2009.02.13-ECE659-L14-notes

2009.02.13-ECE659-L14-notes - ECE 659 Quantum Transport:...

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Unformatted text preview: ECE 659 Quantum Transport: Atom Atom to Transistor Lecture 14: Coherent Transport I Supriyo Datta Spring 2009 Notes prepared by Samiran Ganguly E iℏ ∂ψ n ℏ ∂ψ ∂ψ =− + Uψ → iℏ = ∑ H nmψ m 2 ∂t 2m ∂x ∂t m 2 2 ℏ 2 ∂ 2φ Eφ = − + U φ → φn = ∑ H nmφm 2m ∂x 2 m ka ℏ2 k 2 E= +U 2m E = ε + 2t cos ka dn 1− k 2a2 2 dx 1 ∂ E = dt ℏ ∂ k ψ ∼ eikx e −iEt / ℏ φn = eik na ik ( d m − d n ) E = ∑ H nm e m {ψ ( 0 )} {ψ ( 0 )} = e t a n +1 2 ikna − n 2 n0 e t n −1 ε n d {ψ } dt = [H ] iℏ {ψ } {ψ ( t )} = e − i[ H ]t ℏ {ψ ( 0 )} Matlab can calculate the exponential using expm() function The function of a matrix is found by diagonalizing a matrix by a basis transformation, calculating the function of diagonal elements and then using the inverse transform [ EI − H ]{φ} = 0 In a 1x1 case ( E − ε )φ = 0 E =ε a b f ( a1 ) c d → 0 0 u v → f ( a2 ) w x [ EI − H − Σ]{φ} = {s} < x >= ∫ dxψ *ψ x ∫ dxψ *ψ <x> t Σ H s Coupling with two contacts (complex matrix) Hamiltonian (real matrix) Source matrix E t ε t a {φ } = [G ][ s ] [G ] = [ E I − H − Σ] −1 Eφ = H φ eikx + re − ikx 0 1 2 τ eikx ψ 2 ∼ eik 2 a 3 ψ 3 ∼ eik 3a ∴ψ 3 ∼ ψ 2 eika ε H = t t ε Eψ n = ∑ H nmψ m m Eψ 1 = εψ 1 + tψ 2 + tψ 0 Eψ 2 = εψ 2 + tψ 1 + tψ 3 ψ ε E 1 = ψ 2 t t ψ 1 ε ψ 2 ...
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2009.02.13-ECE659-L14-notes - ECE 659 Quantum Transport:...

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