By means of different polarization terms including full space dependences, Temperature, Magnetostatic coupling and Ampere Field produced in any shape and by any current distribution.

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µMAG Standard Problem #2

µMAG Standard Problem #4


Spin-Transfer Torque in MicromagneticsSpin-Transfer Torque

Slonczewski, 1996

PvR I

TSpin Polarized Current

M

( )3( , )

e

B apl

z s

g jg

L M

µη− ⋅ × ×M P M M P

-. Valid for Spin Valves in ballistic regime-. Mesoscopic Magnetization Vector Field defined as in Maxwell Eqs. -. It can be implemented at the micromagnetic scale

z xy

Py

CuMfixed=P

Mfree

J>0e-

Gilbert Eq. + Slonczewski Spin-Transfer Torque

( )0 3

2 ( , ) ( )B

EffS s

Jd dg

dt M dt d M e

µαγ ⎛ ⎞= − × + × − × ×⎜ ⎟⎝ ⎠

M MM H M M P M M P

( ) ( )[ ] 12323 4314−

+++−= /P/MP)(M, ηη SMg

M Free Layer Magnetization; Heff Effective Field

α Damping Parameter; Ms Saturation Magnetization

γ0 electron gyromagnetic ratio

µB Bohr magneton; J current density

d thickness of free layer; e electron charge

P polarization; η polarization factor

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t

∆ ∆

Lx

Ly (i,j)

Mij

P (usually uniform)

Py

Cu

Py

23

3/ 2

1( , )

(3 )4 (1 )

4

ijij s

gM

ηη

η

⋅ =+ ⋅

− + +

M PM P

⎟⎟⎠

⎞⎜⎜⎝

⎛×+×−=

dt

d

Mdt

d ijij

seffij

ij MMHM

M αγ 0 ( )3( , )

e

B aplij ij ij

z s

g jg

L M

µη− ⋅ × ×M P M M P

Precession + Dissipation(Effective Field)

Spin-Transfer Torque(Slonczweski 96)

Scalar Function (Slonczweski JMMM’96)

n n

n nη + −

+ −

−=

+

Polarization Factor, Py 0.3

0.5 1 1.5 2 2.5 3

0.15

0.2

0.25

0.3

0.35

0.5 1 1.5 2 2.5 3

0.05

0.1

0.15

0.2

Spin-Transfer TorqueAngular dependenceScalar Function

Angular dependence


Spin-Transfer Torque in Micromagnetics

( ) ( )

eff eff

2 3 2 2

'' ( )

2 2( , ) ( ) ( , )

1 1

s

B B

s s

d

dt M

J Jg g

d M e d M e

αγγ

µ µ αα α

= − × − × ×

− × × + ×+ +

MM H M M H

M P M M P M P M P γ’ γ’=γ0/(1+α2)

eff exch ani ext M Amp AF TH= + + + + + +H H H H H H H HHexch Exchange Hani AnisotropyHext ExternalHM MagnetostaticHAF Magnetostatic Coupling between PL and FL

3D simulation of the whole structureHAmp Ampere Field

Computed by means of numerical techniques developed to evaluate solenoidal fields.

HTH Thermal Field

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Snapshot of Amper field and Magnetostatic Coupling (P=1x)

Rectangle

Circle



THERMAL FIELD• No Applied Current

– Thermal effect like additive stochastic field to the effective field– LLG LLLG (stochastic differential equation)

( )zthythxthth HHH ,,, ,,=H

• Garcia-Palacios et al, PRB (55), 1998, 14937. Gaussian Process ξ (zero mean e no spatial and temporal correlation)

( ) 0=tH kth ,( ) ( ) ( )ttDtHtH kllthkth ′−=′ δδ,,

tMVTK

s

Bth ∆∆′+

=γµα

α0

212ξH KB Boltzmann constant, ∆V micromagnetic

cell volume, and ∆t time step.

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THERMAL FIELD• Li et al (PRB 69 – 2004, 134416)• Krivorotov et al , (PRL 93 – 2004, 166603). The experimental data are

well described by current dependent activation barriers that agrees with prediction of the LLG-based models.– The Spin Torque does not contain a fluctuating field.– The fluctuating field is independent by the spin torque.

