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Minimal Model for Transport

µ 2. . . . . . . . . . . . . . . . . . . . . Rate equation g 1,2 , f 1,2. g 1 f 1 + g 2 f 2. g 1 f 1 + g 2 f 2. g 1 f 1 + g 2 f 2. + Broadening D(E). + Electrostatics U (Self-consistent Field). N = dE D(E-U).

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Minimal Model for Transport

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  1. µ2                     Rate equation g1,2, f1,2 g1f1 + g2f2 g1f1 + g2f2 g1f1 + g2f2 + Broadening D(E) + Electrostatics U (Self-consistent Field) N= dE D(E-U) N= dE D(E ) N= g1 g2 g1 + g2 g1 + g2 g1 + g2 g1g2 g1g2 g1g2 µ1 g1 + g2 g1 + g2 g1 + g2 SOURCE DRAIN 2q 2q 2q I1= dE D(E)[f1-f2] I1= dE D(E-U)[f1-f2] I1= [f1-f2] CHANNEL h h h Poisson: U = UL + U0(N-N0) INSULATOR VG VD D(E) I Minimal Model for Transport Silicon / Nanotubes / Molecules (FetToy, CNTbands, MolCtoy)

  2. What can we capture with this model?

  3. mCoxW  I = [(VG-VT)VD-VD2/2]   L      g1f1 + g2f2 N= dE D(E-U) g1 + g2 g1g2 g1 + g2 2q I1= dE D(E-U)[f1-f2] h Classical Theory of MOSFETs g1 = g2 = ħv/L Ballistic FETs: v determined by bandstructure alone Classical FETs: v = mdV/dx (limited by scattering) Ballistic FETs: occupancy determined by D(qV) Classical FETs: CoxA

  4. Deriving Ohm’s Law need 1/R  A/L ! 1/R = G = (2q2/h)T •  1/L (longer channel, slower escape into leads)  1/R  A Still missing 1/L (R indep. of L here)! T  gD D  AL (volume)  1/R  gAL Not there yet ! Where does extra 1/L come from? Missing piece: Scattering inside channel

  5. We can reproduce all classical theories ‘bottom-up’ We can also capture physics notdescribable by classical models

  6. intel.com HW1.1: MOSFET theory

  7. HW 1.2: ThermoEMF E E f1 f2 µ2 µ1 m1 = m2 (no applied bias) T1 >> T2 Which way would current flow?

  8. HW 1.3: Silicon-molecule-metal systems E E f2 f1 Gap Put positive bias on tip, assume levels float halfway What happens to I-V when level enters bandgap?

  9. Beyond Minimal Model 1. Interference 2. Dephasing 3. Correlation

  10. 1. Interference Between Levels Oscillations in magneto- Conductance (‘Shubnikov De-Haas’) Interference between a dot and a channel (‘Fano’) D(E) is an ‘independent level’ model To capture interference, need a matrix version We will see that later (NEGF)

  11. Must go beyond minimal model µ1 µ2 H + U 1. Interference (Rest of the book!) Numbers (e,g,U)  Matrices (H, S, U) Rate equations  NEGF formalism

  12. 2. Dephasing Dissipation Mostly in the contacts Where does dissipation occur? (I2R) ‘Hot’ hole ‘Hot’ electron

  13. 2. Dephasing Vibrational ‘fingerprints’ Current picks up signatures of these vibrations (Inelastic Electron Tunneling Spectroscopy) Expt. Mark Reed (Yale) Electron can lose energy by setting molecule vibrating

  14. 3. Correlation U = UL + U0(N-N0) Adding an electron to a channel raises all its levels by U But an electron should not feel itself !! This should split conductance levels (Coulomb Blockade)

  15. 3. Correlation El-El interactions µ1 µ2 “ Coulomb Blockade “ Levels split for large U0 Metal-insulator transition

  16. 3. Correlation When electrons cooperate Antiferromagnetism Ferromagnetism Superconductivity Superfluidity Quantum Hall Effect Myriad other effects… The very notion of a ‘potential’ U questionable (Ch 4)

  17. To summarize • Minimal Model for current conduction • Ingredients can be measured or calculated • Complications due to quantum interference, scattering and correlation • ‘Minimal’ model already good enough to describe most transport experiments!

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