ECE495N-F08-HW_8

ECE495N-F08-HW_8 - 1 ECE 495N Fall’08 ME118 MWF 1130A –...

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Unformatted text preview: 11/17/08 1 ECE 495N, Fall’08 ME118, MWF 1130A – 1220P HW#8: Due Wednesday Dec.3 in class. Basic equations of coherent transport Γ 1 = i [ Σ 1 − Σ 1 + ] , Γ 2 = i [ Σ 2 − Σ 2 + ] G ( E ) = [ EI − H − Σ 1 − Σ 2 ] − 1 , A ( E ) = i [ G − G + ] = G Γ 1 G + + G Γ 2 G + Density of states [ G n ( E )] = [ G Γ 1 G + ] f 1 + [ G Γ 2 G + ] f 2 Electron density I i ( E ) = q h ( ( Trace [ Γ i A ]) f i − Trace [ Γ i G n ]) Current/energy I ( E ) = q h Trace [ Γ 1 G Γ 2 G + ]( f 1 ( E ) − f 2 ( E )) 2-terminal current In general H has to be replaced with H+U, where U has to be calculated self-consistently from an appropriate “Poisson”-like equation, but you can ignore this aspect in the following problem. Problem 1: Consider a 1-D wire Σ 1 [ ] [H] Σ 2 [ ] modeled with a Hamiltonian having H n , n + 1 = − t = H n , n − 1 (all other elements are zero), such that the dispersion relation is given by E = − 2 t cos ka . Use t = 1 eV. For a lattice with Np points, you could set up the H matrix using the command...
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ECE495N-F08-HW_8 - 1 ECE 495N Fall’08 ME118 MWF 1130A –...

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