Band Structures in Semiconductors Realistic Bandstructures for Semiconductors

Band Structures in Semiconductors

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, selfconsistent 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

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”

“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?

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

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) • 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).

• 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.

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

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

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

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

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

Diamond “Empty Lattice” Bands (Reduced Zone Scheme) E(k) = ħ 2[(kx)2 +(ky)2 +(kz)2]/(2 mo) (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 1 d

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.

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: Eg 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!

Pseudopotential Method (Overview) • Use Si as an example (could be any material, of course). • Electronic structure of an isolated Si atom: 1 s 22 p 63 s 23 p 2 • Core electrons: 1 s 22 p 6 – Do not affect electronic & bonding properties of solid! Do not affect the bands of interest. • Valence electrons: 3 s 23 p 2 – They control bonding & all electronic properties of solid. These form the bands of interest!

Si Valence electrons: 3 s 23 p 2 Consider Solid Si: – As we’ve seen: Si Crystallizes in the tetrahedral, diamond structure. The 4 valence electrons Hybridize & form 4 sp 3 bonds with the 4 nearest neighbors. (Quantum) CHEMISTRY!!!!!!

• 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-.

• 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.

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: d 3 r (ψOk(r))*ψn(r) = 0 (all k, n) βn(k) = d 3 re-ik rψn(r)