  We investigate the three-dimensional compressible Euler-Maxwell system, a
model for simulating the transport of electrons interacting with propagating
electromagnetic waves in semiconductor devices. First, we show the global
well-posedness of classical solutions being a sharp small perturbation of
constant equilibrium in a critical regularity setting, uniformly with respect
to the relaxation parameter $\varepsilon>0$. Then, for all times $t>0$, we
derive quantitative error estimates at the rate $O(\varepsilon)$ between the
rescaled Euler-Maxwell system and the limit drift-diffusion model. To the best
of our knowledge, this work provides the first global-in-time strong
convergence for the relaxation procedure in the case of ill-prepared data.
  In order to prove our results, we develop a new characterization of the
dissipation structure for the linearized Euler-Maxwell system with respect to
the relaxation parameter $\varepsilon$. This is done by partitioning the
frequency space into three distinct regimes: low, medium and high frequencies,
each associated with a different behaviour of the solution. Then, in each
regime, the use of efficient unknowns and Lyapunov functionals based on the
hypocoercivity theory leads to uniform a priori estimates.
