The
Tightbinding
or
LCAO Approach
To
Bandstructure
Theory
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, selfconsistent 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.
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!
QUESTION
• 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).
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.
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:
 Antibonding State
Ψ+ = (ψA + ψ B)/(2)½

 Bonding State
Ψ- = (ψA - ψ B)/(2)½
A Bonding State
(filled, 2 e-. Spin-up  & Spin-down ) &
An Antibonding State
(empty) Qualitatively like
Bond Center
(Equilibrium Position)
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.
Schematic: Atomic Levels Broadening into Bands
In the limit as
a
 p-like
Antibonding States
the atomic levels
for the isolated
atoms come back
 p-like Bonding
States
 s-like Antibonding
a0  material lattice
constant
States
a0
 s-like Bonding
States
Schematic: Evolution of Atomic-Molecular
Levels into Bands
p antibonding

p antibonding

s antibonding

 Fermi 
Energy, EF
Fermi Energy, E
p bonding
F


Isolated Atom
s bonding

s & p Orbital Energies
 Molecule
Solid (Semiconductor) Bands
The Fundamental Gap is on
both sides of EF!
Schematic
Evolution of s & p Levels into Bands at the BZ Center (Si)
Lowest
Conduction
 Band
EG
Atom


Solid
 Fermi Energy
 Highest
Valence Band
Schematic
Evolution of s & p Levels into Bands at the BZ Center (Ge)
Lowest Conduction Band
EG

Fermi Energy
Highest Valence Band
Atom
Solid
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
Schematic
Dependence of Bands & Gaps on Nearest-Neighbor Distance
(from Harrison’s book)
Atom
Semiconductors
Decreasing Nearest Neighbor Distance 
Schematic
Dependence of Bands & Gap on Ionicity
(from Harrison’s book)
Covalent
Bonds
Ionic
Bonds
Metallic
Bonds