  In this paper, variational and Hamiltonian formulations of the
Vlasov--Poisson equations with Maxwell--Boltzmann electrons are introduced.
Structure-preserving particle-in-cell methods are constructed by discretizing
the action integral and the Poisson bracket. We use the Hamiltonian splitting
methods and the discrete gradient methods for time discretizations to preserve
the geometric structure and energy, respectively. The global neutrality
condition is also conserved by the discretizations. The schemes are asymptotic
preserving when taking the quasi-neutral limit, and the limiting schemes are
structure-preserving for the limiting model. Numerical experiments of finite
grid instability, Landau damping, and two-stream instability illustrate the
behavior of the proposed numerical methods.
