Spin currents in non- collinear magnetic tunnel junctions

and metallic multilayers

Peter M Levy

New York University, USA

Long before GMR per se was discovered, there existed, by 1972, another magnetoresistive effect that resembles Current Perpendicular to the Plane Magnetoresistance (CPP-MR); that is tunneling magnetoresistance (TMR). The difference between them is the spacer layer between the magnetic layers. In GMR it’s a nonmagnetic metal whereas for TMR it’s an insulator. The difference is important because it determines the type of conduction process that transmits the current between the magnetic entities [grains or layers]. For a metallic spacer, transmission takes place by conduction electrons at the Fermi level; whereas for an insulating spacer there are no electrons at the Fermi level as the insulator falls in a gap between conduction and valence bands: therefore electrons “tunnel”, in the quantum mechanical sense, between the magnetic entities.

Background to TMR

Conduction electrons have wavefunctions that oscillate between positive and negative amplitudes with a frequency related to the wavelength at the Fermi level, e.g., for a typical 3d transition-metal this is on the order of 1 Å. This is a rapid oscillation so that minute details of the roughness of the interfaces [of this lengthscale] between the spacer and magnetic layers affect the electrical conduction process. Indeed this is why the details of the roughness and diffusion at the interfaces are crucial for predictions of ab-initio calculations of GMR in metallic multilayers. Electrons that tunnel between magnetic entities do not have oscillatory wavefunctions; rather they decay exponentially. In this case details about the interfaces with the magnetic entities are less important. This is the primary reason ab-initio calculation had a far greater success in predicting TMR behavior ..

TMR was first observed in the tunneling between grains in granular nickel films by Gittleman et al in 1972. Michel Jullière was the first to observe it in the more conventional multilayer geometry in 1975 known as Magnetic Tunnel Junctions (MTJ) where he found 14% TMR at low temperatures for Fe(iron)/Ge(germanium) /Co(cobalt); this was followed by Maekawa and Gäfvert’s observations, in1982, of TMR by using nickel, iron and cobalt electrodes across nickel oxide barriers. Then, in 1995, Miyazake and Moodera both observed reproducible TMR in MTJ’s. Their work came at a propitious time when there was increased interest in magnetoresistive elements and it gave rise to a flurry of activity in this field.The first phenomenological models of TMR were provided by Gittleman et al. and Jullière, and theoretical work on MTJ’s was first done by John Slonczewski. Ab-initio calculations came close on the heels of the findings of Miyazake and Moodera and were based on the Landauer-Büttiker formalism of conduction.

This formalism, which is suitable for ballistic transport, was previously used for the contribution of band structure to the GMR in metallic multilayers. Transport in metallic systems is usually described as diffusive; this is in large part due to the oscillatory wave functions at the Fermi surface which are the carriers in metallic structures (of course, impurity scattering is also necessary). However, while the transport in the ferromagnetic electrodes may be diffusive, the tunneling across the insulating barrier is through evanescent states and this part of the conduction can be ballistic, in which case one can apply a Landauer-Büttiker-like analysis to TMR. Also, as tunneling currents are small compared to currents in metals, the role of current-driven charge and spin accumulation do not have a big effect on the resistivity of MTJs, i.e., their neglect does not change one’s predictions for the TMR of MTJs.

€

I p =2eh

Tβ←α μα −Tα ←βμ β[ ]

μα ≈ μ L μ β ≈ μ R

ˆ T β←α ∝ ˆ ρ α ˆ t αβ( )*

ˆ ρ β ˆ t βα( )

Particle current:

€

ˆ t βα =td + tmSz

α /β tmS−α /β

tmS+α /β td − tmSz

α /β

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ t βα = tdˆ 1 + tm

r σ ⋅

r S α /β

Density matrix:

€

ˆ ρ =ρ↑ 0

0 ρ↓

⎛

⎝ ⎜

⎞

⎠ ⎟

Rotated:

€

⇒ρ0 + ρ z cosθ −iρ z sinθ

iρ z sinθ ρ 0 − ρ z cosθ

⎛

⎝ ⎜

⎞

⎠ ⎟

Transmission amplitude:

