  For a natural number $m$, let $\mathcal{S}_m/\mathbb{F}_2$ be the $m$th
Suzuki curve. We study the mod $2$ Dieudonn\'{e} module of $\mathcal{S}_m$,
which gives the equivalent information as the Ekedahl-Oort type or the
structure of the $2$-torsion group scheme of its Jacobian. We accomplish this
by studying the de Rham cohomology of $\mathcal{S}_m$. For all $m$, we
determine the structure of the de Rham cohomology as a $2$-modular
representation of the $m$th Suzuki group and the structure of a submodule of
the mod $2$ Dieudonn\'{e} module. For $m=1$ and $2$, we determine the complete
structure of the mod $2$ Dieudonn\'{e} module.
