"Spin currents in noncollinear magnetic structures: when linear response goes beyond equilibrium states"

The inverse of GMR Current Induced Magnetic Switching

Nanopillar Structure I H I Au F Cu F Cu

AP P

Dimensions ~ 60 nm x 130 nm V F = Co, Py V

SAMPLES MSU Nanopillars I ~130 nm H 2.5 – 6 nm Au Py or Co ~70nm 30 nm Py or Co Cu I Uncoupled F layers: t Cu = 10nm Ferromagnetically-coupled F layers: t Cu = 2.6 nm t Cu H Au Py or Co Py or Co Cu Antiferromagnetically-coupled F layers: t Cu = 10 nm

Hysteretic Uncoupled Co/Cu/Co Nanopillar: l 1 ~ 130 nm; l 2 ~ 70 nm.

295K AP AP

1.56

AP

1.54

MR



5%

1.52

1.50

P P I = 0 P

-1.0

-0.5

0.0

0.5

1.0

R(I) AP H = 0

How can one rotate a magnetic layer with a spin polarized current?

By spin torques: Slonczewski-1996 Berger -1996 Waintal et al-2000 Brataas et al-2000 Stiles et al-2002 By current induced interlayer coupling: Heide- 2001

E field How the electric field drives a spin system out of equilibrium

M

z

m

z 

m z

Normal metal 

mfp



sf z



m

(

z

) Spin Accumulation from left layer

M L

j

M R

z



m

(

z

) Spin Accumulation-left layer-current reversed

M L M R

z

j

Spin Accumulation from left layer 

m

(

z

)

M L

z

Spin Accumulation-left layer-current reversed 

m

(

z

)

M L M R

z

M R

j j How reversal in current directions changes alignment of layers



m

(

z

) Spin Accumulation from central layer

M 0 M R

z

M L

j

One thick-One thin layer

t N

 

N sdl M d

1

M d

2

j e j m M d

2 alone

t pol ari zer

 

F sdl



j e t F M d

1 ||

M d

2

M d

1 

M d

2

j e



First attempt: Spin diffusion equation Spinor current

j

(

x

)  ˆ (

x

) 

ˆ



e

2

ˆ ( 

F

)

ˆ



ˆ



C

0

n

0 ˆ ˆ

    

C

, ˆ

 

m

D

0 ˆ

  

D

,

C

 

C

0

M

d

,

D



D

0

M

d

 

x

Accumulation of spin near interfaces alters equilibrium densities Asya Shpiro et al. Phys. Rev.B

67

,104430 (2003).

j

m

 

j e

M

d

 2

D

0   

m



x

   

M

d

(

M

d

 

m



x

)   Part due to band structure Contribution from diffusive processes

Stationary solutions for spin accumulation Longitudinal  2

m

| | 

x

2 

m

| |  2

sd l

 0 

sd l

 1    

sf

Transverse  2

m

 

x

2 

m

  2

sf



m

  

J

2

M

d

 0 

J



d J



mfp

3  d J 

hv F 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 effect of electric field in electrons.

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

Layer by layer approach to transport in metallic structures: •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.

Conclusion: Attendant to current driven across inhomogeneous media there is charge 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 magnetic structures, 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 by correlations 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 in equilibrium.

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

They can only be induced by the This only happens if the symmetry is lowered by the presence of spin polarized currents.

spin-flip scattering at the interfaces. For the same reason that spin accumulation enters in linear response the 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.



f ss

'   '

k

ˆ

mm

'

T mm

'

ss

' 

f mm

' 

k

ˆ ' , 

k

, 0     '

k

ˆ

mm

''

R mm

'

ss

' 

f mm

' 

k

ˆ '' , 

k

, 0  

T mm

' 

ss R mm

' 

ss

' '  

T mm

'

ss

'

R mm

'

ss

'  

t ms



t



m

'

s

'

r ms



r



m

'

s

'  Note that by matching distribution functions, rather than wavefunctions we lose some 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 calculation Local axis coordinates

Methodology: Boltzmann equation using the layer-by-layer approach Boltzmann equation for spin currents in ferromagnetic metals.