Charge current

€

Ic =2e2V

hTrσ

ˆ T

Spin current

€

Trσ

r σ ˆ T α ←β ≠ Trσ

r σ ˆ T β←α

r T α ≡ Trσ [

r σ ˆ T β←α ]

r T β ≡ Trσ [

r σ ˆ T α ←β ]

r I s =

2e2

h

r T α μα −

r T βμ β[ ]

=2e2

h1

2 μα +μ β( )r T α −

r T β[ ] + eV ⋅1

2r T α +

r T β[ ]{ }

Inelastic scattering; let’s confine ourselves to T=0K:

Only possible to generate magnons when they are emitted by spin current.

€

μα /β ⇒ μα /β − hωqα /βΘ(eV − hωq

α /β )

€

rI s

magnon = −2eh

hωqα /βΘ(eV − hωq

α /β )q∑

r T α −

r T β[ ]

€

rI s =

2eh

r T α μα −

r T βμ β[ ]

Evaluation of sum over magnons

€

hωqα /βΘ(eV − hωq

α /β )q∑ = dωgα /β (ω)hω

0

eV∫

€

rI s

magnon = −eNα /β

i

heV

Emα /β

⎛

⎝ ⎜

⎞

⎠ ⎟ eV( )h

r T iα −

r T iβ[ ]

Interfacial magnons

where superscript i stands for transmission amplitudes for interface magnon production tm

i .

€

rI s =

eh

eV ⋅r T α +

r T β[ ]

Remember the spin current due to elastic scattering is:

Equilibrium spin current

€

rT α −

r T β[ ]∝

r ρ α ×

r ρ β

None other than interlayer exchange coupling

Out of equilibrium spin current

€

rT α +

r T β[ ]∝ ρ β

0 r ρ α + ρα

0 r ρ β

€

(r σ ⋅

r a )(

r σ ⋅

r b ) =

r a ⋅

r b + i

r σ ⋅(

r a ×

r b )

€

ˆ T β←α = td2 ˆ ρ α ˆ ρ β

For tm=0

€

ˆ t βα = tdˆ 1 + tm

r σ ⋅

r S α /β

Spin current

€

€

rI s

α

€

rI s

β

€

rI s

€

rI s =

eh

eV ⋅r T α +

r T β[ ]

Torque on an electrode

€

τ yα ∝ eV td

2sinϑ ρ 0

α ρ zβ

€

τ yβ = τ y

α (α ↔ β )

€

rτ α ≡−h(

r I s −

r I s

α )

r τ β ≡ −h(

r I s

β −r I s)

€

rτ ⊥α =−h (

r I s −

r I s

α )× ˆ α [ ] × ˆ α = −hr I s × ˆ α ( ) × ˆ α

r τ ⊥

β = −h (r I s

β −r I s )× ˆ β [ ] × ˆ β = h

r I s × ˆ β ( ) × ˆ β

There’s something funny about the equilibrium spin current:

€

rI s =

2e2

h1

2 μα +μ β( ) f (ε )∫ dε{ }r T α −

r T β[ ]

r T α −

r T β[ ]∝ ε F td

2 r ρ α ×

r ρ β

€

rI s ∝

2e2

hε

Ftd

2 r ρ α ×

r ρ β

Resolution:

€

H = h2

2mr

∇2 +σ ⋅Sαδ(r − rα )+σ ⋅Sβδ(r − rβ )

2nd order perturbation of the free electron energy due to local moments, i.e., RKKY

€

ΔE = −J(rαβ )r S α ⋅

r S β + iσ ⋅

r S α ×

r S β{ }

2nd order correctionto the energy

Produces precession of conductionelectrons spin

€

dr σ

dt= i

hr

H ⋅r σ ,

r σ [ ]

r H ≡ −iJ(rαβ )

r S α ×

r S β

dr σ

dt∝ J(rαβ )

r σ ×

r S α ×

r S β( )

€

rI s ∝ε

Ftd

2 r ρ α ×

r ρ β ∝ J(rαβ )

r S α ×

r S β( )

