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The Pseudopotential Method Builds on all of this.

Explore the principles behind Si pseudopotential bands and their calculation for quantum chemistry applications using the Pseudopotential Method. Learn about valence electrons, the Orthogonalized Plane Wave Method, and the Pseudo-Hamiltonian concept. Gain insights into solving the Pseudo-Schrödinger Equation for realistic band calculations.

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The Pseudopotential Method Builds on all of this.

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  1. The Pseudopotential MethodBuilds on all of this.

  2. Pseudopotential Bands • A sophisticated version of this (Vnot treated as perturbation!) Pseudopotential Method • Here, we’ll have an overview. For more details, see many pagesin manysolid state or semiconductor books!

  3. 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 indirectband gap!

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

  5. Consider SolidSi: SiValence electrons:3s23p2 • As we’ve seen: Sicrystallizes in the tetrahedral, diamond structure. • The 4 valence electronsHybridize & form 4 sp3 bondswith the 4 nearest neighbors. Quantum CHEMISTRY!!!!!!

  6. 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-.

  7. 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.

  8. 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 statesψn(r)

  9. 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 statesψn(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 statesψn(r)  This requires: d3r (ψOk(r))*ψn(r) = 0 (all k, n) βn(k) = d3re-ikrψn(r)

  10. Given ψOk(r), we want to solve an Effective SchrödingerEquation for the valence e- alone (for the bands Ek): HψOk(r) = EkψOk(r) (1) • In ψOk(r) now replace eikrwith 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

  11. The Effective SchrödingerEquation 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´

  12. (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”

  13. 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!

  14. 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!!

  15. Typical Real Space Pseudopotential: (Direct Lattice)

  16. Typical k Space Pseudopotential: (Reciprocal Lattice)

  17. 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: 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 actual potential V(r)!

  18. Pseudopotential Form Factors:Used as fitting parameters in the empirical pseudopotential method V3sV8s V11s V3aV4a V11a

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

  20. Pseudopotential Bands: Si & Ge Eg Eg Si Ge Both have indirectbandgaps

  21. Pseudopotential Bands: GaAs & ZnSe  Eg  Eg GaAs ZnSe Directbandgap Direct bandgap

  22. 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 • bandstructure diagram. • 3.Have a rough, general ideaabout how • realistic bandsare calculated. • 4.Be able to calculatethe energy bands • for some simple modelsof a solid.

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