  This work is devoted to the optimal decay problem for the Euler-Poisson
two-fluid system, which is a classical hydrodynamic model arising in
semiconductor sciences. By exploring the influence of the electronic field on
the dissipative structure, it is first revealed that the
\textit{irrotationality} plays a key role such that the two-fluid system has
the same dissipative structure as generally hyperbolic systems satisfying the
Shizuta-Kawashima condition. The fact inspires us to give a new decay framework
which pays less attention on the traditional spectral analysis. Furthermore,
various decay estimates of solution and its derivatives of fractional order on
the framework of Besov spaces are obtained by time-weighted energy approaches
in terms of low-frequency and high-frequency decompositions. As direct
consequences, the optimal decay rates of
$L^{p}(\mathbb{R}^{3})$-$L^{2}(\mathbb{R}^{3})(1\leq p<2)$ type for the
Euler-Poisson two-fluid system are also shown.
