Transcript Document

Spin currents in non- collinear
magnetic tunnel junctions
and
metallic multilayers
Peter M Levy
New York University, USA
Background to TMR
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.
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.
Particle current:
2e
I p  T    T  
h
   L
   R
ˆ  tˆ  
ˆ  tˆ 
Tˆ   
*

ˆ  

0
Density matrix:
Rotated:
0 

 
0  z cos

i

sin

z

 

0  z cos 
 iz sin 
Transmission amplitude:

t d  t m Sz / 
tˆ  
 /
 t m S
t m S /  
 /  
t d  t m Sz 
tˆ  t d 1ˆ  t m S  / 
Charge current
2
2e V
Ic 
Tr Tˆ
h
Spin current
Tr Tˆ  Tr Tˆ 
2

T  Tr [Tˆ  ]
T  Tr [Tˆ ]
2e
Is 
T   T  

h
2e

h
2
1
1



T

T

eV

T  T 





2    
2
Inelastic scattering; let’s confine ourselves to T=0K:
Only possible to generate magnons when they are
emitted by spin current.
 /
 /    /   q (eV  q
 /
)
2e
I s  T  T  
h
I
magnon
s

2e
 /
 /
   q (eV  q )T  T 
h q
Evaluation of sum over magnons

q
q
 /
(eV  q
 /
eV
)   dg
 /
( ) 
0
Interfacial magnons
eN  /   eV 
i
i

  /  eV  T   T  
h Em 
i
I
magnon
s
where superscript i stands for transmission amplitudes for
interface magnon production tmi .
Remember the spin current due to elastic scattering is:
e
I s  eV  T T 
h
For tm=0
ˆt  t d 1ˆ  t m S  / 
2
ˆ
ˆ  
ˆ
T  t d 
( a)( b )  a  b  i (a  b )

T
Equilibrium spin current


 T    
None other than interlayer exchange coupling
Out of equilibrium spin current
T  T      

0
0
Spin current
e
I s  eV  T T 
h


Is
Is

Is
   (Is  Is )
Torque on an electrode
   (Is  Is )
ˆ  
ˆ   I s  
ˆ  
ˆ
   (I s  I s )  



ˆ  
ˆ
   (I s  I s )  



I
ˆ  
ˆ


s


 y eV td sin0 z
2


 y   y (   )
There’s something funny about the equilibrium spin current:
2
2e
1

Is 

  f ( )dT  T 



2
h
2
T  T   F t d   
2
2e
2
Is 
 F t d   
h
Resolution:
H
2
    S (r  r )   S (r  r )
2
2m
2nd order perturbation of the free electron energy due to
local moments, i.e., RKKY
E  J(r )S  S  i  S  S 
2nd order correction
to the energy
Produces precession of conduction
electrons spin

d i
 H  , 
dt
H  iJ(r )S  S
d
 J(r ) S  S 
dt
Is  F t d     J(r )S  S 
2
When we focus on spin dependent transmission
ˆt  t m S  / 
 /
Sz
S
t d t m  17
2
 /

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:
Spin dependent elastic tunneling:
1 2 
 y  2 e(eV )sin  t m m  ()  ( ) ,
2
where

m /   N /  (S  /  )2

 ()
2
 ( )  [  ()   ]cos2  2 [  ()   ' ]sin 2  2 .
The only current or bias induced excitations are from
and we have to evaluate
 /  /
S S
at T = 0K  2S

 / 2
so that the inelastic spin-flip contributions to the torque are:

 y  e(eV )sin  t m 
2

 eV    

   m   z 

Em S 



 eV    
 
 
m  z  cos    ()  ( ) ' 






E
S


 m

While for the elastic terms (non spin-flip magnetic as well as
for direct transmission) we found:


 y   y (   )
The new feature for the inelastic contributions to the torque
are that they are not in the same direction for the two
electrodes:



 y   y (   )
Definition of spin torque:
  

 
Elastic
ˆ  
ˆ
I s  
I
s
ˆ  
ˆ
 
Inelastic

Magnons created by hot spin current assist elastic torque on
upstream electrode, but for downstream are in opposite sense.
How does one understand this?
Elastic torque comes from spin current in tunnel junction
being the vector sum of the polarized currents from the
source and drain, i.e., from upstream and downstream
electrodes.
From our calculations we find
When angular momentum is transferred between a spin
current whose polarization is noncollinear to the magnetization
of an electrode, torque is produced. The component of the
vector sum of difference between spin angular momentum
gained by current and that lost by background magnetization
that is transverse to electrode’s magnetization is the torque
created by this exchange of magnons between noncollinear
entities.
At T=0K hot spin currents can only lower the polarization of
electrodes.
Note the sign in definition
of torque due to transfer of
angular momentum
Summarizing:

