Section I
Text:
To describe the spin-wave dynamics, the modified LLG equation including the
spin-torque terms is used,

 
 
 

ˆ p )  bJ m  m
ˆ p,
m
 m  H eff  m  m
 a J m  (m  m
(1)
where Heff is the effective field, γ is the gyromagnetic ratio, α is the Gilbert damping
parameter, aJ represents the strength of the Slonczewski’s torque, and bJ represents the
strength of the fieldlike torque.[1] In our model system, the êx, êy, and êz coordinate
axes are oriented to the length, width, and thickness directions of the spin valve,
respectively. The magnetization of the fixed layer was maintained at the -êx direction,
and that of the free layer was initialized to the êx direction. Letting



ˆp,
ˆ p and H FT  bJ m
H ST  a J m  m
(2)
inserting them into Eq. (1), and after some transformations, one obtains,

 

 
m   0 m  H t  m  (m  H t ) ,
(3)




where γ0=γ/(1+α2), λ=γ0α, and H t  H eff  H ST  H FT . We have specified mp=-êx;
thus, for the free layer,



H t  H k m x eˆ x  D 2 m  H d m z eˆ z  a J m  eˆ x  bJ eˆ x ,
(4)
where Hk and Hd denote the easy-axis and hard-axis anisotropy fields, respectively,
and D is equal to 2A/μ0Ms, with μ0 the permeability of vacuum. The crystalline
anisotropy is neglected and, therefore, Hk and Hd depend solely on the shape of the
free layer and are given by Hk=(Ny-Nx)Ms and Hd=(Nz-Ny)Ms, respectively. Here, Nx,
Ny, and Nz are demagnetization factors along the length, width, and thickness
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directions of the free layer, respectively. The free layer with the size and geometry
specified above gives Nx=0, Ny=0.072, and Nz=0.928 [2] (To calculate these values, the
length of the free layer is approximated as infinity).
For the small-amplitude spin-wave dynamics satisfying the linear approximation,
the variable magnetization can be expanded in a series of plane waves of
magnetization. Micromagnetic simulations reveal that the amplitude of spin waves
decays as a quasiexponential function of the propagation distance. As an
approximation, we assume that the spin waves attenuate exponentially with the
increased propagation distance. Consequently, m can be expanded in the form,


