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.

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

(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:

 “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!!

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).

 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.

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

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

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

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

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)

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)

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

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!

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