2009.02.18-ECE659-L16-notes

2009.02.18-ECE659-L16-notes - ECE 659 Quantum Transport:...

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Unformatted text preview: ECE 659 Quantum Transport: Atom Atom to Transistor Lecture 16: Correlation Function/Current Supriyo Datta Spring 2009 Notes prepared by Samiran Ganguly Γ = i Σ − Σ+ G = [ EI − H − Σ ] G n = G Σin G + Γ1 f1 +Γ 2 f 2 −1 Σ1 Σ2 µ1 H µ2 A = i G − G + [ EI − H − Σ]ψ = s ψ * G n = 2πψψ + = 2π 1 {ψ 1* ψ 2 } ψ 2 * ψ 1ψ 1* ψ 1ψ 2 n G = 2π * ψ 2ψ 1* ψ 2ψ 2 G n = G ( Γ1 f1 + Γ 2 f 2 ) G + A = G ( Γ1 + Γ 2 ) G + G = [ EI − H − Σ ] −1 −1 G −1 = [ EI − H − Σ ] G + = EI − H − Σ + G + − [ G ] = Σ − Σ + −1 −1 The diagonal elements give electron densities ψ = [G ]{s} G n = [G ]{s}{s} [G ] + + G − G + = G Σ − Σ + G + i G − G + = GΓG + A = G + Γ G = G ΓG + = i G − G + Σin 2π Current: Eψ = Hψ + Σψ + s Φ Time dependent Schrodinger equation is then: Φψ + −ψ Φ + I op = iℏ let Φ = Hψ iℏ dψ =Φ dt d (ψψ + ) dt + Gn Gn H ψψ + −ψψ + H I op = iℏ For current we need to look at iℏ d ψ (ψψ + ) = iℏ ddt ψ + +ψ iℏ dψ dt dt I op ( a, a ) = = ∑H b ab G n ba − G n ab H ba dψ + = −Φ + iℏ dt iℏ −t ( G n a +1,a − G n a ,a +1 ) − t ( G n a −1,a − G n a ,a −1 ) iℏ Φψ + −ψ Φ + HG n − G n H = I op = iℏ iℏ d Φ ψ + −ψ Φ + + (ψψ ) = dt iℏ left right µ1 0 1 a 2 3 µ2 dn =0 dt ε −t −t ε −t → −t ε ⋱ ⋱ ⋱ −t −t ⋱ ...
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This note was uploaded on 01/30/2011 for the course ECE 659 taught by Professor Staff during the Spring '08 term at Purdue University.

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2009.02.18-ECE659-L16-notes - ECE 659 Quantum Transport:...

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