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Understanding the Tight-Binding Method for Semiconductor Materials: A Qualitative Approach

This article provides a qualitative overview of the Tight-Binding (LCAO) method, focusing on its application to semiconductor materials. Initially, it delves into the molecular orbitals and their bonding and antibonding states, emphasizing the role of s and p orbitals in the electronic structure of solids. The discussion includes the qualitative review of wavefunctions and energy levels for diatomic molecules, showcasing the formation and characteristics of σ and π bonding and antibonding orbitals. This foundational understanding is crucial for analyzing electronic properties in semiconductors.

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Understanding the Tight-Binding Method for Semiconductor Materials: A Qualitative Approach

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  1. The Tightbinding (LCAO) MethodA Realistic Treatment of Semiconductor Materials!

  2. Tightbinding MethodRealistic 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 SchrödingerEquation That is, it would require us to do some CHEMISTRY!! • What follows is a quick, mostly qualitative review of elementary molecular physics.

  3. s orbitalsarespherically symmetric! Shapes of charge (& probability) densities |ψ|2 for atomic s & p orbitals: p orbitalshavedirectional lobes! The pylobeis along they-axis The pxlobeis along thex-axis The pzlobeis along thez-axis

  4. Ψ forσ antibonding orbital Wavefunctions Ψ & energy levels εfor molecular orbitals in aDiatomic Molecule AB ψsA ψsB An s-electron on atom A bonding with an s-electron on atom B. Ψ forσbonding orbital For ahomopolar molecule (A = B) ε forσ antibonding orbital ε for atomic  s electrons ε for σbonding 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

  5. Wavefunctions Ψ & energy levels εfor molecular orbitals in aDiatomic Molecule AB Ψ forσ antibonding orbital An s-electron on atom A bonding with an s-electron on atom B. Ψ forσbonding orbital For aheteropolar molecule (A  B) ε forσ antibonding orbital ε for atomic s electrons on atoms A & B ε for σbonding 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

  6. 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)

  7. 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)

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