1. Realistic Bandstructures mostly focus on bands in semiconductors.
for Real Materials
Due my interests &
knowledge, we’ll
mostly focus on bands in semiconductors.
But, much of what
we say will be equally
valid for bands in
any crystalline solid.

2. Bandstructure Methods Realistic Methods are all Highly Computational!
REMINDER: Calculational methods fall into 2 general categories which have their roots in 2 qualitatively very different physical pictures for e- in solids (earlier discussion):
1. “Physicist’s ViewPoint”: Start from the “almost free” e- & add a periodic potential in a highly sophisticated, self-consistent manner.
 Pseudopotential Methods
2. “Chemist’s Viewpoint”: Start with the atomic picture & build up the periodic solid from atomic e- in a sophisticated, self-consistent manner.
 Tightbinding/LCAO Methods

3. (Qualitative Physical Picture #1):
Method #1
(Qualitative Physical Picture #1):
“Physicists Viewpoint”: Start with free e- & a add periodic potential.
“Almost Free” e- Approximation
First, it’s instructive to start even simpler, with FREE electrons. Consider Diamond & Zincblende Structures (Semiconductors). Superimpose the symmetry of the reciprocal lattice on the free electron energies:

4.  “Empty Lattice” Approximation It will (hopefully!) teach us
It’s instructive to with FREE electrons. Consider Diamond/Zincblende Structures (Semiconductors). Superimpose the symmetry of the reciprocal lattice on the free electron energies:
 “Empty Lattice” Approximation
Diamond/Zincblende BZ symmetry superimposed on free e “bands”. This is the limit where the periodic potential V  0. But, symmetry of BZ (lattice periodicity) is preserved. Why do this?
It will (hopefully!) teach us
some physics!!

5. Free Electron “Bandstructures” “Empty Lattice Approximation”
Free Electrons: ψk(r) = eikr
Superimpose diamond/zincblende BZ symmetry on the ψk(r). This symmetry reduces the number of k’s needing to be considered. For example, from the BZ, a “family” of equivalent k’s along (1,1,1) is:
(2π/a)(1,  1,  1)
All of these points map the Γ point = (0,0,0) to equivalent centers of neighboring BZ’s. The ψk(r) for these k are degenerate (they have the same energy).

6.  Any linear combination of these also has the same energy
We can treat other high symmetry BZ points similarly. So, we can get
Symmetrized Linear Combinations
of ψk(r) = eikr
for all equivalent k’s.
A QM Result: If 2 (or more) eigenfunctions are degenerate (have the same energy),
 Any linear combination of these also has the same energy
So, consider particular symmetrized linear combinations, chosen to reflect the symmetry of the BZ.

7. Symmetrized, “Nearly Free” e- Wavefunctions for the Zincblende Lattice
Representation Wave Function
Group
Theory
Notation

8. Symmetrized, “Nearly Free” e- Wavefunctions for the Zincblende Lattice
Representation Wave Function
Group
Theory
Notation

9. Symmetrized, “Nearly Free” e- Wavefunctions for the Diamond Lattice
Note: Diamond & Zincblende are different!
Representation Wave Function
Group
Theory
Notation

10. E(k) = ħ2[(kx)2 +(ky)2 +(kz)2]/(2mo)
The Free Electron Energy is:
E(k) = ħ2[(kx)2 +(ky)2 +(kz)2]/(2mo)
So, superimpose the BZ symmetry (diamond/zincblende lattices) on this energy.
Then, plot the results in the reduced zone scheme

11. Zincblende “Empty Lattice” Bands (Reduced Zone Scheme) E(k) = ħ2[(kx)2 +(ky)2 +(kz)2]/(2mo)
1st BZ for the
Zincblende Lattice
(111)
(100)

12. Diamond “Empty Lattice” Bands (Reduced Zone Scheme) E(k) = ħ2[(kx)2 +(ky)2 +(kz)2]/(2mo)
1st BZ for the
Diamond Lattice
(111)
(100)

13. Free Electron “Bandstructures” “Empty Lattice Approximation”
These E(k)
show some features of real bandstructures!
If a finite potential is added:
Gaps will open up at the
BZ edge just as in 1d

14. Calculated (Pseudopotential) Si Bandstructure
GOALS: After this chapter,
you should:
1. Understand the underlying
Physics behind the existence of
bands & gaps.
2. Understand how to interpret
this figure.
3. Have a rough, general idea about
how realistic bands are calculated.
4. Be able to calculate energy bands
for some simple models of a solid.
 Eg
Note: Si has an indirect band gap!

15. Qualitative Comparison:
Si Bandstructures with “Empty Lattice”: (Diamond Structure)
1st BZ for
Diamond
Lattice
Diamond Structure:
“Empty Lattice” Bands
Si: Pseudopotential
Bands