1. The Tightbinding Bandstructure Theory
or LCAO Approach To
Bandstructure
Theory

2. Bandstructures Another qualitative discussion for a while
Recall the beginning of our discussion about band calculations:
Bandstructure Theories are Highly computational!
The theories fall into 2 general categories, which have their roots in 2 qualitatively very different physical pictures for e- in solids (earlier):
“Physicist’s View”: Start from an “almost free” e- & add the periodic potential in a highly sophisticated, self-consistent manner.  Pseudopotential Methods
“Chemist’s View”: Start with atomic e- & build up the periodic solid in a highly sophisticated, self-consistent manner.  Tightbinding or LCAO Methods
Now, we’ll focus on the 2nd method.

3. Atomic Orbitals (LCAO) method.
Method #2 (Qualitative Physical Picture #2)
“A Chemists Viewpoint”
Start with the atomic/molecular picture of a solid.
The atomic energy levels merge to form molecular levels, & merge to form bands as periodic interatomic interaction V turns on.
The Tightbinding or
Linear Combination of
Atomic Orbitals (LCAO) method.
This method gives good bands, especially valence bands! The valence bands are ~ almost the same as those from the pseudopotential method! Conduction bands are not so good!

4. QUESTION ANSWER How can 2 (seemingly) completely different
approaches (pseudopotential & tightbinding) lead to
essentially the same bands? (Excellent agreement
with valence bands; conduction bands are not too good!).
ANSWER
(partial, from YC):
The electrons in the conduction bands are
~ “free” & delocalized. The electrons in the valence bands are ~ in the bonds in r space.  The valence electrons in the bonds have atomic-like character. (So, LCAO is a “natural” approximation for these).

5. The Tightbinding Method
My personal opinion
The Tightbinding / LCAO method gives a much clearer physical picture (than pseudopotential method does) of the causes of the bands & the gaps.
In this method, the periodic potential V is discussed as in terms of an Overlap Interaction of the electrons on neighboring atoms.
As we’ll see, we can define these interactions in terms of a small number of PHYSICALLY APPEALING parameters.

6. First: A Qualitative Diatomic Molecule Discussion
Consider a 2 atom molecule AB
with one valence e- per atom, & a
strong covalent bond. Assume that
the atomic orbitals for A & B, ψA
& ψB, are known. Now, solve the
Molecular Schrödinger
Equation as a function of the A-B
separation. The Results are:
A Bonding State
(filled, 2 e-. Spin-up  & Spin-down ) &
An Antibonding State (empty) Qualitatively like
 Antibonding State
Ψ+ = (ψA + ψ B)/(2)½ 
 Bonding State
Ψ- = (ψA - ψ B)/(2)½
Bond Center (Equilibrium Position)

7. The bonding & antibonding states broaden to become bands.
Tightbinding Method
“Jump” from 2 atoms to 1023 atoms!
The bonding & antibonding states
broaden to become bands.
A gap opens up between the bonding & the antibonding states (due to the crystal structure & the atom valence).
Valence bands: Occupied
 Correspond to bonding levels in the molecular picture.
Conduction bands: Unoccupied
 Correspond to antibonding levels in the molecular picture.

8. Schematic: Atomic Levels Broadening into Bands
In the limit as
a  
the atomic levels
for the isolated
atoms come back
 p-like Antibonding States
 p-like Bonding
States
 s-like Antibonding
States
a0  material lattice
constant
 s-like Bonding States a0

9. Schematic: Evolution of Atomic-Molecular Levels into Bands
p antibonding 
p antibonding 
s antibonding 
 Fermi Energy, EF
p bonding 
 Fermi Energy, EF 
Isolated Atom
s & p Orbital Energies
s bonding 
 Molecule
Solid (Semiconductor) Bands
The Fundamental Gap is on
both sides of EF!

10. Schematic Evolution of s & p Levels into Bands at the BZ Center (Si)
Lowest
Conduction
 Band
EG  
 Fermi Energy
 Highest
Valence Band
Atom Solid

11. Schematic Evolution of s & p Levels into Bands at the BZ Center (Ge)
Lowest Conduction Band
EG 
Fermi Energy
Highest Valence Band
Atom Solid

12. Schematic Evolution of s & p Levels into Bands at the BZ Center (-Sn)
EG = 0
Highest “Valence Band”
Lowest “Conduction Band”
Fermi
Energy
Atom Solid

13. Schematic Dependence of Bands & Gaps on Nearest-Neighbor Distance (from Harrison’s book)
Atom
Semiconductors
Decreasing Nearest Neighbor Distance 

14. Dependence of Bands & Gap on Ionicity (from Harrison’s book)
Schematic
Dependence of Bands & Gap on Ionicity
(from Harrison’s book)
Covalent
Bonds
Ionic
Bonds
Metallic
Bonds