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

[John Denker <jsd@av8n.com>]

On 03/01/2010 01:14 PM, Jeff Loats wrote:
> I am teaching a modern physics course and we are discussing the way in which
> very similar valence electron configuration (2 electrons in a p-shell) leads
> to very different conduction behaviors for carbon, silicon, germanium, etc.
> My solid state is pretty rusty so I am seeking some help.
>
> As we bring many C atoms together, the 2s and 2p bands mix together.
> However, if you bring them even closer together a split occurs, creating the
> valence and conduction bands for C, SI, etc.
>
> I am having trouble finding a good sophomore-level description of what
> causes the splitting to occur. That is, why don't the 2s and 2p bands just
> continue to overlap, forming one huge band? What is the mechanism for the
> later split.

Now that I better understand the question, let's see if
we can make some progress toward answering it.

With one huge exception, I recommend following the road
map laid out in Feynman.  Maybe this is just personal,
since this is how I learned it, at an impressionable age,
but it is a clean and logical approach that doesn't 
require the student to know anything about the Schrödinger
equation.

Executive summary:  the outline goes like this:
 0) Freshman physics:  normal modes of a classical
   mechanical system.  Two weakly-coupled pendulums.
   Low frequency mode:  both pendulums in phase.
   High frequency mode: opposite phase.
   Energy _splitting_ scales like the strength of
   the _coupling_.
 1) two-state system.  hydrogen molecular ion.
   splitting == off-diagonal matrix element,
   depends on overlap integral i.e. inversely on
   separation of the two sites.
 2) multiple sites.  E as a function of k.  width
   of band (width in the E direction) scales like
   the off-diagonal matrix element.  concave up at
   k=0;  concave down at edge of zone.  extreme
   tight binding model ==> only one band, no gap.
 3) multiple bands

In more detail:

One useful thing that we can keep from chapter 13 is
the general shape of the E versus k diagram, figure
13-3.  It is approximately parabolic and concave-up 
near k=0 as we would expect for ordinary particles
of positive mass, whereas near the band edge i.e.
near k=π/b the curve is concave downward, corresponding
to negative mass i.e. holes.  This point can be made
slightly more clearly if you plot more than one cycle
(more than one zone) of the E(k) curve.

Unfortunately, Feynman skips a step.  Chapter 13 only 
does the extreme tight-binding model, i.e  pointlike 
lattice sites, taking no account of anything  happening 
between sites.  This model cannot possibly have more 
than one band, and therefore cannot have a band gap.

At the beginning of chapter 14, he waves his hands and
mumbles a few words, bringing in additional higher bands, 
basically out of thin air.

So we need to fill in this step.

Extreme tight binding model:

   o         o         o         o         o

Each o represents a lattice site.  Specifying the
probability amplitude at each o-site fully specifies
the state of the system.  Makes sense for k in the
range from -π/b to +π/b, where b is the spacing
between o-sites.  E versus k resembles a classical 
particle of positive mass for |k| less than 1/b 
or thereabouts.

Quasi-continuum:

   .........................................

Each dot represents a hyper-lattice site.  Specifying
the probability amplitude at each dot-site fully 
specifies the state of the system.   Makes sense for 
k in the range from -π/d to +π/d, where d is the spacing
between dot-sites.  E versus k resembles a classical 
particle of positive mass for |k| less than 1/d or 
thereabouts.

For clarity of drawing I have drawn d/b = 1/10, but
the physics works better if you imagine a finer grid
of dots, maybe d/b = 1/100, so that there is no doubt
that we have |k| << 1/d for all k of interest, and
therefore we have a classical parabolic E(k) curve.

Fancy tight binding, including some continuum:

   o.........o.........o.........o.........o

This is a combination of the two previous models.  We
have atoms (ion cores) at the o-sites, but now we have
a way of talking about what happens between o-sites.

  You can make a mechanical model of this, using a
  few hundred coupled pendulums.  I've seen it done.

In particular, the point Feynman makes using figure
13-4 is true for his extreme tight binding model but
is *not* true for our fancy tight binding model.

In particular, rather than adjusting the spacing
between ion cores, I choose to adjust the _strength_
of the ion cores.  For starters, if the cores have
no strength i.e. if they are indistinguishable from
the vacuum, then we recover the quasi-continuum model
and recover the purely parabolic E(k) curve out to
large k, large compared to π/b and even large compare
to a modest multiple of π/b.

