SPIN-WAVE ARCHITECTURES - Bio-Inspired and Nanoscale Integrated Computing - page 220

magnetization component My as follows:

" #

exp d 2

2

C exp ð t = tÞ ð y n t Þ

M y ¼

cos ð k 0 y o t þ fÞ; ð 7 :

3 Þ

d 4 þ b 2 t 2 4 ðd 4 þ b 2 t 2 Þ

where C is a constant proportional to the amplitude, t is the decay time, f is the
initial phase, n = qo / q k (k=k 0 ) and b =(1/2) q 2 o / q k 2 (k=k 0 ) are the coefficients
of the first and second order terms, respectively, in the Taylor expansion of the
nonlinear dispersion, o (k). The dispersion relation for spin waves propagating
orthogonally to the magnetization is given by

n o 1 = 2

2 1 exp ð 2kd Þ

o ¼ g 8 p K þð 2 p M s Þ ½ ; ð 7 : 4 Þ

where d is the thickness of the film. In numerical simulations, we used NiFe
material characteristics: A=1.6 10 6 erg.cm 1 ,4 p M s =10 kG, 2K/M s =4Oe,
g =19.91 10 6 rad/s Oe, a =0.0097, known from the literature [1, 18]. Taking the
fitting parameters for the wave packet obtained in [2], we use t =0.6 ns,
k 0 =0.25 m m 1 and d =5.7 m m for d=27 nm. In Figure 7.17, we have show the
results of numerical simulations illustrating spin-wave packet propagation.
The distance between the excitation point and the point of observation is
50 m m. The spin waves produce perturbation in spin orientation perpendicular to
the direction of magnetization, whose amplitude is much less than the saturation
magnetization M y

=

M s 1.

50 μ m

2

0

− 2

0 1 2 3 4 5

Time (ns)

Figure 7.17. Numerical simulations results: propagation of a spin-wave packet.