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Enormous thanks to:

Allan MacDonald Joaquin Fernàndez-Rossier Maxim Tsoi Alvaro Núñez Jairo Sinova


Slonczewksi torque theory and comparison with experiments. Spin accumulation theory and comparison with experiments. Calculation of spin wave properties with Slonczewski torque. Spin transfer without spin conservation.

Experimental Evidence of Spin Transfer

Predicted theoretically by Slonczewksi and Berger in 1996


Myers et al, Science, 285, 867 (1999)

Spin Transfer Torque Picture

<S> v v



The spin of the conduction electron is rotated by its interaction with the magnetization.

This implies the magnetization exerts a torque on the spin. By Conservation of angular momentum, the spin exerts an equal and Opposite torque on the magnetization.

A more careful way of formulating spin transfer is To write down an equation of motion for spin.

Continuity requires density to change when there is a nonzero current divergence

Spin density (magnetization)

Spin current

Current dependent equation of motion for magnetization

Stiles, Phys. Rev. B, 66, 014407 (2002)

Landau-Lifshitz Equation

v v E v v & M = -M × v - M × M M




v v v v v2 ^ E[M] = -M Bapp - KM c + A M

External field Anisotropy Stiffness

Spin Currents in Trilayers




t + t

r + r



Landau Lifschitz with Spin Transfer Torque

When all the spin fluxes are evaluated for the scattering problem, one finds a simple result.

t v ^ ^ ^ Q = N( ST )1,2 = gI 1,2 × 1 × 2



New term in Landau Lifshitz equation

In experiments, one layer is pinned.

Slonczewski, J. Magn. Magn. Mat., 159, L1, (1996).

The modified Landau Lifshitz equation

^ 1 d


v ^ × Beff - × × Beff + gI × × fixed ^ ^ v ^ ^ ^ = -






The damping and spin transfer term fight with each other to determine if M will become aligned or anti-aligned with B (in the simplest geometry).

I c Beff t

Using new L.L. equation to understand experimental data

Thickness (nm)

I c Beff

Tsoi, Phys. Rev. Lett., 93, 036602, (2004)

Ic t

Albert et al., Phys. Rev. Lett., 89, 226802, (2002)

Current B

Kiselev et al., Nature, 425, 380, (2003)


Slonczewski Torque describes experiment well

What about the `W' region?

Stable point of L.L. Equation under current

Bazaliy, Jones, Zhang, Phys. Rev. B, 69, 094421, (2004)


Landau Lifshitz with Slonczewski torque

Spin transfer in continuous case

exp [i ( w ( k ) t - kx ) ]

Spin waves Spin current modifies spin wave spectrum in half metallic case

'( k ) = ( k ) + gJ S k

Static solutions to L.L. equations (like domain walls) move with velocity proportional to j.

Bazaliy et al, Phys. Rev. B, 57, R3213 (1998). Rossier et al, cond-mat 0311522. Tatara et al, PRL, 92, 086601, (2004).

Slonczewski torque term does a good job in describing experiments. There are some experiments which may not be well described by this term. There is another way of understanding spin transfer that may be useful.

Diffusive spin transport in ferromagnets

One can write down macroscopic transport equations For the spin currents. In steady state:

e J ± = µm - µ± ± x l sf 2 J± = -

Steady state "equation of continuity"

± µ ±

e x

Ohm's Law Solve the above for µ up and down, For a fixed J total, subject to continuity, etc.

Valet, Fert, Phys. Rev. B, 48, 7099, (1993).

There's more up current Coming in than going out ­ A net flux of up spin current

This is counterbalanced In steady state By a nonzero µ, Which makes spins flip from Up to down.

F µ = J e ( F lSF )

Explain how this can generate spin waves ­ write down expression for delta mu

Spin flips of conduction electrons



h = µ


= B + Ak 2

Berger, Phys. Rev. B, 54, 9354, (1996) Tsoi, Phys. Rev. Lett., 80, 4281, (1998)

µ must have right sign (>0) to create spin wave

Spin accumulation features

The critical current is inversely proportional to R Low k modes are generally excited

J= µ e ( lsf )

1 Jc R

µ > B + Ak 2


Point contact gives low k modes

Indeed, low k modes are excited...

Rippard et al., Phys. Rev. Lett., 92, 027201, (2004).

Dependence of Ic on temperature

Usually damping increases with temperature*, so Ic would also increase with temperature.

Tsoi, Phys. Rev. B, 69, 100406(R), (2004). *Stutzke et al., Appl. Phys. Lett., 82, 91, (2003)

I c (T ) ?

Spin Accumulation picture seems to do better in this case

Ic and 1/R


Critical current here is inversely proportional to the resistance

The resistance goes up with increasing temperature, so the critical current will go down.

Problems with spin accumulation picture.

How can spin waves explain magnetization reversal? How can spin waves become coherent?

