handout8 - Handout 8 Linear Combination of Atomic Orbitals...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 8 Linear Combination of Atomic Orbitals (LCAO) In this lecture you will learn: • An approach to energy states in molecules based on the linear combination of atomic orbitals C H H H H ECE 407 – Spring 2009 – Farhan Rana – Cornell University Energy Bands and Atomic Potentials in Crystals The potential energy of an electron due to a single isolated atom looks like: 0 x ( ) r V r In a crystal, the potential energy due to all the atoms in the lattice looks like: 0 x ( ) r V r Energy levels The lowest energy levels and wavefunctions of electrons remain unchanged when going from an isolated atom to a crystal The higher energy levels (usually corresponding to the outermost atomic shell) get modified, and the corresponding wavefunctions are no longer localized at individual atoms but become spread over the entire crystal Potential of an isolated atom Potential in a crystal 0 0 Energy bands Vacuum level
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University 0 x ( ) r V r Potential in a crystal 0 Energy bands Vacuum level Failure of the Nearly-Free-Electron Approach • For energy bands that are higher in energy (e.g. 2 & 3 in the figure above) the periodic potential of the atoms can be taken as a small perturbation For higher energy bands, the nearly-free-electron approach works well and gives almost the correct results • For energy bands that are lower in energy (e.g. 1 in the figure above) the periodic potential of the atoms is a strong perturbation For lower energy bands, the nearly-free-electron approach does not usually work very well 1 2 3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University 2 1 1 1 2 1 Energy (eV) Nearly-Free-Electron Approach Vs LCAO for Germanium 2 2 1 1 1 2 1 1 1 2 1 1 Energy (eV) LCAO Energy (eV) 3 4 1 1 1 1 NFA Empirical Pseudopotential • For most semiconductors, the nearly-free- electron approach does not work very well • LCAO (or tight binding) works much better and provides additional insights FBZ (FCC lattice)
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University LCAO: From Hydrogen Atom to Hydrogen Molecule Consider a Hydrogen atom with one electron in the 1s orbital: 0 r ( ) r V r 1s energy level Potential of a Hydrogen atom One can solve the Schrodinger equation: () () () () r E r r V r m r r r r h ψ = + 2 2 2 and find the energy of the 1s orbital and its wavefunction () ( ) r E r H s s s o r r 1 1 1 ˆ φ = 0 r ( ) r s r 1 r V m H o r h + = 2 2 2 ˆ where: o a r o s e a r = 3 1 1 π r Angular probability distribution for the 1s orbital Radial amplitude for the 1s orbital ECE 407 – Spring 2009 – Farhan Rana – Cornell University Linear Combination of Atomic Orbitals (LCAO) Now consider a Hydrogen molecule made up of two covalently bonded Hydrogen atoms sitting at a distance of 2d from each other, as shown: Hamiltonian for an electron is: x d r V x d r V m H ˆ ˆ 2 ˆ 2 2 + + + = r r h The basic idea behind LCAO approach is to construct a trial variational solution in
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This note was uploaded on 04/04/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell University (Engineering School).

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handout8 - Handout 8 Linear Combination of Atomic Orbitals...

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