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Re: HOLES AS CARRIERS

At 10:09 AM 10/7/01 -0400, Ludwik Kowalski wrote:

After being satisfied with the explanation of the "positivity
of holes" I started thinking about it and I am less satisfied.

Good. That's a sign of intelligence, a sign of careful thinking. Don't
believe everything you read.

All materials are made of atoms, systems containing
charged particles (electrons and protons). We learn that
there are no particles called holes, unless the "emptiness",
full of fields, is treated as a set of holes.

A hole represents an emptiness in the sea of carriers, not an emptiness in
the "fields".

A uniform semiconductor, n or p, is electrically neutral. It
remains neutral when a current flows through it due to an
applied difference of potentials. The same is true for metals.

Yes.

But we do not say that a free electron, drifting from point
A to point B, creates a positively charged region near A.

1) Au contraire, sometimes we do say exactly that. If you apply a field to
an isolated chunk of metal, you get polarization fields of exactly this form.

2) In a circuit, we don't say there is a positively charged region left
behind because there is no positively charged region left behind. The
departed charge is replaced by a charge coming in from the upstream circuit
element, to a usually-good approximation.

Remember that Kirchhoff's "laws" are only approximations, usually good at
low frequencies in circuits of typical circuit elements.

What we are saying is that the drift of electrons in
one direction is equivalent to the drift of positive charges.

The drift of electrons _plus a bunch of other complicated phenomena_ is
equivalent to the drift of positive charges.

I am not able to make a good connection between the idea
of donors/acceptors and holes.

Good! They're not the same idea.

If presence of acceptors is equivalent to holes then presence
of donors should also be equivalent to holes, unless a hole
is not simply a place in which neutrality is locally destroyed
for a short period of time.

The premise of this "If" is not satisfied, so we need not worry about where
this thought is leading.

Clearly something is missing in
my mental image of reality. Is it because I am using the
semi-classical way of thinking about tiny particles (accepting
the QM band structure without abandoning the idea of
classical drifting) ?

Good question, but I think the answer is diametrically opposite: I think
the problem comes from not accepting _enough_ of the QM band
structure; see below.

At 10:42 AM 10/7/01 -0500, John Clement wrote:

It think that the distinction between holes and "real" particles may be
resolved by thinking of holes as carriers of net charge. Charge is actually
a property of a particle that allows us to calculate what happens to that
particle in the presence of other charged particles. The hole is a carrier
of net positive charge which physically means that it "looks like" a
positive particle. A hole contains an excess of postive particles (more
protons than electrons). With this model one can understand what is
happening in a semiconductor.

Wow. That strikes my ears as a load of sophistry and double-talk.

Using weasel-words like "net charge" instead of "charge" and "looks like"
instead of "is" doesn't change the physics. Ludwik asked a very well-posed
question about the physics and deserves a physics-based explanation.

With this model one can understand what is happening in a semiconductor.

That sounds like "stone soup" to me. Yes, you can make soup using a stone
as the first ingredient. Yes, you can make a description of what is
happening in a semiconductor using the foregoing description of holes as
one of the ingredients. But you need a lot of other ingredients. Quite a lot.

=================

At 06:00 PM 10/6/01 -0500, Tim Folkerts wrote:
I introduce holes with a couple of (transparent) bottles of shampoo. Take
one that is mostly empty and turn is over - watch the shampoo flow down to
the bottom. Then take a nearly full bottle and turn it over - watch the
bubble rise.

Is something moving up in the second case? Well, not really - shampoo is
flowing down in both cases, but it seems like a completely different
phenomenon. And it is certainly easier to treat the motion of the bubble
(which moves smoothly) rather than the actual shampoo (which has many parts
each moving a short way and then stopping).

This is what we like to see. This is clever. This is a step in the right
direction. This is correct and to the point.

Now, to return to Ludwik's question:

A hole is the oldest example of something that is nowadays called a
"dressed state". The new terminology is nice because it clarifies the
distinction of dressed viewpoint versus undressed (or bare) viewpoint.

The thing that makes a hole a hole is not the mere absence of one electron
at that point. The key is the change of viewpoint. By changing to the
dressed viewpoint, we can describe the motion of the hole _and say nothing_
about all those electrons in the sea of electrons -- they don't exist in
the dressed viewpoint.

The equations that describe the hole have a huge burden: after we have
worked out the motion of the hole, we must be able to transform back to the
undressed viewpoint and be satisfied that the dressed-motion equations
(plus the transformation equations) have correctly accounted for the motion
of the (undressed) vacancy !and! all the (undressed) electrons in the sea.

In particular, note that the hole not only has a funny charge (opposite to
the electron), it has a funny mass (less than zero). For a discussion of
effective mass, look near the end of
http://www.chem.utoronto.ca/staff/GAO/flashed/courses_files/CHM238Yset3.pdf

The bubble in Tim's shampoo has negative (dressed) mass. Note that it goes
!up! when subjected to a gravitational field.


Here are some charts that may help drive home the point:

Electron (q<0, m>0):
+ ========> - applied electric field
<=== force
<=== momentum
<=== velocity
===> conventional current

Proton, or ion in solution (q>0, m>0):
+ ========> - applied electric field
===> force
===> momentum
===> velocity
===> conventional current

Hole in a sea of electrons (q>0, m<0)
+ ========> - applied electric field
===> force
<=== momentum
===> velocity
===> conventional current

Note the funny velocity-versus-momentum relationship for the holes. Also
the funny force-makes-momentum relationship. You may be wondering why the
latter doesn't violate Newton's third law, i.e. violate conservation of
momentum. Positive force makes negative momentum?!!! That sounds bad, but
it turns out to be OK. Analogy: This is like asking how a bat can
increase the energy of a batted ball, when in the bat's frame of reference
the collision is (at best) symmetric or (more likely) lossy. That is: You
get into trouble when you mix viewpoints. The applied electric field (and
the resulting force) originates in the undressed viewpoint. When you
transform it to the dressed viewpoint, you get a force on the hole, but you
also get other terms that dump momentum into the crystal and/or the
uninteresting sea. These other terms don't affect the kinematics of the
hole, but they suffice to ensure that real-world momentum is conserved.