Slide with chart of results and theories

Spin Torque

(I,B) phase diagram ­ switching and precession for pillar Ic proportional to damping? High k spin wave modes excited

Spin accumulation

Spin wave => switching? Ic inversely proportional to R Low k spin wave modes excited

Other formulations of spin transfer:

Shpiro, Levy, Zhang, Phys. Rev. B, 67, 104430 (2003). Heide, Phys. Rev. Lett., 87, 197201, (2001)...

The Slonczewski torque predicts high frequency excitation

Slonczewski considered the point contact geometry to linear order.

Valid for linear order in the transverse magnetization

Slonczewski, J. Magn. Magn. Mat., 195, L261, (1999).

Details of point contact calculation

^ ^ d ^ × ( Beff ) - × d + gI ( x ) × × ^ ^ ^ ^ = - dt dt



I ( x ) = I for x < a I ( x ) = 0 for x > a


Details of point contact calculation

u + ( - b - i + ij )u = 0


Dimensionless, linear L.L. equation with scaled variables

x = Re[u ] y = Im[u ]

Consider harmonic time dependence Linearize the above about Mz=1, write transverse Magnetization as a complex number

( + p )uin = 0

2 2

p 2 = b - - i + ij k 2 = b - - i

( + k )uout = 0

2 2

Do 1-d case

uin ( r ) = sin( pr ) uout ( r ) = eikr p 2 = b - - i + ij k = b - - i


Boundary conditions on u Give the eigenvalue equation:

ptan( p ) = ik

At the critical current, is purely real (recall u has exp(-it) dependence). So find (,j) real so that eq. (1) is satisfied.

For =0 case, a solution is:

= 3.82305

j = 3.35239 ka = 1.95604

Steady state spin waves at the critical current

=3a, as expected. Now investigate with greater current...

For I>Ic, the behavior can be much different.

Time and length Scales increase by an order of Magnitude for I=1.2 Ic. This is sensitive to initial Conditions.

B=.001 Alpha=.01

Ak + B


We may conclude that the Slonczewski torque in point contact geometries need not excited high k spin waves in general.

Outstanding questions:

What determines wavelength and frequency generally? (Current, wall width, etc). Do 2-d case.

Slide with chart of results and theories

Spin Torque

(I,B) phase diagram ­ switching and precession for pillar Ic proportional to damping? High k spin wave modes excited

Spin accumulation

Spin wave => switching? Ic inversely proportional to R Low k spin wave modes excited

The real situation is more complicated than either one!

Future work

Current spreading Demagnetization effects





















Another way to look at spin transfer

Spin transfer without spin conservation

ds/dt=0 for steady state. So the divergence of J equals the precession around the local moment.

Nunez, MacDonald, cond-mat/0403710

Spin transfer without spin conservation

The proposal is to say that the collective coordinate will precess around the nonequilibrium quasiparticle spins, in the same way that the nonequilibrium quasiparticle spins precess around the collective coordinate

v vC v tot = × tr

Can evalute nonequilbrium magnetization directly: with Landauer formalism, or Keldysh formalism, etc. Núñez et al. showed spin torque strong in systems with spin-orbit coupling

Toy model description

Consider ballistic transport, and determine Spin density of transport states. Use Landauer formalism and take incoming Electrons to be eigenstates of 1-d tight-binding Hamiltonian.


^ p2 1 v v = - ( x ) 2m 2

Spin Current

0.02 0.015 0.01 0.005 Qx Qy Qz

0.14 0.12 0.1 0.08 0.06

S x S y S


Spin Density



0.04 0.02 0

0 -0.005 -0.01 -0.015 0

-0.02 -0.04





-0.06 0







The wave function above satisfies the following, As it should for steady state (energy Eigenfunction)


t v Q + S × = 0

Noncollinearity is Essential to spin transfer! In the low k limit, Ein = .01 spin filter: delta = .02

Spin spiral for continuous magnetization

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 80 100 120

S x S y S


Ein = .19, =.22

Spin Torque vs. Theta

1 0.9 0.8 0.7


Spin torque for incoming electrons Polarized up


0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 140 160 180



The above is still in the low k limit, Ein=.19, delta=.21

Future work with this model: Can investigate dynamics for case of no spin conservation...

Movie of M(t)

We can solve for the spin density assuming order parameter dynamics Are adiabatic compared to quasiparticle dynamics. Can then solve for the order parameter dynamics with AA's prescription...

No units labelled! The length scales for L.L. dynamics and Sch. Equation are quite different... How to resolve??


The (I,B) phase space behavior is well described by Slonczewski torque. Other experimental features aren't explained with simple torque model, but an alternative explanation is supplied by spin accumulation Local torque model need not imply high k spin waves in point contact geometries We have a theory and a calculation for systems in which spin is not conserved Can we find other qualitative features of data which simple models can help elucidate?


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