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

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.

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.

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

Again, recognizing that we have where

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

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

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

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

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.

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

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.

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

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

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

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

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

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

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,

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.

Therefore or with

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

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

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

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

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

i.e., Since

With these, we have But

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

This is expected as Cs VG Vsub YS

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

We get

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

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).

Difference between meff and mFE

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

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

meff vs. effective field for four substrate doping concentrations

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.

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