  Similar to Weyl semimetals, in magnetic materials, magnon bands can host Weyl
points, around which the bosonic excitations are called Weyl magnons. Here, we
investigate the Klein tunneling of Weyl magnons, during the process of which
Weyl magnons tunnel through a high potential barrier. Specifically, we study
the magnetization current carried by Weyl magnons in a quasi-one-dimensional
magnetic wire, in the middle of which a gate magnetic field is applied to
generate a potential barrier. The transmission probability is calculated and
the Landauer-B\"uttiker formalism is used to find the magnetization current.
Various types of Weyl magnons are considered, including isotropic, tilted, and
double Weyl magnons. Unlike in Weyl semimetals where fermionic statistics is in
charge and the current oscillates with the gate field as a result of
Fabry-P\'erot resonances, here Bose distribution smears out the oscillations.
We find that the tilting of the Weyl cone causes the decrease of the
magnetization current from Klein tunneling, while for double Weyl magnons,
Klein tunneling is absent in the direction of quadratic dispersion, but is
enhanced in the direction of linear dispersion. Our results show that the
behaviour of the current-voltage characteristics of magnons is rather different
from that of electrons due to different statistics, although the
single-particle properties, such as the transmission probabilities, are the
same for both Weyl bosons and Weyl fermions.
