The slope decreases as you go to higher VGS, because the mobility depend on some

VGS. And, for the device with the Halo

Implants, it goes like this. Initially, it has a high slope region,

and then it kind of behaves like the un-implanted device.

The question is why is there a high slope region over here?

It can be explained roughly as follows, first of all, the center device, because

it is on a lightly doped substrate, has a lower threshold on the side devices.

So, as you raise VGS, the center device turns on first.

But initially the side devices are off. As you keep increasing VGS, eventually

the side devices also turn on, but by then the center device has turned on

strongly. It has a strong, strongly inverted layer.

And initially the resistance of layer is much smaller than that of the two side

devices. So you can neglect this resistance and

consider the entire combination as consisting of two devices in series each

of them with length LH. This means that the device behaves as a

transistor with total length of 2LH. Now, if you remember the current equation

has a W over L, factor in it, so instead of l, we have 2LH, which is much smaller

than L. And because of this large factor, W over

2LH. The current versus VGS initially goes up

like this, but later on, when for even larger VGS values, all three devices are

sufficiently strongly inverted. The whole device behaves essentially as a

single strongly inverted device, with length L.

And then again, this curve approximates the one of the un-implanted device.

Now if you take the slope of these curves, which is the trans conductance

versus VGS, you find that it behaves like this, shown by the broken line for the

un-implanted device. Now to model the transistor, the concept

of average channel doping is used. So what is the average channel doping

here? So let's take the center device multiply

by NA, and then take it to this region, which has a length LH, and multiply by NA

plus LH. Add both of them to NA, LC.

And then divide by the total length. This is the average substrate doping, and

as you can see now it's a function of L. So, rewriting, we can write it like this.

And now we have an effective gamma which is obtained by the classical formula for

gamma but replacing the substrate doping by the effective doping which is shown

over here. And this can be written like this, where

LPE Is this here, but very often it used as a fitting parameter.

And then, you can plug this in, into the, threshold voltage equation to get a VTO

that depends on L, and takes into account the effect of the halo regions.

This is a repeat of the formula that I showed you.

You can see that as L becomes small, gamma becomes large and VTO increases.

This is why as we decrease the channel length, VTO goes up.

We have seen this curve back then when we discussed short channel effects, this is

the main cause of the variation of the threshold.

of, with channel length today. So we have seen how we can approximate

the halo regions, the doping concentration regions, and model the

threshold in a simple way. In the next video we will discuss another

situation where doping concentration varies horizontally and results in the so

called well proximity effect.