When we focus on spin dependent transmission

€

ˆ t βα = tm

r σ ⋅

r S α /β

€

td tm2 ≈ 17

€

Szα /β ⇒ Sα /β

By using this spin dependent amplitude and taking the components of the ensuing spin current transverse to the magnetization of the upstream electrode, the elastic contribution to the torque is:

€

τ yα = 2πe(eV )sinϑ

12

tm2 mβ (−)σ

σ∑ ρσα (−ϑ )ρσ

β ,

where

mα /β ≡ Nα /β (Sα /β )2 h2

(−)σσ∑ ρσ

α (−ϑ )ρσβ =[ (−)σ

σ∑ ρσα ρσ

β ]cos2 ϑ2 −[ (−)σ

σ∑ ρσα ρσ '

β ]sin2 ϑ2.

Spin dependent elastic tunneling:

The only current or bias induced excitations are from

€

σ±

so that the inelastic spin-flip contributions to the torque are:

€

τ yα = πe(eV )sinϑ tm

2 ×

eVEm

α Sα

⎛

⎝ ⎜

⎞

⎠ ⎟m

α ρ↑α ρ z

β +

eVEm

β Sβ

⎛

⎝ ⎜

⎞

⎠ ⎟m

β ρ zα ρ↑

β cosϑ − (−)σσ∑ ρσ

α (−ϑ )ρσ 'β

[ ]

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

and we have to evaluate

€

S±α /β Sm

α /β at T = 0K ⇒ 2Sα /β h2

The new feature for the inelastic contributions to the torqueare that they are not in the same direction for the twoelectrodes:

While for the elastic terms (non spin-flip magnetic as well as for direct transmission) we found:

€

τ yβ = τ y

α (α ↔ β )

€

τ yβ = −τ y

α (α ↔ β )

Elastic

Definition of spin torque:

Inelastic

Magnons created by hot spin current assist elastic torque onupstream electrode, but for downstream are in opposite sense.

€

rτ ⊥α =−h

rI s × ˆ α ( ) × ˆ α

r τ ⊥

β = hr I s × ˆ β ( ) × ˆ β

How does one understand this?

Elastic torque comes from spin current in tunnel junctionbeing the vector sum of the polarized currents from thesource and drain, i.e., from upstream and downstream electrodes.

When angular momentum is transferred between a spin current whose polarization is noncollinear to the magnetizationof an electrode, torque is produced. The component of the vector sum of difference between spin angular momentumgained by current and that lost by background magnetizationthat is transverse to electrode’s magnetization is the torque created by this exchange of magnons between noncollinearentities.

From our calculations we find

At T=0K hot spin currents can only lower the polarization of electrodes.

Note the sign in definitionof torque due to transfer ofangular momentum

Summarizing:

€

rI s =

2eh

r T α μα −

r T βμ β[ ] − hωq

α /βΘ(eV − hωqα /β )

q∑

r T α −

r T β[ ]

⎧ ⎨ ⎩

⎫ ⎬ ⎭

=2eh

r T α −

r T β[ ] 1

2 μα +μ β( ) − hωqiΘ(eV − hωq

i )q∑

i=α ,β∑

⎧ ⎨ ⎩

⎫ ⎬ ⎭

+2eh

r T α +

r T β[ ] ⋅ 1

2 μα −μ β( )

€

rτ ⊥α =−h

rI s × ˆ α ( ) × ˆ α

r τ ⊥

β = hr I s × ˆ β ( ) × ˆ β

€

rτ ⊥α

elasticin same direction as

r τ ⊥

β

elastic

r τ ⊥

α

inelasticin same direction as

r τ ⊥

α

elastic

r τ ⊥

β

inelasticin opposite direction as

r τ ⊥

β

elastic

From experiments on MTJ’s one find that the ratio of the spin torque to the current is relatively flat as one increasesthe bias.

