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

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

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

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

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The Tightbinding (LCAO) MethodA Realistic Treatment of 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.

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

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

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

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)

- 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)