SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

372 CHAPTER 8. FIELD EFFECT TRANSISTORS

underconsideration, can be readily shown to be the gradual channel solution of the depletion
region potential derived previously,

Vpar(x, y)=−VG+

e



[

N(y)−N(0)−yN(a)

]

(8.3.24)

where the following convention has been used.

N(y)≡

∫

N(y)dy (8.3.25)

The above equation readily reduces to the equations in the previous section when N(y) is assumed
to be constant,ND. Hence the total voltage in Region II is given by

V(x, y)=−VG+

e



[

N(y)−N(0)−yN(a)

]

+

2 hEc

π

sin

(πy

2 h

)

sinh

(πx

2 h

)

(8.3.26)

Along the liney=h, equation 8.3.26 reduces to

V(x, h)=Vpar−VG+

2 hEc

π

sinh

(πx

2 h

)

(8.3.27)

The sine function in equation 8.3.26 reflects the inherent symmetry of the structure with a pe-
riod 2 hin they-direction, whereas the hyperbolic function is the standard solution of Laplace’s
Equation in the non-symmetric direction.ThishyperbolicdependenceofVonxiscriticaltothe
operationofFETsinthesaturationregime. The band diagram (and hence the voltage profile) of
a FET biased in the saturation regime is shown in figure 8.10b.
Ascanbeseen,thesolutionofLaplace’sEquationleadstotheelectrostaticformationofa
“collector”region,usingterminologyborrowedfrombipolartransistors. The difference between
this collector formed due to saturation/pinch-off is that voltage has an exponential dependence
on length with applied voltage to conventional depletion regions that follow simple power laws
of depletion depth with voltage. Voltages applied beyondVDS(sat), shown in figure 8.12 and
labeledVdp, are therefore absorbed efficiently within small extensions of Region II. The slope
of theI-V curve in the saturation region is the output conductancegdand is first explained
quantitatively and then analytically.

Qualitative description of the output conductance

For a qualitative description of the output conductance, let us consider what is happening on
the source side of the device. If the source electric field isEs, then the current in the device is

ID=AeμnnsEs

whereeμnns=σsis the conductivity of the source andA=W·his the cross-sectional area
of the device. Under most conditionsσsis not a function of drain biasVD. Hence any increase
inIDcan only result from an increase inEs. Therefore, a large increase inIDwith respect to