The Tightbinding LCAO Method Overview of a Realistic

The Tightbinding (LCAO) Method Overview of 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 looking 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!! • Now, a mostly qualitative review of elementary molecular physics.

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

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

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

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 = (ψs. A+ ψs. B)/(2)½ (occupied; symmetric on exchange of A & B) antibonding orbital Ψ = (ψs. A - ψs. B)/(2)½ (unoccupied; antisymmetric on exchange of A & B)