Transcript Part IX

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!
REMINDER: Calculational 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.
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.
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 the 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
Some Quantum Chemistry!
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
 Antibonding State
Ψ+ = (ψA + ψ B)/(2)½

 Bonding State
Ψ- = (ψA - ψ B)/(2)½
(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 the bonding levels in the molecular picture.
Conduction bands: Unoccupied
 Correspond to the antibonding levels in the molecular picture.
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
a0 
States
 s-like Antibonding
Material Lattice Constant
States
 s-like Bonding
States
a0
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)
EG


Lowest
Conduction
 Band
 Fermi Energy
 Highest
Valence Band
Atom
Solid
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 (α-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