  The work presented in this paper is related to the development of positivity
preserving Discontinuous Galerkin (DG) methods for Boltzmann - Poisson (BP)
computational models of electronic transport in semiconductors. We pose the
Boltzmann Equation for electron transport in curvilinear coordinates for the
momentum. We consider the 1D diode problem with azimuthal symmetry, which is a
3D plus time problem. We choose for this problem the spherical coordinate
system $\vec{p}(|\vec{p}|,\mu=cos\theta,\varphi)$, slightly different to the
choice in previous DG solvers for BP, because its DG formulation gives simpler
integrals involving just piecewise polynomial functions for both transport and
collision terms. Applying the strategy of Zhang \& Shu, \cite{ZhangShu1},
\cite{ZhangShu2}, Cheng, Gamba, Proft, \cite{CGP}, and Endeve et al.
\cite{EECHXM-JCP}, we treat the collision operator as a source term, and find
convex combinations of the transport and collision terms which guarantee the
positivity of the cell average of our numerical probability density function at
the next time step. The positivity of the numerical solution to the pdf in the
whole domain is guaranteed by applying the limiters in \cite{ZhangShu1},
\cite{ZhangShu2} that preserve the cell average but modify the slope of the
piecewise linear solutions in order to make the function non - negative. In
addition of the proofs of positivity preservation in the DG scheme, we prove
the stability of the semi-discrete DG scheme under an entropy norm, using the
dissipative properties of our collisional operator given by its entropy
inequalities. The entropy inequality we use depends on an exponential of the
Hamiltonian rather than the Maxwellian associated just to the kinetic energy.