The normal modes of this system are the states of
definite k.  We connect this to freshman physics,
classical physics, item (0) above.

Now we give some strength to the ion cores.  This
will change the normal modes.  In particular, k
values that are near any integer multiple of π/b
will be strongly perturbed (relative to where they
were in the absence of the ion cores).

Let's think again about the mechanical model.  In
the two-pendulum system, the higher mode was where
the two pendulums were moving in opposite directions.

For our model of the crystal, let's compare the
lowest band to the next-highest band.  At any 
given k-value, the o-sites are doing exactly the
same thing in both bands.  In the low band, the
dot-sites are moving in generally the same direction
as the nearest o-sites, whereas in the high most
of the dot-sites are moving in the opposite direction
from the nearest o-sites.  So the high band has a
higher frequency.

With or without the mechanical version of this model,
this leads to the classic diagram of a parabola with
notches cut out of it:
  http://www.chembio.uoguelph.ca/educmat/chm729/wscells/images/energy~2.gif

Nicer versions of this diagram can be found in 
Ashcroft & Mermin.

If you superimpose multiple copies of this diagram
on top of itself, shifted by multiples of 2π/b, then
you get a good picture of the band structure of the
one-dimensional crystal, band gaps and all.

The difference between a semiconductor versus a 
metal is the not presence versus absence of a gap;
there are tons of gaps in the band structure of a
metal.  The key property of a semiconductor is the 
presence of a gap _right at the Fermi level_ so
that the band below the gap is filled and the band
above the gap is empty.  In a metal, the Fermi
level sits partway up a partially-filled band, and
the existence of gaps far above and far below is
irrelevant.

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

After having said how the explanation goes, let me
say a few words about the explanation does not go:

 -- At the sophomore level of detail, is possible to 
  build a nice model of a semiconductor with a nice
  band gap, using only s-orbitals.  It's true that
  atoms have p-orbitals and we should account for
  that at some point, but mentioning p-orbitals 
  doesn't help explain the band gap.  Mentioning 
  p-orbitals is just gonna confuse the students.

 -- In particular, on 03/01/2010 03:19 PM, Stefan 
  Jeglinski wrote:

> > The valence band 
> > is ... "derived" from the 
> > HOMO, "derived" from the outer occupied shell of the atom in 
> > question. The conduction band is the set of lowest unoccupied 
> > energies, "derived" from the LUMO, "derived" from the outer 
> > unoccupied shell of the atom in question.

  That just can't be right.  An example suffices to
  prove the point:  gray tin is a semiconductor, while
  white tin is a metal.  Same atomic levels, wildly
  different bands structure.

  It's correct to think of the crystal as one big
  macromolecule, with its own HOMO and LUMO.  I
  recommend it, provided the students are up to
  speed on this concept and terminology (which not
  all sophomores are).

  We know from basic observations on molecules, e.g.
  dye molecules, that the HOMO-LUMO splitting is 
  not at all closely related to the atomic level
  splittings of the constituent atoms.

  Even simpler than dye molecules, you can consider
  the hydrogen molecular ion.  Map out the molecular
  levels as a function of separation, as in Feynman
  volume III figure 10-3.  The molecular E(I) - E(II) 
  splitting has got nothing to do with the atomic p-s 
  splitting.  Feynman has not even mentioned the 
  atomic p-orbitals -- which is correct to a good
  approximation, especially when the interproton
  distance is large i.e. when the molecular splitting
  is small.  In any case the point is that you can
  tune the LUMO-HUMO splitting to be anything you
  want (over a wide range) without changing the
  atomic splitting.

> > There is always an energy "gap" between 
> > any energy levels, by definition. 

  Actually not.  There is always an energy difference
  between energy levels, but not all such differences
  are large enough to be called "gaps".

  For a semiconductor the LUMO-HUMO splitting may be
  on the order of a volt ("gap") while in a metal it 
  is more like 1e-23 volts (no "gap" at the Fermi
  level).

  As I understand the question, the point of this 
  thread is to explain where the gap comes from.  The 
  existence of gaps -- and the existence of a gap at
  the Fermi level -- comes from some nontrivial physics;  
  it does not come from a mere definition.