See Jianwei Zhang

et al.

, PRL

93

, 256602 (2004).

Longitudinal

Equations of motion for distribution functions



t f p



v p x



x f p



eE v p

 (   

F

)  

f p

 

p f p



f p

 

sf f p

' Transverse 

t f p

 

v p x



x f p

 

i J p



f p

  

f p

  

p f p

 Transient response is crucial to understanding states off the Fermi surface contribute to conduction in linear response.

Band structure of Co

Charge current Circuit theory Spin current Parallel to magnetization Circuit theory Due to accumulation

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 represent populations, and the off diagonal which are coherences between 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 interact with one another, e.g., see Valet and Fert PRB

48

, 7099 (93).

•

CISP’s

is our way of introducing in a transients steady state calculation that admix excited k states into the ground state so as to arrive at the correct steady state.

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

For example, for transverse distribution function

Connection formulae across interfaces see 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

0.3

0.2

0.1

0.0

0.8

0.7

0.6

0.5

0.4

-0.1

0.8

-20

Spin currents

 =90 0 A=0.05 -15 -10 -5 0 X (nm) 5 10 |j + |(x) j z (x) 15 20 0.7

0.6

0.5

 =90 0 A=0.5 |j + |(x) j z (x) 0.4

0.3

0.2

0.1

0.0

-0.1

-20 -15 -10 -5 0 X (nm) 5 10 15 20

Spin currents

0.2

0.0

-0.2

-0.4

-0.6

-0.8

1.0

0.8

90 0 0.6

0.4

Components referred to local axes j x j y j z j tr -15 -10 -5 0 X (nm) 5 10 15

Transverse components of Spin Current

1.0

//

J m

0.8

d J 0.6

0.4

0.2

90 0  =6nm d J =10nm  sf =250nm  tr

J m

 0.0

-0.2

-0.4

-0.6

-0.8

J z (x) J x (x) J y (x) J + (x) Half width 3.1nm

-1.0

-15 -10 -5 0 X (nm) 5 10 15

Resistance

0.4

0.2

0.0

1.0

0.8

0.6

0 15 30 45 60 75 90 105 120 135 150 165 180 A=0.00 A=0.05 A=0.50 0.4

0.2

0.0

1.0

0.8

0.6

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

Spin torque as a function of angle between layers for three different cases of current induced spin flip (

CISP

) 0.6

0.5

0.4

0.3

0.2

0.1

0.0

0 A=0.5

A=0.05

A=0 30 60 90 Angle 120 150 180

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.

•

Observatio

n: Transmission from

Cu

to

Co

favors majority channel; 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.

  

j e



j e

Spin currents  The magnetization current normalized to ( at times

t

 0.2



sf

,  

B

/

e

)

PJ e

as a function of position

sf

, and 5 

sf

.

From: S. Zhang and P.M. Levy, Phys. Rev. B

65

, 052409 (2002).

Time evolution of spin current for layers 90

0

apart Components referred to global axes

Time evolution of spin current for layers 90

0

apart Components referred to global axes

Time evolution of spin current for layers 90

0

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



mfp

10 9 8  tr (nm) 5 4 6 7 3 2 1 0 0

d J

 

J J

  

h v F J d



Diff J

3  

mfp

5 10 15  J for d J =12nm  Boltz for d J =6.28nm

 J for d J =6.28nm

 J for d J =1nm  Boltz for d J =1nm 20 25 30  mfp (nm) 35 40  Boltz for d J =12nm 45 50

10 9 8 7  tr (nm) 5 6 4 3 2 1 0 0

Dependence on J

 Diff for  mfp =6nm  Boltz for  mfp =6nm 5 10 d J (nm) 15 20 25

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

They have used the Landauer-Keldysh formalism and are capable of finding spin accumulation; both longitudinal as well as transverse.

However, to date their calculations have been done only for the steady state . The new feature of transport in noncollinear structures is that it is necessary to calculate a transient phase during 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 parameter

J

not depend on the current induced spin flip scattering; rather it is set by the exchange 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