6.012 - Microelectronic Devices and Circuits
Fall 2009
The Gradual Channel Approximation for the MOSFET:
We are modeling the terminal characteristics of a
MOSFET and thus want i
D
(v
DS
, v
GS
, v
BS
), i
B
(v
DS
, v
GS
, v
BS
),
and i
G
(v
DS
, v
GS
, v
BS
). We restrict our model to v
DS
≥
0 and
v
BS
≤
0, so the diodes at the source and drain are always
reverse biased; in this case i
B
≈
0. Because of the insulating
nature of the oxide beneath the gate, we also have i
G
= 0,
and our problem reduces to finding i
D
(v
DS
, v
GS
, v
BS
).
The model we use is what is called the "gradual
channel approximation", and it is so named because we
assume that the voltages vary gradually along the channel
from the drain to the source. At the same time, they vary
quickly perpendicularly to the channel moving from the
gate to the bulk semiconductor. In the model we assume
we can separate the problem into two pieces which can be
worked as simple one-dimensional problems. The first
piece is the x-direction problem relating the gate voltage
to the channel charge and the depletion region; this is the
problem we solved when we studied the MOS capacitor.
The second piece is the y-direction problem involving the
current in, and voltage drop along, the channel; this is the
problem we will consider now. To begin we assume that
the voltage on the gate is sufficient to invert the channel
and proceed.
Notice that i
D
(v
DS
, v
GS
, v
BS
) is the current in the
channel; this is a drift current. There is a resistive voltage
drop, v
CS
(y), along the channel from v
CS
= v
DS
at the drain
end of the channel, y = L, to v
CS
= 0 at the source end of
1