1. 1
Spin Torque for DummiesSpin Torque for Dummies
Matt Pufall, Bill Rippard
Shehzu Kaka, Steve Russek, Tom Silva
National Institute of Standards and Technology,
Boulder, CO
J. A. Katine
Hitachi Global Storage Technologies
San Jose, CA
Mark Ablowitz, Mark Hoefer, Boaz Ilan
University of Colorado Applied Math Department
Boulder, CO
3. 3
But first… a puzzle!But first… a puzzle!
We start with a
polarized beam
of spin 1/2 Ag
ions…
… we pass the beam
through a magnetic
field gradient…
… and the beam
“diffracts” into two
beams, polarized
along the axis of
the magnetic field.
Q: What happened to the angular momentum in the original polarization direction???
(For more details, see Feynman’s Lectures on Physics, Vol. III, page 5-1)
The “Stern-Gerlach” experiment
… but
true!!!
Very
strange
…
H
5. 5
Magnets are Gyroscopes!Magnets are Gyroscopes!
2
B
L
e
e
m
µ
γ = − = −
h
e-
r
v
z
2 B
s
µ
γ = −
h
For spin angular
momentum, extra
factor of 2 required.
“The gyromagnetic ratio”
L r p= ×
r r r
angular momentum
for atomic orbit:
L IAµ =
rr
magnetic moment
for atomic orbit:
( )2
ˆ
2
ev
r z
r
π
π
= − ÷
ˆ
2
evr
z= − ˆrmvz=
Classical model
for an atom:
2
e
evr
rm v
= −
z
L
zL
µ
γ B
6. 6
Larmor EquationLarmor Equation
H
M
0T M Hµ= ×
r r r
T
dL
T
dt
=
r
r
L
µ
γ ≡
( )0
dM
T
dt
M H
γ
γµ
=
= ×
r
r
r r
(definition of torque) (gyromagnetic ratio)
Magnetic field exerts torque on magnetization.
Joseph Larmor
Magnets
make me
dizzy!
7. 7
( )0
d
s
T
M M H
M
αµ
=
= × ×
r
r r r
damping torque
0
precession torquepT
M Hµ
=
= ×
r
r r
Landau & Lifshitz (1935):
( )
( )
0
0
p d
s
dM
T T T
dt
M H
M M H
M
γ γ
γ µ
α γ µ
= − = − +
= − ×
− × ×
r
r r r
r r
r r r
α = dimensionless
Landau-Lifshitz
damping parameter
Gyromagnetic precession with energy loss: TheGyromagnetic precession with energy loss: The
Landau-Lifshitz equationLandau-Lifshitz equation
L. Landau and E. Lifshitz, Physik. Z. Sowjetunion 8, 153-169 (1935).
H
Lev Landau
Damping
happens!!
Lev Landau
8. 8
Part 2: Charge transport in magnetic heterostructuresPart 2: Charge transport in magnetic heterostructures
(Magneto-transport)(Magneto-transport)
9. 9
Ferromagnetism in Conductors: Band StructureFerromagnetism in Conductors: Band Structure
D(E)
E
“s-p band”
EF
spin-up spin-down
D(E)
E
“d band”
EF
“majority” states
“minority” states
“Density of States”
“exchange splitting”
s-band states (l = 0) have higher mobility than d-band states (l = 2) in
conductors due to the much lower effective mass of the s-band.
Minority band electrons scatter more readily from the s-p band to the d-band
due to the availability of hole states in the minority d-band.
10. 10
Spin-dependent conductivity in ferromagnetic metalsSpin-dependent conductivity in ferromagnetic metals
“majority
band”
“minority
band”
M
JJ
Conductivities in ferromagnetic conductors are
different for majority and minority spins.
In an “ideal” ferromagnetic conductor, the conductivity for minority spins is zero.
Analogy: Like water flowing through pipes. Large conductivity = big pipe, etc…
11. 11
Concept: interfacial spin-dependent scatteringConcept: interfacial spin-dependent scattering
M
“Majority” spins are preferentially transmitted.
