The Pseudopotential Method Builds on all of this. See YC, Ch. 2 & BW, Ch. 3!

1. The Pseudopotential MethodBuilds on all of this. See YC, Ch. 2 & BW, Ch. 3!

2. The Pseudopotential Method • Given ψOk(r), we want to solve an Effective SchrödingerEquationfor the valence e-alone (for the bands Ek): HψOk(r) = EkψOk(r) (1) In ψOk(r),replace eikrwith more general expression ψfk(r): ψOk(r) = ψfk(r) + ∑βn(k)ψn(r) Put this into (1) & manipulate. This involves Hψn(r)  Enψn(r) (2) (2) = Core e-Schrödinger Equation. The core e- energies & wavefunctions En & ψn(r) are assumed known. H = (p)2/(2mo) + V(r); V(r) True Crystal Potential

3. Solve the Effective SchrödingerEquation for the valence electrons alone (to get the bands Ek): HψOk(r) = EkψOk(r) (1) Much manipulation turns (1), the Effective Shrödinger Equationinto: (H + V´)ψfk(r) = Ek ψfk(r) (3) where V´ψfk(r) = ∑(Ek -En)βn(k)ψn(r) ψfk(r) = “smooth” part of ψOk(r) (needed between the atoms) ∑(Ek -En)βn(k)ψn(r) Contains large oscillations (needed near the atoms, to ensure orthogonality to the core states). This oscillatory part is lumped into an Effective Potential V´

4. (3) is an Effective Schrödinger Equation The Pseudo-Schrödinger Equation for the smooth part of the valence e- wavefunction (& for Ek): H´ψk(r) = Ekψk(r)(4) (The f superscript on ψfk(r) has been dropped). So we finally get a Pseudo-Hamiltonian:H´  H + V´ or H´= (p)2/(2mo) + [V(r) + V´] or H´= (p)2/(2mo) + Vps(r), where Vps(r) = V(r) + V´ The “Pseudopotential”

5. Now, we want to solve The Pseudo-Schrödinger Equation [(p)2/(2mo) + Vps(r)]ψk(r) = Ekψk(r) Of course, we put p = -iħ. In principle, we could use the formal expression for Vps(r) (a “smooth”, “small” potential), including the messy sum over core states from V´. BUT, this is almost NEVERdone!

6. Usually, instead, people either: 1. Express Vps(r) in terms of empirical parameters & use these to fit Ek& other properties • The Empirical Pseudopotential Method or 2.Calculate Vps(r) self-consistently, coupling the Pseudo-Schrödinger Equation [-(ħ22)/(2mo) + Vps(r)]ψk(r) = Ekψk(r) to Poisson’s Equation: 2Vps(r) = - 4πρ = - 4πe|ψk(r)|2  The Self-Consistent Pseudopotential Method Gaussian Units!!

7. A Typical Real Space Pseudopotential(In the Direct Lattice)

8. A Typical k-Space Pseudopotential(In the Reciprocal Lattice)

9. The Pseudo-Schrödinger Equation is [-(ħ22)/(2mo)+Vps(r)]ψk(r) = Ekψk(r) Ek= bandstructure we want Vps(r) is generally assumed to have a  weak effect on the free e- results. But, this is not really true! BUT it is a  justification after the fact for the original “almost free” e- approximation. • Schematically, the wavefunctions will have the form: ψk(r) ψfk(r) + corrections Often:Vps(r) is  weak  Thinking about it like this brings back to the “almost free” e- approximation again, but with Vps(r) instead of the acutal potential V(r)!

10. Pseudopotential Form FactorsFitting parameters in the empirical pseudopotential method V3sV8s V11s V3aV4a V11a

11. Pseudopotential Effective Masses(Γ-point)Compared to experiment! GeGaAs InPInAsGaSbInSb CdTe

12. Pseudopotential Bands of Si & Ge  Eg  Eg Si Ge Both have indirectbandgaps

13. Pseudopotential Bands of GaAs & ZnSe  Eg Eg GaAs ZnSe (Directbandgap) (Directbandgap)

14. Recall thatour GOALS were that after this chapter, you should: 1.Understand the underlying Physics behind the existence of bands & gaps. 2.Understand how to interpret a bandstructurediagram. 3.Have a rough, general idea about how realistic bands are calculated. 4.Be able to calculate the energy bands for some simple models of a solid.