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Spin Torque for DummiesSpin Torque for Dummies

Matt Pufall, Bill RippardShehzu Kaka, Steve Russek, Tom Silva

National Institute of Standards and Technology,Boulder, CO

J. A. KatineHitachi Global Storage Technologies

San Jose, CA

Mark Ablowitz, Mark Hoefer, Boaz IlanUniversity of Colorado Applied Math Department

Boulder, CO

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OutlineOutline

1) Spin dynamics

2) Magnetotransport

3) Spin momentum transfer

4) Spin torque nano-oscillators

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But first… a puzzle!But first… a puzzle!

We start with a polarized beam of spin 1/2 Ag ions…

… we pass the beam through a magnetic field gradient…

… and the beam “diffracts” into two beams, polarized along the axis of the magnetic field.

Q: What happened to the angular momentum in the original polarization direction???

(For more details, see Feynman’s Lectures on Physics, Vol. III, page 5-1)

The “Stern-Gerlach” experiment

… but true!!!

Very strange

…

H

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Part 1: Spin dynamicsPart 1: Spin dynamics

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Magnets are Gyroscopes!Magnets are Gyroscopes!

2B

Le

e

m

e-

r

v

z

2 Bs

For spin angular momentum, extra factor of 2 required.

“The gyromagnetic ratio”

L r p

angular momentum for atomic orbit:

L IA

magnetic moment for atomic orbit:

2 ˆ2

evr z

r

ˆ2

evrz ˆrmvz

Classical model for an atom:

2

e

evr

rm v

zL

zL

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Larmor EquationLarmor Equation

H

M0T M H

T

dLT

dt

L

0

dMT

dt

M H

(definition of torque) (gyromagnetic ratio)

Magnetic field exerts torque on magnetization.

Joseph Larmor

Magnets make me

dizzy!

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0

d

s

T

M M HM

damping torque

0

precession torquepT

M H

Landau & Lifshitz (1935):

0

0

p d

s

dMT T T

dt

M H

M M HM

= dimensionlessLandau-Lifshitzdamping parameter

Gyromagnetic precession with energy loss: The Gyromagnetic precession with energy loss: The Landau-Lifshitz equationLandau-Lifshitz equation

L. Landau and E. Lifshitz, Physik. Z. Sowjetunion 8, 153-169 (1935).

H

Lev Landau

Damping happens!!

Lev Landau

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Part 2: Charge transport in magnetic heterostructures Part 2: Charge transport in magnetic heterostructures (Magneto-transport)(Magneto-transport)

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Ferromagnetism in Conductors: Band StructureFerromagnetism in Conductors: Band Structure

D(E)

E“s-p band”

EF

spin-up spin-down

D(E)

E“d band”

EF

“majority” states

“minority” states

“Density of States”

“exchange splitting”

s-band states (l = 0) have higher mobility than d-band states (l = 2) in conductors due to the much lower effective mass of the s-band.

Minority band electrons scatter more readily from the s-p band to the d-band due to the availability of hole states in the minority d-band.

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Spin-dependent conductivity in ferromagnetic metalsSpin-dependent conductivity in ferromagnetic metals

“majority band”

“minority band”

M

JJ

Conductivities in ferromagnetic conductors are different for majority and minority spins.

In an “ideal” ferromagnetic conductor, the conductivity for minority spins is zero.

Analogy: Like water flowing through pipes. Large conductivity = big pipe, etc…

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Concept: interfacial spin-dependent scatteringConcept: interfacial spin-dependent scattering

M

“Majority” spins are preferentially transmitted.

”Minority” spins are preferentially reflected.

Normal metal Ferromagnet

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Concept: ferromagnets as spin polarizersConcept: ferromagnets as spin polarizers

M

“Majority” spins are preferentially transmitted.

Normal metalFerromagnet

Ferromagnetic conductors are relatively permeable for majority spins. Conversely, they are impermeable for minority spins.

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Concept: spin accumulationConcept: spin accumulation

M

z

I

sM z M M

Non-equilibrium spin polarization “accumulates” near interfaces of ferromagnetic and non-magnetic conductors.

“spin diffusion length”

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Part 3: Spin momentum transferPart 3: Spin momentum transfer

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Non-collinear spin transmissionNon-collinear spin transmission

M

What if the spin is neither in the majority band nor the minority band???

????

????

Is the spin reflected or is it transmitted?

Quantum mechanical probabilities:

2

2

1Pr 1 cos

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Pr 1 cos2

A

B

= +A B

Quantum mechanics of spin:

cos2

sin2

A

B

An arbitrary spin state is a coherent superposition of “up” and “down” spins.

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Spin Momentum Transfer: Small Current LimitSpin Momentum Transfer: Small Current Limit

M

+ =-T

+Tdamp

MPolarizer: At low electron flux,

damping torque compensates spin torque: Magnetization is stable.

e-

M Mf

; 1T

+ =T

Electrons:

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Spin Momentum Transfer: Large Current LimitSpin Momentum Transfer: Large Current Limit

T is driven by spin accumulation in the Cu spacer.

Spin accumulation is proportional to current flowing through the structure.