2e 
 /
 /
I s  T   T    q (eV  q )T  T 

q
h 
1
2e
i
i 
 T  T  2       q (eV  q )


i , q
h
2e
+ T  T  12    
h
ˆ  
ˆ  
   I s  

ˆ
ˆ
I




s 

 elastic in same direction as  elastic
 inelastic in same direction as  elastic
 inelastic in opposite direction as  elastic
From experiments on MTJ’s one find that the ratio of the
spin torque to the current is relatively flat as one increases
the bias.
The (charge) current as a function of bias is:


 t 2  2 t 2      ( )

d
m



4 e2V 
I0 

,

eV
2 eV
  t m  '   ( )  '   ( )
Em
Em


where
 /
3S
Em' /   Em /  S  /    /  kTC
S 1
 ( )   cos2  2   ' sin 2  2 .
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:
t d t m  17,
2


 2.1
S  3 2, and kTC  110meV .
Em  130meV .
The ratio of the spin torque to the current for the upstream
electrode is
eV
eV


1 0.03
1 0.04
 y

y
130 ,
130
( ~ 0)  0.3 sin 
( ~  )  0.36 sin 
eV I 0
eV
I0
2
2
1 0.03
1 0.04
130
130
and the ratio of the spin torque to the current for the
downstream electrode is
eV
eV

1 0.03
1 0.04
 y

130 , y ( ~  )  0.36 sin 
130
( ~ 0)  0.3 sin 
eV I 0
eV
I0
2
2
1 0.03
1 0.04
130
130
This agrees with data as free layer is upstream for forward
bias    .
Reversing polarity we replace    in above expression;
noting that the free layer is now downstream we find the
 torque to current ratio remains relatively flat; in agreement
with the data.

Conclusion:
•Magnon production in magnetic electrodes is able to explain
how the spin torque increases with bias even though the
TMR decreases.
•Our results are for the range + 200 meV; for higher bias one
should take into account the change in barrier profile with
bias.
•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
180 0
150 0
120 0
90 0
60 0
30 0
00

In a magnetic tunnel junction the spin current is
Js= βJe cos [θ/2]
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
J s (x  0)  J e
cos
2
cos   sin 
J
1

'
where  
2
sf
For cobalt  is of the order of 0.02 for cobalt.

The spin current at the interface reaches its maximum
of
cos 2
J s,max (x  0)  J e
2
*
when the angle between the local magnetizations θ* is

1 3
cos  
1 
*
Note: θ* is close to π when λ ≪ 1.

The magnitude of the spin-current in a metallic junction
is enhanced by a factor of λ-1 compared to the bare spin
current 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
j s  limr'  r  dTr [G (r',r)]
r' r
G (r' t',rt)  i * (r' t')(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 field operators * (r' t'), (rt) in the propagators are
found from the equation of motion they obey, i.e., the
Schrödinger equation.

The propagators themselves are found from their equation of
motion.
At the end of the day we arrive at the distribution function
by taking the Wigner transform of the propagators,
f (k,r)   dk'eik'r a*kk' 2, akk' 2,
j s (r) 
 dkTr [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 equation
of motion appropriate for only one k state
Band structure of Co
2- In addition there’s the transmission of information about
out-of-equilibrium distributions from one layer to another. One
has to match functions across layers by using the transmission
and reflection coefficients.
Injection
Propagation
In equilibrium
m and s are spin indices
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.
However, for transverse distribution function for currents
A is the new current induced spin-flip term
Transmission of out-of-equilibrium distributions across
interface
Conventional
fout
Tequil
fout
Noncollinear multilayers one should also consider
fequil
Tout
fout
The following does not enter in linear response:
fout
Tout
fout
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 broken
and this requires one to define new basis states. These
out-of-equilibrium corrections can be sizeable.
Resistance
0
15
30
45
60
75
90
105 120 135 150 165 180
1.0
1.0
A=0.00
A=0.05
A=0.50
Resistance
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0
15
30
45
60
75
90
105 120 135 150 165 180
Angle
Spin torque as a function of angle between layers for
three different cases of current induced spin flip
0.6
A=0.5
A=0.05
A=0
Torque
0.5
0.4
0.3
0.2
0.1
0.0
0
30
60
90
Angle
120
150
180
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 are
fully 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 the
entire multilayer and self-consistently no further corrections
are 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
To improve on whole multilayer solution obtained by
diffusion 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 induced
changes 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 the
equilibrium 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).
Caution: not all steady state solutions are equal. They depend
on what scattering exists in system, e.g. spin-flip relaxation.
The prognosis for a full quantum resolution for existence
of transverse spin currents in the proximity of interfaces
is good.
What can experiments tell us about the existence of
transverse 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.
The smoking gun
• 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.
That’s all for today folks