"Spin currents in noncollinear magnetic

structures: when linear response goes beyond equilibrium

states"

Nanopillar Structure

F

F

Au

Cu

CuP

AP

I

I

V

V

F = Co, PyH

Dimensions ~ 60 nm x 130 nm

The inverse of GMR-Current Induced Magnetic Switching

SAMPLES

MSU Nanopillars

tCu

Py or Co

CuI

Au

I

Py or Co2.5 – 6 nm

30 nm

~130 nm

~70nm

Uncoupled F layers: tCu = 10nm

Ferromagnetically-coupledF layers: tCu = 2.6 nm

H

Antiferromagnetically-coupledF layers: tCu = 10 nm

Cu

Au

Py or Co

Py or Co

H

-6 -4 -2 0 2 4 61.50

1.52

1.54

1.56

-1.0 -0.5 0.0 0.5 1.0

1.50

1.52

1.54

1.56

dV/dI (Ohm)

I (mA)

dV/dI (Ohm)

H (kOe)

Uncoupled Co/Cu/Co Nanopillar: l1 ~ 130 nm; l2 ~ 70 nm.

H = 0

MR 5%

R(I)

P P

AP AP

AP

P

I = 0

AP

P

Is

295K

Hysteretic

How can one rotate a magnetic layer with a spin polarized current?By spin torques:Slonczewski-1996Berger -1996Waintal et al-2000Brataas et al-2000Stiles et al-2002

By current induced interlayer coupling:Heide- 2001

Mz

Efield

z

How the electric field drives a spin system out of equilibrium

δmz

λmfp

λsf

mz

Normal metal

)(zmδ

ML

MR

j

z

Spin Accumulation from left layer

ML

MR

j

z

Spin Accumulation-left layer-current reversed)(zmδ

δm(z)

ML

MR

j

z

Spin Accumulation from left layer

δm(z)

ML

MR

j

z

Spin Accumulation-left layer-current reversed

How reversal in current directions changes alignment of layers

M0

MR

j

z

ML

Spin Accumulation from central layer)(zmδ

1dM

2dM

Ft

tN <<λNsdl

One thick-One thin layer

jeje

tpolarizer>>λsdlF

jmβje

€

M d1 ≠M d

2

Md2 alone Md

1 ||Md2

ˆ C =e2 ˆ N (εF)ˆ D

ˆ C =C0ˆ I +σ ⋅C, D =D0

ˆ I +σ ⋅D,

ˆ n =n0ˆ I +σ ⋅m

C=βC0Md, D= ′ β D0Md

j (x)=ˆ C E(x)−ˆ D ∂ˆ n ∂x

First attempt: Spin diffusion equation

Spinor current

jm =βjeMd −2D0

∂m∂x

−β ′ β Md(Md ⋅∂m∂x

)⎡ ⎣ ⎢

⎤ ⎦ ⎥

Part due to band structure

Contribution from diffusive processes

Accumulation of spin near interfaces alters equilibrium densities

Asya Shpiro et al. Phys. Rev.B67,104430 (2003).

∂2m||

∂x2 −m||

λ2sdl

=0

λsdl = 1−β ′ β λsf

Stationary solutions for spin accumulation

Longitudinal

Transverse

€

∂2m⊥

∂x 2−

m⊥

λ2sf

−m⊥× Md

λ J2

= 0

λ J =dJλ mfp

3π dJ = hvF J

Spin transport in magnetic multilayers

Linear response:

• only electrons close to Fermi surface contribute to conduction

• only equilibrium band structure is necessary to describe effectof electric field in electrons.

The case for spin accumulation; does it enter in linear response?

•Due to screening in metals transport in each layer can be modeled by equilibrium band structure.•Solve for distribution function ( statistical density matrix) in each layer by using the Boltzmann equation.•The distribution functions describing the out of equilibrium transport across layers are connected by the scattering matrices at the interfaces.

Layer by layer approach to transport in metallic structures:

Conclusion:

Attendant to current driven across inhomogeneous media there ischarge and spin redistribution so as to maintain a steady state current.

As seen from the Boltzmann equation this out of equilibrium accumulation enters in linear response.

In the Kubo approach it alters the local electric field seen by the electrons from that externally applied. For magnetic media the effective field for spin channels are different.

This has been sufficient to describe transport in collinear magneticstructures, but it is in sufficient when the magnetic layers are noncollinear?

