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

The Pseudopotential Method Given ψOk(r), we want to solve an Effective Schrödinger Equation for the valence e- alone (for the bands Ek): HψOk(r) = EkψOk(r) (1) In ψOk(r), replace eikr with 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

Effective Potential V´ Solve the Effective Schrödinger Equation for the valence electrons alone (to get the bands Ek): HψOk(r) = EkψOk(r) (1) Much manipulation turns (1), the Effective Shrödinger Equation into: (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´

 The “Pseudopotential” (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”

The Pseudo-Schrödinger Equation [(p)2/(2mo) + Vps(r)]ψk(r) = Ekψk(r) 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 NEVER done!

Empirical Pseudopotential Method Usually, instead, people either: 1. Express Vps(r) in terms of empirical parameters & use these to fit Ek & other properties 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|ψ k(r)|2  Self-Consistent Pseudopotential Method Gaussian Units!!

A Typical Real Space Pseudopotential (In the Direct Lattice)

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

[-(ħ22)/(2mo)+Vps(r)]ψk(r) = Ekψk(r) 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: ψ k(r)  ψ fk(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)!

Pseudopotential Form Factors Fitting parameters in the empirical pseudopotential method V3s V8s V11s V3a V4a V11a

Pseudopotential Effective Masses (Γ-point) Compared to experiment! Ge GaAs InP InAs GaSb InSb CdTe

Pseudopotential Bands of Si & Ge  Eg  Eg Si Ge Both have indirect bandgaps

Pseudopotential Bands of GaAs & ZnSe  Eg  Eg GaAs ZnSe (Direct bandgap) (Direct bandgap)

Recall that our 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 bandstructure diagram. 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.