m  eˆ x  m0 e  x L e  i ( kx t ) ,
(5)
where m0 is the amplitude of the spin wave, fulfilling m0<<1, i.e., m0x  m0y , m0z , and
L is the attenuation length of the spin wave. Replacing Ht and m with the expressions
in Eqs. (4) and (5), respectively, Eq. (3) can then be transformed into,
im0x e  x L e i ( kx t )   0 [ H d m0y m0z  a J (m0y ) 2  a J (m0z ) 2 ][e  x L e i ( kx t ) ]2
 {[ D(ik  1 / L) 2  H k  bJ ]( m0y ) 2  [ D(ik  1 / L) 2  H k  H d  bJ ]( m0z ) 2 }[e  x L e i ( kx t ) ]2
im0y e  x L e  i ( kx t )   0 {a J m0y  [ H k  H d  bJ  D(ik  1 / L) 2 ]m0z }[e  x L e  i ( kx t ) ]
 {[ H k  bJ  D(ik  1 / L) 2 ]m0y  a J m0z }[e  x L e i ( kx t ) ]
(6)
 [ H d m0y (m0z ) 2  a J (m0y ) 2 m0z  a J (m0z ) 3 ][e  x L e i ( kx t ) ]3
im0z e  x L e  i ( kx t )   0 {[ D(ik  1 / L) 2  H k  bJ ]m0y  a J m0z }[e  x L e  i ( kx t ) ]
 {[ H k  H d  bJ  D(ik  1 / L) 2 ]m0z  a J m0y }[e  x L e i ( kx t ) ]
 [ H d (m0y ) 2 m0z  a J (m0y ) 3  a J m0y (m0z ) 2 ][e  x L e i ( kx t ) ]3
2
.
Ignoring the high-order terms [3] in Eq. (6), and after simplification, one finds,
im0y   0 {a J m0y  [ H k  H d  bJ  D(ik  1 / L) 2 ]m0z }  {[ H k  bJ  D(ik  1 / L) 2 ]m0y  a J m0z }
im0z   0 {[ H k  bJ  D(ik  1 / L) 2 ]m0y  a J m0z }  {a J m0y  [ H k  H d  bJ  D(ik  1 / L) 2 ]m0z }
Solving Eq. (7), neglecting small terms proportional to α2, and remembering that
Re[ω]=ωf and Im[ω]=0 as well as 1/(kL)2<<1, we arrive at the following expressions,
 2f  4( 0kD / L) f   02 a J2   02 ( H k  H d  Dk 2  bJ )( H k  Dk 2  bJ )
  02 [( D 2 / L4 )  ( D / L2 )( 2 H k  H d  6 Dk 2  2bJ )]
(8)
and
2 0 Dk
L
 f  a J
f
.
(9)
H k  ( H d / 2)  Dk 2  bJ
References:
[1] aJ=ħJPg(θ)/μ0deMs, where J is the current density, d is the thickness of the free
layer, e is the charge of the electron, and g(θ) measures the spin torque asymmetry
related to the angle θ between magnetization directions of the free and fixed layers
and is usually taken to be 1/2 for simplicity (Refs. 14 and 15 in the article). bJ=ξaJ,
where ξ has a typical value of about 0.1 (Ref. 22 in the article).
[2] M. E. Schabes and A. Aharoni, IEEE Trans. Magn. 23, 3882 (1987); A. Aharoni, J.
Appl. Phys. 83, 3432 (1998).
[3] The treatment is not applicable to the case in which the spin-wave amplitude is
amplified by the suprathreshold current.
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(7)
Section II
Text:
The current flowing into the spin valve from the top electrode will turn in-plane
almost immediately. However, it will not spread out significantly. The reasons are
shown below.
In our model system, the length of the spin valve is 10 μm or more, and the
thickness is only about 30 nm (e.g., free layer 6 nm + spacer 8 nm + fixed/pinning
layers 16 nm). The size of the in-plane spreading should be comparable to the
thickness of the spin valve; otherwise, the current will undergo much bigger
resistance. To spread out significantly, the current must have a larger in-plane
component which traverses the cross section perpendicular to the length direction of
the spin valve. But, the area covered by the top electrode [width of spin valve × length
of top electrode (~ μm)] is much larger than that of the cross section perpendicular to
the length direction of the spin valve [width of spin valve × thickness of spin valve (~
30 nm)]. Therefore, the in-plane spreading must result in thousands [~ (Lte·Lsv)/(dsv)2,
where Lte is the length of the top electrode, Lsv is the length of the spin valve, and dsv
is the thickness of the spin valve] of times larger resistance. As a result, the size of the
in-plane spreading should be only at the scale of the thickness of the spin valve, i.e.,
about 30 nm, which is much smaller than the length of the top electrode.
Consequently, the current can be well confined in the path below the top electrode and
can be regarded as uniform in the perpendicular direction as long as the length of the
top electrode is much larger than the thickness of the spin valve. This condition is
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satisfied by our model system and can be easily realized experimentally.
Schematic of the current distribution (not in proportion to the real spin valve):
According to the argument made above, we can plot the schematic
showing the current distribution. From the schematic, we see that
the in-plane spreading ~ 30 nm is much smaller than the length of
the top electrode ~ μm. Thus, the current can be well confined in
the path below the top electrode and can be regarded to be uniform
in the perpendicular direction.
Remark: In our model system, the top electrode covers the full width of the spin
valve, so the width is not a relevant parameter in discussing current spreading. The
crucial parameters determining the current distribution are the length of the top
electrode, the length of the spin valve, and the thickness of the spin valve.
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