Dealing with that pesky bandgap

November 5, 2012 3:20:01 AM CST

Upon re-reading the last article, I realized I'd left kind of a hole in my argument: I spent all that time talking about Fermi energy levels for atoms and materials, then sort of forgot it when talking about types of materials.

The connection is this: Electrons above a material's Fermi level will live in the conduction band and will be able to move between atoms. Electrons below the material's Fermi level will live in the valence band, and will remain localized.

I also failed to explain that electrons can remain localized within a covalent bond instead of just being localized to a single atom. That kind of localization still lives in the valence band.

.. that's kind of a big deal. Most of the interesting behavior in semiconductors involves electrons hopping between a state of being localized in covalent bonds of the valence band (say that fast a few times) and a state of being fully non-localized in the conduction band.

The silicon bandgap:

Like I said last time, it takes about 1.2v to move an electron from silicon's valence band to its conduction band, but thermal energy can give electrons a bit of a boost.

You can imagine electrons moving between the valence and conduction bands as marbles in a Chinese checkerboard. The marbles are electrons, the dimples are states in the valence band, and the surface of the board is the conduction band. Normally you need an external force to lift a marble out of its dimple and move it somewhere else.

You can imagine the effect of thermal energy as something shaking the board. At absolute zero, there's no shaking at all. As the temperature rises, the shaking gets harder.

At room temperature, the board shakes enough for the marbles to rattle noticeably in their dimples. Occasionally -- though not often -- one will bounce out of its dimple and roll around on the surface of the board until it finds another dimple to fall in.

In silicon, that would be thermal energy exciting a localized electron out of the valence band and into the conduction band, followed by the electron dropping back to the valence band near another atom.

Doping:

A silicon atom has four electrons in its outer shell, and it shares one with each of its four nearest neighbors. In effect, the atom can pretend it has eight electrons (a number that makes atoms happy), four being its own and four being borrowed.

We can mess around with that by putting atoms with different numbers of electrons into the silicon lattice. Those atoms are called 'dopants', and the most interesting behavior comes from combinations of atoms with either three or five electrons in their outer shell.

A dopant with five electrons in its outer shell wants to share one more electron than any of its neighbors want to borrow, and wants to borrow one less than its neighbors want to share. A dopant with three electrons in its outer shell wants to borrow one more electron than its neighbors want to share, and has one less electron to share than its neighbors want to borrow.

This leads to a situation where someone is always unhappy.

When the lattice around a dopant is balanced (every silicon atom has four electrons and shares four electrons), the dopant has either one more or one less electron than it really wants. If the dopant atom is balanced (it has exactly the number of electrons it wants), the lattice around the dopant atom has either one more or one less electron than it really wants.

Electrons and holes:

We describe the presence of an extra electron by dropping the word 'extra' and saying "there's an electron." We describe the lack of an electron by saying, "there's a hole."

Now, even though we know a hole is 'the space where something could be' and not a thing in its own right, it's convenient to pretend that the hole is an actual particle which is the exact opposite of an electron.

We also have to pay attention to who has the electron or the hole.. the dopant or the silicon lattice.

Silicon doped with five-electron dopants is called an 'N-type' semiconductor, where the 'N' stands for 'Negative' (an electron's charge is negative -- long story, but it boils down to Benjamin Franklin guessing wrong and being very persuasive about it). When an N-type dopant is balanced, the lattice around it has an electron. When the lattice around the dopant is balanced, the the dopant has a hole.

Silicon doped with three-electron dopants is called a 'P-type' semiconductor, where the 'P' stands for 'Positive' (holes have a positive charge). When a P-type dopant is balanced, the lattice around it has a hole. When the lattice around the dopant is balanced, the dopant has an electron.

Carrier transport:

Both electrons and holes 'carry charge', so we refer to them generically as 'carriers'. When carriers move from one place to another, that's 'carrier transport'.

There are some interesting nuances to that..

Hole transport:

Remember how I said electricity happens in the conduction band?

I lied again.

It was more of a half truth than a lie actually.. electron transport does happen in the conduction band. Hole transport, on the other hand, happens in the valence band.