• LLG and Fokker–Plank can describe magnetization dynamic.– Solution is Bolzmann distribution with an effective T* and Energy

barrier.• T* is simply the true sample temperature.• The magnetic temperature which is defined through the thermal

fluctuation remain to be T* . – the magnetization configuration of the PL is independent by the

temperature



Combined study

Micromagnetic Tool Time Domain

Micromagnetic Spectral Mapping TechniqueMicromagnetic Spectral Mapping Technique

The procedure consists of performing Fourier transform of the magnetization temporal evolution for each computational cell and then plotting the 2D spatial distribution of the spectral power at the specific frequency of the mode to be analyzed. A grey scale is used to describe the power of the mode for each cell (white small, black large), it is proportional to the sum of the square of the amplitude of Fourier transform of the x, y, and z component of magnetization of that cell and that frequency.

2( , , , ) ( , , , ) hj ftY l m n Y l m n h

h

S x y z f m x y z t e π−=∑ 2

, ,

1( ) | ( , , , ) |T Y l m n

l m n

S f S x y z fN

= ∑M. Grimsditch, et al. Physica B 354, 266, (2004). R. D. McMichael and M. D. Stiles. J. Appl. Phys. 97, 10J901, (2005). G. Finocchio, et al, J. Appl. Phys. 91 08G507 (2006). D. V. Berkov and N. L. Gorn, Phys. Rev. B 72, 094401 (2005) and Phys. Rev. B, 71, 052403 (2005).

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0.00E+000 1.00E-009 2.00E-009 3.00E-009 4.00E-009 5.00E-009

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

My/M

s

Time (s)

0.00E+000 2.00E+010 4.00E+010 6.00E+010 8.00E+010 1.00E+011

0

200

400

600

800

1000

1200

Pow

er

(A.U

.)

Frequency (Hz)

B

0.00E+000 2.00E+010 4.00E+010 6.00E+010 8.00E+010 1.00E+011

0

20

40

60

Pow

er

(A.U

.)

Frequency (Hz)

( , , , )Y l m nS x y z f

2

, ,

1( ) | ( , , , ) |T Y l m n

l m n

S f S x y z fN

= ∑

( , , , )Y l m n peakS x y z f10

2030

405101502.557.510

1020

30

40

02.557.510

10 20 30 40

2

4

6

8

10

12

14

16

58.8 GHz

220 mT+ mw 15.6 GHz


220 mT+ mw 15.6 GHz

1020

30

405

101502.557.5

10

1020

30

40

02.557.5

10

58.8 GHz

1,3

1020

30

4051015800100012001400

1020

30

40

800100012001400

1020

3040 5

10

15010203040

1020

3040

010203040

16 GHz

0,1

40.6 GHz

1,1

10

2030

405

10150

1

2

3

10

2030

40

0

1

2

3

94.3 GHz

1,510

20

3040 5

10150

0.5

1

1020

3040

0

0.5

1

146 GHz

1,7

0.00E+000 5.00E+010 1.00E+011 1.50E+011 2.00E+011

0

5

10

15

20

Pow

er

(A.U

.)

Frequency (Hz)

0,1

1,1

1,31,5 1,7

-. The magnetic normal modes are detected.-. It can be useful also for analyzing other modes compatible with a maximum in the frequency spectrum.-. Also for analyzing switching processes with some intermediate oscillations.

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Reversal Processes

• The Magnetic Random Access Memory is a KEY application of Spin Torque.

• Several strategies have been proposed todecrease the switching time.– FL Magnetic Perpendicular Anisotropy

• G. Consolo, G. Finocchio, L. Torres, M. Carpentieri, L. Lopez-Diaz, B. Azzerboni. Just presented at JEMS 2006.

– Precharging Strategy • T. Devolder et al, App. Phys. Lett., 86, 062505, (2005).

– Combined Action of AC field and SPC • G. Finocchio, I. Krivorotov, M. Carpentieri, G. Consolo, B.

Azzerboni, L. Torres. J. Appl. Phys. 99, 08G507, (2006).


Reversal Processes

z

xy

F

NMfixed=P

Mfree

J>0e-

Our study points to a strategy to decrease the current pulse duration needed for the fast switching process in nanopillar spin valve structures.

We simulate the dynamic behavior of nanomagnets in nanopillar spin valve structures due to the combined action of AC magnetic fields and DC spin polarized current.

