Tightbinding Method Realistic Treatment for Semiconductor Materials!

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Presentation transcript:

The Tightbinding (LCAO) Method A Realistic Treatment of Semiconductor Materials!

Tightbinding Method Realistic Treatment for Semiconductor Materials! For most of the materials of interest, in the isolated atom, the valence electrons are in s & p orbitals. Before at the bands in the solid, lets first briefly & QUALITATIVELY look at the molecular orbitals for the bonding & antibonding states. A Quantitative treatment would require us to solve the Molecular Schrödinger Equation That is, it would require us to do some CHEMISTRY!! What follows is a quick, mostly qualitative review of elementary molecular physics.

s orbitals are spherically symmetric! Shapes of charge (& probability) densities |ψ|2 for atomic s & p orbitals: s orbitals are spherically symmetric! p orbitals have directional lobes! The py lobe is along the y-axis The px lobe is along the x-axis The pz lobe is along the z-axis

Ψ = (ψsA+ ψsB)/(2)½ Ψ = (ψsA - ψsB)/(2)½ Ψ for σ antibonding orbital Wavefunctions Ψ & energy levels ε for molecular orbitals in a Diatomic Molecule AB ψsA ψsB An s-electron on atom A bonding with an s-electron on atom B. Ψ for σ bonding orbital For a homopolar molecule (A = B) ε for σ antibonding orbital ε for atomic  s electrons ε for σ bonding orbital Result: A  bonding orbital (occupied; symmetric on exchange of A & B) Ψ = (ψsA+ ψsB)/(2)½ A  antibonding orbital (unoccupied; antisymmetric on exchange of A & B) Ψ = (ψsA - ψsB)/(2)½

Ψ = (ψsA+ ψsB)/(2)½ Ψ = (ψsA - ψsB)/(2)½ Wavefunctions Ψ & energy levels ε for molecular orbitals in a Diatomic Molecule AB Ψ for σ antibonding orbital An s-electron on atom A bonding with an s-electron on atom B. Ψ for σ bonding orbital For a heteropolar molecule (A  B) ε for σ antibonding orbital ε for atomic  s electrons on atoms A & B ε for σ bonding orbital Result: A  bonding orbital (occupied; symmetric on exchange of A & B) Ψ = (ψsA+ ψsB)/(2)½ A  antibonding orbital (unoccupied; antisymmetric on exchange of A & B) Ψ = (ψsA - ψsB)/(2)½

 bonding orbital:  bonding (+) &  antibonding orbital Charge (probability) densities |Ψ|2 for molecular orbitals in a Diatomic Molecule AB An s-electron on atom A bonding with an s-electron on atom B to get  bonding (+) &  antibonding (-) molecular orbitals.  bonding orbital: Ψ = (ψsA+ ψsB)/(2)½ (occupied; symmetric on exchange of A & B)  antibonding orbital Ψ = (ψsA - ψsB)/(2)½ (unoccupied; antisymmetric on exchange of A & B)

π bonding: Ψ = (ψxA+ ψxB)/(2)½ Combine 2 atomic px orbitals & get π bonding & π antibonding molecular orbitals: π bonding: Ψ = (ψxA+ ψxB)/(2)½ (occupied; symmetric on exchange of A & B) π antibonding: Ψ = (ψxA- ψxB)/(2)½ (unoccupied; antisymmetric on exchange of A & B)