Spin transport in noncollinear magnetic multilayers

Does one have to alter the layer by layer approach to transport in metallic structures that has worked so well for collinear magnetic multilayers?

Yes

How

Current induced coherences

Spin distributions transverse to the magnetization are described bycorrelations between up and down spin states.

In the one electron spin polarized picture of ferromagnetic band structure there are no correlations between spin split bands inequilibrium.

For these correlations to exist they must be induced by the current.

They can only be induced by the spin-flip scattering at the interfaces. This only happens if the symmetry is lowered by the presence ofspin polarized currents.

For the same reason that spin accumulation enters in linear responsethe effect of this current induced coherence also enters linearly.

In a layer by layer approach to calculating the overall conductance or resistance of a multilayer it is necessary to relate the distribution functions for each layer across their interface with an adjacent layer, by using reflection and transmission coefficients.

( ) ( ) ( )+<−>+> ∑∑ += 0,,''ˆ0,,'ˆ0,''ˆ'

''''ˆ'

'''' kkmm

mmssmmkkmm

mmssmmss kfRkfTkf εε

€

Tmm'⇒ ss' = Tmm'ss' = tms∗tm 's'∗

Rmm'⇒ ss' = Rmm'ss' = rms∗rm's'∗

Note that by matching distribution functions, rather than wavefunctions we losesome information, i.e., we lose the coherence of the wavefunction across the multilayer.

If the distance between ferromagnetic layers exceeds the interlayer coupling distance due to the equilibrium RKKY-like coupling then there is a unique direction for the magnetization at each N/F interface for the system in equilibrium. This dictates that in equilibrium the scattering amplitudes are diagonal in spin space when referred to this unique axis, and that the transmission and reflection coefficients can only transmit the component of an incoming spin current that is parallel to the magnetization of the ferromagnetic layer, i.e, there is no transmission of transverse spin currents.

However, when a current flows across a magnetic multilayer the spin accumulation created at one N/F interface is superimposed on other N/F interfaces that are within a spin diffusion length of it. When the layers are noncollinear the symmetry at the N/F interfaces is lowered so that scattering amplitudes contain off diagonal components in spin space, and transverse components of the spin current are transmitted.

Steady state calculationLocal axis coordinates

Methodology: Boltzmann equation usingthe layer-by-layer approach

Boltzmann equation for spin currents in ferromagnetic metals.See Jianwei Zhang et al., PRL 93, 256602 (2004).

Equations of motion for distribution

functions

sf

pp

p

pp

Fppxxppt

ffffveEfvf

ττεεδ ')(

−−

−−=−−∂+∂

p

pp

pp

pxxppt

fff

Jifvf

τ

±±±±±

−−=∂+∂

hm

Longitudinal

Transverse

Transient response is crucial to understanding states off the Fermisurface contribute to conduction in linear response.

Band structure of Co

Parallel to magnetization Due to accumulation

Charge current

Spin current

Circuit theory

Circuit theory

Definition of transverse spin current, in the steady state

•In a statistical density matrix, e.g., the Boltzmann distribution function, there are diagonal matrix elements which representpopulations, and the off diagonal which are coherencesbetween states.

•For noncollinear multilayers one must be mindful of coherences.

•In equilibrium magnetic layers are not magnetically coupled;in the presence of a spin current across a normal spacer the scattering at the opposite interfaces of the spacer interactwith one another, e.g., see Valet and Fert PRB 48, 7099 (93).

•CISP’s is our way of introducing in a steady state calculationtransients that admix excited k states into the ground stateso as to arrive at the correct steady state.

Solution for multilayer is to find distribution function in each layer by using Boltzmann equation. To determineunknown constants one has to match functions across layers by using the transmission and reflection coefficients.

For example, for transverse distribution function

Connection formulae across interfacessee P.M. Levy and Jianwei Zhang, PRB 70, 132406 (2004)

This leads to the “mixing conductance in the conventional view.

