The Pseudopotential Method Builds on all of this.

Pseudopotential Bands A sophisticated version of this (V not treated as perturbation!) Pseudopotential Method Here, we’ll have an overview. For more details, see many pages in many solid state or semiconductor books!

Si Pseudopotential Bands 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!

Pseudopotential Method (Overview) Use Si as an example (could be any material, of course). Electronic structure isolated Si atom: 1s22s22p63s23p2 Core electrons: 1s22s22p6 Don’t affect electronic & bonding properties of solid!  Don’t affect the bands of interest. Valence electrons: 3s23p2 Control bonding & all electronic properties of solid.  These form the bands of interest!

Si Valence electrons: 3s23p2 Consider Solid Si: Si Valence electrons: 3s23p2 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!!!!!!

(Yu & Cardona, in their semiconductor book): Question (Yu & Cardona, in their semiconductor book): 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 (Yu & Cardona): 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-.

 ψk(r) = “plane wave-like”, but (by the QM rule just mentioned) A QM Rule: Wavefunctions for different electron states (different eigenfunctions of the Schrödinger Equation) are orthogonal. “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 the 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)

ψOk(r) = eikr + ∑βn(k)ψn(r) Valence Electron Wavefunction Approximate valence electron wavefunction is: ψOk(r) = eikr + ∑βn(k)ψn(r)  βn = ∑ over all core states n, ψn(r) = core (atomic) wavefunctions (known) βn(k) chosen so that ψOk(r) is orthogonal to all core states ψn(r) 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)

ψOk(r) = ψfk(r) + ∑βn(k)ψn(r) 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) now replace eikr with a 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) is the Core e- Schrödinger Equation. Core e- energies & wavefunctions En & ψn(r) are assumed to be known: H = (p)2/(2mo) + V(r) V(r)  True Crystal Potential

(H + V´)ψfk(r) = Ek ψfk(r) (3) The Effective Schrödinger Equation for the valence electrons alone (to get the bands Ek) is: 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) = the “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!

 The Empirical Pseudopotential Method 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|ψ k(r)|2  The Self-Consistent Pseudopotential Method Gaussian Units!!

Typical Real Space Pseudopotential: (Direct Lattice)

Typical k Space Pseudopotential: (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. This is not really true! BUT it is a  justification after the fact for the original “almost free” e- approximation. Schematically, the wavefunctions will be: 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 actual potential V(r)!

Pseudopotential Form Factors: Used as 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: Si & Ge  Eg  Eg Si Ge Both have indirect bandgaps

Pseudopotential Bands: GaAs & ZnSe  Eg  Eg GaAs ZnSe Direct bandgap Direct bandgap

1. Understand the underlying Physics 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.