We fabricate nanolayer alumina capacitor and apply high electric fields, close to 1 GV m-1, to inject charges in the dielectric. Asymmetric charge distributions have been achieved due to the selectivity of the quantum tunneling process. Namely, the electrons near the Fermi level cannot tunnel intoregions near the cathode, where the total energy would be less than the potential energy. This mechanism exhibits a strong tendency to populate charge traps located near the anode, i.e. the regions where their potential energy is the lowest. Such spatially selective charging of the dielectric allows a permanent bulk charge storage in the dielectric layer, even if the capacitor plates are short-circuited, provided that the temperature is sufficiently low so that the conductivity of the dielectric is negligible. The stored charge can be recovered if the temperature is increased above ~250 K for the dielectric tested, i.e. Al2O3. In our experiments, the total charge stored in the dielectric was up to seven and a half times higher than the charge stored on the capacitor plates. Also, measurements of the breakdown voltage show that the breakdown electric field, i.e. the dielectric strength, is independent of the thickness of the dielectric.