The technology of MEMS atomic cells containing rubidium or caesium vapors in an atmosphere of neon buffer gas has been developed. Two-chamber silicon cells containing an optical cavity, shallow filtration channels and a technical container for a solid-state alkali source have been implemented in a single-step process of anisotropic wet chemical etching. To prevent significant undercutting of the filtration channels during etching of the through silicon cavities, the shapes of the compensating elements at the convex corners of the silicon nitride mask have been calculated and the composition of the silicon etchant has been experimentally found. The sealing of the cells has been carried out by silicon-glass anodic bonding at a temperature of 250 оС. For this purpose the LK5 glass which has an increased ionic conductivity in comparison with the conventional glass Borofloat 33 was used. The best microfabricated cells allowed us to obtain estimates of the relative instability of the coherent population trapping resonance frequency at the level of 5·10-11 at 1 s.