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2009.04.29-ECE659-L41-notes

# 2009.04.29-ECE659-L41-notes - ECE 659 Quantum Transport...

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ECE 659 Quantum Transport: Atom to Transistor Lecture 41: Exchange Interaction SupriyoDatta Spring 2009 Notes prepared by SamiranGanguly

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H U s Σ 2 Σ 1 Σ a a b b t 1 ε 2 ε a b a b t a ε I N+1 1 1 2 2 t b H t a t b ε ε ε = V If orbitals are on the different atoms : anti-parallel If orbitalsare on the same atom: parallel ( ( ( ( ( 29 ( 29 ( 29 ( 29 ( 29 * * * * a r b r r a r a r b r b r ψ ρ = + = + ( ( ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 1 2 2 2 2 1 1 * * * * * * * ab a r b r a r a r b r b r a r a r b r b r b r b r a r a r ψ ψ ψ + N
Correct form: ( ( ( 29 ( 29 1 2 2 1 1 2 a r b r a r b r ψ = - ( ( ( ( ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 * * * * 1 * 2 * * * * a r a r b r b r a r a r b r b r a r b r b r a r a r b r b r a r ψ ψ + = - - 1 2 , r r r r ( 29 ( ( ( ( ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 * * * * 1 , 2 * * * * 0 a r a r b r b r a r a r b r b r r r a r b r b r a r a r b r b r a r r r ρ ρ + = - - = = a b ( ( ( ( ( * * r a r a r b r b r ρ = + Other terms drop out due to a and b being orthogonal ( ( 1 2 * 0 dra r b r = Electrostatic energy: ( 1 2 1 2 1 2 , 4 drdr r r r r ρ πε - In the above case, the “exchange term” is 0, because a and b

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2009.04.29-ECE659-L41-notes - ECE 659 Quantum Transport...

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