In this video we will continue our, the review of basic facts on semiconductors.

We will concentrate on the energy bond model, and how we can describe

concentrations of electrons and holes as functions of energy level.

I'm assuming we're in equilibrium which means that there will be no energy

exchange with the external world, no current flow, no temperature gradients

that would cause heat flow and so on. The semiconductor carriers can be

described by, by what is called Fermi-Dirac statistics which actually for

the cases we will be considering, common temperatures for example, simplify to the

so-called Maxwell-Boltzmann statistics which are shown here.

This is the, full concentration in equilibrium.

N i is the corresponding electron concentration in a purely, in a pure

intrinsic semiconductor. K is Boltzmann's constant.

T is absolute temperature. And e sub f is something called the Fermi

level that characterizes the semiconductor.

And it is different, depending on the doping.

The same quantity appears over here. Now, to show you what the Fermi level is,

for an intrinsic semiconductor. A Fermi level is approximately at the

middle, between the valence band with the conduction band, and it is called e sub i.

So e sub i, basically, it's the Fermi level for an intrinsic semiconductor.

So, if the two are equal. In these formulas you can see here that

the exponent becomes 0, e to the 0 becomes 1, so the hole concentration is n i, and

similarly here the electron concentration is equal to n i.

But ea-, if at other in other situations where you have doping in the

semiconductor, e f becomes different from e i and then these formulas will give you

different concentrations for electrons and holes.

At equilibrium. For example, for an n-type semiconductor.

We have e sub f higher than e sub i. Which means that this exponent now becomes

positive. This becomes something larger than one.

So the concentration of electrons increases above the intrinsic

concentration. Correspondingly here, e i minus e f is

negative. The exponential becomes less than 1.

And the whole concentration becomes less than the corresponding intrinsic

concentration. But if you multiply p0 times n0 you see

that the exponents cancel out and you get p0 times n0 is equal to n i squared which

is a formula we have seen before. Continuing with a L p-type semiconductors

here E sub f is less than E sub i. And therefore e i minus e f is positive,

and the whole concentration becomes high. E f minus e i is negative, and the

electron concentration becomes low. For p-type semiconductors.

Now, in order to simplify the way we write these formulas, and also in order to bring

up some, quantities that we will be using, often, we define the so-called Fermi

potential as the difference between the intrinsic and Fermi levels divided by q.

The units here are units of energy divided by units of charge, so this has units of

potential. And we will call that the Fermi potential,

Phi sub f. Here is phi sub f multiplied by q.

From this definition you can see that for an n-type semiconductor phi sub f is a

negative quantity. For the p-type semiconductor you can see

that phi sub f is a positive quantity. In addition, we will define the so called

thermal voltage, which is k t over q. Phi sub t for us is the thermal voltage.

You probably have already seen this already up here in equation, basic

equations for transistor i v characteristics.

So now in terms of these two quantities we can rewrite these formulas like this.

So in this simple notation, we have the number of holes per unit volume is given

by this, and therefore, if we have phi sub f being positive, as it is for a p-type

semiconductor, the number of holes becomes high, and correspondingly, the number of

electrons becomes low. The product of these two still gives you n

i squared as before. Here I have collected a whole bunch of

values for this quantity, so I'm not going to read them out but you're welcome to

refer to this slide to get the values when needed.

Let us now take the equations in the last three slides.

From those you can derive these quantities.

By the way, I will not be deriving equations, so I'm not going to be going

through the algebra, but I really recommend when you study at home, that

when you see an equation like this you stop and you derive it from earlier

equations in order to consolidate this material.

So, the Fermi potential for p-type and n-type semiconductors is given

correspondingly by this. Where n sub a is the acceptor

concentration. For p-type material, n sub d is the donor

concentration for n-type material. And you can plot phi sub f.

And I'm, here I'm plotting the absolute value of phi sub f, because I want this

plot to be valid both for n-type and p-type semiconductors.

So here is the concentration of impurities, acceptor or donor, depending

on what you have. And here is the Fermi potential.

You can see that there is an one to one correspondence for a given temperature

between doping and Fermi potential. When I say doping, I mean doping

concentration. Doping concentration and Fermi potential.

If you give one for a given temperature, it's like you gave the other.

And you can see for, that for typical impurity concentrations, for example 10 to

the 17th, phi sub f is few tenths of one volt.

Finally let me discuss equilibrium now in the presence of electric fields.

When we have electric fields present, but still we don't have a net exchange of

energy with the external world, still no current flow, we will see situations where

this happens soon. We still have the these equations that I

just showed you before. For the hole concentration and the

electron concentration. Still, the product of n times p is given

to n i squared as before although we may have electric fields present, and it, at

equilibrium, the Fermi level remains constant throughout.

But the Fermi level, very intuitively speaking, is something like the level of

water in your bathtub. If the level of the water is quiet and

nothing is moving you have equilibrium. But if you put your hand inside and you

agitate the water, then there will be waves and momentarily you disturb

equilibrium until things settle and then you one, have one single level at the

surface of water. It's like saying the Fermi level is

constant in a semiconductor, but other bands can bend.

Other, other band levels can bend, so for example, you can have this situation.

The Fermi level is constant. You can see it here, but all of the other

levels, e i, e c, and e v bent, and they bent in order to accommodate the presence

of for electric potentials. For example: going from this point, to

this point there is an increase in e sub c, delta e sub c, and a corresponding

increase in e i, delta e sub i. And these two are equal.

Because potential is energy per charge. And because the charge in an electron is

negative. The change in potential will be in the

opposite direction from the change is e sub c.

So, e sub c is going up here. The corresponding potential is going down.

So e sub c corresponds to potential energy for the electron.

C is electric potential, and you can see that it is going down by this amount.

We will see such diagrams and we will consolidate them when we discuss the

complete semiconductor devices. One final piece of information here.

Let's assume we have a semiconductor like this.

There are two points, one with electro concentration n1.

Another one with electro concentration n2. Psi one two is the electric potential from

point one to point two. If you take the equations I showed you

before, for equilibrium, you have the following.

First of all n times p is always equal to n i squared.

And n i squared by the way is a strong function of temperature.

Now, the ratio of the electronic concentration at these two points is

related simply to the potential between them, psi, 1, 2.

I remind you that, phi sub T, in this equation is the.

Thermal voltage typically about 26, millivolts near room temperatures.

This equation can be derived from the equations I have already shown you.

And I recommend that you do try to derive it at your own pace.

We can also find the ratio of whole concentrations at the same two points.

And the result is like this. The ratio of full concentration, at point

one to that of point two, is given by a similar exponential.

Only here you have psi 2 1. In other words this is the potential of

point two with respect to point one, as opposed to psi 1 2 that you have in the

case of, electron concentrations. So we will be using these equations

directly in our development of the MOS transistor characteristics.

Rather than going all the way back to the equations where these.

We're derived from. In this video we have talked about, a

situation in the semi-conductor in the absence and in the presence of electric

fields, always at the equilibrium. We talked about, electron and hole

concentrations. In such cases.

And in the next video we will continue our discussion of semiconductors including the

case of non-equilibrium.