+ =

M

+ =

T

-T+Tdamp

M

Electrons:

Polarizer:Spin torque exceeds damping torque: Polarizer reacts with changing M. Torque proportional to angle : Unstable!Damping torque

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Transverse torque via spin reorientation/reflectionTransverse torque via spin reorientation/reflection

Consider only reflection events...

For the electron:

ˆ

ˆsin2

ˆ ˆ ˆ2

ˆ ˆ ˆ2

transversetransverse inc

inc

f

ss s y

s

y

m m p

m m m

transverses

incsrefs

AND

Consider only change in angular momentum transverse to magnetization axis. (Equivalent to assuming magnitude of M does not change.)

y

x

where2

ˆ incp s

p̂m̂

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Newton’s Second LawNewton’s Second Law

ˆ ˆ ˆ2trans fs m m m

If…

…then…

ˆ ˆ ˆ2 fM V m m m

…per electron.

For a flowing stream of electrons:

2

ˆ ˆ ˆ2

ˆ2

f

fs

m m mdM I

dt V e

IM M m

eM V

Total moment of “free” magnetic layer

Rate of electron impingement on “free” layer

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The Slonczewski Torque TermThe Slonczewski Torque Term

2ˆ( )

2Sloncewski fs

JT M M m

e M

J. Slonczewski, Journal of Magnetism and Magnetic Materials, vol. 159, page L1 (1996)

J M Mf

efficiency* ~ 0.2 - 0.3

*Accounts for all those messy details: Polarization of ferromagnet, band structure mismatch at interface, spin decoherence, etc…

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The “FAQ Page”The “FAQ Page”

Q: What if the spin is transmitted through the “free” layer rather than reflected?

A: Doesn’t matter. Quantum mechanically, there is an amplitude for both transmission and reflection, but only for spin along the axis of magnetization. The transverse component of spin for the incident electrons is “lost” once the electron wavefunction is split into the transmitted and reflected components. Conservation of angular momentum dictates that the transverse component is transferred to the magnetic layer. This is a purely quantum mechanical phenomenon: There is no classical analog!

Q: Don’t the reflected spins affect the spin accumulation in the spacer layer?

A: Yes, they do. There are several theories that take this “back-action” on the spin accumulation into account. See J. C. Slonczewski, J. Magn. Magn. Mater. 247, 324 (2002); A. A. Kovalev, et al., Phys. Rev. B 66, 224424 (2002); J. Xiao, et al., Phys. Rev. B 70, 172405 (2004); A. Fert, et al., J Magn. Magn. Mater. 69, 184406 (2004).

0 1 0 1cos cos

q q

B B B B

Tsm

t

= constant()

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Back to the puzzle…Back to the puzzle…

A: The quantum equivalent of a card trick. If a card “vanishes” magically from one deck, it must reappear somewhere else. No mechanism for the transfer of angular momentum need be invoked!

The “Stern-Gerlach” experiment, revisited.

Once the linear momentum for the two spin components is split, the transverse angular momentum is “released” to do work on the magnet system.H

“Up” spin current

“Down” spin current

0xS

1

2xS

x

y

z

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Larmor damp smt

dMT T T

dt

0 eff

dMM H

dt

Larmor term: precession

Mz

Mx

M

Heff

Tsmt -Tdamp J ~ 107A/cm2

Slonczewski 1996

Mz

Mx

Mi

MfHeff

Damping term: aligns M with H

02

( )effs

M M HM

Precession

Heff

Spin Torque

Damping

M

Spin Torque Term

2ˆ( )

2B

fs

JgM M m

e M

Spin torque cancounteract damping

mf

Magnetodynamics: Three TorquesMagnetodynamics: Three Torques

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How Can We See This?Torque to current density: must have high current densities to produce large torques

1 mm

10 m

I = 0.1 MA

I = 10 A

100 nm I = 1 mA

We will use nanopillar and nanocontact structures

Typical wire

Size of a human hair

500 atoms across

X

Required Idc Possible

X

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Part 4: Spin torque nano-oscillators

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• Step DC current• Measure DC R, microwave

power output

5 6 7 8 924.8

25.0

25.2

25.4

dV/d

I ()

Current (mA)

9.6 9.7 9.8 9.9 10.0

0.0

0.1

0.2

0.3

0.4

9 mA

8.5 mA 8 mA

7.5 mA

7 mA

6.5 mA

6 mA

5.5 mA

Pow

er (

pW)

Frequency (GHz)

Devices are nanoscale current-controlled microwave oscillators

Au

Cu“Fixed” layer

“Free” layer

~40 nm

0.7 T, = 10o

T = 300K

Nanocontact DynamicsNanocontact Dynamics

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SummarySummary

Magnetization dynamics tutorial: Magnets are gyroscopes.

Magnetotransport tutorial: Magnets are spin filters.

Spin momentum transfer: Back action of spin polarized carriers on magnet.

Spin torque nano-oscillator: Spin torque compensates damping.

An excellent review article!!

M. D. Stiles and J. Miltat, “Spin Transfer Torque and Dynamics,” Topics in Applied Physics 101, 225-308 (2006).