8.3. CURRENT-VOLTAGE CHARACTERISTICS 369
Figure 8.10a shows a schematic diagram of the FET depletion profile whenVDS>VDS(sat).
Beyond pinch-off, the channel does not become any narrower near the drain, since this would
imply a reduction in current and would have to be accompanied by adecrease in the electric
field near the source. Rather, any additional drain voltage is supported by a lateral extension
of the depletion region near the drain. The universality of the analysis alluded to in the title of
this section comes from the similarity of the electrostatics that exists in all FETs to first order
in the saturation regime.Once the device behavior is understood in the saturation region in say
the JFET that is detailed below, the analysis can be readily extended to MOSFETs, HEMTs etc
by merely changing materials and geometrical parameters but keeping the device physics the
same. A very illustrative analysis of the JFET in saturation has been presented by Grebene and
Ghandhi. The detail of their analysis leads to the physical understanding of the universality of
FET electrostatics and I-V behavior and hence deserves consideration. It explains the basis of all
FET design, namely the high aspect ration design, in an elegant, analytical manner. Following
their analysis, it is useful to divide the channel into two separate regions in the direction of current
flow, as shown in figure 8.10. In Region I, near the source, the electric field in the direction of
current flow is small, so the gradual channel approximation is valid. In Region II, near the drain,
the electric field in the direction of current flow is large, so carriers travel at their saturation
velocity. Prior to pinch-off, Region I covers the entire channel, and the current characteristics
are described by the Shockley model.VDS=VDS(sat)represents the onset at which Region II
appears. Beyond pinch-off, Region II continues to become longer. However, the field profile in
Region I remains approximately constant, which implies that the current remains nearly constant
even asVDSis increased.
For the saturation region (Region II), a fundamentally different relationship between voltage
and distance occurs. Here, the charge and electric field (x-component) distributions are shown
schematically in figure 8.11. The voltage distributionV(x, y)in the saturation region is deter-
mined by solving Poisson’s equation.
∇^2 V(x, y)=−ρ(x, y)
=−
eN(y)
(8.3.17)
The solution to this partial differential equation can be divided into a homogeneous solution and
a particular solution such thatV(x, y)=Vhom(x, y)+Vpar(x, y),where
∇^2 Vhom(x, y)=0∇^2 Vpar(x, y)=−eN(y)
(8.3.18)
The solution to the homogeneous part is the solution of a Laplacian is the part of the solution
that isindependentofthedopinginthechannelandisthereforeindependentoftheparticular
formofthegatingmechanismwhichdefinesthevariouscategoriesofFETs, namely a junction
gate for a JFET, an MOS capacitor for a MOSFET and a modulation doped doped structure for a
HEMT. This provides the near-universality to the electrostatics of different FETs. The solution
of the homogeneous part (the Laplacian) is assumed to be of the form
Vhom(x, y)=∑∞
n=1Ansin (αny)sinh(βnx) (8.3.19)