Threshold shifts due to nonuniform doping profiles in surface channel MOSFET's

The threshold shift due to ion implantation may be defined as the shift in curves of
N_{\inv}
versus V
G
, where
N_{\inv}
is the inversion layer carrier density per unit area and V
G
is the gate bias. This definition corresponds to the experimental shift of current versus gate bias curves, because current is proportional to
N_{\inv}
in the linear regime of operation of the MOSFET (metal-oxide-semiconductor field-effect transistor). In addition, the shift in
N_{\inv}
versus V
G
curves is not sensitive to the value chosen for
N_{\inv}
, provided this value is not so low as to fall within the weak-inversion, subthreshold regime. Therefore, this definition of threshold shift avoids the use of arbitrary criteria for threshold, such as 2φ
B
(φ
B
= bulk Fermi level measured from midgap), which have uncertain meaning in implanted structures. Here, the use of
N_{\inv}
versus V
G
curves is made practical by introduction of a simplified method of calculation. It is shown how to evaluate
N_{\inv}
and V
G
without using a numerical solution of Poisson's equation, by invoking a charge-sheet approximation for the inversion layer and a modified depletion approximation for majority carriers. The resulting formulation applies provided the profile does not vary rapidly within a Debye length of the depletion edge, and provided most of the implant extends beyond the inversion layer. Threshold shift is shown to depend primarily upon the zero-order and first-order moments of the excess surface charge the implant has introduced into the depletion region of the device (the dose and centroid of this portion of the implanted charge). For fully depleted implants, a simple equivalent delta-function implant with the same dose and centroid as the real implant can be used to find threshold, and calculations can be made on a programmable pocket calculator. For a partially depleted implant, the moments of the depleted portion of the implant are needed. In addition, built-in junction effects can become significant. Comparison of the delta-function approach with a slightly more complex calculation for Gaussian implants is made. For fully depleted implants, agreement is complete. For partially depleted implants, a good estimate is obtained by introducing the ideas of "effective dose" and of "clamping of the depletion edge."