Free electron gas: always a metal
Wavefunction psik(r) = exp (i k dot r);
energy = Ek = (hbar2/2m) k2
Last lecture - start electrons in crystals
Qualitative ideas and simple principles that help us understand
which materials will be metals vs which will be insulators
Bragg reflection at the zone boundary leads to energy bands with an energy gap
Qualitative interpretation of standing waves and the gap
A material can be an insulator if the Fermi energy is in the gap
The key idea is that one has the just right number of electrons to fill
the lower bands and leave the upper bands empty
Today - more on electrons in crystals
The simple principles that help us understand (and even
predict!) which materials will be metals vs which will be insulators
It is not a accident that some materials have the just right number
of electrons to fill the lower bands and leave the upper bands empty -- and we
can predict which materials can be insulators, which are metals!
General Solution in Fourier space - "Central Equation"
It general not soluble
But we can see general form
More on solution below
Bloch Theorem
Wavefunction always has form psik(r) = exp (i k dot r) x uk(r),
where uk(r) is periodic
Simple exp (i k dot r) variation from one cell to the next
All states can be classified by index n = 1,2,3, ...
and k inside first Brillouin Zone (BZ)
Counting k states
Just like phonons and electrons in box
Result - "one k point per cell"
Relation of sums and integrals:
(1/Nk-points) x Sum over k-points equals (1/VBZ) x Integral over BZ
Nearly free elecron model
Solve as 2 x 2 equation
Always give a gap at the BZ boundary
Leads to energy bands with an energy gap
Rules for Application:
First plot FREE electron bands in BZ
Then introduce effects of potential
Metals vs Insulators
Each band holds 2 electrons
Therefore an crystal with an odd number of electrons
per cell MUST* be a metal! (*Caveat later)
A crystal with an even number of electrons
per cell MAY be an insulator!