A is the new current induced spin-flip term

Interference pattern- y polarization

Interference pattern - x polarization

No spin-flip at interfaces

Spin currents

-20 -15 -10 -5 0 5 10 15 20-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

X (nm)

|j+|(x)

jz(x)

Spin Currents

θ=900 =0.05 A

-20 -15 -10 -5 0 5 10 15 20-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

X (nm)

Spin Currents

θ=900 =0.5 A

|j+|( ) x j

z( ) x

Spin currents

-15 -10 -5 0 5 10 15

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

X (nm)

jx

jy

jz

jtr

spin currents

900

Components referred to local axes

Transverse components of Spin Current//mJ

⊥mJ

-15 -10 -5 0 5 10 15-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

λtr

dJ

Spin Current J

m

( )X nm

900 λ=6nmd

J=10nm

λsf=250nm

Jz( )x

Jx( )x

Jy( )x

J+( )x 3.1Half width nm

Resistance

0 15 30 45 60 75 90 105 120 135 150 165 180

0 15 30 45 60 75 90 105 120 135 150 165 180

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Resistance

Angle

A=0.00 A=0.05 A=0.50

Spin torque as a function of angle between layers for three different cases

of current induced spin flip (CISP)

0 30 60 90 120 150 1800.0

0.1

0.2

0.3

0.4

0.5

0.6

A=0.5 A=0.05 A=0

Torque

Angle

Consequences

• Resistance is lower when one admits transverse currents in ferromagnetic layers.

•Angular variation of resistance and spin torque is changed upon including current induced spin flip, CISP, at interfaces.

•Spin torque is increased for same amount of energy expended when one includes CISP.

•True “mixing” conductance with an effective field component, as well as torque.

•Spatial variation of spin torque and effective field very different.

•Observation: Transmission from Cu to Co favors majoritychannel; penalizes minority channel conduction.

Time dependence of spin transport

Solution is found across entire multilayer by using source terms at interfaces. This obviates any assumptions about the scattering at interfaces; they are built into the Hamiltonian.

The magnetization current normalized to (μB /e)PJ e as a function of position

at times t =0.2τsf,τsf, and 5τsf.

From: S. Zhang and P.M. Levy, Phys. Rev. B65, 052409 (2002).

€

−βje

€

βje

Spin currents

Time evolution of spin current for layers 900 apart Components referred

to global axes

Time evolution of spin current for layers 900 apart Components referred

to global axes

Time evolution of spin current for layers 900 apart Components referred

to global axes

Discontinuity in inhomogeneous solution

Spin currents at steady state

Spin accumulations at steady state

Time dependence of spin torque

Initial Build Up

Spatial Variation

Dependence on mfpλ

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

10

λtr( )nm

λBoltz

for dJ=12nm

λJ for d

J=12nm

λJ for d

J=6.28nm

λBoltz

for dJ=6.28nm

λJ for d

J=1nm

λBoltz

for dJ=1nm

λmfp ( )nm

DiffJ

mfpJJ

FJ

d

J

vhd

λλπλ

λ

=

=

=

3

Dependence on J

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

λtr( )nm

λDiff forλ

mfp=6nm

λBoltz

forλmfp=6nm

dJ ( )nm

Why ab-initio have not found the transverse length scalewe find.

They have used the Landauer-Keldysh formalism and arecapable of finding spin accumulation; both longitudinal aswell as transverse.

However, to date their calculations have been done only forthe steady state. The new feature of transport in noncollinearstructures is that it is necessary to calculate a transient phaseduring which states of opposite spin, off the Fermi surface,are mixed into those on the Fermi surface.

Quite aside from the origin of the current induced spin flip scattering we conclude by presenting its effect on the spin transport in noncollinear magnetic multilayers.• Its very existence assures the continuity of the spin current across the N/F interfaces.• The amount of spin accumulation transverse to the magnetization is proportional to this current induced spin flip scattering.• The length scale for the transverse accumulation does not depend onthe current induced spin flip scattering; rather it is set by the exchange parameter J entering equation of motion for the transverse distribution function.• For a given potential drop the charge current is always larger when this spin-flip scattering is present at the interface,i.e., the resistance is always lower.

• The angular dependence of the resistance, and therefore the spin torque, depends on the amount of this current induced spin flip scattering.

Conclusions

In the conventional approach there is a discontinuity in the spin current at the N/F interfaces, while in our approach there is none.

Therefore the origin of the discontinuity cannot be the band mismatch per se; rather the key difference is that we account for the coherence between states of opposite spin in the ferromagnetic layers that is lost in effective single electron treatments.There is a problem with the layer-by-layer approach to calculating

transport in noncollinear magnetic structures; among other things it overlooks the fact that for these structure there are no pure spin states