It seems a little weird to think of electrons moving between atoms without ever leaving the atoms, but you can do it with clever accounting.

It comes down to the fact that we can only identify electrons in terms of the orbitals they occupy. We can't take the electrons around a silicon atom, name them Groucho, Harpo, Chico, and Zeppo, then say the atom only shares Groucho with its neighbor to the north. The best we can say is that the electron going north always has Groucho's glasses, mustache, and cigar.

If it were possible to do a strict accounting, we'd find that they trade costumes and props while no one is looking.. the one currently going north wearing Groucho's glasses, mustache, and cigar used to be going south with Harpo's wig and bicycle horn.

We also have to remember that the atom is borrowing electrons from its neighbors. I'll call those Larry, Curly, Moe, and Shemp. Those are just as willing to change costumes as the others.

If a silicon atom has a hole, seven of the eight electrons are still there, and they're still trading costumes. If the atom starts off missing a Groucho, that means it's sharing a hole with its neighbor to the north. If we look away then look back, we could find it missing a Curly and sharing a hole with its neighbor to the south.

No electron has moved to the conduction band because this atom still has half-ownership of the hole. The other half-interest in the hole has just moved from north to south.

Hole transport occurs through a combination of sharing and "can't prove it wasn't always there" which is both rigorously supported by the statistical model and strongly reminiscent of a Marx Brothers routine.

Electron transport:

Electron transport works pretty much the same way, but now it's the gag where you look to the left and see Groucho, look to the right and see Groucho, then look back to the left and find that the extra Groucho has disappeared (and is pulling the same gag next atom over).

Recombination:

Like I said earlier, thermal energy can kick an electron into the conduction band, creating both an 'electron' in the 'extra and non-localized' sense of the term, and a hole where it used to be. Those are called 'intrinsic carriers' and there are usually about fifteen billion per cubic centimeter of pure silicon at room temperature.

When an electron in the conduction band encounters the hole where another electron used to be, there's a very strong chance it will lose its extra energy and drop back into the valence band. Both the hole and 'electron' cease to exist, leaving only a burst of photons to mark their collision.

That's called 'recombination'.

The energy emitted as photons can go back to being thermal energy in the silicon lattice, or -- if it has a little more energy -- can leave the silicon in the form of light. That's how LEDs happen.

The distance free holes and electrons can travel before they recombine has a big effect on the electrical properties of the lattice. For pure silicon, it ain't that great.

Dopant concentration:

We can massage the electrical properties of the silicon lattice by adding dopants that plant extra electrons or holes in the material.

The usual concentration is about a trillion dopant atoms per cubic centimeter. That sounds like a lot, but it's only about one dopant atom per hundred million silicon atoms.

Dopants don't just increase the concentration of one carrier in the silicon, they also reduce the concentration of the other. Some of the extra electrons in N-type material recombine with the intrinsic holes, and some of the holes in P-type material recombine with the intrinsic electrons. The total number of charge carriers stays at about fifteen billion per cubic centimeter, but the ratio of holes to electrons changes dramatically.

The math works out so the number of holes times the number of electrons equals the intrinsic carrier density squared. That's roughly 10^20, so if we add 10^15 N-type dopant atoms, we get 10^5 intrinsic holes floating around at any given time.. the electrons would outnumber the holes ten billion to one.

Majority and minority carriers:

The electrons and holes both exist at the same time, they both carry charge. It's just that there are a whole lot more carriers of one kind than the other. The ones supplied by the dopant are most common, and are called the 'majority carriers'. The intrinsic ones currently escaping recombination are called the 'minority carriers'.

The huge imbalance of carriers adds energy to the overall semiconductor lattice, making it easier for electrons to hop up to the conduction band. N-type silicon doped at a concentration of 10^15 atoms per cubic centimeter conducts electricity well enough that its resistance is about 5 ohms per centimeter.

Wrapping up:

If that's all there was to is, semiconductors would be easy. Life gets interesting when you put P-type silicon and N-type silicon together though.

I'll cover that next time.

Random brain cookies:

Chicago, n.: Where the dead still vote ... early and often!