  This article develops a consistent theory of free carrier screening of a
two-dimensional electron gas in the silicon inversion layer in the presence of
stacked layers of dielectric environment-commonly knows as gate stack in
context of field-effect transistors. It is shown that the finite thickness and
of dielectric stacks alters the free carrier screening, a crucial quantity,
which determines screened coulomb interaction in the inversion layer, and
ubiquitously appears in carrier transport theory in semiconductors. Results are
analytical and can be used to accurate prediction Coulomb-interaction limited
mobility in field-effect transistors.