The (charge) current as a function of bias is:

€

rI 0 =

4πe2Vh

td2 + 2 tm

2[ ] ρσ

α ρσβ

σ∑ (ϑ )

+ tm2 eV

Em'α ρ↓

α ρ↑β (ϑ )+

eVEm

'β ρ↑α ρ↓

β (ϑ ) ⎡

⎣ ⎢

⎤

⎦ ⎥

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

,

where

Em'α /β ≡ Em

α /β Sα /β =3Sα /β

Sα /β +1kTC

ρσ (ϑ ) ≡ ρσ cos2 ϑ2 + ρσ ' sin2 ϑ

2.

The ratio of the spin torque to the current for the upstreamelectrode is

We have evaluated the spin torques and charge currents by using the parameters we previously found were able to fit the zero-bias anomaly found for Co/Al2O3/CoFe:

€

td tm2 ≈ 17,

ρ↑

ρ↓≈ 2.1

S = 3 2,and kTC =110meV .

Em ≅130meV .

€

τ yα

I0

(ϑ ~ 0) = 0.3h2

sinϑ1+ 0.03

eV130

1+ 0.03eV130

, τ y

α

I0

(ϑ ~ π ) = 0.36h2

sinϑ1+ 0.04

eV130

1+ 0.04eV130

€

τ yβ

I0

(ϑ ~ 0) = 0.3h2

sinϑ1− 0.03

eV130

1+ 0.03eV130

, τ y

β

I0

(ϑ ~ π ) = 0.36h2

sinϑ1− 0.04

eV130

1+ 0.04eV130

and the ratio of the spin torque to the current for the downstream electrode is

Reversing polarity we replace in above expression; noting that the free layer is now downstream we find thetorque to current ratio remains relatively flat; in agreementwith the data.

This agrees with data as free layer is upstream for forward bias .

€

μα >μβ

€

α↔β

Conclusion:

•Magnon production in magnetic electrodes is able to explainhow the spin torque increases with bias even though theTMR decreases.

•Our results are for the range + 200 meV; for higher bias oneshould take into account the change in barrier profile withbias.

•In and of itself the change in barrier profile with bias cannot explain the data. See PRB 71,024411(2005). But see PRL 97, 237205(2006).

Differences between tunnel barrier and metallic spacer

Primary is lack of equilibrium coupling; its all current driven

For example

In a magnetic tunnel junction the spin current is

Js= βJe cos [θ/2]

€

1800

1500

1200

900

600

300

00

The spin current in the middle of a nonmagnetic spacer between two magnetic layers is parallel to the sum of the magnetizations, and its magnitude is

€

Js (x = 0) = βJe

cosθ2

cosθ + λ sinθ

where λ = 1− ββ '2

λ J

λ sf

For cobalt is of the order of 0.02 for cobalt.

The spin current at the interface reaches its maximum of

when the angle between the local magnetizations θ* is

Note: θ* is close to π when λ ≪ 1.

€

Js,max (x = 0) = βJe

cosθ *

22λ

€

cosθ * = −1− 3λ1− λ

The magnitude of the spin-current in a metallic junction is enhanced by a factor of λ-1 compared to the bare spincurrent βJe cos (θ/2).

This comes from the interplay between longitudinal and transverse accumulations; even though the transverse components of the spin current are absorbed within a region of λJ of the interface.

Out of equilibrium effects control spin transfer in metallic structures

Several approaches:

• Maintain phase coherence (ballistic)

Landauer-Keldysh; see Edwards et al. PRB71, 054407 (2005)

• Discard coherence (diffusive)

Layer-by-layer, e.g., Valet-Fert

Whole potential

To write an out of equilibrium spin current

€

r rj s ∝ lim

r'→r

r ∇ r' −

r ∇ r{ } dεTrσ∫ [

r σ G<(r',r)]

G<(r' t ',rt) = i Ψ*(r' t ')Ψ(rt)out of equilibrium

The devil is in the details

The energy minimum principle does not hold for systems out of equilibrium, even under steady state conditions

For example, one can induce a coherence (to carry transverse spin currents) between states that in equilibrium are not.

The propagators themselves are found from their equation ofmotion.

At the end of the day we arrive at the distribution functionby taking the Wigner transform of the propagators,

€

fαβ (k,r)∝ dk'e ik'⋅r∫ ak +k ' 2,α* ak−k ' 2,β

The field operators in the propagators arefound from the equation of motion they obey, i.e., the Schrödinger equation.

