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Re: [Phys-L] quantum dots



On 04/10/2018 07:03 PM, Larry Smith wrote:

What can we do with the kit in class? Anything beside showing that
the blue/violet laser (405 nm) makes them glow brightly and other
lasers don’t?

Executive summary:
a) Gee whiz, pretty colors.
b) Particle in a box.
Resonant frequency determined by the size of the box,
if we ignore about 90% of what's going on.

==========================
In more detail:

In general, quantum dots drive people crazy, especially
in the introductory class.
++ They are structurally rather simple.
-- They are very hard to make, but that's not our problem.
-- They are hard to understand in any detail, because
you have to keep N different ideas in your head at
the same time.

The all-too-common approach is to pretend that the
observations are explained by only one of the ideas,
ignoring the other N-1 ideas.

I gather these dots are made of CdSeS. The idea of
a compound semiconductor is complicated already, so
for simplicity let's pretend it is isoelectronic
with pure germanium. (It's not really, because the
crystal structure is different, but let's pretend.)
The point is that you can sorta imagine the Ge
crystal to be made of unionized Ge atoms (whereas
in the compound semiconductor the atoms are 100%
ionized for sure). This simplifies the concepts
and the terminology.

Now suppose we put one of the atoms into a highly
excited state, i.e. almost ionized. You could call
it ion+electron, but in some sense ion+electron is
just an excited atom. If you want to get fancy
this is called an /exciton/.

One peculiar thing is that the positive part of
exciton is more mobile than you might have guessed;
it's a /hole/ which can move rather freely through
the lattice. So the exciton is an /electron+hole/
pair.

Another peculiar thing is that diameter of the
exciton is waaaay bigger than you might have guessed.
Hint: Rederive the formula for the Bohr radius,
accounting for the fact that the interaction is
screened by the large dielectric constant constant
of the CdSeS. You find that the effective Bohr
radius is something like 10 nm. That stands in
contrast to the lattice constant, which is only
half a nm. There are a great many other atoms
inside the exciton!

Check the work: If the Bohr radius were not
large compared to the lattice constant, you
would not be sure that the lattice could be
treated as having a well-behaved average
dielectric constant.

There are some devious approximations involved
here. Chemists make these approximations All The
Time. Truly understanding the approximations
is grad-school stuff, not high-school stuff.

For classical waves, like you find in a drumhead
or in a clarinet, there are resonances having
nothing to do with ħ but everything to do with
the physical size of the system. In contrast,
in atomic spectroscopy, e.g. a neon lamp, there
are resonances that depend on ħ but have nothing
to do with the physical size of the lamp. You
can combine these ideas by saying that the atom
is like a particle in a box of size proportional
to ħ, i.e. the Bohr radius, but that leaves us
wondering why the box is that size. (Classical
waves do not explain all of QM.)

In a not-very-small chunk of CdSeS, the behavior
will be dominated by the natural Bohr radius of
the exciton.

HOWEVER, if the chunk is small enough, i.e. tiny
compared to the natural Bohr radius, you can
pretty much forget about the Bohr radius and
just treat the thing as a particle in a box. This
is a tremendous simplification, but it throws out
most of the baby with the bathwater, insofar as
it makes it hard to relate the observations to
ordinary optical spectroscopy.

There is an intermediate regime where the
behavior depends on the natural Bohr radius
*and* on the size of the box. Messy.

Another issue is that the dot is a grotty misshapen
box, so it is hard to quantify the size, let alone
quantify the relationship between box-size and
observed resonant frequency.

Another complication is that the usual introduction
to the idea of electron+hole pair uses band diagrams
that plot E versus k, i.e. energy versus wavenumber,
both of which are quite abstract. Meanwhile, in
addition to the E(k) behavior, the exciton has a
position and a size in non-abstract real 3D space.
The pretense about the non-compound semiconductor
is intended to make it easier to focus on the real
spatial behavior.