0 0ˆ ˆcos( ) sin( )ext h t y h t zϖ ϖ= +H

0 0ˆ ˆcos( ) cos( )ext h t y h t zϖ ϖ= +H

0 1h mT= 10GHzϖ =

(Permalloy (Py) 10nm \ Cu 5nm \ Py 2.5nm)(S1: Lx x Ly x Lz = 60nm x 20nm x 2.5nm)(S2: Lx x Ly x Lz = 120nm x 40nm x 2.5nm)

We used the following Py material parameters:– MS=8.6 105A/m; Hani=0; α=0.02; η=0.3

The nanopillars have been discretized into an array of cubic cells of side of 2.5 nm. Simulations performed using cells of 5x5x2.5 nm3 gave the same results qualitatively and quantitatively the switching times vary by less than 5%. A constant time step of 40 fs has been used: simulations performed using smaller time step values yielded the same results.

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Reversal Processes

x

z

Py 2.5nm

Py 10nm

Cu 5nm

P

Lx=60nmLy=20nm

y

120 nm40 nm

60 nm20 nm

PS APS APS PS

PS APS

APS PS


Reversal Processes

Temporal evolution of <m> from the PS to the APS due to DC SPC, J=-0.8 108A/cm2.

Temporal evolution of <m> from the PS to the APS due to DC SPC + AC field J=-0.8 108 A/cm2+ AC magnetic field

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Reversal Processes

DC SPC + AC DC SPC


Reversal Processes-. Switching without AC field, J= -0.8 108 A/cm2

-. Spatial Power distributionof the main mode, 7.4 GHz

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Reversal Processes-. Switching with AC field, h0=1mT; f=10GHz

-. Spatial Power distributionof the main mode, 7.4 GHz


Reversal Processes

A comparison between temporal evolution of <mx> from the APS to the PS due to DC SPC, J=0.6 108A/cm2.

- The AC field promotes the oscillation of the left part of the structure.- The MSMT (first 1.5 ns) shows that the main mode becomes asymmetric helping the switching and avoiding the possibility of a permanent self-oscillation. - This behavior is also found also for the APS PS transition.

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Reversal Processes

• Employing a combined action of an AC magnetic field and the DC SPC we have predicted a strategy to decrease the switching time for both APS PS and PS APS processes.

• The AC field causes deviation of the y and z components of the magnetization from their initial values faster than when the DC SPC alone is applied, consequently the torque due to the current is larger during the initial steps of the switching process.

• The switching proceeds via a nucleation process. The AC field assisting the magnetization nucleation process for the cells in the left side of the structure S1. Reversal of the polarization vector of the AC field reverses the side of the structure in which the nucleation process takes place thus confirming our conjecture of the field-assisted nature of the nucleation process.


Magnetic Tunnel Junction

z xy

Py

Isolator <1nm – (AlO/MgO) Mfixed=P

Mfree

J>0e-

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Magnetic Tunnel Junction• G. D. Fuchs, I. N. Krivorotov, P. M. Braganca, N. C. Emley, A. G. F.

Garcia, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 86, 152509 (2005). Two Fixed layers – CoFe 8/AlOx 0.8/Py 4/Cu 6/CoFe 5

• L. Berger, J. Appl. Phys. 93, 7693 (2003).

( ) ( )T SVg gθ θ−( ) ( )T SVg gθ π θ+ −

2 1( ) 0.5 (1 cos( ))T T Tg θ η η θ −= +CoFe 8/AlOx 0.8/Py – ηT=0.3 CoFe 8/AlOx 0.8/Py 4 – η=0.35

( ) ( )13 3/2

2 2( ) 4 1 3 cos( ) 4SVg θ η θ η−

⎡ ⎤= − + + +⎣ ⎦

2θ π θ= −


Magnetic Tunnel JunctionMTJs exhibit a TMR that decreases when the applied voltage increases. Current induced switching occurs at a large voltage, where the current must be spin-polarized in order to exert a significant spin-transfer torque. G. D. Fuchs, N. C. Emley, I. N. Krivorotov, P. M. Braganca, E. M. Ryan, S. I. Kiselev, J. C. Sankey, D. C. Ralph, R. A. Burman, and J. A. Katine, Phys.Rev. Lett. 85, 1205 (2004).

