We are interested in the effect of a constant electric field on the electronic structure of a crystal. We model it by a density-functional theory, and derive from this microscopic model macroscopic features of the system by letting the ratio $\varepsilon $ between atomic spacing and the size of the crystal go to zero. Although many aspects are disregarded in this approach, we show that the effect of the electric field is negligible inside the crystal and estimate its vanishing rate with respect to the distance from the boundary of the crystal.
"Screening of an applied electric field inside a metallic layer described by the Thomas-Fermi-von Weizsäcker model." Adv. Differential Equations 7 (7) 847 - 876, 2002. https://doi.org/10.57262/ade/1356651708