1. Long Channel MOS Transistors
The basic cross-section of an MOS transistor is L N+
Sub NA
SiO2 S G D

2. Two-dimensional band diagram of an n-channel MOSFET.
(a): Device configuration.
(b): Flat-band zero-bias equilibrium condition.
(c): Equilibrium condition under a gate bias.
(d): Nonequilibrium condition under both gate and drain biases.

3. We will first consider the inversion charge at the drain end assuming that the vertical electric field is much larger than the horizontal electric field.
That is, we need to solve a 1-D Poisson equation along the vertical direction.

4. \ we have: By assuming mFp to be constant, we have:
Now assuming mFn to be constant along x, we have:
since
\ we have:

5. Again, recognizing that
we have
where

6. In particular, at the surface
with
QB is the bulk (depletion) charge:
Recall that for a MOSCAP, the maximum bandbending
Therefore, near the drain end

7. Also, we will approximate:
Note that xi is usually very small

8. Now, as in the case of MOSCAP, under strong inversion
F can be approximated as
With sheet-charge approximation:

9. Gradual channel approximation is good if
since
Note: #4 implies we can use the equation derived earlier for Qinv!

10. Note the coordinate we use for MOSFETs
Note that gradual channel approximation Þ the Poisson’s equation is roughly one-dimensional and in the x-direction.

11. Therefore, the band diagram along x-direction at y looks like:
First, from equation earlier, we can derive:

12. The equation is too complicated to achieve an analytic equation for the I-V characteristics for MOSFETs. Now, using gradual channel approximation as well as sheet charge approximation for the inversion charges, we can write(if VFB=0):
and
Further, we can approximate QB(y) by depletion approximation.

13. Under the gradual channel approximation, depletion width is like a reversed bias p-n junction with the applied biase V(y)+Vsub and a built-in potential of 2.1fB+VT or:
Hence
and

14. The channel conductance at y is therefore:
i.e.,
But I is constant along the channel in steady-state, which means

15. Carrying out the integration,
Let’s first look at the limiting cases of VD » 0:

16. with
including
we have
with VD » 0, the conducatance is:
This is expected, since in this case

17. \ the conductance is
The transconductance in this case is given by

18. Now, this current equation assumes Qinv(y) ¹0 for any y
Now, this current equation assumes Qinv(y) ¹0 for any y. That is it is only “valid” up to when
That is
where
IDsat can be obtained by putting VDsat into

19. For VD>VDsat, we approximate
This is so called “pinch-off” approximation.
Using this formula for IDsat, we can calculate the saturated regime transconductance:
i.e.,
If NA is small and Cox large,

20. with m as a correcting factor and m ®1/2 when NA is small.
IDsat can also be written as:
If we assume
which is approximately true if the substrate doping is small (why?),
i.e.

21. Therefore or
with

23. If we consider both drift and diffusion, i.e.
The total current is or
and YS is related to VG by

24. Subthreshold Currents in Long Channel MOSFETs
VD=VDD ID
VTH VG

25. Consider the band diagram with an applied VSub, and VG with respect to the substrate (with VFB=0).
EFn
EFp EC EV V Ei fB YS

26. To derive I-V characteriastics, we notice that
We need to find
First
Note: (4) implies that constant along the channel

27. To get QS(0) and QS(L), from ns(0) and ns(L), we use the following
now
and exponential decay rapidly

28. i.e.,
Since

29. With these, we have
But

30. What is
Under weak inversion
since
Recall the depletion capacitance is given by

31. This is expected as Cs VG
Vsub YS

32. Let us now estimate yS by assuming that
i.e., mid-way between the onset of weak inversion and the onset of strong inversion.
Note also that
and

33. We get

34. log(ID) VG
VD > 3VT
A slightly more accurate derivation gives
where, as before

36. Mobility Behavior in the Inversion Layer
Mobility, m, is an important parameter of MOSFETs. Because the conducting carriers are in a confined inversion layer at the Si/SiO2 interface, there are additional scattering centers apart from that of the bulk. Also, due to the two-dimensional nature of the charge, they can have quantum mechanical behavior different from that of the bulk carriers. In general, surface mobility is ~0.6X that of the bulk. Vsat is also smaller. There are two definitions of mobility in MOS systems, meff and mFE (effective and field effective).

37. Difference between meff and mFE

38. Physical mechanisms of surface carrier scattering are:
scattering due to bulk-ionized impurity
scattering due to bulk phonons
scattering due to and , associated with the Si/SiO2 interface
surface acoustic and optical phonons scattering
surface roughness scattering

39. Now, from quantum mechanics, the inversion charge has an approximately Gaussian profile. Thus, from Gauss’ Law, EC EV
EFn
Inversion charge mobility depends on the inversion charge confinement
It was found that

40. meff vs. effective field for four substrate doping concentrations

41. Experimental electron inversion layer mobility vs
Experimental electron inversion layer mobility vs. effective field at two substrate doping levels. VBS =0, -2 and -4 V.

42. For the Ey dependence, it was found that
where
vs, vs and G are fitting parameters.

44. If we consider both drift and diffusion, i.e.
The total current is or
and YS is related to VG by

45. Effects of oxide fixed charges
Recall: if no charges exist inside the SiO2, Eox is constant and
Also
With oxide fixed charges, Gauss law implies
SiO2
Metal Si x
x=0