Re: [Phys-l] Band splitting in Carbon (diamond)...

[John Denker <jsd@av8n.com>]

Comparing carbon to silicon,

On 03/01/2010 09:41 PM, Jeff Loats wrote:

> Their bands are indeed carbon copies (nice pun!) but because of their atomic
> separations diamond ends up being an insulator while silicon is a
> semiconductor. That is what I mean by similar configurations (same band
> structure) but different conduction behaviors.

Aha!  I know what the problem is here.

Consider the following statement:
  "silicon is a semiconductor, but diamond is an insulator"  [1]

I know that every textbook in the world makes such a statement
... but you shouldn't believe everything you read.  They can
coerce that statement to be true by definition, but only by 
using pathologically foolish definitions.

The main difference between the two is the magnitude of the
band gap.
  Silicon: 1.12 eV
  Carbon:  5.5  eV

In principle and in practice, it is better to think of carbon
as being a semiconductor with a large-ish band gap.

What matters is not so much the absolute size of the band gap
but rather the _ratio_ of band gap to kT.  Specifically, making
a semiconductor device out of carbon is essentially the same
as making the corresponding device out of silicon and running
the latter at a 5x lower temperature.  Liquid nitrogen is very 
cheap and provides a temperature 4x below room temperature, 
which is enough to illustrate the point;  liquid helium is 75x 
colder than room temperature, which makes the point with more
than an order of magnitude to spare.

Suppose you *wanted* to operate a semiconductor amplifier at
nitrogen temperature (or helium temperature) ... perhaps 
because you wanted a really nice low noise temperature and/or
you were measuring a tiny thing inside a cryostat and you
wanted to amplify the signal so as to avoid the scenario where
it gets voltage-dividered into oblivion by the capacitance of
the cable that leads out of the cryostat.

Well, if you take a generic small-signal transistor (2N3904 or 
2N3906) and cool it to nitrogen temperature, it doesn't work.  
This is not surprising, because it is a _minority carrier_ 
device and there aren't gonna be any minority carriers when 
the band gap is so large compared to kT.

So, in the immortal words of Henny Youngman:  Don't do it that
way!  Use a majority carrier device, such as a depletion-mode
FET.  
  http://www.google.com/search?q=cryogenic+transistor+amplifier

This is something sophomores can do with their own hands.  
Once they see that silicon is a semiconductor at 77K, it's
obvious that diamond is a semiconductor at room temperature
... just not recommended for minority-carrier devices.

Here's another argument that leads to the same conclusion:
Suppose you wanted an _ultraviolet light emitting diode_.
What are you going to make it out of?  Where are you going
to find something with a big enough band gap?  You don't
need to guess;  the answer is known:
  http://www.google.com/search?q=%22n-type+diamond%22
  http://www.google.com/search?q=%22p-type+diamond%22

There are about ten other arguments that lead to the same
conclusion.  Diamond makes a perfectly terrible insulator.

  If you want to know how to make an actual practical
  insulator, a large band gap is neither necessary 
  nor sufficient.  The physics of real insulators is 
  a bit tricky.  It's related to the often-asked but
  seldom-correctly-answered question of what makes
  white things white.
    http://www.av8n.com/physics/white.htm

So .... you have a choice.
  a) You can try to explain why diamond is categorically
   different from silicon, or
  b) You can explain why diamond is closely analogous to 
   silicon, in principle and in practice.

Option (a) is for crazy people.  Option (b) is lots easier, 
more fun, and more useful.

This has been known for 40 years that I know of.  Maybe in 
another hundred years or so the textbooks will catch up.