Employing Fermi-Dirac (FD) statistics and position-dependent band-gap narrowing the carrier concentrations can be expressed within the rigid-parabolic-band model by
with the gate-contact potential determined by
and
In a constant doping case, A.1 - A.2 reduce to
If Maxwell-Boltzmann (MB) statistics may be applied to minorities,
e.g. holes in 
-type gate, it follows
with the effective intrinsic concentration defined by
denotes neutral electron concentration, 
and
the intrinsic concentration in pure silicon. The equilibrium minority
carrier concentration increases due to total band-gap narrowing
, whereas it decreases
when FD, but not MB statistics is employed for majorities, because
which represents the
deviation of MB statistics from FD statistics is a positive number. For MB
statistics and constant doping we simply have
An analogous expression holds for electrons in 
-type gate.
