In the frame of the long-wavelength Heisenberg model, the
magnonic bandgaps and the selective transmission in a serial
loop structure, made of loops pasted together with segments of
finite length, are investigated theoretically. The loops and the
segments are assumed to be one-dimensional ferromagnetic
materials. Using a Green function method, we obtained
closed-form expressions for the band structure and the
transmission coefficients for an arbitrary value of the number
N of loops in the serial loop structure. It was found that the
gaps originated from the periodicity of the system. The width of
these forbidden bands depends on the structural and
compositional parameters. We also present analytical and
numerical results for the transmission coefficient through a
defective geometry where the length of one finite branch has
been modified. It was demonstrated that the presence of this
defect in the structure can give rise to localized states inside
the gaps. We show especially that these localized states are
very sensitive to the size of the loops and to the periodicity as
well as to the length and the location of the defect branch.
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