Law of mass action involved in impurity level

[xins]

Hei anyone could help out? I am trying to establish an impurity level model.

According to the charge equilibrium, ND+p+p1=NA+NA1+n (p,NA-contribution from valence band, p1,NA1-contribution from impurity band); then come to law of mass action.

How could I include the impurity concentration in the law of mass action: ¨n*p*p1=ni^2¨ or ¨n*(p+p1)=ni^2¨?

Appreciated to your suggestion.

 

[andrewr]

I'm not sure what you mean by "impurity band".

There are two groupings of energy levels -- "bands"; The lower energy one is known as the valence band for it is the one normally bonding the atoms together; and in chemistry those electrons are historically called "Valence" electrons. I'm going to assume you're talking about Silicon?

In that case the S orbitals, (valence band), are all filled with electrons and there is effectively no freedom of motion because they are "packed" too tightly. But, at the same time, P,D,F orbitals and higher energy S orbitals are all empty. That is what causes pure silicon to be a good insulator; so now to your question; when silicon atoms are brought closer together, these empty higher energy orbitals overlap making these conduction bands to deflect to slightly higher or lower energy smearing out the acceptable orbital energy -- but preserving the fact that these energy levels are *all* full or empty unless thermal wiggling of the silicon and impurity atoms randomly "BUMP" an electron or hole into motion. . Essentially *1* new energy level is added to each band when another atom is brought close enough to interact; the effect is to raise and lower the energies of the electrons in the valence band; and to make available more (empty) energy bands in the conduction area. An impurity atom is no different; it does the same thing but with one important difference -- one of it's valence electrons is at a much higher energy level than the rest., or it has an empty valence band equivalent (hole). It is better to think that an impurity bends the band -- rather than to think of it as a distinct band. Consider that Electrons/Holes are typically (room temp) are within five atoms *average* of their "donor" or "acceptor" even when they are trapped by it (not ionized). Its the Bent band made by the donor or acceptor, that is key to semiconductor action. Eg, if silicon was like a saturated wet sponge, heat is similar to heating the sponge so that water will turn into steam in the pores of the sponge.

It's not really removal of a single charge from an atom, but more providing a "place" where electrons can be bumped by hot atoms or be accelerated by hills and valleys; much like a skate park and kids... :)

Just so; these temporary openings (holes/carrier electrons) are merely the shuffling of the spacing of electrons. The more bends and dips in the conduction and valence bands that the impurities make -- the more LOCATIONS that an electron that is found free in "pure" silicon away from an impurity -- could hit as it moves along, or a place where a wiggling atom might knock a very slowly moving electron into high speed..

Charge neutrality is usually written:
$$p_{0}+N_{D}=N_{A}+n_{0}$$

Where n zero and p zero are the electrons and holes that have gained enough extra energy to move about. So charge neutrality is simply saying the *average* number of excited holes and electrons in any given box/volume of silicon must *also* balance out with the number of places that have sufficiently bent bands to make them come into existence (and to destroy them) in a different ratio than pure silicon..

This causes a dynamic balance of extra energy electrons and holes. The "law" of mass action isn't a fundamental law that can be built easily from an understanding of what holes and conduction electrons are. It is a shortcut that has no intuitive model value -- I have yet to see anyone derive it at the undergrad level...
It is best just to look at the law of mass action as talking about the number of excited electrons and holes regardless of whether they came from an "impurity" or from "pure" silicon. The law is simply stated:
$$n_{0}p_{0}=n_{i}^{2}$$
Where n sub i, is the number of excited carriers that would exist in pure silicon. The reasons for why mass action law holds aren't really described. But it may be thought of a crude expectation that *most* of the silicon is still "pure", only a few impurities show up every 50 to 100 atoms in medium to low doped situations; typical. So the silicon is still "pure" everywhere *in-between* impurities -- and since that's the "average" case of silicon; over time those locations "want" to go back to the concentration of carriers found in pure silicon. If there are more holes than usual, then more of them can be destroyed by the same percentage of decay. So, one can estimate (mass action law) how many will make it on average vs. being destroyed.

The product of free electrons and holes (no matter where they came from) has to stay at the intrinsic level of silicon (unless there are so many impurities that the number on the right side of the mass action law is a bad assumption...and that is a specialty area not worried about in simple models...)

All you need to know is that: $$n_{0}p_{0}=n_{i}^{2}$$
Re arrange it, and plug it into the charge neutrality equation I gave and you will arrive at a quadratic equation that you can solve for the (estimated) number of either carrier. Obviously negative amounts of carriers is not a meaningful solution ... good luck.