The Pseudopotential Method Builds on all of this.

Given ψ O k (r), we want to solve an Effective Schrödinger Equation for the valence e - alone (for the bands E k ): Hψ O k (r) = E k ψ O k (r) (1) In ψ O k (r) now replace e ik  r with a more general expression ψ f k (r): ψ O k (r) = ψ f k (r) + ∑β n (k)ψ n (r) Put this into (1) & manipulate. This involves Hψ n (r)  E n ψ n (r) (2) (2) is the Core e - Schrödinger Equation. Core e - energies & wavefunctions E n & ψ n (r) are assumed to be known. H = (p) 2 /(2m o ) + V(r) V(r)  True Crystal Potential

Solve the Effective Schrödinger Equation for the valence electrons alone (to get the bands E k ) is: Hψ O k (r) = E k ψ O k (r) (1) Much manipulation turns (1) (the effective Shrödinger Equation) into: (H + V´)ψ f k (r) = E k ψ f k (r) (3) where V´ψ f k (r) = ∑(E k -E n )β n (k)ψ n (r) ψ f k (r) = the “smooth” part of ψ O k (r) (needed between the atoms) ∑(E k -E n )β n (k)ψ n (r)  Contains large oscillations (needed near the atoms, to ensure orthogonality to the corestates). This oscillatory part is lumped into an Effective Potential V´

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

Now, we want to solve The Pseudo-Schrödinger Equation [(p) 2 /(2m o )+V ps (r)]ψ k (r) = E k ψ k (r) Of course we put p = -iħ  In principle, we could use the formal expression for V ps (r) (a “smooth”, “small” potential), including the messy sum over core states from V´. BUT, this is almost NEVER done!

Usually, instead, people either: 1. Express V ps (r) in terms of empirical parameters & use these to fit E k & other properties  The Empirical Pseudopotential Method or 2. Calculate V ps (r) self-consistently, coupling the Pseudo-Schrödinger Equation [-(ħ 2  2 )/(2m o )+V ps (r)]ψ k (r) = E k ψ k (r) to Poisson’s Equation:  2 V ps (r) = - 4πρ = - 4πe|ψ k (r)| 2  The Self-Consistent Pseudopotential Method Gaussian Units!!

A Typical Real Space Pseudopotential (In the Direct Lattice)

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

The Pseudo-Schrödinger Equation is [-( ħ 2  2 )/(2m o )+V ps (r)] ψ k (r) = E k ψ k (r) E k = bandstructure we want V ps (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 have the form: ψ k (r)  ψ f k (r) + corrections –Often: V ps (r) is  weak  Thinking about it like this brings back to the “almost free” e - approximation again, but with V ps (r) instead of the acutal potential V(r)!

Pseudopotential Form Factors Used as fitting parameters in the empirical pseudopotential method V 3 s V 8 s V 11 s V 3 a V 4 a V 11 a

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

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

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

GOALS 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.