CHAPTER ELEVEN contd…

EXTRINSIC SEMICONDUCTORS 11.3.1 Qualitative effect of Fermi level on ionization fraction

Let us first start with a qualitative understanding of effect of the Fermi level on ionization

fraction, if we add some donor to the semiconductor. If (suppose) Fermi energy is

somewhere near the intrinsic level (we have yet to find out where it will be), then the

fraction of donor ionized will be almost 1 according to 11.6 (EF-ED is large negative

value). If Fermi energy moves to ED level itself then fraction ionized is ½ according to

11.6. So, if EF is at donor level then only half of the donors would have given up their

electron. So one can see that moving the Fermi level can change the ionization fraction.

Same argument can be built for acceptors also.

11.4 Compensation and charge neutrality

Suppose we have added ND donors and NA acceptors (ND>>NA), what is the picture at

0K? Electron from donors can go to acceptors and thereby reduce their overall energy.

Now, if we raise the temperature, then only available electrons (ND-NA) can go to

conduction band. The same argument will hold for NA>>ND.

Further, a material in equilibrium has no charge build up, hence charge neutrality in the

material must be maintained:

11.5 Quantitaive estimation of n and p

We know from Ch. 10, for non-degenrate semiconductor

Since we do not know EF apriori, let us find value of n and p for different conditions of

ionization.

11.5.1Determine n and p when there is complete ionization (case ND >>NA for the

following discussion)

and ,

Inserting this into neutrality condition and using the np product property of

semiconductor (see lecture notes for details), we find the solution for n as

11.8

As you can see, we know all parameters on the right side and hence it is possible to

calculate n and p ( using ). For a given dopant level, a special case is when

, we find that n = or our qualitative picture of compensation

discussed earlier is correct. Similarly, if NA >>ND, it will be p = NA--ND.

From n and p, we can calculate the Fermi level for the extrinsic semiconductor in this

condition using equation from Ch.10.

This point can be easily visualized if we plot the carrier concentration (ln n) of an

extrinsic semiconductor having ND >>NA as a function of 1/T, as shown in Fig. 11.5.

First we plot the ni concentration for this semiconductor as 1/T and slope of this curve

can be used to calculate the bandgap of the semiconductor. In the same plot, we make a

line for ND-NA, so at temperature Tmax, where intrinsic carrier concentration starts

becoming equal to ND-NA, for temperatures below this carrier concentration remain

constant at n = ND-NA as long as all dopants are ionized and is

satisfied. This region is known as the exhaustion region for the extrinsic semiconductor.

Fig. 11.5 Schematic of carrier concentration in an extrinsic semiconductor.

11.5.2 Determine n and p when there is incomplete ionization

If there is incompete ionization then 11.8 reduces to

11.9

Now, 11.9 for a given semiconductor has to be solved numerically as we know all the

expressions, but solution has to be obtained numerically. We can still look at some

special cases.

First case is when , but all donors are not ionized . For this reason, it

is also known as freeze out zone, as all electrons cannot be excited to the conduction

band and this determines the lower limit of temperature where exhaustion zone ends.

Starting from the neutrality condition again we can write

Using the ionization fraction expression and assumption the p <<n:

Rewriting it as

Observe, there is no Fermi level required in this equation. This can be rewritten in the

following form:

11.10

Since we are looking at freeze-out region, we know ND-NA >n (ND >n, as ND>>NA) now

there can be two cases, when n> NA and n < NA.

When n> NA, the Eq. 11.10 reduces to

11.11

So, at Tmin temperature the “ln n” vs 1/T plot will show a slope given by Eq. 11.11

“ “.

When n < NA, the slope will be . So, in Fig. 11.5, around n=NA there will be a slope

change. This now becomes a way to estimate NA

Now, Fig.11.6 ( can be obtained experimentally) can be used to estimate Eg, ED-EC, ND

and NA for a given extrinsic sample.