EE 436 MOSFET current – 1

The electron sheet concentration under the gate is

where φ(y) is the local electrostatic potential in the semiconductor just under the gate.

qns = Cox [VG � φ (y) � VT]

If the drain voltage is also at ground potential, then all parts of the semiconductor (source, body, and drain) are all at φ = 0 and

qns = Cox [VG � VT]

Of course, with VDS = 0, no current will flow. If we increase VDS slightly, say just a few millivolts, we can assume that the electron sheet concentration is still essentially uniform, but a drift current will begin to flow from drain to source. (Electrons are moving from source to drain.)

EE 436 MOSFET current – 2

iD =VDS

Rchannel

Rchannel =ρLA =

LqμnninvWtinv

where ninv is the electron concentration (number per unit volume) of the inversion layer and tinv is the thickness of the inversion layer. Unfortunately, we don’t either of these quantities. However, we do know

ns = ninv · tinv

which is the electron sheet concentration of inversion layer. Putting everything together.

iD = qμnnsWL VDS

= μnCox [VG � VT]WL VDS

Treating the channel inversion layer like a resistor,

EE 436 MOSFET current – 3

However, as VDS increases, the potential along the channel must change correspondingly:

source: φ(0) = 0

drain: φ(L) = VDS

So we have to look smaller. At any point along the channel,

Jn = σE

iDdy = ��qμnninvW

�dφ

iDWtinv

=�qμnninv

� ��dφdy

�

iDdy = ��qμnninvtinvW

�dφ

with continuous variation in between.

iDdy = ��μnCox (VG � φ (y) � VT)W

�dφ

EE 436 MOSFET current – 4

The current must be continuous (Kirchoff’s current law), and so iD is a constant on the LHS of the equation. Then integrating both sides:

iD� L

0dy = �μnCoxW

� VD

0[VG � φ (y) � VT] dφ

iDL = +μnCoxW2 [VG � VT � φ]2

����VD

0

=12μnCox

WL

�(VG � VT � VDS)

2 � (VG � VT)2�

=12μnCox

WL

�(VG � VT)VDS � V2DS

�

Kn =12μnCox

WL

= Kn�(VG � VT)VDS � V2DS