”Minority” spins are preferentially reflected.
Normal metal Ferromagnet
12. 12
Concept: ferromagnets as spin polarizersConcept: ferromagnets as spin polarizers
M
“Majority” spins are preferentially transmitted.
Normal metalFerromagnet
Ferromagnetic conductors are relatively permeable for majority spins.
Conversely, they are impermeable for minority spins.
13. 13
Concept: spin accumulationConcept: spin accumulation
∆M
z
I
( ) sM z M M= + ∆
Non-equilibrium spin polarization “accumulates” near interfaces of
ferromagnetic and non-magnetic conductors.
“spin
diffusion
length”
14. 14
Part 3: Spin momentum transferPart 3: Spin momentum transfer
15. 15
Non-collinear spin transmissionNon-collinear spin transmission
M
θ
What if the spin is neither in the majority band nor the minority band???
????
????
Is the spin reflected or is it transmitted?
Quantum mechanical probabilities:
( )( )
( )( )
2
2
1
Pr 1 cos
2
1
Pr 1 cos
2
A
B
θ
θ
↑ = = +
↓ = = −
= +A B
θ
Quantum mechanics of spin:
cos
2
sin
2
A
B
θ
θ
= ÷
= ÷
An arbitrary spin state
is a coherent
superposition of “up”
and “down” spins.
16. 16
Spin Momentum Transfer: Small Current LimitSpin Momentum Transfer: Small Current Limit
θ
M
+ =
-δT
+
Tdamp
θ
MPolarizer: At low electron flux,
damping torque compensates
spin torque: Magnetization is
stable.
e-
θ
M Mf
; 1Tδ θ θµ
r
=+ =
δT
Electrons:
17. 17
Spin Momentum Transfer: Large Current LimitSpin Momentum Transfer: Large Current Limit
δT is driven by spin accumulation in the Cu spacer.
Spin accumulation is proportional to current flowing
through the structure.
+ =
θ
M
+ =
δT
-δT
+
Tdamp
θ + δθ
M
Electrons:
Polarizer:
Spin torque exceeds
damping torque:
Polarizer reacts with
changing M. Torque
proportional to angle
θ: Unstable!Damping torque
18. 18
Transverse torque via spin reorientation/reflectionTransverse torque via spin reorientation/reflection
Consider only reflection events...
For the electron:
( )
( )
( )
ˆ
ˆsin
2
ˆ ˆ ˆ
2
ˆ ˆ ˆ
2
transverse
transverse inc
inc
f
s
s s y
s
y
m m p
m m m
θ
∆
′∆ = −
′= −
= − × ×
= × ×
r
r r
r
h
h
h
θ
θ
transverses∆
r
incs
rrefs
r
AND
Consider only change in angular momentum
transverse to magnetization axis. (Equivalent to
assuming magnitude of M does not change.)
y′
x′
where
2
ˆ incp s=
r
h
θ ˆp
ˆm−
19. 19
Newton’s Second LawNewton’s Second Law
( )ˆ ˆ ˆ
2
trans fs m m m∆ = × ×
hr
If…
…then…
( ) ( )ˆ ˆ ˆ
2
fM V m m mγ∆ = × ×
r h
…per electron.
For a flowing stream of electrons:
( )
( )( )2
ˆ ˆ ˆ
2
ˆ
2
f
f
s
m m m
dM I
dt V e
I
M M m
eM V
γ
γ
× ×
= ÷ ÷
= × × ÷
hr
r rh
Total moment of
“free” magnetic
layer
Rate of electron
impingement on “free” layer
20. 20
The Slonczewski Torque TermThe Slonczewski Torque Term
2
ˆ( )
2
Sloncewski f
s
J
T M M m
e M
ε
δ
= × ×
r r rh
J. Slonczewski, Journal of Magnetism and Magnetic Materials, vol. 159, page L1 (1996)
J M Mf
δ
efficiency* ~ 0.2 - 0.3
*Accounts for all those messy details:
Polarization of ferromagnet, band
structure mismatch at interface, spin
decoherence, etc…
21. 21
The “FAQ Page”The “FAQ Page”
Q: What if the spin is transmitted through the “free” layer rather than reflected?
