2009.02.16-ECE659-L15-notes

2009.02.16-ECE659-L15-notes - ECE 659 Quantum Transport...

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Unformatted text preview: ECE 659 Quantum Transport: Atom Atom to Transistor Lecture 15: Coherent Transport II Supriyo Datta Spring 2009 Notes prepared by Samiran Ganguly Σ = Σ1 + Σ 2 Γ = i Σ − Σ + G = [ EI − H − Σ ] −1 Σ1 Σ2 A = i G − G + G n = GΓ1G + f1 + GΓ 2G + f 2 q I i = Trace [ Γi A] fi − Γi G n h µ1 H µ2 ( ) E E = ε − 2t cos ka Aeikx + ρ e− ikx τ eikx µ1 0 ε 1 Eψ 1 = εψ 1 − tψ 2 − tψ 0 ka t a 2 3 µ2 Eψ 2 = εψ 2 − tψ 1 − tψ 3 ψ 2 = τ eik 2 a ψ 3 = τ eik 3a ψ 3 = ψ 2 eika Σ1 H Σ2 ψ0 = A+ ρ ψ 1 = Ae + ρ e ika ika − ika 2 ika s1 = −tA (1 − e2ika ) ∴ψ 0 = e ψ 1 + A (1 − e ) = tAeika ( e + ika − e− ika ) 2 2 The second term can be thought of as a source term [ EI − H − Σ]ψ = s Eψ 1 = εψ 1 − tψ 2 − t eikaψ 1 + A (1 − e2ika ) Eψ 2 = −tψ 1 + εψ 2 − tψ 2 eika ψ 1 ε −t ψ 1 −teika E = + ψ 2 −t ε ψ 2 0 −tA (1 − e2ika ) + 0 0 ψ 1 −teika ψ 2 s1 = A ( 2t sin ka ) 2 ( ) E ( k ) = ε − 2t cos ka ℏv ( k ) = s1 2 dE = 2ta sin ka dk 2 2 ℏv =A a Aeikx −te Σ1 = 0 ika 0 0 0 0 Σ2 = ika 0 −te A = dE 2 D(E) f1 ( E ) 2 E dE L 2π L k dk L = dE 2π dE 2π L dk ℏv s1 = 2 dE ℏv f1 ( E ) 2π a Γ1 = i ( −teika + te − ika ) = 2t sin ka ℏv = a Σ1 = −teika a A = dEf1 ( E ) 2π ℏv 2 ...
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2009.02.16-ECE659-L15-notes - ECE 659 Quantum Transport...

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