2009.04.01-ECE659-L30-notes

2009.04.01-ECE659-L30-notes - ECE 659 Quantum Transport...

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Unformatted text preview: ECE 659 Quantum Transport: Atom Atom to Transistor Lecture 30: Spin Density/Current Supriyo Datta Spring 2009 Notes prepared by Samiran Ganguly A property of Pauli spin matrices: σ aσ b = δ I + i ∑ ε abcσ c c Σ2 x G n = ψψ + = H x Σ1 σ x2 = I σ xσ y = iσ z Σ1u Σ1d ε + ε H = 0 0 0 = ε I + εσ z ε0 − ε 0 Σ 2u Σ2d Net spin is given by difference of diagonal elements Trace ( G nσ z ) If we have spins in x polarization: Trace ( G nσ x ) In y direction: 1 0 α iα I + σ z Σ1u = × −i 2 = − 2 2 0 0 If we use spins in x direction: Trace ( G nσ y ) S = Trace ( G nσ ) I s = Trace ( I opσ ) iα I + σ x = Σ 2u 2 2 iα I − σ x Σ1d = − = Σ2d 2 2 Σ1u = − G = [ EI − H − Σ1 − Σ 2 ] E − ε 0 − ε + iα = 0 0 g = u 0 gd gu = gd = G= 1 E − ε 0 − ε + iα 1 E − ε 0 + ε + iα −1 −1 −1 G n = GΓ1u G + = = = ( E − ε 0 ) I − εσ z + iα I α α 2 2 [ g 0 I + g zσ z ][ I + σ x ] g0* I + g *σ z z [ g 0 I + g zσ z ] g0* I + g *σ z + g0*σ x − ig *σ y z z 0 E − ε 0 + ε + iα * ( g 0 g 0 + g z g * ) I z * * α + ( g0 g z + g z g0 )σ z = * * 2 + ( g0 g0 − g z g z )σ x * * −i ( g 0 g z − g z g 0 ) σ y gu + g d g − gd I+ u σz 2 2 g0 gz * Trace ( G n ) = α ( g 0 g 0 + g z g * ) z * Trace ( G nσ z ) = α ( g0 g * + g z g 0 ) z * * Trace ( G nσ x ) = α ( g 0 g 0 − g z g z ) G = g 0 I + g zσ z I +σx Γ1u = α 2 * Trace ( G nσ y ) = α ( g 0 g * − g z g 0 ) z ...
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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.

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2009.04.01-ECE659-L30-notes - ECE 659 Quantum Transport...

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