€

Ψ*(r' t '),Ψ(rt)

€

r rj s(r)∝ dkTrσ∫ [

r σ

r f (k,r)]

Boltzmann equation of motion determines the distribution function

1-Attention must be paid to the different k states in the distribution function. Conventionally for spin split bands there are more than one, but most people use an equationof motion appropriate for only one k state

Band structure of Co

2- In addition there’s the transmission of information aboutout-of-equilibrium distributions from one layer to another. One has to match functions across layers by using the transmission and reflection coefficients.

InjectionPropagation

However, for transverse distribution function for currents

In equilibrium

This leads to the “mixing conductance” in the conventional view,i.e., the transfer of spin current from one spin channel to the other across the interface.

A good example of a DOA mode for propagating transverse waves.

A is the new current induced spin-flip term

m and s are spin indices

fout Tequil fout

fequil foutTout

Transmission of out-of-equilibrium distributions across interface

Conventional

Noncollinear multilayers one should also consider

fout foutTout

The following does not enter in linear response:

To obtain off-diagonal amplitudes requires one to consider the role of out-of-equilibrium spin accumulation created at one interface on a second interface when the magnetic layers are noncollinear, i.e., current-driven symmetry breaking.

This leads to out-of-equilibrium corrections to the scattering amplitudes, or transmission and reflection coefficients.

For spin currents its all about transparency of interfaces to propagating transverse waves.

For collinear structures the out-of-equilibrium corrections are merely changes in population of existing states; they are insignificant.

However, for noncollinear multilayers symmetry is brokenand this requires one to define new basis states. Theseout-of-equilibrium corrections can be sizeable.

ResistanceResistance

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

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

Is it necessary to introduce out-of-equilibrium corrections when using approaches other than the layer-by-layer?

The point is rather that is necessary to do calculations that arefully self-consistent. In the layer-by-layer approach as it has been applied to noncollinear multilayers only the transport within layers is determined self-consistently.

When one solves for the transport using the potential of theentire multilayer and self-consistently no further correctionsare needed to describe steady state spin transport.

For example

Time dependence of spin transport using diffusion equation

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.

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

Time evolution of spin current for layers 900 apart Components referred to global axes

To improve on whole multilayer solution obtained bydiffusion equation.

Use Boltzmann equation with the same source terms

Go fully quantum and use Landauer-Keldysh formalism

•Be sure to maintain phase coherence across layers

•Demand full self-consistency in solutions

•Obtain local densities to compare to semiclassical results

•Most important include spin-flip scattering

Keldysh formalism has been used to find current inducedchanges in the interlayer coupling (RKKY interaction).See R.J.Elliott et al. PRB54,12953 (1996); PRB59, 4287 (1999).

However it was done in the limit that the magnetic layers were paramagnetic, i.e., above the Curie temperature, so that one only has current driven changes along the direction of theequilibrium coupling. It picks up the longitudinal component of the induced effects, but cannot account for transverse terms as there is no time averaged local internal field above the Curie temperature.

A self consistent theory of current induced switching of the magnetization that uses non equilibrium Green’s functions has been recently carried out for a magnetic trilayer structure under conditions in which the current has achieved steady state: D.M. Edwards et al. PRB71, 054407 (2005).

The prognosis for a full quantum resolution for existenceof transverse spin currents in the proximity of interfacesis good.

Caution: not all steady state solutions are equal. They depend on what scattering exists in system, e.g. spin-flip relaxation.

What can experiments tell us about the existence oftransverse spin currents in ferromagnetic layers?

•Dependence of torque on thickness of free layer

•Dynamic exchange coupling induced at microwave frequencies in FMR resonance experiments on magnetic layers separated by a normal metallic spacer. H. Hurdequint experiments in progress; will give us a handle of the transmission of transverse waves across ferro/normal interfaces when layers are noncollinear.

• Perhaps the most direct evidence for effective field component comes from the frequency shift in FMR is done on a trilayer when the subject to a current.

The smoking gun

That’s all for today folks