A recent experimental study on MTJs with an MgOtunnel barrier shows that the polarization value, hT, is a constant function of the bias within a 10% uncertainty. G. Fuchs, J. A. Katine, S. I. Kiselev, D. Mauri, K. S. Wooley, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett., 96, 186603 (2006).

Starting with this experimental observation that a decrease in TMR does not indicate a decrease in the spin-polarization, we consider constant for both positive and negative currents used for our simulations.

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Top left A (t=9.1 ns), top right B (t=10.9 ns),

bottom left C (t=11.14 ns), D bottom right (t=12.5 ns).

Temporal evolution of the three components of the average magnetization from APS to PS switching due to SPC, I=0.25 mA, applied field 26mT. Inset: zoom of the range 10-14 ns.


Magnetic Tunnel JunctionI=0.25 mA, applied field 26 mT. Inset: A plot of the thermal dispersion of average magnetization calculated over 50 reversals where we have included the effects of TB= 77K.

Non ideal shapes (right) used for the micromagnetic simulations computed by SEM images (left) directly. Cells of 2.5x2.5 nm2 has been used for the discretization.

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Temporal evolution of the three components of the average magnetization from APS to PS switching due to SPC, I=0.3 mA, applied field 50mT. Inset: zoom of the range 52-64 ns of the <mX>.

Simple MTJ

The switching occurs via a nucleation process in ~65ns, in the three main steps similar to the case with two fixed layers described above, with timings as follows:

phase 1 (0-30 ns)

phase 2 (30-55 ns)

phase 3 (55-65 ns).

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Average power spectral density of the y-component of the magnetization (a. u.) for the phases 1, 2, and 3 (bottom to top respectively) during the magnetization reversal of previous slide. During these three phases the main peak frequency changes from 7.83GHz to 5GHz. It has been computed by means of MSMT.

Simple MTJ

The switching occurs via a nucleation process in ~65ns, in the three main steps similar to the case with two fixed layers described above, with timings as follows:

phase 1 (0-30 ns)

phase 2 (30-55 ns)

phase 3 (55-65 ns).

fM= 5 GHz

fM=7.83 GHz

fM =6.76 GHz



A comparison among the temporal evolution of the <mX> for APS to PS switching for different values of misalignment of the field with respect to the easy axis (I=0.3 mA, applied field 50mT).

Inset: A plot of the thermal dispersion of average magnetization calculated over 50 reversals where we have included the effects of TB= 77K.

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• Results so far– A misalignment of the applied field with respect to the easy axis does not

qualitatively change the behavior of the switching, but it reduces the switching time because the initial torque is larger and the physical symmetry of the structure is broken by the existence of a non-zero y-component of the applied field.

– A systematic study performed to determine the effect of non-idealities of the shape of the magnetization reversal reveals that the mechanism of inversion changes qualitatively depending on the shape. These non-idealities can either increase or decrease the switching time.

• Future developments– Micromagnetic simulations of Magnetization Dynamics in MTJ with MgO

barrier.– Effect of the voltage dependence of polarization in the fast reversal processes.– Direct comparison to the time-domain measurement of magnetoresistance in

MTJ.


Exchanged Biased Samples

y

x

130 nm

60 nmz

x

Py – 4nm

Cu – 8 nm

Py – 4nm

IrMn

Easy Axis

45°

Exchange Biased fiel of fixed layer –Ir20Mn80 (8 nm)

• I. Krivorotov et al, Science 307, pp. 228-231 (2005).

• Switching Time as function of the temperature.

• Effect of non-idealities in the shape (T=0K)

• Effect of exchanged biased angle (T=0K)

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• Experimental data

Dynamic Regime Switching Regime – Signal Minus Baseline



• Simulation

Switching Regime

Current applied at t=1 ns

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• Dependence on Exchanged biased Angle

I=-3.8mA I=-4.8 mA



I=-6.0 mA

• Dependence on Exchanged biased Angle

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I=-3.8 mA

• Dependence on non-idelities in the shape (30°)


Acknowledges

The authors are thankful to M. Carpentieri, G.Consolo, I. Krivorotov, J. Sankey, G. Fuchs, O. Ozatay, R.A.Buhrman, D.C.Ralph, for the helpful discussion, and to S. Bonsignore, S. Siragusa, A. Romeo and F. Bruschetta for the software support.

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Thank You for Your attention!

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