1. Band Structures in Semiconductors

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

3. “The Empty Lattice Approximation”
Method #1 (Qualitative Physical Picture #1):
“Physicists View”:
Start with free e- & a add periodic potential.
The “Almost Free” e- Approximation
First, it’s instructive to start even simpler, with FREE electrons. Superimpose the symmetry of the diamond & zincblende lattices on the free electron energies:
“The Empty Lattice Approximation”

4. “The Empty Lattice Approximation”
Diamond & Zincblende BZ symmetry superimposed on the free e- “bands”. This is the limit where the periodic potential V  0. But, the symmetry of BZ (lattice periodicity) is preserved.
Why do this?

5. The Empty Lattice Approximation” It will (hopefully!) teach
Why do this?
It will (hopefully!) teach
us some physics!!

6. Free Electron “Bandstructures” “The Empty Lattice Approximation”
Free Electrons: ψk(r) = eikr
Superimpose the 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).

7. If 2 (or more) eigenfunctions are degenerate
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 eigenfunctions also has the same energy
So, we consider particular symmetrized linear combinations, chosen to reflect the symmetry of the BZ.

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

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

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

11. 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 (for diamond/zincblende lattices) on this energy.
Then, plot the results in the reduced zone scheme

12. Zincblende “Empty Lattice” Bands (Reduced Zone Scheme) E(k) = ħ2[(kx)2 +(ky)2 +(kz)2]/(2mo)
(111)
(100)

13. Diamond “Empty Lattice” Bands (Reduced Zone Scheme) E(k) = ħ2[(kx)2 +(ky)2 +(kz)2]/(2mo)
(111)
(100)

14. 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

15. Pseudopotential Bandstructures
A highly computational version of this (V is not treated as perturbation!)
 Pseudopotential Method
Here, we’ll just have an overview. For more details, see many pages in BW, Ch. 3 &YC, Ch. 2.

16. After this chapter, you should: Si has an indirect band gap!
The Pseudopotential Bandstructure of Si Note the qualitative similarities of these to the bands of the empty lattice approximation.
Recall our 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.
Note
Si has an indirect band gap!
 Eg

17. Pseudopotential Method (Overview)
Use Si as an example (could be any material, of course).
Electronic structure of an isolated Si atom:
1s22s22p63s23p2
Core electrons: s22s22p6
Do not affect electronic & bonding properties of solid!
 Do not affect the bands of interest.
Valence electrons: s23p2
They control bonding & all electronic properties of solid.
 These form the bands of interest!

18. Si Crystallizes in the tetrahedral,
Si Valence electrons: 3s23p2
Consider Solid Si:
As we’ve seen:
Si Crystallizes in the tetrahedral,
diamond structure.
The 4 valence electrons Hybridize & form 4 sp3 bonds with the 4 nearest neighbors.
 (Quantum) CHEMISTRY!!!!!!

19. Question (from YC): Why is an approximation which begins with the “nearly free” e- approach reasonable for these valence e-?
They are bound tightly in the bonds!
Answer (from YC): These valence e- are “nearly free” in sense that a large portion of the nuclear charge is screened out by very tightly bound core e-.

20. The Next approximation: “Almost Free”
A QM Rule: Wavefunctions for different electron states (different eigenfunctions of the Schrödinger Equation) are orthogonal.
A “Zeroth” Approximation to the valence e-:
They are free  Wavefunctions have the form
ψfk(r) = eikr (f  “free”, plane wave)
The Next approximation: “Almost Free”
 ψk(r) = “plane wave-like”, but (by the QM rule just mentioned) it is orthogonal to all core states.

21. Orthogonalized Plane Wave Method
“Almost Free”  ψk(r) = “plane wave-like” & orthogonal to all core states
 “Orthogonalized Plane Wave (OPW) Method”
Write valence electron wavefunction as:
ψOk(r) = eikr + ∑βn(k)ψn(r)
∑ over all core states n, ψn(r) = core (atomic) wavefunctions (known)
 βn(k) are chosen so that ψOk(r) is orthogonal to all core states ψn(r)
The Valence Electron Wavefunction
ψOk(r) = “plane wave-like” & orthogonal to all core states
Choose βn(k) so that ψOk(r) is orthogonal to all core states ψn(r)
 This requires: d3r (ψOk(r))*ψn(r) = 0 (all k, n)
 βn(k) = d3re-ikrψn(r)