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The Pseudopotential Method Builds on all of this.

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!.

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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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