A: Doesn’t matter. Quantum mechanically, there is an amplitude for both transmission
and reflection, but only for spin along the axis of magnetization. The transverse
component of spin for the incident electrons is “lost” once the electron wavefunction is
split into the transmitted and reflected components. Conservation of angular momentum
dictates that the transverse component is transferred to the magnetic layer. This is a
purely quantum mechanical phenomenon: There is no classical analog!
Q: Don’t the reflected spins affect the spin accumulation in the spacer layer?
A: Yes, they do. There are several theories that take this “back-action” on the spin
accumulation into account. See J. C. Slonczewski, J. Magn. Magn. Mater. 247, 324
(2002); A. A. Kovalev, et al., Phys. Rev. B 66, 224424 (2002); J. Xiao, et al., Phys. Rev.
B 70, 172405 (2004); A. Fert, et al., J Magn. Magn. Mater. 69, 184406 (2004).
( )
0 1 0 1cos cos
q q
B B B B
ε θ
θ θ
+ −
= +
+ −
Tsmt
ε = constant
ε (θ )
22. 22
Back to the puzzle…Back to the puzzle…
A: The quantum equivalent of a card trick. If a card “vanishes” magically from one deck,
it must reappear somewhere else. No mechanism for the transfer of angular momentum
need be invoked!
The “Stern-Gerlach” experiment, revisited.
Once the linear
momentum for the two
spin components is split,
the transverse angular
momentum is “released”
to do work on the
magnet system.
H
“Up” spin current
“Down” spin current
0xS =
1
2
xS = +
x
y
z
23. 23
( )Larmor damp smt
dM
T T T
dt
γ= + +
r
r r r
0 eff
dM
M H
dt
γµ= ×
r
r r
Larmor term:
precession
My
Mz
Mx
M
Heff
Tsmt ≅ -Tdamp
J ~ 107
A/cm2
Slonczewski 1996
My
Mz
Mx
Mi
Mf
Heff
Damping term:
aligns M with H
0
2
( )eff
s
M M H
M
αγµ
− × ×
r r r
Precession
Heff
Spin Torque
Damping
M
Spin Torque Term
2
ˆ( )
2
B
f
s
Jg
M M m
e M
µ ε
δ
+ × ×
r r
Spin torque can
counteract damping
mf
Magnetodynamics: Three TorquesMagnetodynamics: Three Torques
24. 24
How Can We See This?
Torque ∝ to current density: must have high current densities
to produce large torques
1 mm
10 µm
I = 0.1 MA≈
I = 10 A≈
100 nm I = 1 mA≈
We will use nanopillar and nanocontact structures
Typical wire
Size of a human hair
500 atoms across≈
X
Required Idc Possible
X
26. 26
• Step DC current
• Measure DC R, microwave
power output
5 6 7 8 9
24.8
25.0
25.2
25.4
dV/dI(Ω)
Current (mA)
9.6 9.7 9.8 9.9 10.0
0.0
0.1
0.2
0.3
0.4
9 mA
8.5 mA8 mA
7.5 mA
7 mA
6.5 mA
6 mA
5.5 mA
Power(pW)
Frequency (GHz)
Devices are nanoscale
current-controlled
microwave oscillators
Au
Cu
“Fixed” layer
“Free” layer
~40 nm
0.7 T,
θ = 10o
θT = 300K
Nanocontact DynamicsNanocontact Dynamics
27. 27
SummarySummary
Magnetization dynamics tutorial: Magnets are gyroscopes.
Magnetotransport tutorial: Magnets are spin filters.
Spin momentum transfer: Back action of spin polarized carriers on magnet.
Spin torque nano-oscillator: Spin torque compensates damping.
An excellent review article!!
M. D. Stiles and J. Miltat, “Spin Transfer Torque and Dynamics,”
Topics in Applied Physics 101, 225-308 (2006).
