SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

444 CHAPTER 9. FIELD EFFECT TRANSISTORS: MOSFET

Also

ΔVox =ΔEox·dox=

sΔEsdox

ox

=

sΔEs

Cox

(9.3.8)

whereCoxis the oxide capacitanceperunitarea (=ox/dox). Thus

VGS=Vfb+ψs+

sΔEs

Cox

=Vfb+ψs+

Qs

Cox

(9.3.9)

Let us evaluate the threshold voltageVTapplied to the gate at which strong inversion starts in
the channel. A reasonable approximation when inversion just occurs, the charge in the channel
isessentiallyduetothedepletioncharge(=eNaW)sincethetotalfreechargeisstillsmall.
This is because even though the maximum mobile charge at inversion is equal to the bulk charge
concentration,Na, its concentration drops off exponentially with band bending and hence the
areal charge density is much smaller than the depletion chargeeNaW. Using the relation
between the depletion widthWand the potentialVs,

W=

(

2 s|ψs|

eNa

) 1 / 2

(9.3.10)

the areal charge density (Qs=eNaW) becomes (usingψs(inv)=2φF)

Qs=(2seNa|ψs|)^1 /^2 =(4seNa|φF|)^1 /^2 (9.3.11)

This gives, from equation 9.3.10,

VT=VGS(ψs=+2φF)=Vfb+2φF+(4esNa|φF|)^1 /^2

1

Cox

(9.3.12)

Oncetheinversionconditionissatisfied,thedepletionwidthdoesnotchangesincethelarge
densityoffreecarriersinducedafterinversionstartspreventfurtherdepletion as all additional
applied voltage is dropped across the oxide since small changes in semiconductor band bending
cause exponential increases in the inversion charge. The maximum depletion width is given by
usingψs=+2φFin equation 9.3.11 as

Wmax=

(

4 s|φF|

eNa

) 1 / 2

(9.3.13)

Using the above equation and equation 9.3.6, the field at the surface at the onset of strong inver-
sion is

Es=

(

4 eNa|φF|

s

) 1 / 2

(9.3.14)